Physical Modeling of Multiphase flow. Boltzmann method

Transcription

1 with lattice Boltzmann method Exa Corp., Burlington, MA, USA Feburary, 2011

2 Scope

3 Scope Re-examine the non-ideal gas model in [Shan & Chen, Phys. Rev. E, (1993)] from the perspective of kinetic theory

4 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

5 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

6 Lattice Boltzmann from kinetic theory

7 Outline Lattice Boltzmann from kinetic theory

8 General Principles 1 Lattice Boltzmann can be obtained from continuum kinetic theory based on two observations: 1. For hydrodynamics, only the leading moments of the distribution function matter explicitly. 2. If (and only if) distribution is a finite Hermite expansion, leading moments 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 moments preserved. 1 Shan et al, J. Fluid Mech., 550, 413(2006)

9 Isomorphism between moments and discrete velocities Hydrodynamic moments f (ξ)ξ m dξ. Gauss-Hermite quadrature: For polynomial p: ω(ξ)p(ξ)dξ = d w i p(ξ i ). i=1 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 through quadrature. N + M order of quadrature

10 Projection into low-dimensional Hilbert space Take the continuum BGK equation with body force: f t + ξ f + g ξf = 1 f f (0) τ Expand f in Hermite series (orthogonal projection) f (ξ) =ω(ξ) n=0 1 n! a(n) H (n) (ξ) where ω(ξ) = Construction by Gram-Schmidt process: a (n) = f (ξ)h (n) (ξ)dξ Evaluate the truncated equation on discrete velocities 1 ( 2π) D exp ξ2 2

11 Projection into low-dimensional Hilbert space Take the continuum BGK equation with body force: f t + ξ f + g ξf = 1 f f (0) τ Expand f in Hermite series (orthogonal projection) f (ξ) =ω(ξ) n=0 1 n! a(n) H (n) (ξ) where ω(ξ) = 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 1 ( 2π) D exp ξ2 2

12 Projection of the Maxwell distribution The (dimensionless) Maxwellian: f (0) 1 (ξ) = exp ξ2 (2πθ) D/2 2θ Construction by Gram-Schmidt process: f (0) (ξ) =ρω 1+u ξ + 1 (u ξ) 2 u 2 + (θ 1)(ξ 2 D) + 2 Differences from low-mach number expansion: ξ and u scaled with sound speed, universal on any lattice

13 Projection of the Maxwell distribution The (dimensionless) Maxwellian: f (0) 1 (ξ) = exp ξ2 (2πθ) D/2 2θ Construction by Gram-Schmidt process: f (0) (ξ) =ρω 1+u ξ + 1 (u ξ) 2 u 2 + (θ 1)(ξ 2 D) + 2 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.

14 Projection of the Maxwell distribution The (dimensionless) Maxwellian: f (0) 1 (ξ) = exp ξ2 (2πθ) D/2 2θ Construction by Gram-Schmidt process: f (0) (ξ) =ρω 1+u ξ + 1 (u ξ) 2 u 2 + (θ 1)(ξ 2 D) + 2 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

15 Projection of the Maxwell distribution The (dimensionless) Maxwellian: f (0) 1 (ξ) = exp ξ2 (2πθ) D/2 2θ Construction by Gram-Schmidt process: f (0) (ξ) =ρω 1+ (θ 1)(ξ2 D) + u ξ + 1 (u ξ) 2 u 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

16 Projection of the body force 2 The body force term: g ξ f.let: f (ξ) =ω(ξ) n=0 Body-force term has the following expansion: g ξ f = ω(ξ) n=1 1 n! a(n) H (n) (ξ) 1 n! ga(n 1) H (n) (ξ). Due to conservations of mass and momentum, up to second moments: g ξ f = g ξ f (0) 2 Martys, Shan & Chen, Phys. Rev. E, 58, 6855(1998)

17 Projection of the body force 2 The body force term: g ξ f.let: f (ξ) =ω(ξ) n=0 Body-force term has the following expansion: g ξ f = ω(ξ) n=1 1 n! a(n) H (n) (ξ) 1 n! ga(n 1) H (n) (ξ). Due to conservations of mass and momentum, up to second moments: 2 Martys, Shan & Chen, Phys. Rev. E, 58, 6855(1998) g ξ f = g ξ f (0) (0) (ξ u) gf. θ (ξ u) gf (0) represents a body-force.

18 Outline Lattice Boltzmann from kinetic theory

19 New insight Necessary and sufficient conditions on equilibrium distribution and underlying 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

20 New insight Necessary and sufficient conditions on equilibrium distribution and underlying 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 Grad 13-moments) Navier-Stokes: asymptotically truncated Boltzmann Contains (not approximates) compressible Navier-Stokes Asymptotically approaches to Boltzmann-BGK

21 LB compressible flow solver 3 Figure: Flow past a 15 wedge (Static pressure). Ma=1.8. Shock angle: Theory 51,Simulation Nie et al, AIAAPaper ,(2009)

22 Outline Lattice Boltzmann from kinetic theory Lattice Boltzmann from kinetic theory

23 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 F(x, x ). x All mass collapses to singular point. Needs repulsive hard-sphere. But a potential over distance would be in-practical. 4 Shan & Chen, Phys. Rev. E, 47, 1815,(1993)

24 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 F(x, 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 density when 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). 4 Shan & Chen, Phys. Rev. E, 47, 1815,(1993)

25 Intuitions 4 Lattice Boltzmann from kinetic theory 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 F(x, 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 density when 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 4 Shan & Chen, Phys. Rev. E, 47, 1815,(1993)

26 Features Non-ideal gas equation of state: p = ρθ + Gψ 2 (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

27 Outline Lattice Boltzmann from kinetic theory Lattice Boltzmann from kinetic theory

28 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 ξ f, a: a mean-field interaction force field Recent progresses: How is a computed? How does a enter into LB dynamics? 5 Martys, Int. J. Mod. Phys. C, 10, 1367(1998) 6 He, Shan & Doolen, Phys. Rev. E, 57, R13,(1998)

29 Determination of the mean force field Two philosophically different starting points: LB as a discrete model F = Gψ(x) i w( e i 2 )ψ(x+e i )e i = G ψ ψ ψ ( 2 ψ)+ Guaranteed momentum conservation LB as a discrete approximation of a continuum theory F = V (ρ), where finite difference of is needed

30 Determination of the mean force field Two philosophically different starting points: LB as a discrete model F = Gψ(x) i w( e i 2 )ψ(x+e i )e i = G ψ ψ ψ ( 2 ψ)+ Guaranteed momentum conservation LB as a discrete approximation of a continuum theory F = V (ρ), where finite difference of is needed Postulation: V, thereisaψ. Caution: Finite difference operator might not commute with some differential operators

31 Outline Lattice Boltzmann from kinetic theory Lattice Boltzmann from kinetic theory

32 Thermodynamics What does it mean? True temperature in EoS, Inter-molecular potential 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 7 Sbragaglia et al, J. Fluid Mech., 628, 299,(2009) 8 He & Doolen, J. Stat. Phys., 107, 309,(2002)

33 Thermodynamics What does it mean? True temperature in EoS, Inter-molecular potential 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 7 Sbragaglia et al, J. Fluid Mech., 628, 299,(2009) 8 He & Doolen, J. Stat. Phys., 107, 309,(2002)

34 Outline Lattice Boltzmann from kinetic theory Lattice Boltzmann from kinetic theory

35 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) 9 Kupershtokh, Computers and Math. Appl., 59, 2236,(2010) 10 Yu & Fan, J. Comp. Phys., 228, 6456,(2009) 11 Falcucci et al, Commun. Comput. Phys., 2, 1071,(2007),Shan,Phys. Rev. E, 77, ,(2008)

36 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

37 Thank you!