Two-Phase Flow
Marcus HerrmannCenter for Turbulence Research
Stanford University
2006 CTR Summer Program
Tutorial
August 2nd 2006 Two-Phase Flow Tutorial 2
Overview
bull Introduction
bull Modeling two-phase flow
ndash Schemes assuming the phase interface geometry
ndash Schemes tracking the phase interface geometry
bull Tracking the interface
bull Coupling the interface to the flow solver
bull Sub-grid scale modeling
ndash Coupling
bull Summary
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August 2nd 2006 Two-Phase Flow Tutorial 3
What is Two-Phase Flow
bull Matter commonly occurs in one of three phases
ndash solid
ndash liquid
ndash gas
bull Any flow involving two of the three phases is a two-phase flow
Examples
ndash dust storms
ndash sediment transport in rivers
ndash flash floods
ndash clouds
bull The focus in this tutorial will be on liquidgas flows
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August 2nd 2006 Two-Phase Flow Tutorial 4
LiquidGas Flows
Liquidgas interfaces occur in a wide variety of
natural phenomena and technical processes
bull Ocean waves
bull Geysers
bull Inkjets
bull Deposition and coating
bull Firefighting
bull Pest control
bull Tire splash
bull Combustion devices
ndash SCRAM jets
ndash Direct injection IC-engines
ndash Gas turbines
ndash Cryogenic rocket engines
wave breaking
LOX+GH2 cold jet (Mayer et al 01)
truck tire splash
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August 2nd 2006 Two-Phase Flow Tutorial 5
Challenges of Modeling LiquidGas Flows
bull Phase interface separating the liquid from the gas is extremely thin
discontinuity
bull Density change across the phase interface is large
bull Phase interface exerts a localized surface tension
force on the liquid
bull Phase transition
bull Topology changes
bull Vast range of time and length scales are common
bull Example Atomization of a liquid jet in a turbulent environment
atomization in combustion devices
airwater 816
airmagma 4000
airliquid steel 10000
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August 2nd 2006 Two-Phase Flow Tutorial 1
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
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August 2nd 2006 Two-Phase Flow Tutorial 2
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
divide and conquer
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops
forming a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
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August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
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August 2nd 2006 Two-Phase Flow Tutorial 5
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Modeling the Atomization Process
Lasheras et al 1998
track the phase interface
assume the drop geometry
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August 2nd 2006 Two-Phase Flow Tutorial 1
Modeling Secondary Atomization
bull If one can assume that
ndash liquid has simple geometry drops = solid spheres
ndash liquid density gtgt gas density
ndash volume loading is small ie inter-drop distance is large
ndash drop size is smaller than grid size
ndash effect of shear on droplet motion can be neglected
Point-particle approach
with
particle position
particle velocity
gas velocity
liquid density
gas density
gravitational accel
particle diameter
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August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
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August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
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August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
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August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
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August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
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August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
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August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
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August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
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August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
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August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
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August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
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August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
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August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
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August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
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August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
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August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
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August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
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August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
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August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
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August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
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August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
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August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
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August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
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August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
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August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
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August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
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August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
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August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
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August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 2
Overview
bull Introduction
bull Modeling two-phase flow
ndash Schemes assuming the phase interface geometry
ndash Schemes tracking the phase interface geometry
bull Tracking the interface
bull Coupling the interface to the flow solver
bull Sub-grid scale modeling
ndash Coupling
bull Summary
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August 2nd 2006 Two-Phase Flow Tutorial 3
What is Two-Phase Flow
bull Matter commonly occurs in one of three phases
ndash solid
ndash liquid
ndash gas
bull Any flow involving two of the three phases is a two-phase flow
Examples
ndash dust storms
ndash sediment transport in rivers
ndash flash floods
ndash clouds
bull The focus in this tutorial will be on liquidgas flows
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August 2nd 2006 Two-Phase Flow Tutorial 4
LiquidGas Flows
Liquidgas interfaces occur in a wide variety of
natural phenomena and technical processes
bull Ocean waves
bull Geysers
bull Inkjets
bull Deposition and coating
bull Firefighting
bull Pest control
bull Tire splash
bull Combustion devices
ndash SCRAM jets
ndash Direct injection IC-engines
ndash Gas turbines
ndash Cryogenic rocket engines
wave breaking
LOX+GH2 cold jet (Mayer et al 01)
truck tire splash
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August 2nd 2006 Two-Phase Flow Tutorial 5
Challenges of Modeling LiquidGas Flows
bull Phase interface separating the liquid from the gas is extremely thin
discontinuity
bull Density change across the phase interface is large
bull Phase interface exerts a localized surface tension
force on the liquid
bull Phase transition
bull Topology changes
bull Vast range of time and length scales are common
bull Example Atomization of a liquid jet in a turbulent environment
atomization in combustion devices
airwater 816
airmagma 4000
airliquid steel 10000
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August 2nd 2006 Two-Phase Flow Tutorial 1
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
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August 2nd 2006 Two-Phase Flow Tutorial 2
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
divide and conquer
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops
forming a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
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August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
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August 2nd 2006 Two-Phase Flow Tutorial 5
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Modeling the Atomization Process
Lasheras et al 1998
track the phase interface
assume the drop geometry
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August 2nd 2006 Two-Phase Flow Tutorial 1
Modeling Secondary Atomization
bull If one can assume that
ndash liquid has simple geometry drops = solid spheres
ndash liquid density gtgt gas density
ndash volume loading is small ie inter-drop distance is large
ndash drop size is smaller than grid size
ndash effect of shear on droplet motion can be neglected
Point-particle approach
with
particle position
particle velocity
gas velocity
liquid density
gas density
gravitational accel
particle diameter
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August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
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August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
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August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
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August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
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August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
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August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
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August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
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August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
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August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
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August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
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August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
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August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
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August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 3
What is Two-Phase Flow
bull Matter commonly occurs in one of three phases
ndash solid
ndash liquid
ndash gas
bull Any flow involving two of the three phases is a two-phase flow
Examples
ndash dust storms
ndash sediment transport in rivers
ndash flash floods
ndash clouds
bull The focus in this tutorial will be on liquidgas flows
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August 2nd 2006 Two-Phase Flow Tutorial 4
LiquidGas Flows
Liquidgas interfaces occur in a wide variety of
natural phenomena and technical processes
bull Ocean waves
bull Geysers
bull Inkjets
bull Deposition and coating
bull Firefighting
bull Pest control
bull Tire splash
bull Combustion devices
ndash SCRAM jets
ndash Direct injection IC-engines
ndash Gas turbines
ndash Cryogenic rocket engines
wave breaking
LOX+GH2 cold jet (Mayer et al 01)
truck tire splash
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August 2nd 2006 Two-Phase Flow Tutorial 5
Challenges of Modeling LiquidGas Flows
bull Phase interface separating the liquid from the gas is extremely thin
discontinuity
bull Density change across the phase interface is large
bull Phase interface exerts a localized surface tension
force on the liquid
bull Phase transition
bull Topology changes
bull Vast range of time and length scales are common
bull Example Atomization of a liquid jet in a turbulent environment
atomization in combustion devices
airwater 816
airmagma 4000
airliquid steel 10000
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August 2nd 2006 Two-Phase Flow Tutorial 1
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
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August 2nd 2006 Two-Phase Flow Tutorial 2
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
divide and conquer
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops
forming a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
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August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
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August 2nd 2006 Two-Phase Flow Tutorial 5
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Modeling the Atomization Process
Lasheras et al 1998
track the phase interface
assume the drop geometry
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August 2nd 2006 Two-Phase Flow Tutorial 1
Modeling Secondary Atomization
bull If one can assume that
ndash liquid has simple geometry drops = solid spheres
ndash liquid density gtgt gas density
ndash volume loading is small ie inter-drop distance is large
ndash drop size is smaller than grid size
ndash effect of shear on droplet motion can be neglected
Point-particle approach
with
particle position
particle velocity
gas velocity
liquid density
gas density
gravitational accel
particle diameter
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August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
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August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
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August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
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August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
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August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
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August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
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August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
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August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
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August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
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August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
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August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
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August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
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August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
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August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
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August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
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August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
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August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
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August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
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August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
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August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
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August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
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August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
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August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
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August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
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August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
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August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
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August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
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August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
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August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 4
LiquidGas Flows
Liquidgas interfaces occur in a wide variety of
natural phenomena and technical processes
bull Ocean waves
bull Geysers
bull Inkjets
bull Deposition and coating
bull Firefighting
bull Pest control
bull Tire splash
bull Combustion devices
ndash SCRAM jets
ndash Direct injection IC-engines
ndash Gas turbines
ndash Cryogenic rocket engines
wave breaking
LOX+GH2 cold jet (Mayer et al 01)
truck tire splash
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Challenges of Modeling LiquidGas Flows
bull Phase interface separating the liquid from the gas is extremely thin
discontinuity
bull Density change across the phase interface is large
bull Phase interface exerts a localized surface tension
force on the liquid
bull Phase transition
bull Topology changes
bull Vast range of time and length scales are common
bull Example Atomization of a liquid jet in a turbulent environment
atomization in combustion devices
airwater 816
airmagma 4000
airliquid steel 10000
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
divide and conquer
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops
forming a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Modeling the Atomization Process
Lasheras et al 1998
track the phase interface
assume the drop geometry
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Modeling Secondary Atomization
bull If one can assume that
ndash liquid has simple geometry drops = solid spheres
ndash liquid density gtgt gas density
ndash volume loading is small ie inter-drop distance is large
ndash drop size is smaller than grid size
ndash effect of shear on droplet motion can be neglected
Point-particle approach
with
particle position
particle velocity
gas velocity
liquid density
gas density
gravitational accel
particle diameter
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
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August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
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August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
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August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
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August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
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August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
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August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
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August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
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August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
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August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
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August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
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August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
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August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
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August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
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August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
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August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
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August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
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August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
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August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
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August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
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August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
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August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
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August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
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August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
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August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
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August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
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August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
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August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
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August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 5
Challenges of Modeling LiquidGas Flows
bull Phase interface separating the liquid from the gas is extremely thin
discontinuity
bull Density change across the phase interface is large
bull Phase interface exerts a localized surface tension
force on the liquid
bull Phase transition
bull Topology changes
bull Vast range of time and length scales are common
bull Example Atomization of a liquid jet in a turbulent environment
atomization in combustion devices
airwater 816
airmagma 4000
airliquid steel 10000
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
divide and conquer
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops
forming a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Modeling the Atomization Process
Lasheras et al 1998
track the phase interface
assume the drop geometry
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Modeling Secondary Atomization
bull If one can assume that
ndash liquid has simple geometry drops = solid spheres
ndash liquid density gtgt gas density
ndash volume loading is small ie inter-drop distance is large
ndash drop size is smaller than grid size
ndash effect of shear on droplet motion can be neglected
Point-particle approach
with
particle position
particle velocity
gas velocity
liquid density
gas density
gravitational accel
particle diameter
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 1
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
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Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
divide and conquer
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
mah
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops
forming a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
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Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Modeling the Atomization Process
Lasheras et al 1998
track the phase interface
assume the drop geometry
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Modeling Secondary Atomization
bull If one can assume that
ndash liquid has simple geometry drops = solid spheres
ndash liquid density gtgt gas density
ndash volume loading is small ie inter-drop distance is large
ndash drop size is smaller than grid size
ndash effect of shear on droplet motion can be neglected
Point-particle approach
with
particle position
particle velocity
gas velocity
liquid density
gas density
gravitational accel
particle diameter
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August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
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August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
mah
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August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
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August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
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August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
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August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
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August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
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August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
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Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
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August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
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Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
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August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
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Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 2
Atomization in Combustion Devices
bull Fuel is typically injected as a liquid
bull Combustion occurs only in the
gaseous phase
Atomize liquid to enhance evaporation
bull Challenges
ndash phase interface is a discontinuity
ndash density contrast is high O(100)
ndash surface tension forces
ndash frequent topology changes
ndash range of scales from cm to μm
ndash large phase interface surface area
ndash phase transition
ndash interaction with turbulence
bull How can one modelsimulate this
divide and conquer
Marmottant amp Villermaux 2002
Lasheras et al 1998
coaxial atomization
mah
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops
forming a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
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August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
mah
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August 2nd 2006 Two-Phase Flow Tutorial 5
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Modeling the Atomization Process
Lasheras et al 1998
track the phase interface
assume the drop geometry
mah
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August 2nd 2006 Two-Phase Flow Tutorial 1
Modeling Secondary Atomization
bull If one can assume that
ndash liquid has simple geometry drops = solid spheres
ndash liquid density gtgt gas density
ndash volume loading is small ie inter-drop distance is large
ndash drop size is smaller than grid size
ndash effect of shear on droplet motion can be neglected
Point-particle approach
with
particle position
particle velocity
gas velocity
liquid density
gas density
gravitational accel
particle diameter
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August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
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August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
mah
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August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
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August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
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August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
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August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
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August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
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August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
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August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
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August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
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August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
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August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
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August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
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August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
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August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
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August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
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August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
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August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
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August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
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August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
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August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
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August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops
forming a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Modeling the Atomization Process
Lasheras et al 1998
track the phase interface
assume the drop geometry
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Modeling Secondary Atomization
bull If one can assume that
ndash liquid has simple geometry drops = solid spheres
ndash liquid density gtgt gas density
ndash volume loading is small ie inter-drop distance is large
ndash drop size is smaller than grid size
ndash effect of shear on droplet motion can be neglected
Point-particle approach
with
particle position
particle velocity
gas velocity
liquid density
gas density
gravitational accel
particle diameter
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling the Atomization Process
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Lasheras et al 1998
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Modeling the Atomization Process
Lasheras et al 1998
track the phase interface
assume the drop geometry
mah
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August 2nd 2006 Two-Phase Flow Tutorial 1
Modeling Secondary Atomization
bull If one can assume that
ndash liquid has simple geometry drops = solid spheres
ndash liquid density gtgt gas density
ndash volume loading is small ie inter-drop distance is large
ndash drop size is smaller than grid size
ndash effect of shear on droplet motion can be neglected
Point-particle approach
with
particle position
particle velocity
gas velocity
liquid density
gas density
gravitational accel
particle diameter
mah
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August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
mah
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
mah
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August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
mah
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August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
mah
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August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
mah
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August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
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August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
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August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
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August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
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August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
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August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 5
bull Split atomization into primary amp secondary
atomization
bull Primary atomization
ndash Initial breakup of liquid jets or sheets into large and
small structures (ligamentsdrops) close to the
injection nozzle
ndash Complex interface topology of mostly large scale
coherent liquid structures
ndash Total phase interface surface area is small
bull Secondary atomization
ndash Subsequent breakup into ever smaller drops forming
a spray
ndash Simple geometry of small scale liquid drops
ndash Total phase interface surface area is large
ndash Volume loading is small
Modeling the Atomization Process
Lasheras et al 1998
track the phase interface
assume the drop geometry
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Modeling Secondary Atomization
bull If one can assume that
ndash liquid has simple geometry drops = solid spheres
ndash liquid density gtgt gas density
ndash volume loading is small ie inter-drop distance is large
ndash drop size is smaller than grid size
ndash effect of shear on droplet motion can be neglected
Point-particle approach
with
particle position
particle velocity
gas velocity
liquid density
gas density
gravitational accel
particle diameter
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 1
Modeling Secondary Atomization
bull If one can assume that
ndash liquid has simple geometry drops = solid spheres
ndash liquid density gtgt gas density
ndash volume loading is small ie inter-drop distance is large
ndash drop size is smaller than grid size
ndash effect of shear on droplet motion can be neglected
Point-particle approach
with
particle position
particle velocity
gas velocity
liquid density
gas density
gravitational accel
particle diameter
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 2
Modeling Secondary Atomization
bull Consider more realistic physics by relaxing some of the assumptions
for example
ndash allow drops to be ellipsoidal [Helenbrook amp Edwards 02]
ndash take droprsquos internal circulation into account [Helenbrook amp Edwards 02]
bull Phase transition models
ndash use Spalding mass and heat transfer numbers and Clausius-Clapeyronrsquos
vapor-pressure equilibrium relation for
ndash apply convective correction factors for high Reynolds numbers
latent heat of evaporation
liquid specific heat
effective heat transfer
coefficient
particle mass
particle temperature
mah
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August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
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August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
mah
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August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
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August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
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August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
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August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
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August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
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August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
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August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
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August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
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Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
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August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
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August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
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August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
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August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
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August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
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August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
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August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 3
Modeling Secondary Atomization
Coupling to gas phase
bull Source term in continuity and mixture fraction
equation
bull Source term in momentum equations
bull Interpolation operator from particle position to cv-centroids
local grid size
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 4
Example
bull Glass particles injection into swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 91]
ndash Simulation [Apte et al 03]
bull 16 million hexas 11 million particles
bull D10 = 45 μm log-normal size distribution
bull Re = 26200
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 5
Example
Mean Axial Velocity
Mean Swirl Velocity
Mean Radial Velocity
RMS of Axial Velocity
RMS of Radial Velocity
RMS of Swirl Velocity
LES Apte et al IJMF 2003 Experiments Sommerfeld amp Qiu 1991
Gas phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 6
Example
Mean Axial Velocity RMS of Axial Velocity
Mean Radial Velocity RMS of Radial Velocity
Mean Swirl Velocity RMS of Swirl Velocity
Mean Particle Diameter RMS of Particle Diameter
Particle phase
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 7
Evaporation Examplebull Isopropyl alcohol into non-swirling flow in coaxial geometry
ndash Experiment [Sommerfeld amp Qiu 98]
ndash Simulation [Moin amp Apte 06]
bull 15 million hexas 075 million particles
bull droplet size distribution from experiment
bull Re = 21164
Fuel Mass
Fraction
Axial
Velocity
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 8
Evaporation Example
Diameter
Axial Velocity
Particle Mass
Flux
bullbullbull Experiments
mdashmdash LES
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 9
Breakup Modelingbull TAB (Taylor Analogy Breakup) Model [Taylor 63]
ndash analyze droplet distortion by spring-mass system
bull external force = droplet drag force
bull spring force = surface tension force
bull damping force = droplet viscous force
ndash initiate breakup when distortion xR gt 05
ndash determine children drop size from energy conservation
ndash applicability low Weber numbers
bull Wave Breakup Model [Reitz 87]
ndash assume drops breakup due to Kelvin-Helmholtz instability
ndash applicability high Weber numbers
bull Stochastic Breakup Model [Apte et al 03]
ndash use Fokker-Planck equation for drop radius
ndash obtain breakup frequency and critical radius from balance of aerodynamic
and surface tension forces
ndash children drop sizes from Fokker-Planck equation
ndash applicable to turbulent flows
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 10
Simulating Secondary Atomization
bull Lagrangian point particle tracking
ndash Number of drops can be huge order tens of millions large computational cost
ndash Drops are typically confined to a relatively small region of the whole
computational domain load balancing difficult
bull Alternative strategies
ndash Hybrid particle parcel technique
bull group drops of similar size location and properties into a single parcel
bull solve Lagrangian equations for averaged properties of the parcel
ndash Solve for PDFs of particle properties instead of individual drops
bull pdf transport equations
bull solve for moments of the pdf and presume the pdf shape
bull DQMOM
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 1
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
P = 11 MPa
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 2
Breakup Examplebull Liquid jet injected into chamber
ndash Experiment [Hiroyasu amp Kudota 74]
ndash Simulation [Moin amp Apte 06]
bull 04 million uniform cells
bull initial droplet size = jet diameter
bull stochastic breakup and hybrid particleparcel technique
Penetration Depth
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 3
Problems of Spray Models
bull Results are sensitive to the initial drop size distribution
ndash need to have experimental data of initial drop size distribution
ndash tune initial drop size distribution to give experimental data down-stream
ndash guess initial distribution
ndash assume initial drop size = injector (violates assumptions in model derivation)
bull Represent coherent liquid core by collection of particles of equal mass
bull Need predictive capability of initial drop size distribution
primary atomization modeling
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 4
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 5
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 6
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 8
Moving Grids
bull Represent phase interface by grid
nodes on the interface
bull Move interface grid nodes by Lagrangian
transport
can result in large grid deformations
re-griding necessary
bull Successful for small interface deformations
bull Topology changes difficult
bull Normal interface movement (phase change)
difficult
[Scardovelli amp Zaleski 1999
from J Magnaudet and coworkers]
[Scardovelli amp Zaleski 1999
from McHyman 1984]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 9
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 10
Marker Particles
bull Track phase interface by Lagrangian
marker particles in a fixed grid
bull Phase interface can be reconstructed by
polynomials through neighboring marker
particles
phase interface geometry is very accurate
(normal curvature)
need to keep connectivity information of
markers
topology changes are difficult
bull Normal interface movement (phase change) difficult
bull Does provide sub-grid phase interface resolution
[Scardovelli amp Zaleski 1999]
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 11
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 12
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 13
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 15
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 16
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 17
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
m
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 18
Volume of Fluid
bull Represent phase interface by liquid volume fraction in each cell
bull Move phase interface by solving PDE
How
ndash standard advection schemes have too much dispersion
use artificial compression to preserve jump in
ndash reconstruct interface geometry and perform geometric flux calculation
bull calculate normal direction to interface
bull assume interface in each cell is planar
bull find plane normal to m that has liquid cell volume
PLIC
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 19
Volume of Fluid
bull Geometric flux calculation
ndash Eulerian
ndash Lagrangian
bull perform directional operator splitting
bull advect planar interface by linearly
interpolated velocities in each cell
bull calculate change in liquid volume in
cell and neighbors
bull CFL number 05
F wetted cell face area
n cell face normal
[Gueyffier et al 1999]
F
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 20
Volume of Fluid
bull Problems in practical applications
ndash Volume of Fluid method is not exactly volume preserving
ndash gt 1 or lt 0 possible
ndash is not = 1 in liquid or = 0 in gas ( = or = 1- ) wisps
ndash lower order geometric interface reconstruction yields flotsam
ndash interface curvature not easily calculated
height function approach PROST (quadratic interface reconstruction)
ndash combining geometric flux calculation and normal interface movement
(phase change) difficult
mah
Rectangle
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 1
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 2
Tracking Interfaces with Level Sets
bull Level set equation
bull Define
level set scalar for a 2D circle
bull G for G G0 is arbitrary
ndash usually chosen to be signed distance function with
G0
= 0
ndash but also smeared Heaviside function with G0
= 05
[Olsson amp Kreiss 05]
velocity
phase change vel
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 3
Solving the Level Set Equation
bull Level set transport equation is a Hamilton-Jacobi PDE
bull Front can develop corners in finite time
Example front with phase
change
need the weak solution to the PDE
bull Use upwind-biased WENO schemes for Hamilton-Jacobi equations
with appropriate flux functions
ndash 5th-order WENO with RoeLLF flux function
bull Use higher order TVD Runge-Kutta schemes for time advancement
ndash 3rd-order TVD Runge-Kutta
[Sethian 96]
weak solution
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 4
Level Set Toolbox
bull Interface geometric properties
bull Liquid volume and phase interface surface area
bull For numerical purposes use smeared out versions usually
with
but no convergence under grid refinement [Engquist et al 05]
Instead use [Engquist et al 05]
and
with
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 5
bull How to extend a quantity defined on to the whole domain (redistribution)
ie
Level Set Toolbox
bull How to keep G a distance function (reinitialization) ie
[Sussman et al 94]
with [Peng et al 99]and
with
bull Pros and cons of PDE based reinitialization and redistribution
+ easy to implement and parallelize for domain decomposition
- costly due to pseudo-time iteration
- tend to move the interface and smooth
[Peng et al 99]
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 6
Level Set Toolbox
bull Alternatives to PDE based reinitialization and redistribution
ndash Fast Marching Method (FMM)
bull discretize and
bull solve resulting quadratic equations moving along characteristics
bull characteristics = normal vectors pointing away from the interface
bull use values on the interface as Dirichlet boundary conditions
Pros and Cons of FMM
+ low operation count O(N log N)
- domain decomposition parallelization complicated and parallel efficiency
dependent on surface geometry [Herrmann 03]
ndash Fast Sweeping Method (FSM)
bull similar to FMM but instead of following characteristics perform directional back
and forth sweeps along the coordinate axis
Pros and Cons of FSM
+ higher operation count than FMM
- easier domain decomposition parallelization but efficiency dependent on interface
geometry
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 7
Tracking InterfacesCommon numerical methods
bull Moving grids
+ very accurate for small ndash very complex
deformations ndash topology changes amp normal movement difficult
bull Marker particles
+ accurate ndash very complex in 3D
ndash topology changes by manual intervention
challenging in 3D
ndash normal interface movement not handled
automatically
bull Volume-of-Fluid (VoF)
+ good volume conservation ndash interface geometry reconstruction challenging
ndash normal interface movement not straightforward
bull Level sets
+ simple interface geometry ndash not inherently volume conserving
reconstruction
+ normal interface movement
handled automatically
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 8
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
Volume Conservation amp Level Sets
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 9
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D regularized vortex sheet rollup
target solution standard level set
Vortex Sheet Rollup
target solution
particle corrected
level set
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 10
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Example 2D de-singularized vortex sheet rollup
target solution
Vortex Sheet Rollup
particle corrected
level set
curvature distribution
initial conditions
t = 3
target solution
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 11
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 12
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution standard level set
target solution
RLSG
initial conditions
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 13
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles [Enright et al 02]
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution refine level set grid independent
of flow solver grid
Refined Level Set Grid Method
bull Example 2D regularized vortex sheet rollup
Vortex Sheet Rollup
target solution
target solution RLSG
initial conditions
t = 3
curvature distribution
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 14
Volume Conservation amp Level Sets
bull Numerically the volume of each fluid is not inherently preserved
ndash Couple level set to volume preserving scheme for example VoF or marker
particles (Enright et al 02)
bull Use secondary scheme to identify and correct local errors in level set
bull Problem local correction can introduce large errors in higher derivatives of G
curvature
bull Delocalization techniques only partially successful [Coyajee et al 04]
ndash Increase fidelity of level set solution
bull Volume error ~ grid resolution
refine level set grid independent of flow solver grid
Refined Level Set Grid Method
bull Efficiency two-layer narrow band approach
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 15
Refined Level Set Grid Method
bull Front on a flow solver grid
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 16
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 17
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 18
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 19
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 20
Refined Level Set Grid Method
bull Front on a flow solver grid
bull Introduce equidistant Cartesian
super-grid (blocks)
bull Activate (store) only narrow
band of blocks
bull Active blocks consist of an equi-
distant Cartesian fine G-grid
bull Activate (store) only
narrow band of fine G-grid
Solve and store all level set equations only on active cells of G-gridcost is only O(N2) not O(N3) high resolution x
Glaquo x
fs
Efficient domain decomposition parallelization straightforward
Fast and accurate Cartesian solution techniques for HJ-PDEs can be used
(5th order WENO FMM)
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 21
RLSG Results Vortex in a Box
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 512 x 512 cells max
Flow solvervolume fraction
0 t 3
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 22
RLSG Results Vortex in a Box
512x512
t = 3
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells max
Flow solvervolume fraction
0 t 3
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 23
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 24
512x512
t = 3
RLSG Results Vortex in a Box
Total Volume
128x128
t = 3
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 25
Sphere in a Deformation Field
Flow solver
bull 643 cells
RLSG
bull 2563 cells max
RLSG
G = G0 iso-surface
0 t 15
initial conditions
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 26
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 27
Sphere in a Deformation Field
RLSG 643
RLSG 1283
RLSG 2563
t = 15
2563
t = 15
1283
Flow solver 643 cells
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 28
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 29
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 30
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 31
Jump conditions at the phase interface
bull In most applications the phase interface can be treated as a
discontinuity
bull Jump conditions at the phase interface
bull Alternatives resolve change of quantities through the interface
ndash Navier-Stokes-Korteweg equations
ndash Cahn-Hilliard equations
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 32
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 33
Ghost Fluid Method
bull Applicable to finite difference methods
bull Define a ghost fluid ghost air in
liquid ghost liquid in air
bull For ghost fluid state
ndash extrapolate all quantities that jump
across the interface
ndash keep all quantities that have zero
jump
bull For all gradients nearacross the interface use only airghost air or
liquidghost liquid states
controls dispersion errors for quantities that have non-zero jump
bull For higher order perform Taylor expansion within one fluid and apply
jump conditions at the interface
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 34
bull Applicable to finite volume methods
bull All cells with part of the interface
reconstruct in-cell gas and liquid
state from
bull All single phase cells close to
interface apply jump conditions to define ghost state
bull For all gradients nearacross the interface use only the same phase
(true reconstructed or jumped)
eliminates dispersion errors for quantities that have non-zero jump
In-Cell Reconstruction
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 35
Coupling to Flow Solver
bull Approaches
ndash Interface conforming grids
bull Deform grid such that cell faces conform to phase interface then use
jump conditions across the interface as boundary conditions
ndash One fluid approach on non-conforming grids
bull Ghost fluid method [Fedkiw et al 99]
bull In-cell reconstruction technique [Smiljanovski 96]
bull Assume gas and liquid are one fluid with change in density and
composition at the phase interface
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 36
bull Marker particles
ndash interpolate gradient of density to grid
ndash solve Poisson system for
bull For finite volume Navier-Stokes solver
bull Flow solver liquid volume fraction
ndash VoF tracks directly
ndash Level set
One Fluid Approach
bull Navier-Stokes equations for two-phase incompressible flows in non-conservative form
[van der Pijl et al 2004]
liquid density viscosity
gas density viscosity
surface tension force
or
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 37
One Fluid Approach
bull Surface tension force
ndash Marker Particles
ndash Continuum Surface Force (CSF) [Brackbill et al 92]
ndash Continuum Surface Stress (CSS) [Zaleski et al 94]
surface element
surface tension coeff
surface mean curvature
surface normal
surface element edge
surface edge tangent
delta function at surface
marker function
VoF level set
VoF
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 38
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
single time step t = 10-3
[Francois et al 06]
|umax| = 0146
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 39
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
VoF and level set methods [Lafaurrie et al 94]
marker methods [Tryggvason]
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial 40
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
ndash remedy introduce viscosity to control errors
ndash better balanced force algorithm [Young et al 02 Francois et al 06]
after 1 time step exact curvature
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull Test case
ndash 8 x 8 box circle radius R = 2
ndash inviscid = 73
ndash time step t = 10-6
bull Inviscid stationary drop (circle) with surface tension should remain motionless forall time
bull But numerical errors introduce spurious currents
ndash discrete imbalance between surface tension forces and pressure gradients
balanced force algorithm errors reduced to machine accuracy zero
ndash errors in surface tension force evaluation = errors in curvature evaluation
make curvature calculation as accurate as possible
bull Marker Particles polynomial reconstruction of the surface
bull VoF height function approach or PROST
bull Level Set higher order gradient approximation or RLSG
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
bull Level set methods calculate curvature at nodes
introduces O( x) error for interface curvature
bull Interface projected curvature
1 Calculate curvature at nodes
2 Locate closest point on interface for each node xj
ndash approximate G in each cell by tricubic polynomial
ndash apply two-step Newton algorithm to find closest point j (Chopp 01)
3 Calculate curvature at interface j by trilinear interpolation
4 Assign interface curvature to node
2nd order converging curvatures
bull use as Dirichlet bc for reinitialization
Level Set Curvature Calculation
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
10-6
10-5
10-4
10-3
10-2
101
102
103
n
L(|
-
ex|)
2nd
orderLoo
L1
L2
Error in curvature evaluation
Convergence under grid refinement
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 40x40 fs-grid single time step 1
2 = 1
Spurious current magnitude
RLSG Convergence under G-grid refinement
10-6
10-5
10-4
10-3
10-2
10-1
101
102
103
n
Loo(|
p-p
ex|)
2nd
orderpmax
ptot
ppart
10-10
10-9
10-8
10-7
10-6
101
102
103
n
L(|
u|)
2nd
orderLoo
L1
L2
Error in pressure jump
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Spurious Currents Inviscid Static Drop
bull 20x20 fs-grid 2000 time step 1
2 = 1000
Spurious current kinetic energy
Convergence under
G-grid refinement
10-8
10-6
10-4
10-2
0 5 10 15 20t
Ekin
20x40
20x80
20x160
20x20
10-6
10-5
10-4
10-3
101
102
n
max
L2(u
)
2nd
order
Spurious current magnitude
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull Compare to initial value linear theory (Prosperetti 81)
ndash box size 2 x 2
ndash wave length = 2 initial amplitude A0 = 001
ndash no gravity
ndash surface oscillations caused by surface tension forces
ndash theory for contrasting densities but equal
bull Use level set interface tracking and one fluid approach
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 grid refinement
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
4thorder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1632x3264x64
theory
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
10-7
10-6
10-5
10-4
10-3
101
102
n
erro
r
2nd
order
1storder
-001
-0005
0
0005
001
0 5 10 15 20n
A
16x1616x3216x64
theory
16x128
Damped Surface Wave
bull1
2 = 1 1=
2 = 006472 fs-grid 16x16 RLSG refine G-grid only
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
10-4
10-3
10-2
101
102
n
erro
r
2nd
order
1storder
-0015
-001
-0005
0
0005
001
0015
0 50 100 150 200 250 300t
A
16x1632x3264x64
theory
128x128
Damped Surface Wave
bull1
2 = 1000 1=
2 = 0 grid refinement
t
oscillation amplitude vs time convergence
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Introduce spatial (volume) filters into the Navier-Stokes equations
ndash additional surface tension force term
bull Standard volume filters not applicable to level set equation (Oberlack et al 01)
phase interface based filters are required
ndash Mean position (Pitsch 05)
ndash Sub-filter length scale
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
box size
L x L periodic
surface filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
surface filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Volume filter based on surface parameterization
introduces new small scales
bull Alternative Heaviside based filtering
ndash Apply standard LES volume filter
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
Heaviside filter
box size
L x L periodic
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Consistent Surface Filters for LES
bull Problem How to define for Heaviside filter
ndash If based on
for planar surface
ndash If based on
for planar surface but measures volume not length scale
bull Alternative Use first and second moment of sub-filter
PDF of surface coordinates
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Sub-Grid Scale Physics
bull Large Surface Structure model
+ filtered Navier-Stokes equations with
bull How to close these terms
ndash perform DNS and gain insight
use consistent filters to analyze DNS results
ndash use Refined Level Set Grid approach
spray transfer velocity
production
dissipation
turbulent trans
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
The RLSG Approach for Sub-Grid Modeling
bull Solve Navier-Stokes equations on (implicit) filter scale
bull Solve level set equations on RLSG with DNS resolution
ndash feasible on massively parallel machines since O(N2)
bull Close all phase interface related unclosed terms in Navier-Stokes
equations by explicit filtering of the RLSG solution
bull Caveat must ensure that DNS phase interface behaves correctly
need to reconstruct structured on RLSG grid from LES information
on filter scale
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Modeling Primary Atomization in LES
bull Phase interface geometry is partially resolved by LES grid
track large scale interface geometry + model sub-grid physics
Large Surface Structure (LSS) model
bull Couple LSS model to secondary atomization spray model
bull Track the phase interface
bull Couple to flow solver
bull Model sub-grid scale physics
bull Couple to spray model
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale
v
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
v
lost mass
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
RLSG Coupling to Spray Models
bull Prerequisite for most spray models
ndash RLSG provides sub-flow solver resolution
identify amp transfer all separated liquid structures with
bull When does breakup occur in level set methods (VoF)
ndash inherent breakup length scale mass is lost
ndash RLSG with and
identify amp transfer all RLSG liquid structures thinner than
ndash but is this physical
not really but this is how spray models represent coherent liquid structures
bull RLSG data yield drop center mass and momentum
rArr input conditions for Lagrangian spray model
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Vortex in a Box with Spray Coupling
X
Y
0 02 04 06 08 10
02
04
06
08
1
Flow solver
bull 64 x 64 cells
RLSG
bull 128 x 128 cells maxbull drop transfer = 4
Flow solvervolume fraction
0 t 3
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
128x128
t = 3
Vortex in a Box with Spray Coupling
128x128
t = 3
drops
128x128 without amp with drops
=4 LS V
=4 total V
no drops
=4 drop V
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
Flow solver
bull 643 cells
RLSG
bull 1283 cells maxbull = 4
RLSG
G = G0 iso-surface
0 t 15
initial condition
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Sphere in a Deformation Field with Spray
1283 no drops
1283 =4 total V
Flow solver 643 cells
1283 =4 LS V
1283 =4 drop V
=4
t = 15
1283
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
August 2nd 2006 Two-Phase Flow Tutorial
Acknowledgments
bull S Apte
bull F Ham
bull V Moureau
bull E vd Weide
bull D Kim
bull O Desjardins
bull E Knudsen
