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


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



Conclusions

Lattice Boltzmann from kinetic theoryHow it worksNew insights

Modeling non-ideal gas in lattice BoltzmannAn intuitive modelStatistical physicsThermodynamic consistencyOther issues

Conclusions



Conclusions

How it worksNew insights

Outline



Conclusions



Conclusions


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)



Conclusions


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



Conclusions


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



Conclusions


Projection of the Maxwell distribution

The (dimensionless) Maxwellian:

f (0)(ξ) =1

(2πθ)D/2exp

�−ξ2

2θ

�


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



Conclusions


Projection of the Maxwell distribution

The (dimensionless) Maxwellian:

f (0)(ξ) =1

(2πθ)D/2exp

�−ξ2

2θ

�


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



Conclusions


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)



Conclusions


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)



Conclusions


Outline



Conclusions



Conclusions


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



Conclusions


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)



Conclusions

An intuitive modelStatistical physicsThermodynamic consistencyOther issues

Outline



Conclusions



Conclusions


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)



Conclusions


Intuitions 4


F(x, x�) ∼ − Gψ(ρ(x))ψ(ρ(x�))


ρ∆u = τ�

x�

F(x, x�).



� Essence: Mean-field interaction force field

4Shan & Chen, Phys. Rev. E, 47, 1815, (1993)



Conclusions


Intuitions 4


F(x, x�) ∼ − Gψ(ρ(x))ψ(ρ(x�))


ρ∆u = τ�

x�

F(x, x�).



� Essence: Mean-field interaction force field4Shan & Chen, Phys. Rev. E, 47, 1815, (1993)



Conclusions


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



Conclusions


Outline



Conclusions



Conclusions


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)



Conclusions


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



Conclusions


Outline



Conclusions



Conclusions


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)



Conclusions


Outline



Conclusions



Conclusions


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)



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



Conclusions

Thank you!


Physical Modeling of Multiphase flow with lattice ... - LBM.pdf · Physical Modeling of Multiphase...

Documents

Transcript of Physical Modeling of Multiphase flow with lattice ... - LBM.pdf · Physical Modeling of Multiphase...