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Lattice Boltzmann

- Karin Erbertseder
- Ferienakademie 2007

Outline

- Introduction
- Origin of the Lattice Boltzmann Method
- Lattice Gas Automata Method
- Boltzmann Equation
- Explanation of the Lattice Boltzmann Method
- Comparison between Lattice Boltzmann Method and

Navier-Stokes-Equations - Applications

Introduction

- Computational Fluid Dynamics
- (CFD)
- solution of transport equations
- simulation of mass, momentum and energy transport

processes - Applications
- automotive, ship and aerospace industry,

material science, - Advantage
- prediction of flow, heat and mass transport
- fundamental physical understanding
- optimization of machines, processes,

source www.ansys.com

Introduction

Experiment vs. Simulation

- Experiment
- measurement often difficult or impossible
- expensive and time consuming
- parameter variations extremely expensive
- measurement of only a few quantities at

predefined locations

- Numerical Simulation
- compliance of similarity rules is no problem
- less expensive and faster
- easy parameter variation
- provides detailed information on the entire flow

field

Introduction

General Procedure

Flow Problem

Solution of the Problem

mathematical model, measured data

Conservation Equations

Visualization Analysis Interpretation

discretization, grid generation

software, computer

Algebraic System of Equations

Numerical Solution

algorithms

Introduction

- Macroscopic Methods
- e.g. Navier-Stokes fluid simulation (FDM, FVM)
- Mesoscopic Methods
- e.g. Lattice Boltzmann method
- Microscopic Methods
- e.g. molecular dynamics

Lattice Gas Automata (LGA)

- Cellular Automata (CA)
- idealized system where space and time are

discrete - regular lattice of cells characterized by a set

of boolean state variables 1 or 0 particle at a

lattice node - Lattice Gas Automata (LGA)
- special class of CA
- description of the dynamics of point particles

moving and colliding in a discrete space-time

universe

Lattice Gas Automata (LGA)

streaming step flow simulation by moving

representative particles one node per time step

collision step

source www.cmmfa.mmu.ac.uk

Lattice Gas Automata (LGA)

- advantages
- stability
- easy introduction of boundary conditions
- high performance computing due to the intrinsic

parallel structure

- disadvantages
- statistical noise
- lack of Galilean invariance
- velocity dependent pressure

motivation for the transition from LGA to

LBM removal of the statistical noise by

replacing the Boolean particle number in a

lattice direction with its ensemble average

density distribution function all disadvantages

are improved or vanish

Boltzmann Equation (BE)

- definition
- description of the evolution of the single

particle distribution f in the phase space by a

partial differential equation (PDE) - particle distribution function f (x,?,t)
- probability for particles to be located within a

phase space control element dxd? - about x and ? at time t where x and ? are the

spatial position vector and the particle velocity

vector - macroscopic quantities, like density or

momentum, by evaluation the first moments of the

distribution function

Boltzmann Equation (BE)

collision term interaction between the molecules

time variation

spatial variation

effect of a force acting on the particle

velocity vector of a molecule

force per unit mass acting on the particle

position of the molecule

f f (x, ?, t) distribution function

The collision term is quadratic in f and has a

complex integrodifferential expression simplifica

tion of the collision term with the

Bhatnagar-Gross-Krook (BGK) model

Lattice Boltzmann Method (LBM)

assumptions - neglect of external forces - BGK

model (SRT single-relaxation-time

approximation) - velocity discretization using

a finite set of velocity vectors ei

- movement of the particles only along the

lattice vectors - modeling of the fluid by

many cells of the same type - update of all

cells each time step - storage of the number

of particles that move along each of the

lattice vectors particle distribution

function f

velocity discrete Boltzmann equation

Lattice Boltzmann Method (LBM)

- common lattice nomination DXQY

number of distinct lattice velocities

number of dimensions

model for two dimensions

f4 4 3 f3 2 f2

D2Q9 - most common model in 2D - 9 discrete

velocity directions - eight distribution

functions with the particles moving to the

neighboring cells - one distribution function

according to the resting particle

e4 e3 e2

e5

f5 5

1 f1

e1

e6 e7 e8

f6 6 7 f7

8 f8

source J.Götz 2006

Lattice Boltzmann Method (LBM)

models for three dimensions D3Q15 D3Q19 D3Q2

7 small range of good compromise highest

stability between the two computational

models effort

19 distribution functions one stationary

velocity in the center for the particles at

rest 6 velocity directions along the Cartesian

axes

12 velocities combining two coordinate

directions resting particles dont move in the

following time step, but changing amount of

resting particles due to collisions

source J.Götz 2006

Lattice Boltzmann Method (LBM)

next step calculation of the density and

momentum fluxes in the discrete velocity

space starting point velocity discrete

BE equilibrium distribution function for D2Q9

model

discrete particle velocity vector

weighting factor

4/9 i 0 1/9 i 1, 3, 5, 7

1/36 i 2, 4, 6, 8

lattice speed with the lattice cell size x

and the lattice time step t

Lattice Boltzmann Method (LBM)

calculation of the density and the

momentum density momentum

Discretization discretization in time and space

leads to the lattice BGK equation

dimensionless relaxation time

point in the discretized physical space

Lattice Boltzmann Method (LBM)

- lattice BGK equation is solved in two steps
- collision step
- streaming step
- collision step
- interpretation as many particle collisions
- calculation of the equilibrium distribution

function for each cell and at each time step from

the local density ? and the local macroscopic

flow velocity u using the equations of the slide

before

values after collision and propagation, values

entering the neighboring cell data for the next

time step

distribution values after collision

Lattice Boltzmann Method (LBM)

- streaming step
- streaming of the particles to their neighboring

cells according to their velocity directions - lattice vector 0 no change of its particle

distribution function in the streaming step

particle distribution before stream step

particle distribution after stream step

source J.Götz 2006

LBM Parametrization

- standard parameters describing a given fluid flow

problem - size of a LBM cell ?x m
- fluid density ? kg/m3
- fluid viscosity ? m2/s
- fluid velocity u m/s
- strength of the external force g m/s2
- lattice time step ?t, lattice density ?,

lattice cell size ?x constant during simulation

- no multiplications with real world values of

the time step, the density, the lattice size are

necessary

LBM Parametrization

- calculation of the dimensionless lattice values
- lattice viscosity
- lattice velocity
- lattice gravity
- relation of all lattice values to the physical

ones - calculation of the physical time

step restricted time step depending on the

maximal lattice velocity

lattice viscosity, lattice velocity, lattice

gravity are dimensionless lattice velocity of

0.3 means that the fluid moves 0.3 lattice cells

per time step

LBM Parametrization

- calculation of the lattice viscosity ?
- Calculation of the relaxation time
- fluid velocity v is given calculation of

the relaxation time needed for a simulation with

the formula above - due to stability reasons

relaxation time

speed of sound 1/v3

LBM Boundary Treatment

- no-slip
- no movement of the fluid close to the

boundary each cell next to a boundary has the

same amount of particles moving into the boundary

as moving into the opposite direction

zero velocity (along the wall and in wall

direction) - reflection of all distribution functions at the

wall in the - opposite direction
- sourceN.Thürey 2005

LBM Boundary Treatment

- free slip
- reflection of the velocities normal to the

boundary - boundaries with no friction (zero velocity only

in wall direction) - inflow
- given velocities calculation of the

distribution function based on the equilibrium

function (only on special type) - outflow
- several different types

sourceN.Thürey 2005

LBM Boundary Treatment

- periodic
- particles that leave the domain through the

periodic wall reenter the domain at the

corresponding periodic wall copying the PDFs

leaving the domain to the corresponding cells

during the streaming step

source C.Feichtinger 2006

Navier-Stokes Equations (NSE)

- description of the macroscopic behavior of an

isothermal fluid - conservation of mass
- incompressible fluid (? constant)
- momentum equation

velocity in i-direction (i 1,2,3 for x,y,z)

density

viscous stress tensor

advection pressure momentum

forces acting due to molecule

upon the fluid movement

Comparison between LBM and NSE

Navier-Stokes Equations Lattice Boltzmann

Method

- second order partial differential equations
- non-linearity quadratic velocity terms
- need to solve the Poisson equation for pressure

calculation - global solution for all lattice cells grid

generation needs longer than simulation

- set of first order partial differential equations
- linear non linear convective term becomes

a simple advection - pressure through an equation of state
- regular square grids
- kinetic-based easy application to micro-scale

fluid flow problems - complicate simulation of stationary flow problems

Applications

- Java-Simulation

Applications

source N.Thürey

Applications

- metal foam simulation

sourceN.Thürey 2005 - D3Q19 model
- free-surface model
- filled with fluid
- interface contains both liquid and gas
- gas not considered in fluid simulation
- computation of the fill level of a cell by

dividing by the density of this cell (0 empty

cell ? filled cell) - transformation of fluid and gas cells into

interface cells and vice versa

Applications

source N.Thürey

Thanks For Your Attention

any questions ?

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