Lattice Boltzmann Method and Its Lattice Boltzmann Method and Its Applications in Multiphase FlowsApplications in Multiphase Flows

Xiaoyi He

Air Products and Chemicals, Inc.

April 21, 2004

OutlineOutline

Lattice Boltzmann methodKinetic theory for multiphase flowLattice Boltzmann multiphase modelsApplicationsConclusions

A Brief History of Lattice A Brief History of Lattice Boltzmann MethodBoltzmann Method

Lattice Gas Automaton (Frisch, Hasslacher, Pomeau,, 1987)

Lattice Boltzmann model (McNamara and Zanetti (1988)

Lattice Boltzmann BGK model (Chen et al 1992 and Qian et al 1992)

Relation to kinetic theory (He and Luo, 1997)

Lattice Boltzmann BGK ModelLattice Boltzmann BGK Model

eq

aaaaa

fftxftttexf

),(),(

• fa: density distribution function; • : relaxation parameter• f eq: equilibrium distribution

aaa

aa

aaa

eqa

efuf

RT

u

RT

ue

RT

uef

,

2)(2

)(1

2

2

2

Kinetic Theory of Multiphase FlowKinetic Theory of Multiphase Flow

BBGKY hierarchy

functionon distributi particle-two

potentialular intermolec

functionon distributi particle-single

:)r,,r,(

:)(

:

)()()(

2211)2(

12

22121

)2(

1 111

f

rV

f

drdrVf

fFft

fr

Intermolecular InteractionIntermolecular Interaction

}:{

}:{

122

121

for theory fieldMean

for theory Enskog

drrD

drrD

-0.5

0

0.5

1

1.5

2

0 1 2 3

r/d

VLennard-Jones potential

Interaction models

Model for Intermolecular RepulsionModel for Intermolecular Repulsion

uCuCCTCTu

fb

drdrVf

I

eq

D

)2

5(:2

5

2ln)

2

5(

5

3)ln()(

)(

222

0

22121

)2(

1

1

1

For D1 (repulsion core)

Model for Intermolecular AttractionModel for Intermolecular Attraction

For D2 (attraction tail), by assuming

fVdrdrVf

I m

D1

2

1 22121

)2(

2 )(

)r,( )r,()r,,r,( 22112211)2( fff

We have

Model for Intermolecular AttractionModel for Intermolecular Attraction

Vm is the mean-field potential of intermolecular attraction

dr

dr

drrVr

drrVa

)(6

1

)(2

1

2

22 aVm

where

Control phase transition

Control surface tension

For small density variation:

dr

m drrVrV )()(

Kinetic Model for Multiphase FlowKinetic Model for Multiphase Flow

Boltzmann equation for non-ideal gas / dense fluid

functionon distributi particle-single:

)()( 1

f

fVIfFft

fm

Kinetic Model for Multiphase FlowKinetic Model for Multiphase Flow

Mass transport equation

0)(

ut

20

220

)1(),(

2),(

)()(

abRTTp

Tpp

pFuut

u

Momentum transport equation

Chapman-Enskog expansion leads to the following macroscopic transport equations:

Kinetic Model for Multiphase FlowKinetic Model for Multiphase Flow

Comments on momentum transport equation

1. Correct equation of state

2. Thermodynamically consistent surface tension

drT

2

2),(

3. Thermodynamically consistent free energy (Cahn and Hillary, 1958)

interfacein energy free excess : )(2 )W(dW

Kinetic Model for Multiphase FlowKinetic Model for Multiphase Flow

Energy transport equation

ITpP

u

uTuPuet

e

]2

),([

)](2

1)([:

:)(:)(

220

Kinetic Model for Multiphase FlowKinetic Model for Multiphase Flow

Comments on energy transport equation

1.Total energy needs include both kinetic and potential energies, otherwise the pressure work becomes:

2. Last term is due to surface tension and it is consistent with existing literature (Irving and Kirkwood, 1950)

upubRT )1(

LBM Multiphase Model Based on LBM Multiphase Model Based on Kinetic TheoryKinetic Theory

Temperature variations in lattice Boltzmann models;Discretization of velocity space;Discretization of physical space;Discretization of temporal space.

Temperature in Lattice Boltzmann Temperature in Lattice Boltzmann MethodMethod

Non-isothermal model model is still a challenge – Small temperature variations can be modeled

– Need for high-order velocity discretization

Isothermal model is well developed

0

2

20

2

00

2

0

2)(2

)()

2

3

2(1

)1(

RT

u

RT

u

RT

u

RTf

TT

aeq

Isothermal Boltzmann Equation for Isothermal Boltzmann Equation for Multiphase FlowMultiphase Flow

)2

)(exp(

)2(

)()()(

2

RT

u

RTf

fVRT

ufffFf

t

f

Deq

eqm

eq

Discretization in Velocity SpaceDiscretization in Velocity Space

Constraint for velocity stencil

Further expansion of f eq

3 2, 1, 0, n for , exactdf eqn

RT

u

RT

u

RT

u

RTf eq

2)(2

)(1)

2exp(

2

2

22

5 ..., 1, 0, n for ,)2

exp(2

exactdRT

n

Discretization in Velocity SpaceDiscretization in Velocity Space

RT

u

RT

ue

RT

uef aa

aeq

a 2)(2

)(1

2

2

2

9-speed model 7-speed model

a: weight coefficients

Discretization in Physical and Discretization in Physical and Temporal SpacesTemporal Spaces

Integrate Boltzmann equation

eqam

aeq

aaaaa fV

RT

tue

t

fftxftttexf

)(

/),(),(

• Discretizations in velocity, physical and temporal spaces are independent in principle; • Synchronization simplifies computation but requires

• Regular lattice• Time-step constraint:

RTtx 3/

Further Simplification for Nearly Further Simplification for Nearly Incompressible FlowIncompressible Flow

Introduce an index function :

)()()(

),(),( uRT

uefftxftttexf a

eqaa

aaa

)]())0()(())(([

)(),(),(

uGFu

uegg

txgtttexg

s

a

eqaa

aaa

)(2

)(2

1

GFRT

geRTu

RTpugp

f

saa

a

a

ApplicationsApplications

Phase SeparationRayleigh-Taylor instabilityKelvin-Helmholtz instability

Phase SeparationPhase Separation

Van der Waals fluid

T/Tc = 0.9

Rayleigh-Taylor Instability (2D)Rayleigh-Taylor Instability (2D)

Re = 1024single mode

RT instability (2D)

Single mode

Density ratio: 3:1

Re = 2048

270.0/ AgWuT

RT instability (2D)

Multiple mode

Density ratio: 3:1

hB /Agt2 = 0.04

RT instability (3D)

single mode

Density ratio: 3:1

Re = 1024

61.05.0/ AgWuT

RT instability (3D)

single mode

Density ratio: 3:1

Re = 1024

Cuts through spike

RT instability (3D)

single mode

Density ratio: 3:1

Re = 1024

Cuts through bubble

KH instability

Effect of surface tension

Re = 250

d1/d2 = 1

Ca = 0.29

Ca = 2.9

Other Applications Other Applications

Multiphase flow in porous media (Rothman 1990, Gunstensen and Rothman 1993);

Amphiphilic fluids (Chen et al, 2000) Bubbly flows (Sankaranarayanan et al, 2001);Hele-Shaw flow (Langaas and Yeomans, 2000).Boiling flows (Kato et al, 1997);Drop break-up (Halliday et al 1996);

Challenges in Lattice Boltzmann Challenges in Lattice Boltzmann Method Method

Need for better thermal models;Need for better model for multiphase flow

with high density ratio; Need for better mode for highly

compressible flows;Engineering applications …

ConclusionsConclusions

Lattice Boltzmann method is a useful tool for studying multiphase flows;

Lattice Boltzmann model can be derived form kinetic theory;

It is easy to incorporate microscopic physics in lattice Boltzmann models;

Lattice Boltzmann method is easy to program for parallel computing.

Thank You!Thank You!

AcknowledgementAcknowledgement

Raoyang Zhang, ShiyiChen, Gary Doolen

Xiaowen Shan