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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