Lattice Boltzmann methhod slides

Lattice Boltzmann Method and ItsLattice Boltzmann Method and Its
Applications in Multiphase FlowsApplications in Multiphase Flows
Xiaoyi He
Air Products and Chemicals, Inc.
April 21, 2004

OutlineOutline
 Lattice Boltzmann method
 Kinetic theory for multiphase flow
 Lattice Boltzmann multiphase models
 Applications
 Conclusions

A Brief History of LatticeA 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
τ
δδ
eq
aa
aaa
ff
txftttexf
−
−=−++ ),(),(
• fa: density distribution function;
• τ: relaxation parameter
• f
eq
: equilibrium distribution
∑∑ ==






−
⋅
+
⋅
+=
a
aa
a
a
aa
a
eq
a
efuf
RT
u
RT
ue
RT
ue
f
ρρ
ρω
,
2)(2
)(
1
2
2
2

Kinetic Theory of Multiphase FlowKinetic Theory of Multiphase Flow
 BBGKY hierarchy
functionondistributiparticle-two
potentialularintermolec
functionondistributiparticle-single
:)r,,r,(
:)(
:
)()()(
2211
)2(
12
2212
1
)2(
1 111
ξξ
ξ
ξ
ξ ξξ
f
rV
f
drdrV
f
fFf
t
f
r ∇⋅
∂
∂
=∇⋅+∇⋅+
∂
∂
∫∫

Intermolecular InteractionIntermolecular Interaction
}:{
}:{
122
121
fortheoryfieldMean
fortheoryEnskog
drrD
drrD
>−
<−
-0.5
0
0.5
1
1.5
2
0 1 2 3
r/d
V
Lennard-Jones potential
Interaction models

Model for Intermolecular RepulsionModel for Intermolecular Repulsion












⋅∇−+∇+





∇−+∇⋅−
−Ω
=∇⋅
∂
∂
= ∫∫
uCuCCTCTu
fb
drdrV
f
I
eq
D
)
2
5
(:2
5
2
ln)
2
5
(
5
3
)ln()(
)(
222
0
2212
1
)2(
1
1
1
χρξ
ρχχ
ξ
ξ
ξ
For D1 (repulsion core)

Model for Intermolecular AttractionModel for Intermolecular Attraction
For D2 (attraction tail), by assuming
fVdrdrV
f
I m
D
1
2
1 2212
1
)2(
2 )( ξξ ξ
ξ
∇⋅∇=∇⋅
∂
∂
= ∫∫
)r,()r,()r,,r,( 22112211
)2(
ξξξξ 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
κ
ρκρ 2
2 ∇−−= aVm
where
Control phase transition
Control surface tension
For small density variation:
∫>
=
dr
m drrVrV )()(ρ

Kinetic Model for Multiphase FlowKinetic Model for Multiphase Flow
Boltzmann equation for non-ideal gas / dense fluid
functionondistributiparticle-single:
)()( 1
f
fVIfFf
t
f
m ξξξ ∇⋅∇+=∇⋅+∇⋅+
∂
∂

Mass transport equation
0)( =⋅∇+
∂
∂
u
t
ρ
ρ
2
0
22
0
)1(),(
2
),(
)()(
ρρχρρ
ρ
κ
ρκρρ
ρρκρρ
ρ
abRTTp
Tpp
pFuu
t
u
−+=
−∇−=
∇∇⋅∇+Π⋅∇+∇−=⋅∇+
∂
∂
Momentum transport equation
Chapman-Enskog expansion leads to the following macroscopic
transport equations:

Comments on momentum transport equation
1. Correct equation of state
2. Thermodynamically consistent surface tension
drT∫ 



+=Ψ
2
2
),( ρ
κ
ρψ
3. Thermodynamically consistent free energy
(Cahn and Hillary, 1958)
interfaceinenergyfreeexcess:)(2 )W(dW ρρρκσ ∫=

Energy transport equation
ρρκρ
κ
ρκρρ
ρρρρκ
λ
∇∇+−∇−=
∇⋅∇−∇∇∇+
Π∇+∇⋅∇+∇−=⋅∇+
∂
∂
ITpP
u
uTuPue
t
e
]
2
),([
)](
2
1
)([:
:)(:)(
22
0

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( ρχρ

LBM Multiphase Model Based onLBM 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 BoltzmannTemperature 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
2
0
2
00
2
0
2)(2
)(
)
2
3
2
(1
)1(
RT
u
RT
u
RT
u
RT
f
TT
a
eq ξξ
θ
ξ
ω
θ

Isothermal Boltzmann Equation forIsothermal Boltzmann Equation for
Multiphase FlowMultiphase Flow
)
2
)(
exp(
)2(
)(
)()(
2
RT
u
RT
f
fV
RT
uff
fFf
t
f
D
eq
eq
m
eq
−
−=
∇⋅
−
+
−
−=∇⋅+∇⋅+
∂
∂
ξ
π
ρ
ξ
τ
ξ ξ

Discretization in Velocity SpaceDiscretization in Velocity Space
Constraint for velocity stencil
Further expansion of f eq
32,1,0,nfor, ==∫ exactdf eqn
ξξ






−
⋅
+
⋅
+−=
RT
u
RT
u
RT
u
RT
f eq
2)(2
)(
1)
2
exp(
2
2
22
ξξξ
ρ
5...,1,0,nfor,)
2
exp(
2
==−∫ exactd
RT
n
ξ
ξ
ξ

Discretization in Velocity SpaceDiscretization in Velocity Space






−
⋅
+
⋅
+=
RT
u
RT
ue
RT
ue
f aa
a
eq
a
2)(2
)(
1
2
2
2
ρω
9-speed model 7-speed model
ωa: weight coefficients

Discretization in Physical andDiscretization in Physical and
Temporal SpacesTemporal Spaces
Integrate Boltzmann equation
eq
am
a
eq
aa
aaa fV
RT
tue
t
ff
txftttexf ∇⋅
−
+
−
−=−++
δ
δτ
δδ
)(
/
),(),(
• Discretizations in velocity, physical and temporal spaces are
independent in principle;
• Synchronization simplifies computation but requires
• Regular lattice
• Time-step constraint: RTtx 3/ =δδ

Further Simplification for NearlyFurther Simplification for Nearly
Incompressible FlowIncompressible Flow
Introduce an index function φ:
)()(
)(
),(),( u
RT
ueff
txftttexf a
eq
aa
aaa Γ∇⋅
−
+
−
−=−++ φψ
τ
δδ
)]())0()(())(([
)(),(),(
ρψ
τ
δδ
∇Γ−Γ−+Γ
⋅−+
−
−=−++
uGFu
ue
gg
txgtttexg
s
a
eq
aa
aaa
∑
∑
∑
++=
−∇⋅−=
=
)(
2
)(
2
1
GF
RT
geRTu
RTpugp
f
saa
a
a
ρ
ρ
φ

ApplicationsApplications
Phase Separation
Rayleigh-Taylor instability
Kelvin-Helmholtz instability

Phase SeparationPhase Separation
Van der Waals fluid
T/Tc = 0.9

Rayleigh-Taylor Instability (2D)Rayleigh-Taylor Instability (2D)
Re = 1024
single 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 /Agt
2
= 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 ApplicationsOther 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 BoltzmannChallenges in Lattice Boltzmann
MethodMethod
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.

AcknowledgementAcknowledgement
Raoyang Zhang, ShiyiChen, Gary Doolen
Xiaowen Shan

Lattice Boltzmann methhod slides

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