Analytical and numerical solutions of the regularized 13 moment equations for rarefied gases

Transcription

1 xxxx Analytical and numerical solutions of the regularized 13 moment equations for rarefied gases Henning Struchtrup University of Victoria, Canada Manuel Torrilhon ETH Zürich Peyman Taheri University of Victoria, Canada

2 The task of finding continuum approximations conservation laws for mass, momentum, energy = 5equationsforρ, v i, θ=rt Dρ Dt +ρ v k x k =0 ρ Dv i Dt +ρ θ x i +θ ρ x i + [ σik x k ] =0 [ ] 3 2 ρdθ Dt +ρθ v k qk v k + +σ kl x k x k x l =0 closure problem findadditionalequationsforpressuredeviatorσ ij andheatfluxq i

3 Boltzmann equation and moments Boltzmann equation some moments idealgaslaw: p=ρθ f t +c f k = 1 x k Kn S(f) e.g. BGK-model: S(f)= 1 τ (f f M) massdensity ρ=m fdc momentumdensity ρv i =m c i fdc energydensity ρu= 3 2 ρθ= m 2 C 2 fdc pressuretensor p ij =pδ ij +σ ij =m C i C j fdc heatfluxvector q i = m 2 C 2 C i fdc generalmoments u a i 1 i n =m C 2a C i1 C in fdc peculiarvelocity:c i =c i v i equilibriumphasedensity(maxwell): f E = ρ m Kn= 1 2πθ 3 ] exp [ C2 2θ meanfreepathl 0 : Knudsennumber ˆ=smallnessparameter macroscopiclengthscalel

4 Bulk reduction methods with boundary conditions Kn= mean free path macroscopic length scale goal: Replace Boltzmann eq with simplified models for Knudsen number Kn< 1 Chapman-Enskog expansion in powers of Kn = Euler, Navier-Stokes-Fourier[Enskog 1917, Chapman 1916/17] = Burnett, super-burnett(unstable)[burnett 1935, Bobylev 1981] = augmented Burnett(stable)[Zhong et al. 1993] = hyperbolic Burnett(stable) [Bobylev 2007/08] Grad smomentmethod(choiceofmomentsnotrelatedtokn) [Grad1949] = Euler, 13 moments, 26 moments, etc. (discontinuous shocks) Regularization of 13 moment equations(based on Kn orders) = linearr13eqs [Karlinetal. 1998] = Regularized Burnett[Jin& Slemrod 2001] = Consistent order ET[Müller et al. 2003] = CombinedGrad/CE= R13eqs [HS&M.Torrilhon2003/04] = Orderofmagnitudemethod= R13eqs [HS2004]

5 R13 equations (non-linear) [HS& MT 2003, HS 2004] (Euler/NSF/Grad13/R13) Dρ Dt +ρ v k =0 x k ρ Dv i Dt +ρ θ +θ ρ [ ] σik + =ρg i x i x i x [ k ] 3 2 ρdθ Dt +ρθ v k qk v k + +σ kl =0 x k x k x l ] [ ] Dt +4 q i v j v k u 0 ijk +2σ k i +σ ij + = ρθ 5 x j x k x k x k [ Dσij [ ] σij µ +2 v i x j [ Dqi Dt +5 2 σ θ ik σ ik θ lnρ +θ σ ik + 7 x k x k x k 5 q v k i + 7 x k 5 q v i k + 2 ] x k 5 q v k k x [ i + σ ij σ jk + 1 wik w 2 ] +u 0 v j ijk x k 2 x k 6 x i x k w 2 = σ ijσ ij 12 µ ρ p u 0 ijk= 2 µ p wij= 1 4 σ k i σ j k 7 ρ [ θ q k [ θ σ ij x k σ ij lnρ ] x k 2 q θ lnρ v i k θq k +θσ ij x k x k x k + 4 ] x k 5 q v j i x [ k θ q i θ lnρ +q i θq i + 5 x j x j x j 7 θ µ p Chapman-Enskog expansion of R13 Euler/ NSF/ Burnett/ super-burnett ( σ k i v j x k +σ k i v k = 5 [ 2 ρθ qi κ + θ ] x i 2 )] x j 3 σ v k ij x k

6 R13 equations(linear, dimensionless) [HS& MT 2003] (Euler/NSF/Grad13/R13) ρ t + v k =0 x k v i t + ρ x i + θ x i + σ ik x k =G i 3 θ 2 t + v k + q k =0 x k x k σ ij t +4 5 q i x j 2Kn x k σ ij x k +2 v i x j = σ ij Kn q i t + σ ik 12 x k 5 Kn q i 2Kn q k + 5 x k x k x i x k 2 θ = 2 q i x i 3Kn

7 Semi-linearized R13 for shear flow (dimless) [PT, MT& HS 2009] includequadraticcontributionsin σ 12, velocity problem temperature problem the rest 2dq 1 16 σ 12 5dx 2 15 Knd2 dx 2 2 dv 1 dx 2 dσ 12 d x Knd2 q 1 dx 2 2 dσ 12 dx 2 =G 1 + dv 1 dx 2 = 1 Kn σ 12 = 2 1 3Kn q 1 dσ dx 2 7 σ dσ dx Kndσ 12dv dx 2 dx 2 35 Knσ d 2 v σ dv σ 11 dx 2 3 Knd2 6 5 σ dv σ 22 dx 2 5 Knd2 dx 2 2 dx Knd2 σ 22 dx Kn2 d dx Kn2 d dx 2 dx 2 2 dq 2 dv 1 = σ 12 dx 2 dx dθ = 2 1 2dx 2 3Kn q 2 ) = 1 Kn σ 22 ( dσ12 dx 2 dv 1 dx 2 d(ρ+θ+σ 22 ) =0 ( dx 2 ) dσ12 dv 1 dx 2 dx 2 = 1 Kn σ 11

8 Boundary conditions for moments [MT& HS 2008] kineticbcforoddfluxes (atleftandrightboundary) slip σ nt = χ [ 2 PV t χ πθ 5 q t+ 1 ] 2 u0 tnn jump q n = χ [ 2 2P(θ θ W )+ 5 2 χ πθ 28 w nn w kk+ 1 2 θσ nn 1 ] 2 PV2 t w tn =+ χ [ 2 PθV t 1 2 χ πθ 2 θu0 tnn 11 ] 5 θq t PVt 3 +6P(θ θ W )V t u 0 nnn=+ χ [ χ πθ 5 P(θ θ W) 1 14 w nn w kk 7 5 σ nn 3 ] 5 PV2 t u 0 ttn +1 2 u0 nnn = χ [ ( 2 θ σ tt + 1 ) 2 χ πθ 2 σ nn + 1 ( w tt + 1 ) 14 2 w nn 1 ] 2 PV2 t mass conservation M = L/2 L/2 ρdx withv t =v t vt W, P = ( ρθ+ 1 2 σ nn 1 w nn 28 θ indices n, t indicate normal/tangential components Semi-linear channel flow = well-posed problem! kineticbcforr13pioneeredby[gu&emerson2007],buttoomanybcleadtospuriouswalllayers H-Theorematwallinlinearcase[HS&MT2007] ) w kk θ

9 AnalyticalsolutionforCouetteflow [PT,MT&HS2008] velocity problem v 1 = C 1 Kn x q 1 temperature problem σ 12 =C 1 q 1 =C 2 sinh θ=c 3 2C2 1 15Kn 2x2 2 2C 4 5 cosh σ 22 =C 4 cosh [ ] 5 3Kn x 2 [ ] 5x2 + 32C 2 6Kn 35 5 σ 12cosh [ 5x2 6Kn ] 6C C σ 12cosh q 2 = C2 1 Kn x 2+ 2C 2 5 σ 12sinh [ ] 5x2 3Kn [ ] 5x2 3Kn 0 [ ] 5x2 3Kn superpositions of bulk and Knudsen layer contributions

10 Analytical solution for Couette flow: R13 vs. DSMC [PT, MT& HS 2008]

11 Analytical solution for Poiseuille flow [PT, MT& HS 2008] velocity problem v 1 =C 1 G 1 2Kn 0 x q 1 σ 12 =G 1 x 2 q 1 = 3G 1Kn 2 +C 2 cosh [ ] 5x2 3Kn temperature problem θ=c 4 G2 1x 4 2 1x Kn 2+488G2 525 q 2 = G2 1x 3 2 3Kn 6G 1KnC 2 5 sinh 5 2C 3 5 cosh [ ] 5x2 + 2C 2 3Kn 5 σ 12cosh [ ] 5x G 1KnC 2 6Kn 375 [ ] 5x2 3Kn cosh [ ] 5x2 + 32C 2 3Kn 35 5 σ 12sinh [ ] 5x2 3Kn σ 22 = 6G2 1 5 x2 2 84G2 1Kn G 1KnC 2 25 cosh [ ] [ ] 5x2 5x2 +C 3 cosh 12C 2 3Kn 6Kn 5 5 σ 12sinh [ ] 5x2 3Kn superpositions of bulk and Knudsen layer contributions

12 Force driven Poiseuille flow [PT, MT& HS 2008] R13 equations exhibit temperature dip [Tij& Santos 1994/98, Xu 2003] θ= C 4 G2 1 Kn 2 [ ] y Kn2 y 2 2 [ 5y ] C 3 5 cosh 6Kn +C G 1Kncosh +C σ 12sinh superposition of bulk solution Knudsen layers [ 5y [ 5y 3Kn 3Kn ] ] Kn=0.072,0.15,0.4,1.0

13 R13to2ndorderinthebulk (shearflowgeometry)[hs&mt2008] conservation laws + Navier-Stokes-Fourier σ 12 y =ρ G ( p+kn 2 σ ) 22 q 2 1, =0, y y = σ v 12 y, σ 12= µ v 1 y, q 2= 15 4 µ θ y second order contributions = 12 µθ p jumpandslipbc σ 11 = 8 5 σ 12 σ 12 p, σ 22 = 6 5 q 2 y +56 q 2 q 2 5 p +10θ p σ 12 σ 12, R22 = 16 5 σ 12 σ 12 p µθ p, q 1 = 3 µθ σ 12 2 p y +7 σ 12 q 2 2 p q 2 y +128 q 2 q 2 75 p p σ θ 12 σ 12, m 122 = µθ p σ 12 y +32 σ 12 q 2 45 p V= v i v W i Kn T = θ θ W Kn = 2 χ 1 πθ χ 1 2 πθ 2 = 2 χ 2 χ 2 σ 12 p n 2 1 Kn q 1 5 p 1Kn m p q 2 2p n KnV2 1 θkn σ 22 4 p 1 Kn 60 p 5 Kn R p secondorderjumpandslipbccombinetheabove v i vi W = 2 χ 1 πθ σ 12 Kn χ 1 2 p n Kn2µθ σ 12 p 2 y 19Kn2 σ 12 q 2 18 p 2 θ θ W = 2 χ [ 2 πθ q 2 Kn χ 2 2 2p n ( ) ] 2 35 Kn2µθ q 2 π 2 χ1 p 2 y +Kn θ σ 12 σ 12 Kn 2178 q 2 q p p 2 χ 1

14 ForcedrivenPoiseuilleflow Knudsenminimum [HS&MT2008] linearized Navier-Stokes with 2nd order slip(values for α and β vary between authors) σ 12 y =G 1, σ 12 = v y averagemassflux J= vdy J NS = G 1 12Kn J R13 = G 1 12Kn [ 1+ 6 π 2 α Kn + 12 β Kn 2] [ 1+6 π 2 ( π ) π v, v v W =αkn 2 y n 2 βkn 2 2 v y 2 15 Kn Kn Kn ( (1+5Kn) coth 5 6Kn ) 1+10Kn ) )] comparisonsuggests α=1.046, β=0.823

15 Absorption heating (similar to Knudsen minimum) [HS& MT 2008] gasheatedbyradiation: gasatrest,wallsatθ W,energyabsorbedS averagerelativetemperature E= θ θ W S dy Fourier and R13 (second order jump condition) E F = 1 1 π 45Kn +1 4 E R13 = Kn Kn π Kn ( 7π+160Kn π 2 ( [ ] +384Kn2) coth +2 ) 15π 62Kn

16 Thermal transpiration flow [PT& HS 2009] flowdrivenby T-gradientinwallKn=0.09,0.18,0.35,0.53 mass flow, heat flux, velocity : R13, linear Boltzmann[Ohwada, Aoki]

17 Thermal transpiration flow [PT& HS 2009] temperature profile and other non-linear effects: R13 prediction

18 Thermal transpiration flow [PT& HS 2009] half space problem influence of accommodation coefficient χ NSF/ linearized Boltzmann/ R13

19 Cylindrical flows [PT& HS 2009] velocity problem (linear) analytical solution σ rφ r +2σ rφ [ r ] =0 2 q φ r 5r +v φ = 1 σ rφ r Kn r [ ] qφ r r +q φ = 5 r 9Kn 2q φ v φ = C 1 1 2Knr +C 4r 2 5 q φ σ rφ = C 1 r 2 q φ =C 2 I 1 [ 5r 3Kn ] [ ] 5r +C 3 K 1 3Kn Knudsen layers are Bessel functions

20 Cylindrical flows [PT& HS 2009] velocity problem Kn = 0.08, Kn = 0.447, accommodation coefficients χ[aoki, Garcia]

21 Cylindrical flows [PT& HS 2009] temperature problem Kn = 0.03, Kn = 0.2 (numerical solution)

22 OscillatingCouetteflow [PT,AR,MT&HS2009] v w =v w 0 sinωt, Kn= µ 0 ρ 0 θ0 H, StokesnumberSt=H ρ0 ω µ 0

23 Oscillating Poiseuille flow [PT, AR, MT& HS 2009] G=G 0 sinωt, Kn= µ 0 ρ 0 θ0 H dimensionlessfrequencyw= ωh θ0

24 Dispersion and Damping [HS& MT 2003] phase speed and damping measured by Meyer and Sessler proper Knudsen number for oscillation Kn Ω =ω R13allowsproperdescriptionclosetonaturallimitKn Ω =1

25 Shocks: Comparison with DSMC results [MT& HS 2004] Success of R13

26 Switching criteria for hybrid codes [D.Lockerby, J.Reese, HS 2009] hybrid Boltzmann/NSF solvers: usensffor small Kn,Boltzmannfor large Kn requires local Knudsen number to distinguish domains usual choice: gradient Knudsen number (mean free path λ) Kn G = λ ρ dρ dx not too bad: for strongly non-linear flow(steep gradients, shocks etc.) problem: Kn G 0forlinearflow(microflows,ultrasound) goal: local Knudsen number for linear and non-linear regime

27 Switching criteria for hybrid codes [D.Lockerby, J.Reese, HS 2009] Switch Boltzmann/R13 = NSF Step 1: computeρ,v i,θ,σ ij,q i fromboltzmann/r13 Step 2: computeσ (NSF) ij = µ v i x, q (NSF) j i Step 3: local Knudsen number as deviation from NSF σ ij σ (NSF) ij Kn σ = σ (NSF) ij = κ θ x i fromboltzmann/r13 q i q (NSF) i, Kn q = q (NSF) i with q i = q i q i = q 2 1 +q2 2 +q2 3 σ ij = 1 2 σ iiσ jj σ ij σ ij = 1 2 σ ijσ ij = σ σ 11σ 22 +σ σ2 12 +σ2 13 +σ2 23

28 Switching criteria for hybrid codes [D.Lockerby, J.Reese, HS 2009] Switch Boltzmann/R13 = NSF Example I: Shock structure with Burnett/R13 NSF and Burnett/R13 in shock(leading term) σ (NSF) 11 = 4 3 µdv dx, σ(b) 11 = Aµ2 p ( ) 2 dv dx local Knudsen number Kn (shock) σ = 3 4 σ(b) σ(nsf) 11 = Aµ 2 ( dv ) 2 p dx 4 3 µ( ) dv dx = 3A 4p µdv dx =αmaλ ρ dρ dx. similar to gradient Knudsen number

29 Switching criteria for hybrid codes [Lockerby, Reese, HS 2009] Switch Boltzmann/R13 = NSF Example II: Nonlinear shear flow with second order hydrodynamics R13/Burnett(to second order in Kn) σ 12 = µ dv dy, σ 11= 8 σ 12 σ 12 5 p, σ 22 = 6 σ 12 σ 12 5 p, q 1 = 7 σ 12 q 2 2 p, q 2 = 15 4 µdθ dy local Knudsen numbers Kn (shear) σ = σ 12 p = ˆαMaλ v dv dy Kn (shear) q = 7 2 σ 12 p = ˇαMaλ v dv dy similar to gradient Knudsen number

30 Switching criteria for hybrid codes [Lockerby, Reese, HS 2009] Switch Boltzmann/R13 = NSF Example III: Linear Poiseuille flow with R13 equations R13 (drivingforcef,globalknudsennumberkn) ( 3 ( ) 1 1 σ 12 =Fy, v=f 2Kn 4 y2 + 1 π 25 (1+5Kn) 1 2 cosh Kn+ cosh [ 5 ] tanh 5 6Kn [ ] 59 y Kn [ ] 5 6Kn ) local Knudsen number σ ij σ (NS) ij Kn σ = σ (NS) ij with σ (NSF) 12 = Kn v 1 y =Fy+F 5 (1+5Kn) 5 [ tanh 5 6Kn [ ] sinh 5 y 9Kn cosh[ 5 6Kn ] ]

31 Switching criteria for hybrid codes [D.Lockerby, J.Reese, HS 2009] Switch NSF = Boltzmann/R13 Step 1: computeρ (NSF),v (NSF) i,θ (NSF), andσ (NSF) ij,q (NSF) i from NSF Step 2: insert NSF result into R13 to compute mismatch σ (R13) ij = µ [ 2p v i + Dσ ij p x j Dt +σ v k ij + 4 q i v j +2σ k i + m ] (NSF) ijk x k 5 x j x k x k q (R13) i = 3 [ µ 5 2p 2 p θ + Dq i x i Dt +5 2 σ θ ik +θ σ ik lnρ θσ ik + 7 x k x k x k 5 q v i k + x k Step 3: local Knudsen number as deviation from NSF σ (R13) ij σ (NS) ij Kn σ = σ (NS), Kn q = ij q (R13) i q (F) i q (F) i identifies non-linear rarefaction effects identifies linear bulk effects, can t identify Knudsen layers, ] (NSF)

32 Switching criteria for hybrid codes [Lockerby, Reese, HS 2009] Switch NSF = Boltzmann/R13 Example: linear shear flow with driving force F NSF reduce to R13 reduce to dσ (R13) 12 dy feednsfintor13 dσ (NS) 12 dy =F, σ (NS) 12 = Kn dv dy =F, σ (R13) 12 = Kn dv dy +52 σ Kn2d2 dy +9 v σ Kn3d3 dy Kn4d4 dy 4 σ (R13) 12 = Kn dv dy 5 v v 3 Kn3d3 dy Kn5d5 dy 5=σ(NS) Kn2dF dy 48 F 25 Kn4d3 dy 3 local Knudsen number Kn σ =Kn 2 5 df 3dy Kn2d3 F Fdy dy 3

33 Regularized 13 moment equations rational derivation from Boltzmann equation third order in Knudsen number( super-burnett) linearly stable phase speeds and damping of ultrasound waves agree to experiments smoothshockstructuresforallma,agreetodsmcfor Ma<3 H-theorem for linear case, including boundary conditions theory of boundary conditions Knudsen boundary layers in good agreement to DSMC accurate Poiseuille flow, second order slip conditions accurate thermal transpiration flow define local Knudsen number Future work 2-D/3-D/transient simulations increased understanding of BC for non-linear case RXY equations for polyatomic gases and mixtures