The Regularized 13 Moment Equations for Rarefied and Vacuum Gas Flows

Transcription

1 xxxx The Regularized 13 Moment Equations for Rarefied and Vacuum Gas Flows Henning Struchtrup Peyman Taheri Anirudh Rana University of Victoria, Canada Manuel Torrilhon RWTH Aachen

2 Goal: Accurate and efficient modelling of vacuum flows Microscopic theory: Boltzmann Equation microscopicvariable: distributionfunctionf(x i,t,c i )(7independentvariables!) Direct Numerical Solutions are accurate, but numerically expensive Direct Simulation Monte Carlo is powerful(molecules, reactions), but expensive Macroscopic transport equations ApproximationtoBoltzmann limitedrangeofvaliditykn n 1 collective behavior described by finite 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) heatflux,... fast deterministic solutions analytic solutions give deeper insight into processes

3 Order of magnitude method [HS 2004] Kn= mean free path macroscopic lengthscale of process Step by step derivation of equations from Boltzmann eq. O ( Kn 0) :Euler O ( Kn 1) :Navier-Stokes-Fourier O ( Kn 2) :Grad13 O ( Kn 3) :regularized13momentequations(r13) stableequationsatallorders[hs&mt2003] accessible for arbitrary interaction potentials[hs 2005]

4 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 mijk +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 R ik + 1 ] v j +m ijk x k 2 x k 6 x i x k = 5 [ 2 ρθ qi κ + θ ] x i = σ ijσ ij 12 µ [ θ q k v k +θσ kl + 7 ] ρ p x k x l 2 q θ θ p k q k x k p x k R ij = 4 1 7ρ σ k iσ j k 24 [ µ θ q i θ +2q i + 5 ( 5 p x j x j 7 θ v j v k σ k i +σ k i 2 ) x k x j 3 σ v k ij θ x k p q i m ijk = 2 µ [ θ σ ij θ +σ ij + 4 ] p x k x k 5 q v j θ p i σ ij x k p x k Chapman-Enskog expansion of R13 Euler/ NSF/ Burnett/ super-burnett p x j ]

5 Euler/NSF/Grad13/R13 (linearized)[hs&mt2003] t ρ+ρ 0 v=0 ρ 0 t v+ p+ σ=ρ 0 G 3 2 ρ 0 t θ+p 0 v+ q=0 t σ+ 4 5 q +2p 0 v = p 0 σ+ 2 µ 0 µ 0 3p 0 [ σ+ 65 ( σ) ] t q+ σ+ 5 2 θ= 2 p 0 q+ 6 µ [ 0 q+2 ( q) 3µ 0 5p 0 ] φ symmetric tracefree tensors

6 Boundary conditions for moments [MT& HS 2008] derived from Maxwell boundary conditions for Boltzmann eq. kineticbcforoddfluxes (atleftandrightboundary) slip σ tn = χ [ 2 P ( ) v t v W 1 t + 2 χ πθ 5 q t+ 1 ] 2 m tnn n n jump q n = χ [ 2 2P(θ θ W ) 1 2 χ πθ 2 PV θσ nn ] 28 R nn n n m ttn = χ [ χ πθ 14 R tt+θσ tt 1 5 θσ nn+ 1 5 P(θ θ W) 4 5 PV2 + 1 ] 150 m nnn =+ χ [ χ πθ 5 P(θ θ W) 3 5 PV2 7 5 θσ nn ] 14 R nn n n R tn =+ χ [ 2 Pθ ( ) v t v W 11 t 2 χ πθ 5 q t 1 ] 2 θm tnn PV 3 +6PV(θ θ W ) = purely local BC, well-posed problem! H-Theorematwallinlinearcase [HS&MT2007] 2ndorderBCforNSFinlimitO ( Kn 2) [HS&MT2009] withv t =v t vt W, v n =0, P =ρθ+ 1 2 σ nn θ 1 R nn 28 θ indices n, t: normal/tangential components [Gu&Emerson2007]: kineticbcforr13,buttoomanybcleadtospuriouswalllayers n n n n

7 ForcedrivenPoiseuilleflow [PT,MT&HS2008] R13 equations exhibit temperature dip [Tij& Santos 1994/98, Xu 2003] Analytical solution for all moments 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

8 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

9 Flow through cylindrical pipes [PT& HS 2010]

10 Flow through cylindrical pipes [PT& HS 2010] Poiseuille flow, dimensionless, pressure gradient NSFwith1stand2ndorderBC,R13 v z =C 1 + 4Kn r2 2 5 q z, σ rz = 2 r, q z=c 2 J 0 ( 5 9 ) r Kn Kn

11 Flow through cylindrical pipes [PT& HS 2010] Transpiration flow, dimensionless, temperature gradient τ NSF, R13 v z =C q z, σ rz =0, q z =C 2 J 0 ( 5 9 ) r Kn 15Kn 4 τ

12 Flow through cylindrical pipes [PT& HS 2010] Simultanous Poiseuille and transpiration flow, dimensionless, gradients, τ v z =C 1 + 4Kn r2 2 5 [ C 2 J 0 ( 5 9 zeromassflowrate 1 0 v zrdr=0 = C 1 thermomolecular pressure difference p 2 p 1 = ( ) γ T2 T 1 ] r )+ 32 Kn Kn 154 Knτ data points: [Sharipov 1996]

13 Lid-drivencavityflow [AR&HS(inprep.)] velocitystreamlinesandstresscontours Kn=0.1,v lid =30 m s numerical solution at steady state, cells, matlab, not optimized = <5minutes Drag δ= 1 Kn NSF Varoutisetal R

14 Lid-drivencavityflow [AR&HS(inprep.)] temperaturecontoursandheatfluxstreamlineskn=0.1,v lid =30 m s numerical solution at steady state, cells, matlab, not optimized = <5minutes

15 2D transpiration flow: Heated bottom plate [AR& HS(in prep.)] qualitativecomparisontodsmcforn 2,Kn=0.1/0.2

16 The R13 Equations: A useful tool for simulation and understanding of micro- and vacuum flows in the transition regime! rational derivation from Boltzmann equation analytical and numerical solutions of boundary value problems describe all rarefaction effects(linear/ non-linear) quantitativeaccuracyfor Kn 0.5 qualitativemeaningfuldescriptionfor Kn 1 smallest set of equations with Knudsen layers methods can be generalized: e.g., R26[Gu-Emerson] = larger Kn

17 ... stilllotstodo Applications of R13 for technical processes: CFD Numerical solutions: 2-D/3-D/transient pipeflows,heatexchangers,externalflow,knudsenpumps,... Development of equations for polyatomic gases gas mixtures phononheattransferinnanowires plasmas, radiation Optimize number of moments 13/26/45/... Hybridmethods: NSF-R13-RXY Boltzmann Variable regions with local Knudsen number as switching parameter

18 Are there meaningful macroscopic approximations? Burnett/super-Burnett equations (Chapman-Enskog expansion) unstable[bobylev 1982], stable alternatives available[bobylev, Söderholm, Zhong] boundary conditions not clear incomplete Knudsen layers, spurious oscillations Grad moment equations unclearhowmanyandwhichmomentsmustbeused boundary conditions not clear(for non-linear eqs) noknudsenlayerswith13moments 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 smoothshockstructuresforallma,agreetodsmcforma<3 reproduceallrarefactioneffectsforkn 0.5 accessible for fast analytical and numerical solutions same methods = R26 eqs[gu&emerson 2007]

19 Flow through cylindrical pipes [PT& HS 2010] Onsager symmetry E P energyfluxduetopressureforce M T massfluxduetotemperatureforce