Simulation of floating bodies with lattice Boltzmann - FAU · Simulation of floating bodies with lattice Boltzmann ... – Hydrostatic floating stability ... Simulation of floating

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Simulation of floating bodies with

lattice Boltzmann

bySimon Bogner,

17.11.2011,

Lehrstuhl für Systemsimulation, Friedrich-Alexander Universität Erlangen

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Simon Bogner - Lehrstuhl für Systemsimulation - Friedrich-Alexander Universität Erlangen-Nürnberg

Simulation of floating bodies with lattice Boltzmann

■ Lattice Boltzmann Method (LBM)– Kinetic origin of the lattice BGK Method

■ Multiphase Flow– 3 Phases: Liquid, gas and solid (rigid bodies)– Cell conversion scheme– Simulations with waLBerla and pe

■ Floating Bodies– Hydrostatic floating stability– Evaluation of forces

■ Outlook & Conclusion– Further applications

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Kinetic Origin of Lattice Boltzmann

■ Navier-Stokes equation for fluids– Continuum assumption: Macroscopic Variables are defined at every

point in space inside the medium– Behavior of the flow is described as equation of the macroscopic

variables

■ Boltzmann:– Microscopic assumption: Fluid is made up of particles (molecules)– Particles of mass m, defined by position and velocity.– Statistical mechanics: Description of kinetic behavior by means of

probabilistic methods (Large number of particles!)– Kinetic particle distribution function– is the number of particles within the volume element

around and velocity within around .x d v vd x

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Boltzmann Equation

■ Boltzmann: Macroscopic variables are moments of the statistical distribution function

■ Boltzmann Equation:

– LHS: transport term– RHS: collision term (hidden)

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BGK Collision Model

■ Collision term:– Describes the changes in the particle motion due to collisions– Boltzmann: Time-independent two-body collisions (“Stosszahl

Ansatz”)■ Bhatnagar Gross Krook – Model

– H-Theorem: Thermodynamic systems strive towards a state of Equilibrium (Entropic behavior)

– Equilibrium state solution given by a Maxwell-Boltzmann distribution

(distribution of particle velocities in thermodynamic equilibrium)

■ BGK - Equation

– Collision term: Linear BGK - relaxation towards equilibrium

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From the BGK Equation to Lattice Boltzmann

■ BGK Equation

– is function of

■ Discretization of velocity space– Restriction of particle velocities to finite set .– Set must span a discrete lattice (~grid) of cells.– Lattice velocities “connect” the cells.– Discrete set of particle distributions functions at each cell– D3Q19 – Model shown in figure

{c⃗i}

f ( x⃗ , v⃗ )

{ f i( x⃗ , t)}

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Lattice Boltzmann Scheme

■ Discretized equilibrium function

– Discrete approximation of around (low Mach number expansion) .

– is the lattice speed of sound (model dependent constant).

(… Skip lots of lots of mathematics … )Skip lots of lots of mathematics … )

u⃗=0f eq

c s

■ Stream and Collide Algorithm– Streaming:

– Collision:

– Dimensionless lattice relaxation time is related to the viscosity– Lattice Boltzmann scheme

τ

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Moments of the Lattice Boltzmann Model

■ Local macroscopic quantities are moments of the discrete particle distribution functions (PDFs):

■ Approximates Navier-Stokes in the incompressible limit.(See Hänel, D. Molekulare Gasdynamik; Succi, S. Lattice Boltzmann Equation for Fluid Dynamics and Beyond)

■ Mesoscopic Method

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Three - Phase Flow

■ Liquid-Gas-Solid Simulation– Free Surface Flows

● Everyday life: water & air● 2 immiscible fluids (a liquid and a gas)● Examples: river, bubbles & foam,

– Particulate Flows (Rigid Bodies)● Suspensions (e.g., paint, blood, colloids)

– Rigid Bodies in Free Surface Flow● Non-deformable Newtonian body physics● Examples: Ship, Weizenbier

Pictures taken from “Physics of Continuous Matters” (Benny Lautrup), and Wikipedia.

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Liquid-Gas-Solid Lattice Boltzmann

■ Boltzmann method used to simulate the liquid phase■ Different cell types control the system behavior■ Figure: floating box in discrete lattice

– Gas, liquid, and solid cells represent the three phases– Interface cells model the free surface boundary– Computation uses a flag field to store the cell type of each cell– Flag field is updated dynamically

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■ Boltzmann method used for the liquid phase■ Interaction with other phases via boundary conditions■ Three phase transitions:

– Liquid-gas boundary (free surface)– Liquid-solid boundary (obstacle walls)– Solid-gas (no boundary for LBM scheme!)

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Liquid-Gas Boundary

■ Free surface boundary

– Boundary treatment is done at theinterface cells.

– PDFs only in liquid and interfacecells

– No PDFs defined in gas cells!

■ Free Surface Boundary Condition– Construct PDFs pointing towards liquid phase

from streamed PDFs according to

where incorporates the gas pressure, and is the local flow velocity.

– No tangential stresses at free surface boundary

ρG=1/cs2⋅pG

u⃗( x⃗ )

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Liquid-Gas Boundary

■ Free Surface Boundary– Second moment of distribution functions:

– Split sum for momentum flux and stress tensor, respectively

– For equilibrium all stresses vanish (S=0), and p is the gas pressure.

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Liquid-Solid Boundary

■ Particle reflection at obstacles– Bounce back rule realized as a modified

stream step (figure)– Reflection is given by

with boundary velocity .– Flow velocity near wall equals boundary

velocity (no slip)

u⃗w

■ Momentum transfer Elastic collision of PDFs at the surface. Change in momentum

Force exerted locally onto boundary. Momentum exchange method: Calculate boundary stress directly from all PDF

reflections at a given surface.

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Multiphase Flow – Cell Conversion Scheme

■ Lattice configuration

■ Cell state (liquid, gas, interface, obstacle) stored as flag value in each cell

■ May change during simulation■ Free surface movement■ Rigid body movement

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Free Surface Movement

■ Free Surface Flow Model

– Volume of Fluid approach– Interface cells: Additional fill value stores the amount of

liquid in a cell, such that

■ Mass Exchange

– Mass balance is calculated during the stream step according to

– Fill level changes according to free surface movement

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Free Surface Movement

■ Free Surface Flow Model

– Mass balance is calculated during the stream step according to

– Interface cell converts to liquid, if .– Interface cell converts to gas, if .– May trigger further conversions to close interface layer

(assure valid boundary!).

– No direct conversions from liquid to gas or vice versa!

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Obstacle Movement

■ Obstacles are mapped to the lattice– Cell is treated as obstacle, if the center of the cell is

inside of the object shape (e.g., box shape, sphere shape, …)

■ Obstacle movement calculated from the fluid stresses– Physics engine calculates movement

from given surface stresses.

– Lattice has to be updated accordingto obstacle movement.

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Conversion Scheme with Obstacles

■ Free Surface: No direct transition between liquid and gas.

■ Remaining transitions are from obstacle to fluid, and back.

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Conversion Scheme with Obstacles

■ Consider a spherical particle with rightwards movement.

■ Direct conversions from fluid into obstacle (continuous lines) regardless of fluid state.

■ Conversions from obstacle back to fluid are critical (broken lines).

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Cell Conversion Algorithm

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Cell Conversion Algorithm

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Simulations with waLBerla and pe

■ Widely Applicable Latttice Boltzmann from Erlangen■ p.e. - Rigid body physics engine■ Software projects of the “Lehrstuhl für Systemsimulation”,

University of Erlangen-Nürnberg

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Showcase 1

■ 4096 Particles dropped into a Basin■ Spherical particles, radius 6 lattice units■ Red particles are heavier, green ones more lightweight■ Computed on 32 woodcrest processes■ ~ 3 days computation time

Watch online: http://youtu.be/leORsCgdRQM

http://youtu.be/leORsCgdRQM

launchParticles.sh

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Showcase 2

■ Bubbe Rise in Particle Array■ Freely floating spherical particles (neutral material

density)

Watch online: http://youtu.be/MTOiDjcVuXU

http://youtu.be/MTOiDjcVuXU

launchBubble.sh

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Floating Stability - Literature

■ Floating Positions of Rigid Bodies

– Application: Hydrostatic Floating stability (as known from marine engineering)

– J.M.J. Journée and W.W. Massie: Offshore Hydromechanics, www.shipmotions.nl(some pictures and formulae taken from this book)

– Captain D.R. Derrett and C.B. Barrass: Ship Stability for Masters and Mates– Simon Bogner, Ulrich Rüde. Liquid-gas-solid flows with lattice Boltzmann –

Simulation of floating bodiesICMMES proceedings 2011 (submitted article under review)

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Floating Bodies

■ Buoyancy Force– Archimedes: Lifting force equals the weight of the displaced fluid

mass

– Force acts at the center of buoyancy B– Partial Immersion: B is different from the center of gravity G

■ Floating behavior of half immersed cube– Assumption: Equilibrium of buoyancy and weight (vertical balance)– Unstable and stable equilibrium

(Equilibrium position for cube of specific density 0.5)

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Floating Bodies – Rotational Stability

■ Unstable Equilibrium of Cube:

– Construction of B as the center of gravity of the immersed trapezoid

● Elongate each of the parallel sides (u and v) by its opposite in opposed directions

● Connect the newly obtained endpoints● Intersection with middle line of the parallel

sides gives B

– Horizontal displacement of G versus B

– Result: Rotational moment in the direction of heel.

➔ Upright position is unstable➔ 45° position is stable

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Floating Bodies – Righting Moment

■ Heel from stable floating position

– Righting Moment opposes the heeling moment

– Shift of masses from to (shift of B)– Metacenter : Intersection of line

with the corresponding line of the upright position.

M H

N ϕ

Bϕ+α⋅ρ g⃗∇=Bϕ−α⋅F⃗ B

■ Righting Stability Moment

– (Momentum: )– Important for the stability of offshore

structures (ships, barges, ..)

M S=ρ g∇⋅GZ= F⃗ B⋅GZ

z e z i

lever arm× force

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Floating Structures – Stability Formula

■ Wall-sided structure– Parallel side walls in upright position– Immersed and emerged volume parts

are wedges with triangular front side

■ Scribanti Formula

– Compute the metacenter for a given angle of heel.

– is the moment of inertia of the water plane.

– From follows and the righting moment

■ Example: Floating Box

I T

N ϕ

BN ϕ Bϕ

M S=ρ g∇⋅GZ

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Floating Structures – Floating Box

■ Stability of Floating Box– For a cuboid, the Stability Formula can

be written as

with

– L, B and T are the length, width and draft of the box

■ Stability Curve– Righting moment at a given angle of

heel– Cube (b:h = 4:4): negative righting

moment; upright position unstable– Increased width means more stability

α

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Floating Structures – Evaluation of Torque

■ Validation of righting moment of box structures

■ Half immersed box

■ Angle of heel 0°..30°

■ Tested b:h ratios 6:4 (a) and 5:4 (b)

■ Resolution of box in lattice units:– 24x16, 48x32, 96x64 (a)

– 20x16, 40x32, 80x64 (b)

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Floating Structures – Evaluation of Torque

■ Ideal Stability Curve Versus Simulation

– Higher relative errors in (b) because of lower floating stability.– Convergence to ideal curve

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Floating Structures – Convergence Test

■ Check for convergence of three-phase system (ideal floating positions)

■ Equilibrium for cube of density 0.5■ For cube of density 0.25, the stability curve

shows has a root at 26.57°(~“Angle of Loll”)

■ Same angle for density 0.75

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Floating Structures – Convergence Test

■ Demonstration of convergence to ideal floating positions■ Cubes of density 0.25, 0.5, and 0.75■ Low resolution of particles: 16x16 lattice units

Watch online: http://youtu.be/5F-qHsPIrYE

http://youtu.be/5F-qHsPIrYE

launchStability.sh

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Conclusion

■ Liquid-Gas-Solid Method so far...

– Arbitrary shaped rigid bodies or particles– Free surface flows– Ready for parallel computation (tested on woodcrest cluster)

■ Outlook

– Further development for bubbly flows (foams), like, e.g., flotation processes or chemical reactors

– Surface tension and contact line behavior with particles– Further validation, e.g., floating objects motion in waves, bubble-

particle interaction, …– ...– ...

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Thanks for Listening

Thank you for your attention!

Have a nice and pleasant evening.

http://www10.informatik.uni-erlangen.de

http://www10.informatik.uni-erlangen.de/

Documents

Simulation of floating bodies with lattice Boltzmann - FAU · Simulation of floating bodies with lattice Boltzmann ... – Hydrostatic floating stability ... Simulation of floating