Introduction to DNS of Multiphase Flows-Igtryggva/CFD-Course2010/2010...Introduction to DNS! of! Multiphase Flows-I! Grétar Tryggvason! Spring 2010! Computational Fluid Dynamics I!

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Computational Fluid Dynamics I!

Introduction to DNS!of!

Multiphase Flows-I!

Grétar Tryggvason!Spring 2010!


Motivation!

Governing Equations!

The “One-Fluid” Approach!

Numerical approach!

Advecting the Marker Function!!VOF; LS; FT; Other; Tests!

Surface Tension!

Topology Changes!

Outline!

Computational Fluid Dynamics I!Motivation and Goals!

School of fish!Splash!

Microstructure!Cavitation!

Atomization!

Bubbly Flow!


Vf nf

�

x f s,t( )

�

ρ1,µ1, k1,…

�

ρ0,µ0, k0,…

�

ρ2,µ2, k2,…

Systems composed of different phases and materials, separated by a sharp interface whose location changes with time!

Evolving Heterogeneous Continuum Systems!

Motivation and Goals!

χ1=0!

χ1=1!

Phase 0!Phase 1!


Dynamics of Heterogeneous Continuum Systems!

•  Systems composed of different phases and materials, separated by a sharp interface whose location changes with time!

•  The physics is well described by continuum theories!•  The systems are sufficiently large so that simulations

resolving the smallest and the largest scales are impractical!•  There are good reasons to believe that the behavior of the

smallest scales is—in some sense—universal!•  The goal is to use fully resolved numerical simulations of the

small scale behavior to help understand how the large and the small scale motion are coupled and to develop “closure” models!

Motivation and Goals!Computational Fluid Dynamics I!

Example: The “two-fluid model”

�

∂∂tεpρp + ∇ ⋅ εpρpup( ) = ˙ m p

�

∂∂t

εpρpup( ) + ∇ ⋅ εpρpupup( ) = −εp∇pp

+∇ ⋅ εpµpDp( ) + εpρpg + ∇ ⋅ εpρp < uu >( ) + Fintinterfacial forces

Reynolds stresses

The goal is to simulate accurately the smallest continuum scales for multiphase systems that are sufficiently large so that meaningful averages can be obtained and the results used to help generate insight and closure models for engineering tools

Equations for the average motion of each constituent

Motivation and Goals!

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DNS allows us to compute directly the average evolution and properties of the mixture, including slip velocity, most probable configuration, change of composition, effective conductivity, etc. Quantities of interest range from simple volume averages to more sophisticated measures of the phase distribution:

�

εi = 1Vol

χ i(x,t)dvV∫

�

f i = 1εiVol

χ i(x,t) fi(x,t)dvV∫Volume fraction of phase i Volume average of fi

�

1Vol

nndaS∫

1Vol

nnnn daS∫

Characterization of the mixture Structure functions, turbulent quantities, etc.

Motivation and Goals!Computational Fluid Dynamics I!

BC: Birkhoff and boundary integral methods !for the Rayleigh-Taylor Instability!

65ʼ Harlow and colleagues at Los Alamos: !The MAC method!

75ʼ Boundary integral methods for Stokes flow and potential flow !

85ʼ Alternative approaches (body fitted, unstructured, etc.)!

95ʼ Beginning of DNS of multiphase flow. Return of the “one-fluid” approach and development of other techniques!

CFD of Multiphase Flows—one slide history!

From: B. Daly (1969)!

Numerical Method!


Governing Equations


∂∂t

ρdvV∫ = − ρu ⋅nds

S∫

∂∂t

ρudv∫ + ρu(u ⋅n)ds∫ = ρ f dv∫ + nTds∫

Control volume V

Control surface S

Conservation of mass

Conservation of momentum

�

T = −p + λ∇⋅u( )I + 2µD

�

D =12

∇u +∇uT( )

Stress Tensor

Deformation Tensor

�

Dij =12

∂ui∂x j

+∂uj∂xi

⎛

⎝ ⎜ ⎞

⎠ ⎟



Incompressible flows:

Navier-Stokes equations (conservation of momentum)

�

DρDt

= 0

DρDt

+ ρ∇⋅u = 0

�

∇ ⋅u = 0

�

ρ ∂u∂t

+ ρu∇u = −∇p + ρ fb + ∇ ⋅ µ ∇u + ∇Tu( )�

∂ρu∂t

+ ∇ ⋅ ρuu = ρ f + ∇ ⋅T

�

∂ρ∂t

+ ∇ ⋅ (ρu) = 0

Conservation of mass :

Governing Equations!Computational Fluid Dynamics I!

�

ρ ∂u∂t

+ u∂u∂x

+ v ∂u∂y

⎛

⎝ ⎜

⎞

⎠ ⎟ = −∂p

∂x+ ∂∂x

µ ∂u∂x⎛ ⎝ ⎜

⎞ ⎠ ⎟ +

∂∂y

µ2

∂v∂x

+ ∂u∂y

⎛

⎝ ⎜

⎞

⎠ ⎟ + fx

∂u∂x

+∂v∂y

= 0

The two-dimensional Navier-Stokes Equations in component form

�

ρ ∂v∂t

+ u∂v∂x

+ v ∂v∂y

⎛

⎝ ⎜

⎞

⎠ ⎟ = −∂p

∂y+ ∂∂y

µ2

∂v∂x

+ ∂u∂y

⎛

⎝ ⎜

⎞

⎠ ⎟ +

∂∂y

µ ∂v∂y⎛

⎝ ⎜

⎞

⎠ ⎟ + fy

Incompressibility


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The conservation equations for mass and momentum apply to any flow situation, including flows of multiple immiscible fluids. Each fluid generally has properties that are different from the other constituents and the location of each fluid must therefore be tracked. We usually also have additional physics that must be accounted for at the interface, such as surface tension.

The “regular” conservation equations can be extended to handle these situations by using generalized functions

Sharply stratified flows


�

∂H∂t

+ u ⋅ ∇H = 0

Identify each fluid by a marker function H

The marker moves with the fluid and is updated by integrating the following advection equation in time

�

H =1 in fluid10 Otherwise⎧ ⎨ ⎩

Updating H—in spite of its apparent simplicity—is one of the hard problems in CFD!



In addition to advect the marker function accurately, we must often account for physics unique to the interface. The most common example is surface tension.

Surface Tension

m

t

n u const.

v const.

xv

xu

�

k = −∇ ⋅n�

xu = ∂x∂u; xv = ∂x

∂v

�

n = xu × xvxu × xv

�

x u,v( ) = x u,v( ),y u,v( ),z u,v( )( )Tangent vectors

Normal to the surface

It can be shown that: kn = lim

δA→0m∫ ds

Definition of a surface

and


The “One-Fluid” approach!


H (x, y,t) = δ (x − x ')δ (y − y ')da 'A(t )∫

∇H = ∇[δ (x − x ')δ (y − y ')]da 'A∫

= − ∇ '[δ (x − x ')δ (y − y ')]da 'A∫

= − δ (x − x ')δ (y − y ')S∫ nds'

= − δ (x − x ')δ (y − y ')S∫ nds'

= − δ (s)δ (n)S∫ nds'

= −δ (n)n

A!

S!

∂H∂t

+u ⋅∇H = 0

H=1! H=0!

�

δ(x − x')δ(y − y') = δ(s)δ(n)Using:!

n s

x

y


The conservation equations are solved on a regular fixed grid and the front is tracked by connected marker points. !The “one fluid” formulation of the conservation equations is the starting point for several numerical methods, including MAC, VOF, level sets, and front tracking.!

�

ρ ∂u∂t

+ ρ∇ ⋅uu = −∇p + f + ∇ ⋅ µ ∇u + ∇Tu( ) + σF∫ κnδ x − x f( )da

�

∇ ⋅u = 0

�

DρDt

= 0; DµDt

= 0

Conservation of Momentum!

Conservation of Mass!

Equation of State:!

Singular interface term!

Incompressible flow!

Oscillating drop: pressure!

Constant properties!


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The “one-fluid” formulation implicitly contains the proper interface jump conditions. Integrating each term across a small control volume centered at the interface:!

�

ρ DuDt

dvδV∫ = − ∇pdv

δV∫ + fdvδV∫ + ∇ ⋅ µ ∇u + ∇Tu( )dvδV∫ + κσ nδ n( )dv

δV∫=0! =0!

�

p[ ]n

�

µ ∇u + ∇Tu( )[ ]n

�

κσ n

Jump Condition:!

�

−p + µ ∇u + ∇Tu( )[ ]n = −κσn


�

u = H1u1 + H2u2P = H1p1 + H2p2ρ = H1ρ1 + H2ρ2

Write:

Substitute into the momentum equation

�

H1 ………( ) + H2 ………( ) + δ x f( ) ………( ) = 0

Interface conditions

Momentum equation in phase 2

Momentum equation in phase 1

=0 =0 =0

We can also show that the “one-fluid” formulation contains the equations written separately for each fluid and the jump conditions:



Numerical Solutions


Work with the finite volume approximation

�

u = 1V

udVV∫

Dc =

1V

∇ ⋅µ ∇hu+∇hTu( )dV

V∫ =

1V

µ ∇hu+∇hTu( )nds

S∫

Ac =

1V

∇ ⋅ uu( )dVV∫ =

1V

u u ⋅n( )dsS∫

∂∂t

ρudv∫ + ρu(u ⋅n)ds∫ =

ρ f dv∫ + µ ∇u +∇Tu( )nds∫ + σκ nδ n( )dvV∫

Discretize each term

Discretization in time


∇h ⋅ui , jn +1 = 0


�

ui, jn+1 −ui, j

n

Δt= −A i, j

n − 1ρn (∇h p +Di, j

n + fσn ) + fb

n

Summary of discrete vector equations

No explicit equation for the pressure!

Evolution of the velocity—first order explicit in time:

Constraint on velocity

�

u = 1V

udVV∫

�

Dc = 1V

∇ ⋅ µ ∇hu +∇hTu( )dV

V∫

= 1V

µ ∇hu +∇hTu( )nds

S∫

�

A c = 1V

∇ ⋅ uu( )dVV∫ = 1

Vu u ⋅n( )ds

S∫


�

ui, jn+1 −ui, j

tmp

Δt= − 1

ρn ∇h pi, j�

ui, jtmp −ui, j

n

Δt= −A i, j

n + fbn + 1

ρn Di, jn + fσ

n( )The momentum equation is solved in two steps

and Projection Method

To derive an equation for the pressure we take the divergence of the second equation and use . The result is:

�

∇h ⋅ui, jn+1 = ∇h ⋅ui, j

tmp −Δt ∇h ⋅1ρn ∇h pi, j

⎛

⎝ ⎜

⎞

⎠ ⎟

0

�

∇h ⋅1ρn ∇h pi, j

⎛

⎝ ⎜

⎞

⎠ ⎟ = 1

Δt∇h ⋅ui, j

tmp

∇h ⋅ui , jn +1 = 0


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1. Find a temporary velocity using the advection and the diffusion terms only:

2. Find the pressure needed to make the velocity field incompressible

3. Correct the velocity by adding the pressure gradient:

�

ui, jn+1 = ui, j

tmp − Δtρn ∇h pi, j�

∇h ⋅1ρn ∇h pi, j

⎛

⎝ ⎜

⎞

⎠ ⎟ = 1

Δt∇h ⋅ui, j

tmp�

ui, jtmp = ui, j

n + Δt −A i, jn + fb

n + 1ρn Di, j

n + fσn( )⎛

⎝ ⎜

⎞

⎠ ⎟

4. Update the marker function to find new density and viscosity

Discretization in time Computational Fluid Dynamics I!

Advecting the marker function!


�

∂f∂t

+ u ⋅ ∇f = 0

Introduce a numerical marker function f, approximating H. The advection of f is governed by:!

Integrating this equation in time, for a discontinuous initial data, is one of the hard problems in computational fluid dynamics!!

Advecting the Marker Function Computational Fluid Dynamics I!

The sharp marker function H can be approximated in several different ways for computational purposes. Below we show a smoothed marker function, I, the volume of fluid approximation, C, and a level set representation, Φ. !

Advecting the Marker Function


In 1D, using upwind and LW.!The solution quickly deteriorates.!Modern advection methods help, but not completely.!


Finite Thickness! “Zero” Thickness!

Marker Function!Marker Points!

Level Set! Volume of Fluid!Phase Field!Front Tracking!

Methods that use the “one-field” or “weak” formulation of the Navier-Stokes equations!


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Volume of Fluid!


Upwind

To advect a discontinuous marker function, first consider 1D advection. Using simple upwind leads to excessive diffusion due to averaging the function over each cell, before finding the fluxes!



One-dimensional Volume-Of-Fluid!

Since the marker function only takes on two values, 0 and 1, the advection can be made much more accurate by “reconstructing” the function in each cell before finding the fluxes, integrated over time:!


While VOF works extremely well in one-dimension, there are considerable difficulties extending the approach to higher dimensions. The basic problem is the “reconstruction” of the interface in each cell, given the volume fraction in neighboring cells. !

In the SLIC method the interface was taken to be perpendicular to the advection direction.!In the Hirt/Nichols method the interface was taken to be parallel to one axis. !In PLIC the interface is a line with arbitrary orientation.!

Once the interface has been reconstructed, the marker function is advected by geometric considerations!


Original!SLIC!

PLIC!Hirt/Nichols VOF!


Level Set Methods!

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Identify the interface as a “level-set” of a smooth function!

Advect the level set function by!

use!

to get!


The level set function can be arbitrarily smooth. To identify each fluid it is necessary to construct a marker function with a narrow transition zone!

�

φ

�

I φ( )

The marker function can be generated by (for example):!

The delta function is generated as the derivative of the marker function !


The level set function for two circles, as a distance function!


To keep the interface shape the same, it is necessary to “reinitialize” the level set function. This is usually done by making it a distance function.!

At each time step, solve:!

For most applications, the shape of the level set functions must remain the same close to the interface!


Front-Tracking Methods!


The method has been used to simulate many problems and extensively tested and validated.!

See also Tryggvason et al. (2002) for details, tests and applications!

Tracked front to advect the fluid interface and find surface tension!

Fixed grid used for the solution of the Navier-Stokes equations. Relatively standard explicit finite volume fluid solver!

Front Tracking (Unverdi & Tryggvason, 1992) !

Numerical Method!

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�

φl = φijk∑ wijk

The velocities are interpolated from the grid:!

The front values are distributed onto the grid by!

φijk = φl∑ wijkΔslh3

On the front: per length!On the grid: per volume!

the weights wijk can be selected in several different ways!

Interpolating from grid!

Tracked Front!

Finite Difference Grid!

Computational Fluid Dynamics I!Numerical Method!


Other Methods for the

Advection of the Marker

Function!


The CIP (Constrained Interpolation Polynomial) !Method (Yabe)!In addition to advecting the marker function f, its derivative is advected by fitting a third order polynomial through the function and its derivatives.!Start with!

Introduce!In 1D, the advection of the derivative is given by!

Therefore, the derivative is translated with velocity u, just as the function. In 2D splitting is used to separate translation and deformation!

f and g given!

New f and g!


The CIP method results in very accurate advection and for a sharp interface it greatly reduces overshoots, but does not eliminate them completely!


The phase field Method (Jacqmin)!Solve modified Navier-Stokes equations, developed by thermodynamic considerations at the microscale!

and!

The energy function can take several different forms, for example, if:!

and!Then it can be shown that surface tension and interface thickness are:!

and

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Several other methods have been developed to improve the performance of those described here, including hybrid methods, such as Particle Level Set and VOF-LS, as well as methods that capture the interface more sharply, such as the Ghost Fluid Method and the Immersed Interface Method.

Similar approach has also been used to capture rigid and elastic boundaries, both moving and stationary.


Standard Tests for advection!


Zalasakʼs test: A notched circular blob is advected by a solid body rotation, measuring how the blob deteriorates!


High order advection (PPM)!

Level Set! PLIC!Markers!

From W. J. Rider and D.B. Kothe. Reconstructing volume tracking. Los Alamos National Laboratory Report la-ur-96-2375. Technical report, 1996!



Surface Tension


Singular interface forces are approximated by a smoothed delta function that becomes “more singular” as the smoothing is reduced

�

k = −∇ ⋅n

�

∂H∂t

+ u ⋅ ∇H = 0

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Parasitic Currents!

The regular grid induces a small anisotropy. These currents are typically small in immersed boundary methods.!

Stationary drop!

Weak parasitic !currents!

Numerical Method—Surface Tension!

�

∇h p + fσn = 0

At steady state the pressure gradient should be balanced by surface tension!


Surface tension can be added in several ways:!


�

fij = σκ( )ij∇Iij Curvature and normal vector computed directly on the fixed grid—marker function!

Curvature and normal vector computed on the front, distributed to the fixed grid—our original approach!

Curvature computed on the front, distributed to the fixed grid. Normal vector computed on the grid!

�

fij = σκ( )ijf∇Iij

f

�

fij = σκ( )ijf∇Iij

The last approach seems to combine the accuracy of tracking with the possibility of balancing pressure exactly!


For the solution of the Navier-Stokes equations, we need the net force on each front segment:!

δ f f = σκndsΔS∫ =

σ s2 − s1( ) 2D

σ s × nds 3DS∫

⎧⎨⎪

⎩⎪κn = ∂s

∂s-s2!

s1!

s!n!

m=s x n!


σκ( )ij =± δ fp( )wij

p

front∑

Δs pwijp

front∑δ f f = σκnds

ΔS∫ ≈ σκnΔs Distribute to the grid using!

The force can be distributed directly to the fixed grid. Or, we can distribute only the magnitude and find the normal on the grid !


Changing the interface

topology for 3D flow!


In general, the interface separating two fluids will undergo topology changes where two regions of one fluid coalesce, or one region breaks in two. Of those, the coalescence problem appears to be the harder one.!

In their simples implementation, explicit tracking method never allow coalescence and method based on a marker function always coalesce two interfaces that are close.!

In reality, films between two fluid interfaces take a finite time to drain and rupture only when the thickness is sufficiently small so the film is unstable to non-continuum attractive forces. In general this draining can not be resolved and must be modeled.!

Topology Changes!Computational Fluid Dynamics I!

Binary Collision of Drops

Topology changes in Front Tracking computations—other work!•  Nobari and Tryggvason (1996) coalescence only!•  Glimm and collaborators (FronTier)!•  Shin and Juric: Contour Reconstruction!•  Hua, Stene and Lin: Front tracking!

Topology Changes!

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Effect of resolution on the coalescence of two drops. In the last column (the third simulation) the criterion for coalescence of drops is the same as the first simulation (first column). Here the non-dimensional numbers of the flow are: !Re=272., !We= 23., !α =40., λ = 20. !

64 × 32 × 32! 128 × 64 × 64! 128 × 64 × 64!

Topology Changes!Computational Fluid Dynamics I!

G. Agresar, J.J. Linderman, G. Tryggvason, and K.G. Powell. An Adaptive, Cartesian, Front-Tracking Method for the Motion, Deformation and Adhesion of Circulating Cells. J. Comput. Phys. 1998

The basic approach described here can be used with advanced methods developed for constant density fluids, including higher order advection methods and adaptive mesh refinement (AMR)

Adaptive Mesh Refinement


Although methods based on the one fluid formulation have been used successfully for many problems, several challenges remain. These are slowly being eliminated

•  High Reynolds numbers: high order advection methods and non-conservative form of the advection terms

•  Continuity of the viscous stresses: interpolation using the harmonic mean

•  Solution of the pressure equation/slow convergence at high density ratios: More advanced fast solver

•  Parasitic currents: increased smoothing helps—or separate computations of the curvature and the normal

Computational Fluid Dynamics I!DNS of Bubbly Flow

Direct Numerical Simulations of

Multiphase Flows

G. Tryggvason R. Scardovelli

S. Zaleski

Cambridge 2009

TO APPEAR 2010

Documents

Introduction to DNS of Multiphase Flows-Igtryggva/CFD-Course2010/2010...Introduction to DNS! of! Multiphase Flows-I! Grétar Tryggvason! Spring 2010! Computational Fluid Dynamics I!