Lattice Boltzmann

1
Lattice Boltzmann

• Karin Erbertseder
• Ferienakademie 2007

2
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

3
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
4
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

5
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
6
Introduction

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

7
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

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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
9
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
10
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

11
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
12
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
13
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
14
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
15
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
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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
17
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
18
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
19
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

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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
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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
22
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

23
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
24
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
25
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
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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
  

27
Applications

• Java-Simulation

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Applications
source N.Thürey
29
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

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Applications
source N.Thürey
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Thanks For Your Attention
any questions ?