Physical Modeling of Multiphase flow with lattice ... - LBM.pdf · Physical Modeling of Multiphase...
Transcript of Physical Modeling of Multiphase flow with lattice ... - LBM.pdf · Physical Modeling of Multiphase...
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
Physical Modeling of Multiphase flow with latticeBoltzmann method
Xiaowen Shan ([email protected])
Exa Corp., Burlington, MA, USA
Feburary, 2011
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
Scope
� Re-examine the non-ideal gas model in [Shan & Chen, Phys. Rev.E, (1993)] from the perspective of kinetic theory
� Focus on the modeling of underlying physics mechanism
� Review some recent progress
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
Scope
� Re-examine the non-ideal gas model in [Shan & Chen, Phys. Rev.E, (1993)] from the perspective of kinetic theory
� Focus on the modeling of underlying physics mechanism
� Review some recent progress
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
Scope
� Re-examine the non-ideal gas model in [Shan & Chen, Phys. Rev.E, (1993)] from the perspective of kinetic theory
� Focus on the modeling of underlying physics mechanism
� Review some recent progress
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
Scope
� Re-examine the non-ideal gas model in [Shan & Chen, Phys. Rev.E, (1993)] from the perspective of kinetic theory
� Focus on the modeling of underlying physics mechanism
� Review some recent progress
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
Lattice Boltzmann from kinetic theoryHow it worksNew insights
Modeling non-ideal gas in lattice BoltzmannAn intuitive modelStatistical physicsThermodynamic consistencyOther issues
Conclusions
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
Outline
Lattice Boltzmann from kinetic theoryHow it worksNew insights
Modeling non-ideal gas in lattice BoltzmannAn intuitive modelStatistical physicsThermodynamic consistencyOther issues
Conclusions
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
General Principles 1
Lattice Boltzmann can be obtained from continuum kinetic theory basedon two observations:
1. For hydrodynamics, only the leading moments of the distributionfunction matter explicitly.
2. If (and only if) distribution is a finite Hermite expansion, leadingmoments and discrete function values are isomorphic.
Continuum BGK discretized in velocity space by:
� Project continuum BGK into a low-dimensional Hermite space
� Evaluate at discrete velocities with corresponding momentspreserved.
1Shan et al, J. Fluid Mech., 550, 413 (2006)
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
Isomorphism between moments and discrete velocities
� Hydrodynamic moments�f (ξ)ξmdξ.
� Gauss-Hermite quadrature: For polynomial p:
�ω(ξ)p(ξ)dξ =
d�
i=1
wip(ξi ).
� If f is an order-N Hermite series
f (ξ) = ω(ξ)N�
n=0
1
n!a(n)(x, t)H(n)(ξ)
first M moments exactly given by discrete values throughquadrature.
� N +M ≤ order of quadrature
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
Projection into low-dimensional Hilbert space
� Take the continuum BGK equation with body force:
∂f
∂t+ ξ ·∇f + g ·∇ξf = −
1
τ
�f − f (0)
�
� Expand f in Hermite series (orthogonal projection)
f (ξ) = ω(ξ)∞�
n=0
1
n!a(n)H(n)(ξ) where ω(ξ) =
1
(√2π)D
exp
�−ξ2
2
�
Construction by Gram-Schmidt process:
a(n) =�
f (ξ)H(n)(ξ)dξ
� Evaluate the truncated equation on discrete velocities
� Two terms require special attension
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
Projection into low-dimensional Hilbert space
� Take the continuum BGK equation with body force:
∂f
∂t+ ξ ·∇f + g ·∇ξf = −
1
τ
�f − f (0)
�
� Expand f in Hermite series (orthogonal projection)
f (ξ) = ω(ξ)∞�
n=0
1
n!a(n)H(n)(ξ) where ω(ξ) =
1
(√2π)D
exp
�−ξ2
2
�
Construction by Gram-Schmidt process:
a(n) =�
f (ξ)H(n)(ξ)dξ
� Evaluate the truncated equation on discrete velocities
� Two terms require special attension
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
Projection of the Maxwell distribution
The (dimensionless) Maxwellian:
f (0)(ξ) =1
(2πθ)D/2exp
�−ξ2
2θ
�
Construction by Gram-Schmidt process:
f (0)(ξ) = ρω
�1 + u · ξ +
1
2
�(u · ξ)2 − u2 + (θ − 1)(ξ2 − D)
�+ · · ·
�
Differences from low-Mach number expansion:
� ξ and u scaled with sound speed, universal on any lattice
� Orthogonal expansion. No assumption of small Mach number.
� Temperature included
� Zero-th term corresponds to temperature
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
Projection of the Maxwell distribution
The (dimensionless) Maxwellian:
f (0)(ξ) =1
(2πθ)D/2exp
�−ξ2
2θ
�
Construction by Gram-Schmidt process:
f (0)(ξ) = ρω
�1 + u · ξ +
1
2
�(u · ξ)2 − u2 + (θ − 1)(ξ2 − D)
�+ · · ·
�
Differences from low-Mach number expansion:
� ξ and u scaled with sound speed, universal on any lattice
� Orthogonal expansion. No assumption of small Mach number.
� Temperature included
� Zero-th term corresponds to temperature
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
Projection of the Maxwell distribution
The (dimensionless) Maxwellian:
f (0)(ξ) =1
(2πθ)D/2exp
�−ξ2
2θ
�
Construction by Gram-Schmidt process:
f (0)(ξ) = ρω
�1 + u · ξ +
1
2
�(u · ξ)2 − u2 + (θ − 1)(ξ2 − D)
�+ · · ·
�
Differences from low-Mach number expansion:
� ξ and u scaled with sound speed, universal on any lattice
� Orthogonal expansion. No assumption of small Mach number.
� Temperature included
� Zero-th term corresponds to temperature
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
Projection of the Maxwell distribution
The (dimensionless) Maxwellian:
f (0)(ξ) =1
(2πθ)D/2exp
�−ξ2
2θ
�
Construction by Gram-Schmidt process:
f (0)(ξ) = ρω
�1 +
(θ − 1)(ξ2 − D)
2+ u · ξ +
1
2
�(u · ξ)2 − u2
�+ · · ·
�
Differences from low-Mach number expansion:
� ξ and u scaled with sound speed, universal on any lattice
� Orthogonal expansion. No assumption of small Mach number.
� Temperature included
� Zero-th term corresponds to temperature
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
Projection of the body force 2
The body force term: g ·∇ξf . Let:
f (ξ) = ω(ξ)∞�
n=0
1
n!a(n)H(n)(ξ)
Body-force term has the following expansion:
g ·∇ξf = −ω(ξ)∞�
n=1
1
n!ga(n−1)
H(n)(ξ).
Due to conservations of mass and momentum, up to second moments:
g ·∇ξf = g ·∇ξf(0)
≡(ξ − u) · gf (0)
θ.
(ξ − u) · gf (0) represents a body-force.
2Martys, Shan & Chen, Phys. Rev. E, 58, 6855 (1998)
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
Projection of the body force 2
The body force term: g ·∇ξf . Let:
f (ξ) = ω(ξ)∞�
n=0
1
n!a(n)H(n)(ξ)
Body-force term has the following expansion:
g ·∇ξf = −ω(ξ)∞�
n=1
1
n!ga(n−1)
H(n)(ξ).
Due to conservations of mass and momentum, up to second moments:
g ·∇ξf = g ·∇ξf(0)
≡(ξ − u) · gf (0)
θ.
(ξ − u) · gf (0) represents a body-force.2Martys, Shan & Chen, Phys. Rev. E, 58, 6855 (1998)
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
Outline
Lattice Boltzmann from kinetic theoryHow it worksNew insights
Modeling non-ideal gas in lattice BoltzmannAn intuitive modelStatistical physicsThermodynamic consistencyOther issues
Conclusions
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
New insight
� Necessary and sufficient conditions on equilibrium distribution andunderlying lattice (velocity sets).
� Systematic framework for analyzing LB models� Puzzles solved: Galilean invariance, bulk viscosity, thermodynamic
sound, . . .� LB model for compressible flows� LB models beyond Navier-Stokes� LB models with generic (velocity-independent) multi-relaxation times
What is lattice Boltzmann?
� Moment space truncated Boltzmann-BGK (compared with Grad13-moments)
� Navier-Stokes: asymptotically truncated Boltzmann� Contains (not approximates) compressible Navier-Stokes� Asymptotically approaches to Boltzmann-BGK
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
New insight
� Necessary and sufficient conditions on equilibrium distribution andunderlying lattice (velocity sets).
� Systematic framework for analyzing LB models� Puzzles solved: Galilean invariance, bulk viscosity, thermodynamic
sound, . . .� LB model for compressible flows� LB models beyond Navier-Stokes� LB models with generic (velocity-independent) multi-relaxation times
What is lattice Boltzmann?
� Moment space truncated Boltzmann-BGK (compared with Grad13-moments)
� Navier-Stokes: asymptotically truncated Boltzmann� Contains (not approximates) compressible Navier-Stokes� Asymptotically approaches to Boltzmann-BGK
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
How it worksNew insights
LB compressible flow solver 3
Figure: Flow past a 15◦ wedge (Static pressure). Ma=1.8. Shock angle:Theory 51◦, Simulation 51.5◦.
3Nie et al, AIAA Paper 2009-139, (2009)
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Outline
Lattice Boltzmann from kinetic theoryHow it worksNew insights
Modeling non-ideal gas in lattice BoltzmannAn intuitive modelStatistical physicsThermodynamic consistencyOther issues
Conclusions
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Intuitions 4
Non-ideal gas: inter-molecular interaction.� Attractive force between nearest neighbors
F(x, x�) ∼ − Gρ(x)ρ(x�)
� Increment local momentum in collision accordingly:
ρ∆u = τ�
x�
F(x, x�).
� All mass collapses to singular point. Needs repulsive hard-sphere.But a potential over distance would be in-practical.
� Introduce pseudo-potential ψ to reduce attraction at high densitywhen inter-molecules distance is small.ψ(ρ) ∼ ρ at ρ � 1, and ψ(ρ) = const. at ρ � 1. An obvious(unrealistic) choice: ψ = 1− exp(−ρ).
� Essence: Mean-field interaction force field
4Shan & Chen, Phys. Rev. E, 47, 1815, (1993)
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Intuitions 4
Non-ideal gas: inter-molecular interaction.� Attractive force between nearest neighbors
F(x, x�) ∼ − Gψ(ρ(x))ψ(ρ(x�))
� Increment local momentum in collision accordingly:
ρ∆u = τ�
x�
F(x, x�).
� All mass collapses to singular point. Needs repulsive hard-sphere.But a potential over distance would be in-practical.
� Introduce pseudo-potential ψ to reduce attraction at high densitywhen inter-molecules distance is small.ψ(ρ) ∼ ρ at ρ � 1, and ψ(ρ) = const. at ρ � 1. An obvious(unrealistic) choice: ψ = 1− exp(−ρ).
� Essence: Mean-field interaction force field
4Shan & Chen, Phys. Rev. E, 47, 1815, (1993)
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Intuitions 4
Non-ideal gas: inter-molecular interaction.� Attractive force between nearest neighbors
F(x, x�) ∼ − Gψ(ρ(x))ψ(ρ(x�))
� Increment local momentum in collision accordingly:
ρ∆u = τ�
x�
F(x, x�).
� All mass collapses to singular point. Needs repulsive hard-sphere.But a potential over distance would be in-practical.
� Introduce pseudo-potential ψ to reduce attraction at high densitywhen inter-molecules distance is small.ψ(ρ) ∼ ρ at ρ � 1, and ψ(ρ) = const. at ρ � 1. An obvious(unrealistic) choice: ψ = 1− exp(−ρ).
� Essence: Mean-field interaction force field4Shan & Chen, Phys. Rev. E, 47, 1815, (1993)
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Features
� Non-ideal gas equation of state: p = ρθ + Gψ2(ρ)/2
� Multiphase with any number of components
� Equilibrium solved in one-component system
� Phase transition, solubility, and mass transports
� Non-local momentum conservation.
Issues:
� No exact energy conservation (athermal)
� Equilibrium unsolved in multi-component systems
� Unstable at high density ratio
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Outline
Lattice Boltzmann from kinetic theoryHow it worksNew insights
Modeling non-ideal gas in lattice BoltzmannAn intuitive modelStatistical physicsThermodynamic consistencyOther issues
Conclusions
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Modeling interaction in kinetic theory
From continuum kinetic theory
� Correlation ignored in Boltzmann equation, has to be modeled
� Long-range interaction from the second equation in BBGKY 5
� Enskog equation for dense gases
� Both formally lead to a mean-field Vlasov-Enskog term 6
a ·∇ξf , a: a mean-field interaction force field
Recent progresses:
� How is a computed?
� How does a enter into LB dynamics?
5Martys, Int. J. Mod. Phys. C, 10, 1367 (1998)
6He, Shan & Doolen, Phys. Rev. E, 57, R13, (1998)
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Determination of the mean force field
Two philosophically different starting points:
� LB as a discrete model
F = −Gψ(x)�
i
w(|ei |2)ψ(x+ei )ei = −G
�ψ∇ψ +
1
2ψ∇(∇2ψ) + · · ·
�
Guaranteed momentum conservation
� LB as a discrete approximation of a continuum theory
F = ∇V (ρ), where finite difference of ∇ is needed
� Postulation: ∀V , there is a ψ.
� Caution: Finite difference operator might not commute with somedifferential operators
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Determination of the mean force field
Two philosophically different starting points:
� LB as a discrete model
F = −Gψ(x)�
i
w(|ei |2)ψ(x+ei )ei = −G
�ψ∇ψ +
1
2ψ∇(∇2ψ) + · · ·
�
Guaranteed momentum conservation
� LB as a discrete approximation of a continuum theory
F = ∇V (ρ), where finite difference of ∇ is needed
� Postulation: ∀V , there is a ψ.
� Caution: Finite difference operator might not commute with somedifferential operators
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Outline
Lattice Boltzmann from kinetic theoryHow it worksNew insights
Modeling non-ideal gas in lattice BoltzmannAn intuitive modelStatistical physicsThermodynamic consistencyOther issues
Conclusions
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Thermodynamic consistency
Thermodynamics
� What does it mean? True temperature in EoS, Inter-molecularpotential energy ⇔ kinetic energy (latent heat).
� Applications: Boiling, Liquid cooling of electronics
� Difficulty: Conserve momentum and energy in discrete dynamics.
� Must have a correct thermal ideal-gas model first
� A viable approach: energy conservation 7
Consistencies:
� Consistent with ideal-gas thermodynamics
� Consistent with continuum kinetic theory (mean-field)
� Lack of energy conservation is a discrete artifact (nearest-neighbor) 8
7Sbragaglia et al, J. Fluid Mech., 628, 299, (2009)
8He & Doolen, J. Stat. Phys., 107, 309, (2002)
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Thermodynamic consistency
Thermodynamics
� What does it mean? True temperature in EoS, Inter-molecularpotential energy ⇔ kinetic energy (latent heat).
� Applications: Boiling, Liquid cooling of electronics
� Difficulty: Conserve momentum and energy in discrete dynamics.
� Must have a correct thermal ideal-gas model first
� A viable approach: energy conservation 7
Consistencies:
� Consistent with ideal-gas thermodynamics
� Consistent with continuum kinetic theory (mean-field)
� Lack of energy conservation is a discrete artifact (nearest-neighbor) 8
7Sbragaglia et al, J. Fluid Mech., 628, 299, (2009)
8He & Doolen, J. Stat. Phys., 107, 309, (2002)
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Outline
Lattice Boltzmann from kinetic theoryHow it worksNew insights
Modeling non-ideal gas in lattice BoltzmannAn intuitive modelStatistical physicsThermodynamic consistencyOther issues
Conclusions
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An intuitive modelStatistical physicsThermodynamic consistencyOther issues
Density ratio, instability, surface tension
� Stability at high-density ratio Acoustic CFL number > 1 9
� Equilibrium depends on τ Inaccurate forcing 10
� Inflexible surface tension coefficient Multi-range interaction 11
� Immiscible multi-component No such thing in reality
� . . . ?
Remaining issues:
� Equilibrium in multiple component system
� Stability at small viscosity (high Reynolds number multiphase flow)
9Kupershtokh, Computers and Math. Appl., 59, 2236, (2010)
10Yu & Fan, J. Comp. Phys., 228, 6456, (2009)
11Falcucci et al, Commun. Comput. Phys., 2, 1071, (2007), Shan, Phys. Rev. E, 77, 066702, (2008)
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
An updated view:
� A velocity-space discretization of Vlasov-Enskog
� Consistent with ideal-gas thermodynamics
� Exact discrete conservations with general interaction potential
� Energy conservation did not survive discretization
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow
Lattice Boltzmann from kinetic theoryModeling non-ideal gas in lattice Boltzmann
Conclusions
Thank you!
Xiaowen Shan ([email protected]) Physical Modeling of Multiphase flow