Introduction to Lattice Boltzmann Method and Application

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Introduction to Lattice Boltzmann Method and Application to Fluid Flow and Heat Transfer

Problems (A Basic Approach to Computation)

Presented by,C. Jayageeth

Tutor: Prof. S. C. Mishra

Department of Chemical EngineeringIndian Institute of Technology Guwahati

INDO-GERMAN WINTER ACADEMY 2007

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Outline

Introduction

Recent developments in the LBM

Example problems

Analysis of phase-change problems

Implementation of LBM on non-uniform lattices

Parallelization of LBM

Conclusion

References

LATTICE BOLTZMANN METHOD

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Introduction

Lattice Boltzmann method (LBM) is based on Lattice Gas (LG) methodhistorically developed from lattice gas automata (which evolved from Cellular Automata developed by Stanislaw Ulam and John Von Neumann while working on the study of growth of crystals and self-replicating systems.)

Discrete Boltzmann equation is solved to simulate the flow of a Newtonian fluid with collision models such as Bhatnagar-Gross-Krook (BGK) Model.

LBM is based on microscopic models and mesoscopic kinetic equations that incorporate the essential physics of microscopic or mesoscopic processes so that the macroscopic averaged properties obey the desired macroscopic equations.

The first LB model is HPP (Hardy, Pomeau and de Pazzis, 1973) model. It lacked rotational invariance which made the model highly anisotropic.

FHP (Frisch Hasslacher Pomeau ) model was introduced in 1986 based on Hexagonal grid.


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The lattice gas automaton is constructed as a simplified, fictitious molecular dynamic in which space, time, and particle velocities are all discrete.

Evolution equation of the LGA:

where, = Boolean variable (particle distribution function in LBM)

= local particle velocities

= Collision Operator

The relaxation term is known as the Bhatnagar-Gross-Krook (BGK) collision operator (Bhatnagar et al 1954). In this lattice BGK (LBGK) model, the local equilibrium distribution is chosen to recover the Navier-Stokes macroscopic equations.

( , 1) ( , ) ( ( , )), ( 0,1,.., )i i i in t n t n t i M+ + = +Ω =x e x x


( , ), ( 0,1,.., )in t i M=x

ie

( ( , ))i n tΩ x

Introduction

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Lattice Boltzmann Equations (LBE): An Extension of LG Automata

Where, = Particle distribution function in the ith direction = collision operator which represents the rate of

change of resulting from collision.In the LBM, space is discretized in a way that is consistent with the kinetic equation, i.e. the coordinates of the nearest neighbor points around x are x+ ei .The density and momentum density u are defined as particle velocity moments of the distribution function, ,

ui i ii i

f fρ ρ= =∑ ∑ e

( , ) ( , ) ( ( , )), ( 0,1...., )i i i if t t t f t f t i M+ Δ + Δ = +Ω =x e x x

if( ( , ))i f tΩ x

if

ρ ρif


Reference:Chen,S. and Doolen, G.D.,” Lattice Boltzmann Method For Fluid Flows”, Annu. Rev. Fluid Mech. 1998. 30:329–64

Introduction

14/12/07 6LATTICE BOLTZMANN METHOD

is required to satisfy conservation of total mass and totalmomentum at each lattice:

Kinetic equation:

Where, = Particle distribution function denoting no. of particles= Lattice node position in space

Using the single time relaxation of the Bhatanagar-Gross-Krook(BGK) approximation, the discrete Boltzmann equation given by

Where, = Relaxation time= Equilibrium distribution function

iΩ0, 0i i i

i i

Ω = Ω =∑ ∑ e

( , ) ( , ) 1, 2,3,....,ii i i

f r t e f r t i bt

∂+ ⋅∇ = Ω =

∂

rr r

ifrr

(0)( , ) 1( , ) ( , ) ( , )ii i i i

f r t e f r t f r t f r tt τ

∂ ⎡ ⎤+ ⋅∇ = − −⎣ ⎦∂

rr r r r

(0)ifτ

Introduction

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After discretization

This is the LB equation with the BGK approximation that describes the evolution of the particle distribution function

Lattice Boltzmann Equation has the following ingredients:An Evolution equation with discretized time and phase space of which configuration space is of a lattice structure and momentumspace is reduced to a small set of discrete momentaConservation constraintsA proper equilibrium distribution function which leads to the Navier-Stokes equations (Chapman-Enskog analysis)

(0)( , ) ( , ) ( , ) ( , )i i i i itf r e t t t f r t f r t f r tτΔ ⎡ ⎤+ Δ + Δ = − −⎣ ⎦

r r r r r

if


Introduction

14/12/07 8LATTICE BOLTZMANN METHOD

Lattices used in LBMDepending upon geometry lattices are used.

D1Q2 and D1Q3 are the lattices used in 1-D geometries.D2Q7 and D2Q9 are the lattices used in 2-D geometries.D3Q15 and D3Q19 are the lattices used in 3-D geometries.

1-D Planar medium: D1Q2 LatticeLattice velocities:

Weights associated with directions:

Lattice center

2er1er 1er2er

x1n N= +1n =

xΔ

2xΔ

xΔ

1f2f 1f2fTW TE

Lattice

1 2, x xe et t

Δ Δ= = −Δ Δ

r r

1 212

w w= =

Introduction

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For 2-D Rectangular medium:D2Q9 Lattice

Lattice velocities:


6 2 5

1

84

3

7

0

6er5er2er

3er

1er

7er

8er4er

1f

2f

3f

4f

5f6f

7f8f

xΔ

y xΔ = Δ

0 (0,0)e =r

1,3 ( 1,0).e c= ±r

2,4 (0, 1).e c= ±r

5,6,7,8 ( 1, 1).e c= ± ±r

049

w = 1,2,3,419

w =

5,6,7,8136

w =0

1m

ii

w=

=∑LATTICE BOLTZMANN METHOD

x yct t

Δ Δ= =Δ Δ

Introduction

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D3Q15 lattice

For 3-D geometry: D3Q15 LatticeLattice velocities:


(0,0,0), 0,( 1,0,0) ,(0, 1,0) ,(0,0, 1) , 1,...,6,( 1, 1, 1) , 7,...,14

ae c c cc

ααα

=⎧⎪= ± ⋅ ± ⋅ ± ⋅ =⎨⎪ ± ± ± ⋅ =⎩

r

2 , 0,91 , 1,...,6,91 , 7,...,1472

wα

α

α

α

⎧ =⎪⎪⎪= =⎨⎪⎪ =⎪⎩

x y zct t t

Δ Δ Δ= = =Δ Δ Δ


Reference: Premnath, K.N., and Abraham.J, “Three-dimensional multi-relaxation time (MRT) lattice-Boltzmann models for multiphase flow “, Journal of Computational Physics

,Vol. 224( 2), 10 June 2007, 539-559

Introduction

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Particle InteractionsCollision: Calculation of new distribution function

Propagation: Streaming of distribution function

Relaxation Time:In heat transfer problems, the relaxation time for the D1Q2 lattice is computed from

, = Thermal diffusivity

for the D2Q9 and the D3Q15 lattices is given by:

( ) ( )( , ) ( , ) ( , ) ( , )new eqi i i i

tf r t f r t f r t f r tτΔ⎛ ⎞⎡ ⎤= − −⎜ ⎟⎣ ⎦⎝ ⎠

r r r r

( )( , ) ( , )newi i if r e t t t f r t+ Δ + Δ =r r r


Reference: http://www.science.uva.nl/research/scs/projects/lbm_web/lbm.htmlD.A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction, Springer-Verlag, Berlin-Heidelberg, 2000.

2

32

i

teατ Δ

= +r

2 2i

teατ Δ

= +r

Introduction

α

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Application of LBM

Fluid Flow Heat Transfer

Compressible & Incompressible flows

Multiphase flows

Multi-component Flows

Porous Media Flows

Conduction

Convection

Radiation

LBM


Introduction

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Recent developments in LBM

Authors Work DoneShan Rayleigh-Benard Convection in a square

cavityHo et al. Non-Fourier Heat Conduction problem in a

planar mediumJiaung et al. Solidification of a planar mediumChatterjee & Chakraborty Solid-liquid phase transitions in the

presence of thermal diffusionMishra & Lankadasu Solved energy equation of transient

conduction and radiation heat transfer in a planar medium with or without heat generation. DTM was used to compute radiative information.


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Authors Work Done

Mishra et al. Solved energy equation of transient conduction-radiation heat transfer in a 2-D square enclosure. Collapsed Dimension Method (CDM) was used to compute radiative information.

Raj et al. Solidification of a semi-transparent planar layer was studied. DTM was used to compute radiative information

Gupta et al. Used the concept of variable relaxation time in the LBM and solved energy equation of a temperature dependent transient conduction and radiation heat transfer in a planar medium. DOM was used for determination of radiative information.



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Advantages of LBMSimple calculation procedureSimple and efficient implementation for parallel computationEasy and robust handling of complex geometriesHigh Computational performance with regard to stability and accuracyDue to particulate nature and local dynamics, it is easy to deal with complex boundaries, incorporating microscopic interactions, and parallelization of algorithm.

Drawbacks of LBMModels must satisfy Galilean invarianceReduce statistical noise



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Example Problems


Two-dimensional fluid–solid conjugate heat transfer problem.

Governing equations:Continuity equation:Momentum equation:

Energy equation for fluid:

Energy equation for solid:

Interface conditions :non-slip on the solid wall, temperature and heat flux continuities

Reference: Wang, J., Wang,M., Li, Z., “A lattice Boltzmann algorithm for fluid–solid conjugate heat transfer”, International Journal of Thermal Sciences, 2006

0u∇⋅ =2

f f Pt

ρ ρ μ∂+ ⋅∇ = −∇ + ∇

∂u u u u

2( )p f f fTc T k T Qt

ρ ∂⎛ ⎞+ ⋅∇ = ∇ +⎜ ⎟∂⎝ ⎠u &

2( )p s s sTc k T Qt

ρ ∂⎛ ⎞ = ∇ +⎜ ⎟∂⎝ ⎠&

,int ,int ,f su u= ,int ,int ,f sT T=,int ,int

f sf s

T Tk kn n

∂ ∂=

∂ ∂

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LBM MethodologyEvolution equation for fluid flowD2Q9 lattice Boltzmann model with single relaxation time collision operator (BGK model) is used to solve the NS equation for fluid flow

, = Kinematic viscosity

Evolution equation for heat transfer

( )1( , ) ( , ) ( , ) ( , )eqf r e t t t f r t f r t f r tα α α α αντ⎡ ⎤+ Δ + Δ − = − −⎣ ⎦

r r r r r

2 2( )

2 4 2

( ) 31 3 92

eqf

e ef w

c c cα α

α α ρ⎡ ⎤⋅ ⋅

= + + −⎢ ⎥⎣ ⎦

u u ur r

23 0.5vt

xτ ν Δ

= +Δ

, ue f

ffα αα

αα αα

ρ = = ∑∑ ∑

( )1 0.5( , ) ( , ) ( , ) ( , ) 1eq

g g p

Qg r e t t t g r t g r t g r t wcα α α α α ατ τ ρ

⎛ ⎞⎡ ⎤+ Δ + Δ − = − − + −⎜ ⎟⎣ ⎦ ⎜ ⎟

⎝ ⎠

&r r r r r

2

3 0.52g

p

kc c t

τρ

= +Δ2 p

t QT gcα

α ρΔ

= +∑&


Example Problems

ν

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Boundary conditionsFluid flow :The pressure boundary condition is implemented at the inlet and outlet by introducing an adapted “counter-slip” approachUnknown parameters ux and ρ’, are determined by the inlet fluid density,

2 2

1 52 2

2

8 2

1 1' 1 3 3 , ' 1 3 39 36

1 ' 1 3 336

x x x x

x x

u u u uf f

c cc c

u uf

c c

ρ ρ

ρ

⎛ ⎞ ⎛ ⎞= + + = + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎛ ⎞

= + +⎜ ⎟⎝ ⎠

0 2 4 3 6 72( )1x

in

f f f f f fu

ρ+ + + + +

= −

0 2 4 3 6 72

6[ ]'

(1 3 3 )x x

f f f f f fu u

ρ ρ+ + + + +

= −+ +

2

2

2

2 2 2

2

2 2 2

u =0

u u3 3 9 3 u =1,2,3,42 2 c 2 c 2 c

u u 3 u3 6 4.5 =5,6,7,8c c 2 c

eq

w Tc

e eg w T

e ew T

α

α αα α

α αα

α

α

α

⎧⎪−⎪⎪ ⎡ ⎤⋅ ⋅⎪= + + −⎨ ⎢ ⎥

⎣ ⎦⎪⎪ ⎡ ⎤⋅ ⋅⎪ + + −⎢ ⎥⎪ ⎣ ⎦⎩

r r

r r

Equilibrium distribution function:


Example Problems

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Heat transfer:Dirichlet B.C: Unknown distribution functions (North wall, for example)

are g4 , g7 , and g8 , which can be obtained by the equilibrium distribution of the local T0

Tp is the sum of known populations coming from the internal nodes and nearest wall nodes:

Thus the unknown distributions areNewman B.C: (North wall, for example) The local T0 is

At the inlet, the unknown distributions are g1, g5 and g8, which can be obtained from the equilibrium distribution of the local T0

0 3 3 1.5 swall p

p

QT T T t

cρ= − + Δ

&

0 1 3 2 5 6pT g g g g g g= + + + + +

0g w Tα α=

0 3 3( 0.5)

gp

g

dTT Tc dy

τχ

τ= −

−2 5 6pT g g g= + +


Example Problems

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LiquidSolid

Inte

rface

Half lattice division treatment for the fluid–solid interface.


Tp is the sum of known populations coming from the internal nodes 0 2 4 3 6 7PT g g g g g g= + + + + +

Example Problems

0 2 2

6 6 32 3 3

in p f pT T t Q cT

u c u cρ− + Δ

=+ +

&

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Problem1: Transient Conduction-Radiation Heat Transfer Problems

Reference: Mishra, S.C., Roy, H.K., “Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method”,Journal of Computational Physics, July 2005


Example Problems

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OBJECTIVE:

Establish the suitability of LBM-FVM scheme in solving transient conduction-radiation heat transfer and compare with the FVM-FVM scheme

2-D Enclosure

To study the effect scattering albedo


Example Problems

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Problem FormulationIn the absence of convection and heat generation, for a homogeneous medium, the energy equation is given by

Where, = Density= Specific heat= Thermal conductivity= Radiative heat flux

The radiative information is calculated using FVM and the energy equation is solved using LBM.

2p R

Tc k T qt

ρ ∂= ∇ −∇⋅

∂r

ρpc

kRqr

Rq∇⋅r


Example Problems

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Radiative information can be calculated using:Discrete Transfer Method (DTM)Collapsed Dimension Method (CDM)Discrete ordinate Method (DOM)Finite Volume Method (FVM)Monte Carlo Method

Based on the known intensity distributions, radiative informationrequired for the energy equation is computed from

Where, = Extinction coefficient= Scattering albedo= Stefan–Boltzmann constant (5.670x10-8 W/m2 K4)= Incident Radiation

Rq∇⋅r

4

(1 ) 4RTq Gσβ ω ππ

⎛ ⎞∇ ⋅ = − −⎜ ⎟

⎝ ⎠

r

βωσG


Example Problems

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In case of heat transfer problems, the temperature is obtained after summing over all direction

Equilibrium distribution function for heat conduction problems is given by

Summation over all directions gives

Incorporating the volumetric radiation obtained from FVM into LBE we get the modified form of LBE:

if

0

( , ) ( , )b

ii

T r t f r t=

= ∑r r

( ) ( , ) ( , )eqi if r t wT r t=

r r

( )

0 0 0

( , ) ( , ) ( , ) ( , )b b b

eqi i i

i i i

f r t wT r t T r t f r t= = =

= = =∑ ∑ ∑r r r r

( )( , ) ( , ) ( , ) ( , )eqi i i i i i R

p

t tf r e t t f r t f r t f r t w qcτ ρ

⎛ ⎞Δ Δ⎡ ⎤+ Δ = − − − ∇⋅⎜ ⎟⎣ ⎦ ⎜ ⎟⎝ ⎠

r r r r r r


Example Problems

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y

xSouth wall

North wall

Y xΔ

xΔ

yΔyΔ

X

Wes

t wal

l

Eas

t wal

l

FVM CV

LBM CV


Schematic representation of 2-D Uniform Lattices

Example Problems

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LBM-FVM AlgorithmInitialization: Known Temperature Field

First Iteration: Calculate the equilibrium distribution function ( ) ( ,0)eq

if rr

Using Known Initial Temperature Field, Calculate using FVMRq∇⋅

r

Calculate particle distribution function: ( , )i if r e t t t+ Δ + Δr r

Propagation and new temperature field: ( , )T r tr

Check Convergence and modify locally to satisfy B.C

Compute from new temperature field for every lattice

if

( ) ( ,0)eqif rr


Example Problems

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Results of simulation(a) (b)


Example Problems

Comparison of non-dimensional temperature T/TW in a 2-D enclosure at different instants for (a) scattering albedo = 0.0, and (b) = 0.9.ξ ω ω

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Problem 2: Phase-change problem with volumetric radiation

Reference: Parida, P.R., Raj, R. , Prasad, A., Mishra, S.C. , “Solidification of a semitransparent planar layer subjected to radiative and convective cooling”, Journal of Quantitative Spectroscopy & Radiative Transfer 107 (2007) 226–235


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The process of solidification takes place at a fixed temperature Tm marked by the formation of a two-phase region.

Since the medium is considered semitransparent one, volumetric radiation also plays a role, and it appears in the governing energy equation.

Where, = Enthalpy == Density

L is the latent heat and is the liquid fraction. In the solid region, In the liquid region, In the mushy-zone, . It is to be noted that solid fraction

2

2Rqh Tk

t xxρ

∂∂ ∂= −

∂ ∂∂

lcT f L+

22

2 2

1l Rf qh h Lt x x x

αρ

⎛ ⎞∂ ∂∂ ∂= − −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

lf0lf =1lf =

0 1lf< < 1s lf f= −

hρ



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In the solid-, mushy- and liquid-zones, the liquid fraction and enthalpy are, respectively, related as

Initial condition :

Boundary conditions:

DOM was used to compute the radiative information

Where, = Absorption coefficient= Refractive Index= Incident Radiation

lf

0,

,

1,

s

sl s l

l s

l

h hh hf h h hh h

h h

⎧ <⎪

−⎪= ≤ ≤⎨ −⎪⎪ >⎩

0( ,0)T x T=4 4

4 4

(0, ) ( ) ( )

( , ) ( ) ( )W W

E E

T t h T T T T

T D t h T T T T

εσ

εσ∞ ∞ ∞

∞ ∞ ∞

= − + −

= − + −

424R

aq Tn Gx

σκ ππ

⎛ ⎞∂= −⎜ ⎟∂ ⎝ ⎠

aκnG



= Convective heat transfer coefficient

= Emissivity

h∞

ε

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LBM Formulation: Modified LBM equation:

With non-dimensional quantities defined in the following way

Using the above dimensional quantities the energy equation and LB equation are written as:

2(0)

2( , ) ( , ) ( , ) ( , ) ( )( ) l i Ri i i i i i

p

f tw qtn x e t t t n x t n x t n x t L twc xx

ατ ρ

∂ Δ ∂Δ ⎡ ⎤+ Δ + Δ = − − − Δ −⎣ ⎦ ∂∂r r r r r

*3 4

3

, , , , ,4 4

4 1 ,

m R

m m m m

m ml s

h cT qx T kx H ND T cT D T T

D T cTtf f H St StcD L

θ ψσ σ

σζρ

−= = = = =

⎛ ⎞= = − = × =⎜ ⎟⎝ ⎠

( )2

* * * * * * * *(0) ** *2 *( , ) ( , ) ( , ) ( , ) i l

i i i i i iN w fn x e n x n x n x w

St x xζζ ψζ ζ ζ ζ ζ ζ ζ

τΔ ∂Δ ∂⎛ ⎞⎡ ⎤+ Δ + Δ = − − − − Δ⎜ ⎟⎣ ⎦ ∂ ∂⎝ ⎠

r r r r r

22

*2 *2 *lfH H NN

Stx x xψ

ζ∂∂ ∂ ∂

= − −∂ ∂ ∂ ∂

( )*

2 4 * 4* (1 ) ,

4 mGD n G G T

xψ β ω θ σ π

π⎛ ⎞∂

= − − =⎜ ⎟∂ ⎝ ⎠



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Comparison of the LBM and the FVM results for (a) solid fraction and (b) non-dimensional temperature at different instants .

Results of simulation

sfζθ

a b



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Problem 3: Transient Conduction-Radiation Heat Transfer Problem using non-uniform lattices

Reference: Mondal, B. and Mishra, S.C., “Lattice Boltzmann method applied to the solution of the energy equations of the transient conduction and radiation problems on non-uniform lattices”, International Journal of Heat and Mass Transfer, 2007


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Conduction and radiation heat transfer in a 1-D planar medium:Non-uniform lattices were generated using the following expression:

Where, = Location of the lattice center in the LBM= Total number of lattices= Cluster Value ( ) , :Uniform lattices

For 2-D enclosure:

Where, = total number of lattices/control volumes in the x and y direction

max max

1 ( 1)sinnn C nxn n

ππ

⎧ ⎫⎛ ⎞− −⎪ ⎪= +⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭

0 1C≤ ≤ 0C =

nxmaxnC

1 ( 1)sin

1 ( 1)sin

xn

x x

yn

y y

Cn nxN N

Cn nyN N

ππ

ππ

⎧ ⎫⎛ ⎞− −⎪ ⎪= +⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭⎧ ⎫⎛ ⎞− −⎪ ⎪= + ⎜ ⎟⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭

,x yN N



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Eas

t wal

lY

1 / 2xΔ2 / 2xΔ

1xΔ 2xΔ

FVM

CV

LBM

CV

x

y

South wall

North wall

Wes

t w

all

ixΔ

jyΔ

XSchematic representation of 2-D non-uniform lattices



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Results of simulation

Comparison of centerline temperature in a 2-D square enclosure at different instants for conduction–radiation parameter N = (a) 0.01 (b) 1.0.

a b



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Parallelization of LBMDomain decomposition, or domain partitioning.For domain partitioning, the entire computational domain is divided into several sub-domains.

One dimensional domain decomposition with 3 PUs.Before a new iteration can start, each PU has to send required cells (dark grey cells) to other PUs. Additionally, each PU receives an update of its halo (light grey cells).

Reference: Körner,C., Pohl,T., Rüde,U., Thürey,N., and Zeiser, T., “ Parallel Lattice Boltzmann Methods for CFDApplications”


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Streaming step of LBM needs to access data from neighboring cells which might be located in adjacent sub-domains.

A common way to approach the necessary data exchange is the introduction of a so-called halo (also known as ghost cell layer) at the sub-domain interfaces, i.e. where two sub-domains are adjacent.

Halos hold copies of the values of neighboring domains, and naturally they must be updated at specific times when the original value has been changed.

Depending on the type of data dependencies the halo must consistof a single or more layers.

For the streaming step of the LBM, only nearest neighbor communication is necessary.



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With the halo, the algorithm can proceed in each sub-domain concurrently and can perform a time step in parallel. However, between time steps, the halos must be exchanged with the neighboring sub-domains.

Since data is exchanged in larger blocks, the startup overhead for communication on cluster parallel machines is better amortized.

A simple one-dimensional domain partitioning for a two dimensional rectangular grid of cells.



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For a 3D grid, a sub-domain in the interior will not only have to communicate with the six sub-domains adjacent to its faces, but also to the twelve neighboring sub-domains with which it shares an edge (for the D3Q19 model).

To alleviate the cost of these communication steps, sophisticated schemes can be devised.

The communication of data across an edge can be accomplished by,e.g., sending the data as part of the messages between face neighbors and using one intermediate neighbor as a mailman.

This, however, can become quite complicated if the dependencies between neighboring cells are more complex



14/12/07 42

Conclusion

LBM relates microscopic particle dynamics to macroscopic properties.

LBM formulation eliminates the non-linearity involved in the governing PDEs of fluid flow and heat transfer problems.

Clear understanding of the physical phenomenon.

Has been successfully applied in the simulation of fluid flow and heat transfer problems.

Can be easily implemented on a computer and is suitable for parallel computation. Applicable to different platforms.

Lattice Boltzmann Method has the potential to become a versatilenumerical tool to study complex problems (Electrohydrodynamics, hydrodynamics , simulating ocean circulation etc.)


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References

Chen,S. and Doolen, G.D.,” Lattice Boltzmann Method For Fluid Flows”, Annu. Rev. Fluid Mech. 1998. 30:329–64

Premnath, K.N., and Abraham.J, “Three-dimensional multi-relaxation time (MRT) lattice-Boltzmann models for multiphase flow “, Journal of Computational Physics ,Vol. 224( 2), 10 June 2007, 539-559

D.A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction, Springer-Verlag, Berlin-Heidelberg, 2000.

Wang, J., Wang,M., Li, Z., “A lattice Boltzmann algorithm for fluid–solid conjugate heat transfer”, International Journal of Thermal Sciences, 2006

Mishra, S.C., Roy, H.K., “Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method”,Journal of Computational Physics, July 2005


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Parida, P.R., Raj, R. , Prasad, A. Mishra, S.C. , “Solidification of a semitransparent planar layer subjected to radiative and convective cooling”, Journal of Quantitative Spectroscopy & Radiative Transfer 107 (2007) 226–235

Mondal, B. and Mishra, S.C., “Lattice Boltzmann method applied to the solution of the energy equations of the transient conduction and radiation problems on non-uniform lattices”, International Journal of Heat and Mass Transfer, 2007

Körner,C., Pohl,T., Rüde,U., Thürey,N., and Zeiser, T., “ Parallel Lattice Boltzmann Methods for CFD Applications”

http://www.science.uva.nl/research/scs/projects/lbm_web/lbm.html


References

Documents

Introduction to Lattice Boltzmann Method and Application