A GAS KINETIC SCHEME FOR HYBRID SIMULATION OF PARTIALLY RAREFIED FLOWS

Transcription

1 Progress in Flight Physics ) DOI: /eucass/ A GAS KINETIC SCHEME FOR HYBRID SIMULATION OF PARTIALLY RAREFIED FLOWS S. Colonia, R. Steijl, and G. Barakos School of Engineering, University of Liverpool Liverpool L69 3GH, U.K. Approaches to predict ow elds that display rarefaction e ects incur a cost in computational time and memory considerably higher than methods commonly employed for continuum ows. For this reason, to simulate ow elds where continuum and rare ed regimes coexist, hybrid techniques have been introduced. In the present work, analytically de ned gas-kinetic schemes based on the Shakhov and Rykov models for monoatomic and diatomic gas ows, respectively, are proposed and evaluatedwiththeaimtobeusedinthecontextofhybridsimulations. This should reduce the region where more expensive methods are needed by extending the validity of the continuum formulation. Moreover, since forhigh-speedrare edgas owsitisnecessarytotakeintoaccountthe nonequilibrium among the internal degrees of freedom, the extension of the approach to employ diatomic gas models including rotational relaxation process is a mandatory rst step towards realistic simulations. Compared to previous works of Xu and coworkers, the presented scheme is de ned directly on the basis of kinetic models which involve a Prandtl number correction. Moreover, the methods are de ned fully analytically insteadofmakinguseoftaylorexpansionfortheevaluationoftherequired derivatives. The scheme has been tested for various test cases and Mach numbers proving to produce reliable predictions in agreement with other approaches for near-continuum ows. Finally, the performance of the scheme, in terms of memory and computational time, compared to discrete velocity methods makes it a compelling alternative in place of more complex methods for hybrid simulations of weakly rare ed ows. NOMENCLATURE c x c x f F m Particlevelocityin x-direction c x u x,particlevelocity uctuation particle distribution function mf molecular mass The authors, published by EDP Sciences. This is an Open Access article distributed under the terms of the Creative Commons Attribution License Article available at or

2 PROGRESS IN FLIGHT PHYSICS p ρtnondimensional equilibrium pressure) p r ρt r nondimensionalrotationalpressure) p t ρt t nondimensionaltranslationalpressure) q x Totalheat uxin x-direction qx r Rotationalheat uxin x-direction qx t translationalheat uxin x-direction R gas constant t Time variable T Total temperature T r Rotationaltemperature T t translationaltemperature u x velocityin x-direction W conservative variables vector x spatial coordinates vector Z r collisionnumber c x velocitystep t time step x Spatial step ζ Rotational degrees of freedom ā 0 1, c x, c 2,0) T ā 1 0,0,1,1) T λ Particle mean free path µ Viscosity ρ Density τ particle collision time φ generic macroscopic variable φ nondimensional generic macroscopic variable ACRONYMS AUSM Advection Upstream Splitting Method BGK BhatnagarGrossKrook BTE Boltzmann Transport Equation CE ChapmanEnskog DSMC Direct Simulation Monte Carlo DVM Discrete Velocity Method ES Ellipsoidal-Statistical GKS Gas-Kinetic Scheme GKS τ GKSwithmodi ed τ MD Molecular Dynamics MPC Modular Particle-Continuum MPI Message Passing Interface 302

3 NONEQUILIBRIUM AND RAREFIED FLOWS M C MultiphysicsCode NS NavierStokes TVD Total Variation Diminishing UGKS Uni ed GKS 1 INTRODUCTION At intermediate altitudes7090 km), the ow around hypersonic aircraft can be characterized as mainly continuum with localized areasgenerated by the rapid expansioninthewakeofthevehicleaswellasbystronggradientsinshockwaves and boundary layers) that display rarefaction e ects. The ow conditions near the vehicle surface and in the wake determine the drag and the heat transferred to the vehicle and its payload. Therefore, it is important that these regions are simulated using appropriate physical models. When the gradients of the macroscopicvariablesbecomesosteepthattheirlengthscaleisofthesameorderas the average distance travelled by molecules between collisions, the number of impacts is not enough to drive the uid towards a local thermodynamic equilibrium. Attheseconditions,the owcannolongerbeconsideredacontinuumandthe transport terms in the NavierStokesNS) equations fail since the constitutive relation is not valid. The mathematical model at molecular level is the Boltzmann transport equationbte)[1]andfortheregionsofthe ow eldwherehighlynonequilibrium e ects occur, the direct simulation Monte CarloDSMC) method[2] is typically employed to statistically estimate the solution of the BTE. Alternatively, adiscretevelocitymethoddvm)[3,4]canbeusedtosolveakineticmodel approximation of the BTE[57]. In previous works[811], hybrid techniques have been introduced to simulate ow elds where continuum and rare ed regimes coexist. In these methods, the more expensive approaches, such as DSMC or DVM, are employed only where needed and is coupled with a nite-volume scheme for the NS equations used where the ow is continuum. A hybrid technique couples two simulation methodsbymeansofinformationexchangebetweenthepartsofthe owdomain.in recent works, this has been achieved using an overlap region where ow state variables or numerical uxes are exchanged between the two models[12] or employing a bu er region where the two models are blended at equation level[13]. Recent works on these methods focused on rare ed high-mach ow can be found in[1416]. An alternative approach is the uni ed gas-kinetic schemeugks)[17, 18] which uses a nite-volume method where the numerical uxes are based on the solutionoftheshakhovmodel[6]foramonoatomicgasortherykovmodel[7] for a diatomic gas with rotational nonequilibrium. Where the ow is underresolved, by accounting for the pressure jump in the de nition of the collision 303

4 PROGRESS IN FLIGHT PHYSICS time, additional numerical viscosity is added resulting in a shock thickness of theorderofthecellsize[19]. ThisallowstheUGKStosimulate owsinboth rare ed and continuum regimes. In the present work, two GKS methods, analytically-de ned on the basis of the ChapmanEnskogCE) expansion of nondimensional Shakhov and Rykov models, are proposed for the simulation of weakly rare ed ows. The derivatives of the equilibrium function and the time derivatives of the primitive variables are analytically de ned, employing the compatibility condition of the kinetic model forthelatter. InpreviousworksfromXuandcoworkers[20,21],similarGKS are de ned using the CE solution of the BhatnaganGrossKrookBGK) model with rotational nonequilibrium and a scaling of the energy numerical ux[19] to correct the Prandtl number. Moreover, in those schemes, the required derivatives are expressed in terms of Taylor series where the coe cient are calculated by meansofpropertiesoftheemployedbgkmodel. TheproposedGKS,dueto the use of the CE expansion, is limited to near-continuum regions but is simpler thantheugks[17,18].however,thevalidityoftheapproachcanbeextended considering a modi ed collision time[22]. Also, this correction, in the present work, is de ned fully analytically for both schemes. Based on a literature survey of related works, the authors believe that the proposed GKS represents an e cient method, relative to DVM, capable of modeling complex diatomic gas ows with moderate rarefaction e ects but with signi cant rotational nonequilibrium. As such, the proposed approach is a novel alternative to the DVM for a range of practically relevant ows. Moreover, the update of the nonequilibrium distribution function, as used in the UGKS, is neglected reducing the memory cost of the approach. Thus, the GKS method represents a viable option to reduce the cost of hybrid simulations by reducing the region where the expensive method is needed and extending the validity of the continuum formulation. The schemes are built in a computational framework described in section 2 that also includes a DVM for the kinetic Boltzmann equations successfully employed for di erent monoatomic cases[23]. The framework has been recently improved[24] with the addition of the Rykov model and an Ellipsoidal-Statistical ES) model[25] for diatomic gases with rotational nonequilibrium. In section 3, the mathematical de nition of the Rykov model and of the proposed GKS are described. Finally, in section 4, some preliminary results where the presented GKSiscoupledwithaDVMareshownandanassessmentofreliabilityand performance of the GKS is presented. 2 MULTIPHYSICS CODE ThemethodsusedinthepresentworkarebuiltinthemultiphysicscodeM C) developed at the University of Liverpool[23, 26, 27]. Multiphysics code is a com- 304

5 NONEQUILIBRIUM AND RAREFIED FLOWS putational framework designed for simulations of complex ows, where di erent mathematical ow models are employed for di erent regions of the ow domain depending on the ow physics. The NS equations represent the baseline level of mathematical models used. For the continuum ow solver based on the compressible NS equations, a cell-centered block-structured nite-volume method isemployedusingtheausm + /upadvectionupstreamsplittingmethod)for the convective uxes[28]. For low-speed ow analysis, the framework further includes a Lattice Boltzmann Method as well as a Vortex-In-Cell method for vortex-dominated incompressible ows. In the present work, emphasis is placed on the simulation of hypersonic, partially-rare ed ows for which mathematical models at a more detailed level of physics than the compressible NS equations are required. For ows with strong nonequilibrium and rare ed e ects, the framework includes Molecular Dynamics MD) methods as well as deterministic DVM for a range of kinetic Boltzmann equations. The Shakhov and ES models are included for monoatomic gas ows, while the Rykov model and a polyatomic ES-BGK model were implemented for diatomic gas ow simulations[24]. The kinetic models are discretized using a discrete velocity method within a nite-volume method for multiblock structured grids; second order total variation diminishingtvd) time marching is employed. The velocityphase) space is discretized using either a uniformly spaced method with the trapezoidal rule for the evaluation of the moments of the distribution functions or a Gauss-quadrature method with modi ed Hermite polynomials. For the present high-speed ow cases, the uniform velocity space with the trapezoidal rule is the preferred approach and was used exclusively. The memory and CPU time requirements are considerable and for this reason, an e cient parallel implementation involving two levels of parallelism was conceived. In this parallelization, the phase space as well as the ow domain are distributed over the processes. First, the phase space is partitioned in regular subspaces, each to be assigned to separate processes within separate message passing interfacempi) communicators. The overall number of processes is then divided by the number of partitions to obtain the required number of communicators. The mesh-blocks in physical space are then distributed over these communicators to obtain an equal distribution of the cells. An important factor in the performance is the number of processes assigned to the velocity space discretization. Limiting the size of these communicators will minimize the overhead in collective operations required for the evaluations of the moments of the distribution functions. However, this implies an increasing number of communicators over which the mesh-blocks are divided, potentially creating a load imbalance in physical space. The best performance is obtained if the number of velocity space partitions is chosen such that the load imbalance in physical space is typically less than 10%. The parallel performance of the coupled continuum/kinetic solver was investigated as part of a European Union funded projectprace Preparatory 305

6 PROGRESS IN FLIGHT PHYSICS Figure 1 Pressure distribution at Mach 8 a); hybrid setup, kinetic- ow domains around leading-edges b); and ow solution in kinetic- ow domain c) at Mach 8 waverider, length = λ, Radius LE = 2.5λ, T = 116 K, T wall = 2.5T, Z r is determined by Eq. 4), µ is determined by Eq. 6), and specular wall boundary condition 306

7 NONEQUILIBRIUM AND RAREFIED FLOWS Figure2 Strong scaling on SuperMUC LRZ, Munich) at the conditions of Fig. 1: 1 linear;2 and3 full leading-edge, thin layer, velocity space discretization 32 3 divided in and partitions, respectively; and4 full leading-edge, thick layer, velocity space discretization 24 3 divided in partitions Access), providing access to the SuperMUC computer at the Leibniz RechenzentrumLRZ)inMunich. Intheproject,2048to8192coresofIntel R Xeon R processorswereusedforarangeoftestcases. Theexampleshownhereisfor the Mach 8 ow of a diatomic gas around the waverider geometryleading-edge diameter5λandbodylength λ)showninfig.1a. Amultiblockmesh with867blocksand cellswasused. Themultiphysicssimulationsinvolved a region around the leading-edges which were simulated at the kinetic level; in the case shown in Fig. 1b, the kinetic- ow domain comprises 300 blocks with 575,000 cells. A reduced spatial extent with about half the blocks was also considered, corresponding to the thin layer results in Fig. 2. Velocity space discretizationsas24 3 and32 3 wereuseddividedin8 4 4or8 8 4partitions. The ow demonstrates signi cant thermodynamic nonequilibrium in the kinetic- ow domain around the leading-edges as can be seen from the rotational temperature relaxation in Fig. 1c. The strong scaling for these cases areshowninfig.2,showinga85percentparallele ciencyfrom2048to8192 processes. Forthecouplingbetweenthecontinuumsolverandthekineticsolver,M C employs state-based or ux-based hybrid techniques. In the state-based couplingfig. 3a), over an overlap region, the macroscopic variables are obtained from the microscopic solution while the latter is constructed from the macroscopic state. In the ux-based coupling, instead, at a designed interface, the numerical uxesforonesolverareobtainedfromtheotherasshowninfig.3b, while for hybrid NS-DSMC simulations, the choice of a state-based coupling is the preferableone due to the lowerscatteringerrorthat it involves[12]. In the current deterministic DVM kinetic solver, the statistical scatter is absent, creating more exibility in the used coupling technique. In the literature, 307

8 PROGRESS IN FLIGHT PHYSICS Figure 3 Coupling techniques, adapted from [12]: a) state-based coupling; and b) ux-based coupling the domain decomposition is generally done during the simulation on the basis of a continuum breakdown parameter, for example, as done by other researchers in[12, 29]. At the moment, this feature is still under development inm CandtheDVMdomainhastobede nedbytheuserinaninput le. However, the framework can perform a recon guration of the di erent regions throughout the calculation when the domain de nition le is modi ed by the user. 3 FLOW MODELS AND NUMERICAL SCHEMES 3.1 Nondimensional Rykov Model Consideringthe owofadiatomicgas,letusassumethatthegastemperature isnottoohigh,sothatthevibrationaldegreesoffreedomarenotexcited,and nottoolow,sothattherotationaldegreesoffreedomcanbeconsideredtobe fully excited. In this case, the particle distribution function fx, c, t, ζ), which describesthestateofthegas,willbeafunctionnotonlyofthespatialcoordinatex,theparticlevelocityc,andthetime t,butalsooftherotationaldegrees of freedom ζ. The Rykov model represents an extension of the Shakhov model where also rotational nonequilibrium is considered and has been proved to be a reliable kinetic approximation, up to the heat uxes moments of the BTE, forthiskindof ow[7,18,30,31]. Sincetherotationaldegreesoffreedomare considered fully excited, ζ is reduced by the model and the second distribution function is obtained. 308

9 NONEQUILIBRIUM AND RAREFIED FLOWS Employing the following nondimensional variables: ρ= ρ ρ ; T= T T ; u= c 2RT ; t= t µ p 1 ; x= µ= µ µ ; q= x ; p= 2RT µ p 1 p ρ RT ; q ρ 2RT ) 3/2; τ= τ µ p 1, the nondimensional distribution functions of the model result F 0 = F 0 ρ 2RT ) 3/2; F F 1 1= mrt ρ 2RT ) 3/2; thus,fortherykovmodelwrittenintermsof F= mf,oneobtains F 0 t +c F 0 = F eq 0 F 0 τ ; ) 1 1Zr F0; t F eq F 1 t +c F 1 eq F1 = F 1 τ ) 1 1Zr F eq 0 = 1 F Z 0+ r 1 = 1 F r Z 1+ r F1; t [ r )] F0= r F M T) ω qx t c x c 2 0 p T T 5 ; 2 [ )] F0= t F M T t ) 1+ 8 qx t c x c 2 5 ; 15 p t T t T t 2 [ ) ] F1 r= TF MT) ω qx t c x c 2 0 p T T 5 +4ω 1 1 δ) qr i c i ; 2 pt [ ) ] F1 t= T rf M T t ) 1+ 8 qx t c x c δ) qr i c i 15 p t T t T t 2 p t T r ; 1) where ρ F M T)= πt) 3/2 exp ) c 2 T andthetotalcollisiontime τisexpressedas µ t /p t withtheviscositydetermined from the translational temperature. In a system of colliding particles, energy is transferred between the various internal modes. These collisions tend to drive the internal energy distributions towards their respective equilibrium state and the number of them necessary to 309

10 PROGRESS IN FLIGHT PHYSICS push a particular mode to the equilibrium is the collision number, Z, associated tothatmode[32].therykovmodelisbasedontheassumptionthatthefraction of collisions involving the excitation of the rotational degrees of freedom, Z r,isagivenconstantorafunctionofthe owtemperatures. Severalworks provideanexpressionof Z r asafunctionofthetemperatureinthe ow eld. Probably, the rst attempt to appear in the literature is the theoretical work of Parker[33] where, employing an empirical nonimpulsive model and assuming coplanar collisions and zero initial rotational energy, the following approximate expression is obtained: Z Par r = Z r ) ) 1/2+ 1+ π 3/2 /2 T / T) π 2 /4+π ) T / T ) 2) where T =91.5KisthecharacteristictemperatureoftheintermolecularpotentialandZ r ) =23.5isthelimitingvaluesuggestedin[34].WhileParker s expression2) is derived involving a large number of simplifying assumptions, the overall dependence on the temperature is in agreement with the more rigorous treatment of[35]. However, this expression does not involve any dependence on the di erent translational and rotational temperatures. Thus, in the recent literature, models derived from data tting, either from numerical or experimental results, have been employed. In[7, 30, 31], the following expression for the collision number is presented to be used with the Rykov model: Zr Ryk = 3 4 π ψ T) 9 ) ) 2 T T Tt Tt r ) T 1/6 T+8 T t 1K 1K where ψ T)= T [ ]) 1/6 exp 1.17 T 1 ; T= T t T. Analternativeexpressionfor Z r T t, T r )derivedfrommdsimulationscanbe found in[36]: Zr Val = a 1 Tt 1K ) 1/4 + a 2 Tt 1K ) 1/4 ) Tt a 3 [ 1K 1000 T 1 b 1 )] r T t 4) where a 1 = ; a 2 = ; a 3 = ;and0<b 1. Itis important to notice that considering the moments of the Rykov model collision term,therelaxationprocessinthemodelisdescribedas ρt T r )/Z r τ)while 310

11 NONEQUILIBRIUM AND RAREFIED FLOWS in[33,36],jeansequationisconsideredleadingto ρt t T r )/Z r τ).thismeans that the collision number in the Rykov model results in Z r =0.6Z Par/Val r. Fortheviscositylaw,Rykovandhiscoworkers[7,30,31]suggest otherwise, a simpler power law µt t )=µt ) T 2/3 ψ T) ; 5) ) ω Tt µt t )=µt ) 6) T with an exponential factor of 0.72[37] can be employed. Tomakethesystem1)complete,thevaluesoftheconstants δ, ω 0,and ω 1 needtobedetermined. In[38], ω 0 =0.2354and ω 1 =0.3049or ω 0 =0.5 and ω 1 =0.286aregivenfordiatomicgases. Bothpairsofvalueshavebeen successfullyemployedin[30,31,38,39]with δ 1 =1.55.Inthepresentwork,the values ω 0 =0.5and ω 1 =0.286areemployed. Thedimensionlessmacroscopicquantitiescanbeobtainedfrom F 0 and F 1 by means of the following formulae: ρ= + ρt r = F 0 dc; ρu x = + F 1 dc; + qx t = c x + c x F 0 dc; 3 2 ρt t+ ρu 2 x= + c 2 F 0 dc; 5 2 T=3 2 T t+ T r ; p t = ρt t ; p=ρt; c 2 2 F 0 dc; q r x = + c x 2 F 1 dc. 3.2 Gas-Kinetic Scheme for Near-Continuum Flows BasedontheRykovModel Integrating in time the nondimensional reduced Rykov model system1) and takingthemomentsā 0 =1, c x, c 2, 0) T of F 0 n+1 i andā 1 =0, 0, 1, 1) T of F 1 n+1 i, the following can be obtained for the update of the nondimensional macroscopic variables: 311

12 PROGRESS IN FLIGHT PHYSICS W n+1 i =Wi n + 1 t n+1 ) ā 0 [cx m F 0 ] i 1/2 [c x m F 0 ] i+1/2 dt cx x m t n + 1 x m t n+1 t n ā 1 wherethesourcetermis t 2 S n+1 i +S n i [cx m F 1 ] i 1/2 [c x m F 1 ] i+1/2 ) dt cx + t 2 ) = t 2 0,0,0, S n+1 i +S n i ρt n+1 i T r n+1 i ) Z r τ n+1 + ρt n i T r n ) T i) Z i r τi n and the velocityphase) space is discretized using a uniformly spaced method with the trapezoidal rule for the evaluation of the moments of the distribution functions. As previously shown in[40, 41], reconstructing the time-dependent distributionfunctionsatthecell-faces,i.e., F 0 i±1/2 and F 1 i±1/2,consistently onthebasisofthecesolutionoftherykovmodel: [ F 0 = F eq 0 τ FM T t ) +c F MT t ) t F 1 = F eq 1 τ [ Tr F M T t )) t for a well-resolved ow, it is possible to obtain: F 0 = F eq 0 τ FM T t ) F M T t ) + c x t F 1 = F eq 1 τ Tr F M T t )) t ] ; +c T rf M T t )) ) + t F MT t ) t + c x T r F M T t )) ] ; ) + t T rf M T t )) t where the derivatives are obtained analytically with the derivative of the Maxwellian de ned as follows: [ ) ] F M α = F 1 ρ M ρ α +1 c 2 T T 3 T 2 α +2c x u x ; T α F eq 0 = F0 t + F 0 r F0 t ; F eq 1 = F1 t Z + F 1 r F1 t. r Z r The time derivatives of the macroscopic variables can be obtained in terms of the space derivatives by means of the compatibility condition for the Rykov model: + [ ) )] FM T t ) F M T t ) Tr F M T t ) T r F M T t ) ā 0 + c x +ā 1 + c x dc t t =S; ) 7) 312

13 NONEQUILIBRIUM AND RAREFIED FLOWS then, + [ ] F M T t ) T r F M T t ) ā 0 +ā 1 dc t t =S + which,asexplainedin[40,41],leadsto ) F M T t ) T r F M T t ) ā 0 c x +ā 1 c x dc 8) W R t =S Q R where W R = ρ, ρu x, ) T 5 2 ρt+ ρu2 x, ρt r ; Q R = ρu x 1 2 ρt t+ ρu 2 x 5 2 ρt. 9) tu x + ρu 3 x+ ρt r u x ρt r u x Introducing the CE solutions7) of the Rykov model1) with a modi ed collision time τ : F 0 = F eq 0 τ FM T t ) +c F ) MT t ) ; t ) F 1 = F eq 1 τ Tr F M T t )) T r F M T t )) + c x t backinthemodelequation,itispossibletoobtain τ DF M T t )+τ τd 2 F M T t )=τdf eq 0 ; τ DT r F M T t ))+τ τd 2 T r F M T t ))=τdf eq 1. } 10) Since the di erence in the relaxation rate between translational and rotational processesisinheritedinthecollisionnumber, Z r,letusde neasinglemodi ed collision time by taking the moment relative to the total heat ux. Thus, multiplyingthe rstofeqs.10)by c xc 2 andthesecondoneby c xandaddingthe two resulting equations, one obtains: τ = τ DF eq 0 + DFeq 1 ) DF M T t ) + D {T r F M T t )} )+τ D 2 F M T t ) + D 2 {T r F M T t )} ). 313

14 PROGRESS IN FLIGHT PHYSICS Let us simplify the numerical evaluation of this closure of the CE expansion by neglecting the terms relative to the Prandtl number correction introduced in the Rykov model; then, τ = τ 1+τ D2 F M T t ) + D 2 ) 1 {T r F M T t )} 11) DF M T t ) + D {T r F M T t )} where DF M T t ) = 5 ρt 2 ) t ρt ux t t + u x D 2 F M T t ) = 5 2 ρt 2 ) t + 2 ρt 2 ) ) t u x 2 t 2 + T t 5 ρ u x ρ t t +5 2 u u x x t + ρ ) u x t + ρ 5 T t u x t t +5 2 u Tt u x x t + T t u x t ) 2 5 ux + ρt t t 2 u 2 u x x t +8 u x u x t + ) u x ; ρt r T t ) +ρt ux r t + u x D {T r F M T t )} = 1 2 D 2 {T r F M T t )} = 2 ρt r T t ) t + T r 2 ρ u x t t +2u ρ u x x t +2u ρ x T t T r + ρt r + ρ 2 T r t u x t +2u T r x t 2 u x + 2 ρt r T t u x ) x 2 u x +2u x 2 u x + ρt r 2 t +2u x t +4 u x t ) ) 11 ρ + 2 T t+5u 2 u x x ) + ) 11T t +5u 2 ) T t u 12) x x 11 4 T t+ 5 ) 2 u x 2 u2 x 2 x ) ) 2 +16u x ux ) u x ; u x t + T t +2u 2 x ) ρ T r u x t + T t +2u 2 ) ρ x ) Tt u x u2 x u x + 2 x +4u x ux ; ) u x ) u x ) 2 ). 13) The secondderivatives 2 / t) can be expressedin terms of only spatial derivatives thanks to the compatibility conditions8) 314

15 NONEQUILIBRIUM AND RAREFIED FLOWS 2 W R t = S 2 Q R 2 14) wherew R andq R arede nedineqs.9),whilethesecondtimederivativeof the mean velocity can be obtained considering the second-order compatibility conditions for the translational part: where 2 W S t = 2 Q S 2 ; 2 W S 2 t = 2 + W S = Q S = āc x 2 F M T) t + ρ, ρu x, ρu x, āc 2 x dc 2 F M T) 2 dc= 2 Q S x t 2 Q S 2 ) T 3 2 ρt tu x + ρu 2 x ; 1 2 ρt t+ ρu 2 x, 5 Q S= 2ρu x, ρt t +2ρu 2 x) T; Q S = ) T 2 ρt tu x + ρu 3 x. 1 2 ρt t+ ρu 2 x, 2 2 ρt tu x + ρu 3 x 15) 16) ) T. 17) as discussed in[40, 41]. Finally,similarto[20,21],since DF M T) and D 2 F M T) willbesensitive to numerical errorsespecially, close to equilibrium regions where they tend to vanish, and to impose that the physical stress inside the shock layer should be largerthanthestressinthensequation),alimiterisneeded. Inthecurrent work, the following nonlinear limiter is used: τ = τ 1+max 0.5, min exp c Kn)), τ D2 F M T t ) + D 2 ))) 1 {T r F M T t )}. 18) DF M T t ) + D {T r F M T t )} whereknisthelocalknudsennumberbasedonthegradientsofthemacroscopic variables de ned as in[12, 42] ρ Kn=λmax, u x, T t, T r, T ) t T r 19) ρ u x T t T r 2T r andthefunction fkn)= exp ckn))isusedtoobtainasmoother transitionfrom τ = τto τ =2τ.Here,theparameter c= iscalculated inordertohaveatleastastarting τ =1.1τatKn=

16 PROGRESS IN FLIGHT PHYSICS 3.3 Di use Wall Boundary Conditions The gas evolution at a solid boundary is modeled assuming that particles hit the wall with a distribution function according to the ow conditions whereas they are re ected with: a Maxwellian distribution according to the wall state for fully accommodation boundaryviscous wall); the same distribution function for specular re ection boundaryinviscid wall);and a combination of di use and specular boundaries depending on the accommodation coe cient σ. Therefore, the nal gas distribution function at the wall can be written as F= σf M u>0 + F u<0 +1 σ)f u>0 where u <0and u >0representthevelocitiesofparticleshittingthewalland re ectedbythewall,respectively,while F M and FaretheMaxwellianandthe nonequilibrium distribution functions at the wall. The uid state at the wall can be extrapolated from the domain. In the present work, fully accommodated wallshavebeenemployed,i.e., σ=1. 4 RESULTS AND DISCUSSION 4.1 Assessment of the Gas-Kinetic Scheme ComputationalandmemorycostisamajordrawbackforDVMinhybridsimulationsandtheuseoftheGKStoreducetheextensionofthedomainwhere Table 1 Test cases details GKS/Hybrid Test T, T M λ wall, case K K Z r µ Solver Shock Eq. 6) GKS/GKS τ /DVM Shock Eq. 6) GKS/GKS τ /DVM Plate mm Eq. 3) Eq. 5) GKS/DVM Wedge mm Eq. 3) Eq. 5) GKS/DVM Cylinder 12 Diameter/100; Eq. 3)Eq. 5) Diameter = 80 mm Waverider 8 Length/ ), T Eq. 4)Eq. 6) Diameter GKS LE/

17 NONEQUILIBRIUM AND RAREFIED FLOWS it is strictly required represents a preferable alternative. To support this assertion, di erent test cases for a wide range of Mach numbers have been considered. Details of the simulations are reported in Table 1 and wall boundary conditions are considered fully accommodated. As shown in Figs. 4a and 5a, in contrast to traditional single-/multitemperature NS approaches[44, 45], the GKS is able to resolve shock structures with and without rotational nonequilibrium. However, due to the continuum formulation, it predicts steeper shocks relative to Figure 4 Nondimensional ρ 1), T t 2), and T r 3) normalized pro les, i.e., φ φ )/φ φ ), for normal shocks in nitrogen. Velocity space cells size is equal to 0.5u. The DSMC results signs) are from [43]; curves refer to GKS based on Rykov model; viscosity power law with exponential 0.72: a) M = 2.8 and Z r = 4.2; and b) M = 10 and Z r = 5 Figure 5 Nondimensional ρ 1), T t 2), and T r 3) normalized pro les, i.e., φ φ )/φ φ ), for normal shocks in nitrogen. Modi ed collision time τ is de ned by Eqs. 11)19) and velocity space cells size is equal to 0.5u. The DSMC results are from [43]; curves refer to GKS τ based on Rykov model; viscosity power law with exponential 0.72: a) M = 2.8 and Z r = 4.2; and b) M = 10 and Z r = 5 317

18 PROGRESS IN FLIGHT PHYSICS Figure6 Mach 4 nitrogen ow around a 25 degree wedge; velocity space cells size is equal to 0.5u : temperatures di erence, GKS solution Figure7 Mach 8 waverider, GKS solution: temperatures di erence DSMC solutions. Shock-structure predictions in better agreement with DSMC resultscanbeobtainedwheneq.11)isemployedtomodifythecollisiontime asshowninfig.5a. Ontheotherhand,employingthemodi edcollisiontime slightly a ects the accuracy in the subsonic region of the shock. The main differences between GKS and DSMC results can be observed for the higher Mach numbercasesinfigs.4band5b. ThisisprobablyduetotheuseoftheCE expansion which does not allow to represent the typical bi-modal behavior of the distribution function across shock waves at very high Mach numbers> 4). The capability of the GKS to resolve multiple temperatures across shock structures,asshownalsoinfigs.6and7forthewedgeandwaveridercases, respectively, makes possible the reduction of the domain where more complex approaches, such as the DVM, are necessary. Indeed, as it is possible to observe fromfigs.811,thegksresultsprovetobeingoodagreementwithdvmand DSMC calculations with the biggest di erences between GKS and DVM, de ned here as = φ DVM φ GKS 100%, φ DVM occurring when the local Knudsen numbersee Eq.19)) is much higher than the commonly employed threshold, i. e., 0.05[47], for the continuum breakdown. Furthermore, in both two-dimensional cases, the quantities at the wall are pre- 318

19 NONEQUILIBRIUM AND RAREFIED FLOWS Figure 8 Mach 4.89 nitrogen ow around a at plate; velocity space cells size is equal to 0.5u : a) T t; b) T r;1 DVM;2 DSMC; and3 GKS based on Rykov model. The DSMC calculations 2) and experimental data signs) are reported in [46]; Kn x = λ /x = 0.24; x = 5 mm Figure9 The GKS and DVM results di erence a) and local Kn see Eq. 19)) b). Mach 4.89 nitrogen ow around a at plate; velocity space cells size is equal to 0.5u Figure10 Mach 4 nitrogen ow around a 25 degree wedge; velocity space cells size is equal to 0.5u : GKS and DVM results di erence a) and local Kn see Eq. 19)) b) 319

20 PROGRESS IN FLIGHT PHYSICS Figure11 Mach 4 nitrogen ow around a 25 degree wedge:1 CFD;2 hybrid; 3 DSMC;4 GKS based on Rykov model; and5 DVM based on Rykov model. Velocityspacecellssizeisequalto0.5u. TheDSMC,hybridapproach, andcontinuum results reported are from [11] Figure 12 Mach 8 waverider, GKS solution: Mach contours. Refer color plate, p. XXVIII.) dictedwithadi erencelessthan5% relative to DVM calculations and Fig.11balsoshowsagoodagreement between GKS and hybryd method results. Regarding the waverider case, asexpected,thegkspredictsa ow eld with high thermal nonequilibrium around the waverider nose where also important viscous interactione ectsoccurandintheboundary layer over the upper body surface Figs.7and12). Forthecylindertestcase,Fig.13 shows density and temperatures pro- lesforthesectionat45degreerespective to the symmetry plane. The GKS correctly predicts the shock position as well as quantities and gradientsatthewallincomparisonto the results of the Modular Particle- ContinuumMPC) method of[42]; however, a thinner shock is predicted. A better shock structure prediction can be obtained using the hybrid approach as can beobservedinfig.14. Here,forthehybridapproach,theDVMisemployed inaregionaroundthebowshockwhiletherestofthedomainissimulatedusing the GKSFig. 15). The information exchange between the two solvers has 320

21 NONEQUILIBRIUM AND RAREFIED FLOWS Figure13 The ρ a) and T t solid curves) and T r dashed curves) pro les b) at 45. Mach 12 nitrogen ow around a cylinder; velocity space cells size is equal to 0.5u. The MPC results 1) are from [42]; curves2 refer to GKS based on Rykov model Figure14 The ρ a) and T t solid curves) and T r dashed curves) pro les b) at 45. Mach 12 nitrogen ow around a cylinder; velocity space cells size is equal to 0.5u. TheMPC results 1)are from [42]; curves2 refer togksbased on Rykovmodel/DVM been handled employing the state-based couplingsee Fig. 3a), described in section2withanoverlapregionextensionofabout10cells.thismeansthatacross the overlap region, the distribution function employed as boundary condition forthedvmisreconstructedusingtheceexpansionseeeq.7)),wherethe macroscopic variables and the relative derivatives are de ned by the GKS solutions; vice versa: the macroscopic variables and the relative derivatives needed forthegks uxesareobtainedfromthedvmsolution. Thedi erencescan be still observed between the hybrid results from the present work and the ones in[42]. Thereasonofthesediscrepanciescanbefoundinthedi erentmethods employed. Indeed, in[42], a DSMC method is used where rarefaction e ect aredominantwhileinthepresentwork,advm,whichsolvesonlyasimpli ed modelofthebtewherethecollisiontime τ doesnotdependontheparticle 321

22 PROGRESS IN FLIGHT PHYSICS Figure 15 The DVM and GKS domains. Mach 12 nitrogen ow around a cylinder; velocity space cells size is equal to 0.5u Figure16 TheMach contours. Mach12 nitrogen ow around a cylinder; velocity space cells size is equal to 0.5u velocities, is employed. When employed in a hybrid simulation, as it is possible tonoticefromfig.16,thetransitionbetweenthegksandthedvmsolvers at the interface is naturally smooth, no blending of the two solutions has been appliedintheoverlapregion,duetothecommonrootoftheapproaches.this andtheextendedvalidityofthethegksalsosuggestareductionofthehybrid simulations sensitivity to the positioning of the interfaces. However, the latter point requires further investigations. 4.2 Computational and Memory Cost The GKS is implemented so that the particle velocity space dimensions depend onthelocalstate,whiledueitscomplexity,thisfeatureisnotavailableinthe DVMforthekineticBoltzmannequationsinM C.Toperformafaircomparison of the computational time for the two approaches, constant velocity spaces wereused. TherunshavebeenperformedonIntel R Xeon R processorsatthe University of Liverpool cluster Chadwick and solutions have been considered convergedwhenthe L 2 -normoftheupdatebetweentwoconsecutivesolutions islowerthan10 7 forthenormalshockcasesand10 8 forthetwo-dimensional cases.furthermore,itneedstoberemindedthatthehaloexchangeinthegks involvesonlythe owstatewhilethedvmneedstoexchangethefullvelocity spaceandthisrepresentsanadvantageofthegksrelativetothedvmandthe full UGKS when a parallel calculation is performed. In Table 2, the computa- 322

23 NONEQUILIBRIUM AND RAREFIED FLOWS Table 2 Test cases details and computational time Test M Physical Velocity Time, Solver Cores Iterations case cells cells min Shock DVM Shock GKS Plate DVM Plate GKS Wedge DVM Wedge GKS tionaltimeforthestudiedcasesisreportedandthegksisfoundtobefrom50% to90%fasterthanthedvm.thisispartiallyduetothelowernumberofiterationsneededingeneralbythegks,butmainlytothetimeneededperiteration, around710timessmallerthanthedvm.finally,intermsofthememorycost ofthegks,thelatterisdrasticallyreducedcomparedwiththedvmandthe fullugks.indeed,inthedvmandtheugks,thevaluesofthedistribution functionneedtobestoredforeachphysicalcellinthefullvelocityspacewhile thegksbeingemployedinthecontextofacontinuumsolverrequiresonlythe storage of the primitive variables. Thus,employingtheGKSinplaceoftheDVMwherethe uidisnearthermal equilibrium, the performance of the hybrid solver can be improved in both memory and CPU time requirements, this without compromising the accuracy as shown in subsection CONCLUDING REMARKS Inthepresentwork,ananalyticalGKShasbeendevelopedandaddedtothe framework presented in[23, 26, 27]. The prediction of ow elds where rare ed and continuum regions coexist requires the solution of two models; the NS equationsandthebte.sincethemethodstosolvethebteareexpensive, the reduction of the region where this is strictly required could improve the performance of hybrid simulations. For these reasons, a GKS for near-continuum regime has been proposed. TheschemehasbeentestedforvariouscasesandMachnumbersprovingto produce reliable predictions for near-continuum ows. Moreover, the GKS also proved to be capable of solving more complex three-dimensional ow elds and to couple naturally with a DVM based on the same kinetic model. Regarding the computational time, when compared with a kinetic DVM solver, the nearcontinuumgkssolverwasfoundtobebetween50%and90%fasterthanthe 323

24 PROGRESS IN FLIGHT PHYSICS former. Furthermore, due to the lower number of variables that need to be stored, thegksislessexpensiveintermsofmemorythanthedvmandthefullugks. ThisprovesthatGKScanbeaviablewaytoimprovetheperformanceofhybrid simulations maintaining an acceptable level of reliability when used in place of more complex methods for weakly rare ed ows. ACKNOWLEDGMENTS The nancial support by the University of Liverpool is gratefully acknowledged. We acknowledge PRACE for awarding us access to SuperMUC at the Leibniz RechenzentrumLRZ) in Munich, Germany. The work has also made use of the University of Liverpool cluster Chadwick and the N8 HPC facilities provided and funded by the EPSRC and N8 consortium coordinated by the Universities of Leeds and Manchester. REFERENCES 1. Vincenti, W., and C. Kruger Introduction to physical gas dynamics. New York, NY: John Wiley & Sons, Inc. 538 p. 2. Bird, G TheDSMCmethod. CreateSpace. 300 p. 3. Baranger, C., J. Claudel, N. H erouard, and L. Mieussens Locally re ned discrete velocity grids for deterministic rare ed ow simulations. AIP Conference Proceedings. 1501: Titarev, V., M. Dumbserc, and S. Utyuzhnikov Construction and comparison of parallel implicit kinetic solvers in three spatial dimensions. J. Comput. Phys. 256: Bhatnagar, P., E. Gross, and M. Krook A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 943): Shakhov, E Generalization of the Krook kinetic relaxation equation. Fluid Dyn. 35): Rykov, V A model kinetic equation for a gas with rotational degrees of freedom. Fluid Dyn. 106): Bourgat, J.-F., P. Le Tallec, and M. Tidriri Coupling Boltzmann and Navier Stokes equations by friction. J. Comput. Phys. 127: Le Tallec, P., and F. Mallinger Coupling Boltzmann and NavierStokes equation by half uxes. J. Comput. Phys. 136: Glass, C., and T. Horvarth Comparison of a 3-D CFD-DSMC solution methodology with a wind tunnel experiment. NASA/TM Wang, W.-L., Q. Sun, and I. Boyd Towards development of a hybrid DSMC- CFD method for simulating hypersonic interacting ows. 8th AIAA/ASME Joint Thermophysics and Heat Transfer Conference. St. Louis, MI. 324

25 NONEQUILIBRIUM AND RAREFIED FLOWS 12. Schwartzentruber, T A modular particle-continuum numerical algorithm for hypersonic non-equilibrium ows. The University of Michigan. PhD Diss. 13. Degond, P., S. Jin, and L. Mieussens A smooth transition model between kinetic and hydrodnamic equaions. J. Comput. Phys. 209: Degond, P., G. Dimarco, and L. Mieussens A multiscale kinetic- uid solver with dynamic localization of kinetic e ects. J. Comput. Phys. 229: Deschenes, T., and I. Boyd Extension of a modular particle-continuum method to vibrationally exited, hypersonic ows. AIAA J. 499): Degond, P., and G. Dimarco Fluid simulations with localised Boltzmann upscaling by direct simulation Monte-Carlo. J. Comput. Phys. 231: Xu, K., and J.-C. Huang An improved uni ed gas-kinetic scheme and the study of shock structures. IMA J. Appl. Math. 76: Liu, S., Y. Pubing, K. Xu, and C. Zhong Uni ed gas-kinetic scheme for diatomic molecular simulations in all ow regimes. J. Comput. Phys. 259: Xu, K A gas-kinetic BGK scheme for the NavierStokes equations and its connection with arti cial dissipation and Godunov method. J. Comput. Phys. 171: Xu, K., and E. Josyula Continuum formulation for non-equilibrium shock structure calculation. Commun. Comput. Phys. 13): Xu, K., X. He, and C. Cai Multiple temperature kinetic model and gaskinetic method for hypersonic non-equilibrium ow computations. J. Comput. Phys. 227: Xu, K Regularisation of the ChapmanEnskog expansion and its description of shock structures. Phys. Fluids 144):L Steijl, R., and G. Barakos Computational uid dynamics of partially rare ed ows with coupled kinetic Boltzmann/NavierStokes methods. ECCOMAS. Vienna, Austria. 24. Colonia, S., R. Steijl, and G. Barakos Kinetic models with rotational degrees of freedom for hybrid methods. ECCOMAS: 6th European Conference on Computational Fluid Dynamics. Barcelona, Spain. 25. Andries, P., P. LeTallec, J.-P. Perlat, and B. Perthame The Gaussian-BGK Model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B Fluid. 196): Steijl, R., and G. Barakos Coupled NavierStokes-molecular dynamics simulations using a multi-physics ow simulation framework. Int. J. Numer. Meth. Fl. 62: Steijl, R., and G. Barakos Coupled NavierStokes/molecular dynamics simulations in nonperiodic domains on particle forcing. Int. J. Numer. Meth. Fl. 69: Colonia, S.,R.Steijl,andG.Barakos Implicitimplementation oftheausm + and AUSM + up schemes.int.j.numer.meth.fl. 7510): Deschenes, T Extension of a modular particle-continuum method for nonequilibrium, hypersonic ows. The Univeristy of Michigan. PhD Diss. 325

26 PROGRESS IN FLIGHT PHYSICS 30. Rykov, V., V. Titarev, and E. Shakhov Numerical study of the transverse supersonic ow of a diatomic rare ed gas past a plate.comp.math.math.phys. 471): Rykov, V., V. Titarev, and E. Shakhov Shock wave structure in a diatomic gas based on a kinetic model. Fluid Dyn. 432): Boyd, I Rotational-translational energy transfer in rare ed nonequilibrium ows. Phys. Fluids A Fluid 23): Parker, J Rotational and vibrational relaxation in diatomic gases. Phys. Fluids 24): Bird, G Moleculargasdynamicsandthedirectsimulationofgas ows. 2nd ed. Oxford engineering science ser. Clarendon Press. 484 p. 35. Lordi, J., and R. Mates Rotational relaxation in nonpolar diatomic gases. Phys. Fluids 94): Valentini, P., C. Zhang, and T. Schwartzentruber Molecular dynamics simulation of rotational relaxation in nitrogen: Implications for rotational collision number models. Phys. Fluids 24):106101: Jones, F NBS TN-1186 interpolation formulas for viscosity of six gases: Air, nitrogen, carbon dioxide, helium, argon, and oxygen. U.S. Department of Commerce/National Bureau of Standards. 38. Larina, I., and V. Rykov Kinetic model of the Boltzmann equation for a diatomic gas with rotational degrees of freedom. Comp. Math. Math. Phys. 5012): Larina, I., and V. Rykov Computation of a rare ed diatomic gas ows through a plane microchannel. Comp. Math. Math. Phys. 524): Colonia, S., R. Steijl, and G. Barakos Kinetic models and gas kinetic schemes for hybrid simulation of partially rare ed ows. AIAA Atmospheric Flight Mechanics Conference, AIAA Science and Technology Forum. Kissimmee, FL. 41. Colonia, S., R. Steijl, and G. Barakos Kinetic models and gas kinetic schemes for hybrid simulation of partially rare ed ows.aiaaj. 544): doi: /1.J Deschenes, T., T. Holman, and I. Boyd E ects of rotational energy relaxation in a modularparticle-continuum method. J. Thermophys. Heat Tr. 252): Alsmeyer, H Density pro les in argon and nitrogen shock waves measured by the absorption of an electron beam. J. Fluid Mech. 743): Candler, G On the computation of shock shapes in nonequilibrium hypersonic ows. 27th Aerospace Sciences Meeting. Reno, Nevada. 45. Gno o, P., N. Roop, and L. Judy Conservation equations and physical models for hypersonic air ows in thermal and chemical nonequilibrium. NASA/TP Tsuboi, N., and Y. Matsumoto Experimental and numerical study of hypersonic rare ed gas ow over at plates. AIAA J. 436): Boyd, I., G. Chen, and G. Candler Predicting failure of the continuum uid equations in transitional hypersonic ows. Phys. Fluids 71):