Regularized 13 Moment Equations for Maxwell and Hard Sphere - PowerPoint PPT Presentation

xxxx Regularized 13 Moment Equations for Maxwell and Hard Sphere molecules Henning Struchtrup Anirudh Rana Alireza Mohammadzadeh University of Victoria, Canada Manuel Torrilhon RWTH Aachen, Germany

The case for moment equations • Microscopic theory: Boltzmann Equation — microscopic variable: distribution function f ( x i , t, c i ) (7 independent variables!) — Direct Numerical Solutions are accurate, but numerically expensive — Direct Simulation Monte Carlo is powerful (molecules, reactions), but expensive • Macroscopic transport equations — Approximation to Boltzmann ⇒ limited range of validity Kn n ¿ 1 — collective behavior described by fi nite number N of macroscopic variables, e.g. ρ ( x i , t ) — density, v i ( x i , t ) — velocity, T ( x i , t ) — temperature σ ij ( x i , t ) — stress, q i ( x i , t ) — heat fl ux , ... — fast deterministic solutions — explicit equations, analytic solutions give deeper insight into processes

Are there meaningful macroscopic continuum approximations? Burnett/super-Burnett equations (CE expansion) [Burnett, 1935] • unstable [Bobylev 1982], stable alternatives available [Bobylev, Söderholm, Zhong] • boundary conditions not clear • incomplete Knudsen layers, spurious oscillations Grad moment equations [Grad 1949] • unclear how many and which moments must be used • boundary conditions not clear (for non-linear eqs) • no Knudsen layers with 13 moments Regularized 13 moment equations [Struchtrup & Torrilhon, since 2003] • derived from Boltzmann equation (order of magnitude method) • third order in Knudsen number (CE expanison = ⇒ super-Burnett) • complete theory of boundary conditions • linear equations: stable, H-theorem (incl. boundary conditions) • phase speeds and damping of ultrasound waves agree to experiments • smooth shock structures for all Ma, agree to DSMC for Ma < 3 • reproduce all rarefaction e ff ects for Kn . 0 . 5 • accessible for fast analytical and numerical solutions • same methods = ⇒ R26 eqs [Gu&Emerson 2007]

Order of magnitude method [HS 2004] mean free path Kn = macroscopic lengthscale of process Step by step derivation of equations from Boltzmann eq. ¡ Kn 0 ¢ • O : Euler ¡ Kn 1 ¢ • O : Navier-Stokes-Fourier ¡ Kn 2 ¢ • O : Grad 13 ¡ Kn 3 ¢ : regularized 13 moment equations (R13) • O • stable equations at all orders [HS & MT 2003] • accessible for arbitrary interaction potentials [HS 2005, HS&MT 2013 ] = ⇒ today

Motivation: Dependence of moments on molecular interaction model Couette fl ow at Kn=0.25, DSMC with VHS, VSS, Maxwell Hydrodynamic quantities not much a ff ected: velocity temperature shear stress normal heat fl ux

Motivation: Dependence of moments on molecular interaction model Couette fl ow at Kn=0.25, DSMC with VHS, VSS, Maxwell Hydrodynamic quantities not much a ff ected: velocity temperature shear stress normal heat fl ux Rarefaction quantities (Burnett etc.): visible dependence on molecular model parallel heat fl ux pressure normal stress

Moment method in kinetic theory kinetic equation: Knudsen number ε ∂ f ∂ f = 1 ∂ t + c k ε S ( f ) ∂ x k moment method: replace kinetic equation with equations for moments Z u A = ψ A ( c i ) fdc A = 1 , . . . , N moment equations: multiply kinetic equation with ψ A ( c i ) and integrate ∂ u A ∂ t + ∂ F Ak = 1 ε P A ∂ x k fl uxes: F Ak = R ψ A ( c i ) c k fd c , productions: P A = R ψ A ( c i ) S ( f ) d c Question 1: which moments: ψ A ( c i ) = ?? Question 2: how many moments: N = ?? Question 3: how to close the equations: F Ak ( u B ) = ?? , P A ( u B ) = ?? Q3: Grad closure; Q1, Q2, Q3: Order of Magnitude method

Order of magnitude method [HS 2004] Step 1: • set up moment system for arbitrary number of moments N • close with Grad method Step 2: • Chapman-Enskog expansion to fi nd leading ε − order of moments • linear combination of moments such that number of moments at given ε − order is minimal • repeat for next order of magnitude Step 3: • use ε − orders to rescale equations for new moments • use scaling for model reduction to a given order of accuracy

Step 1: Grad closure for linear moment equations hydrodynamic moments Z Z Z ρ u = 3 2 ρθ = 1 C 2 fd c ρ = fd c , ρ v i = c i fd c , 2 Z Z q i = 1 C 2 C i fd c p ij = p δ ij + σ ij = C i C j fd c , 2 general moments: a = 0 , 1 , . . . , N n , n = 0 , 1 , 2 , . . . h· · · i indicates tracefree tensors Z u a C 2 a C h i 1 C i 2 · · · C i n i fd c i 1 ··· i n = with u 0 = ρ , u 0 i = 0 , u 1 = 3 ρθ = 3 p , u 0 ij = σ ij , u 1 i = 2 q i Grad-type distribution: ¸ Ã ! ∙ N n X X − C 2 ρ λ a j 1 ··· j n C 2 a C h j 1 C i 2 · · · C j n i f | G = 3 exp 1 + √ 2 θ 2 πθ n =0 a =0 coe ffi cients λ b i 1 ··· i n from inversion of Z N m X i 1 ··· i n θ a + b + n (2 ( a + b + n ) + 1)!! u a C 2 a C h i 1 C i 2 · · · C i n i f | G d c = ρ n ! λ b i 1 ··· i n = (2 n + 1)!! b =0

Step 1: Grad closure for linear moment equations (dimless) Conservation laws ∂ρ ∂ t + ∂ v k ∂ v i ∂ t + ∂ρ + ∂θ + ∂σ ik 3 ∂θ ∂ t + ∂ v k + ∂ q k = 0 = G i = 0 , , ∂ x k ∂ x i ∂ x i ∂ x k 2 ∂ x k ∂ x k equations for higher moments (renumbered), a = 1 , . . . , N N N X X w a u b ∂ ˜ ∂ ˜ − (2 a + 3)!!2 ( a + 1) ∂ q k = − 1 R (1) C (0) ˜ ˜ k w b ∂ t + ab ˜ ab ∂ x k 3 ∂ x k ε b =1 b =1 X N X N u a u b w a ∂ ˜ ∂ ˜ + 1 ∂ ˜ − (2 a + 3)!! ∂σ ik + (2 a + 3)!! a ∂θ = − 1 R (2) ˜ C (1) ˜ i u b ik ∂ t + ab ˜ i ab ∂ x k 3 ∂ x i 3 ∂ x k 3 ∂ x i ε b =1 b =1 u a X N u a u a ∂ ˜ ∂ ˜ ∂ t + ∂ ˜ ∂ v h i + 2 + 2 (2 a + 3)!! = − 1 ij ijk h i C (2) ˜ u b ab ˜ ij ∂ x k 5 ∂ x j i 15 ∂ x j i ε b =1 X N u b X N u a u a ∂ ˜ ∂ ˜ + ∂ ˜ + 3 = − 1 h ij ijk ijkl R (2) C (3) ˜ ˜ u b ab ˜ ijk ab ∂ t ∂ x l 7 ∂ x k i ε b =1 b =1 etc. w a = ˜ u a − ˜ u a ˜ E : non-equilibrium part of scalar moments R ( n ) ˜ ab : coe ffi cients from Grad closure (closure coe ffi cients only for b = N ) C ( n ) ˜ ab : production matrices: collision term + Grad closure [Gupta & Torrilhon, RGD28] ε : Knudsen number

Step 2: O of M method: CE expansion Chapman-Enskog expansion of moments : for now, interested in zeroth and fi rst order φ = φ 0 + εφ 1 + ε 2 φ 2 + · · · only vector and 2-tensor moments have fi rst order contributions : ∂ v h i ∂θ u b u b ˜ i, 1 = − κ b , ˜ ij, 1 = − μ b ∂ x i ∂ x j i h C (1) i − 1 h C (2) i − 1 X N X N ab (2 b + 3)!! b ab (2 b + 3)!! 2 ˜ ˜ κ a = , μ a = 3 15 b =1 a =1 Navier-Stokes-Fourier μ∂ v h i κ ∂θ q NSF σ NSF = − ˆ , = − 2ˆ i ij ∂ x i ∂ x j i κ = ε κ 1 μ ˆ μ = ε (dimless), Prandtl: Pr = 5 κ = 5 heat conductivity: ˆ 2 , viscosity ˆ 2 ˆ κ 1 ¡ ε 0 ¢ ⇒ ρ , v i , θ are O = ¡ ε 1 ¢ u b u b = ⇒ ˜ i , ˜ ij are O ¡ ε 2 ¢ ⇒ all others are at least O =

Step 2: O of M method: Reconstructing variables to fi rst order: linear dependence ( a = 2 , . . . , N ) ∂θ = 2 κ a u a ˜ i, 1 = − κ a q i, 1 ∂ x i κ 1 ∂ v h i = μ a u a ˜ ij, 1 = − μ a σ ij, 1 ∂ x j i μ 1 de fi ne second order variables ( a = 2 , . . . , N ) i − 2 κ a w a u a ˜ i = ˜ q i κ 1 ij − μ a w a u a ˜ ij = ˜ σ ij μ 1 ¡ ε 0 ¢ ⇒ ρ , v i , θ are O = ¡ ε 1 ¢ = ⇒ q i , σ ij are O ¡ ε 2 ¢ = ⇒ all others are at least O moments describe collective behavior ij depend on interaction C ( n ) w a w a = ⇒ new moments ˜ i , ˜ ab

Step 2: O of M method: Next CE expansion ¡ ε 2 ¢ higher moments to leading order O w b = − ζ b ε∂ q k ˜ ∂ x k ∙ ¸ i = − ϑ b ε∂σ ik q i + 5 ε ∂θ w b ˜ − η b 2 Pr ∂ x k ∂ x i ∙ ¸ ij = − ϕ b ε ∂ q h i σ ij + 2 ε∂ v h i w b ˜ − φ b ∂ x j i ∂ x j i ijk = − ξ b ε∂σ h ij u b ˜ ∂ x k i coe ffi cients depend on interaction, closure " # h C (0) i − 1 X N X N κ b − (2 a + 3)!!2 ( a + 1) ˜ R (1) ˜ ζ c = 2 ab κ 1 3 ca a =1 b =1 " N ¸# ∙ μ 2 ∙ a (2 a + 3)!! ¸ ∙ ¸ h D (1) i − 1 h D (1) i − 1 X N X X N μ c − (2 a + 3)!! − κ a , η b = 2 − 5 κ a ab − κ a ˜ ˜ ˜ , ˜ D (1) C (1) ˜ C (1) ˜ R (2) ϑ b = − 5 ab = ac 1 b μ 1 3 κ 1 μ 1 κ 1 3 κ 1 κ 1 ba ba a =2 c =1 a =2 ∙ κ a ¸ ∙ (2 a + 3)!! ¸ ∙ ¸ h D (2) i − 1 h D (2) i − 1 X N X N ϕ b = 4 − μ a , φ b = 2 − μ a ab − μ a ˜ ˜ , ˜ D (2) C (2) ˜ C (2) ˜ ab = 1 b 5 κ 1 μ 1 μ 1 15 μ 1 μ 1 ba ba a =2 a =2 h C (3) i − 1 X N X N ξ b = 3 μ c ˜ ˜ R (2) ac 7 μ 1 ba a =1 c =1