Effect of viscosity on slip boundary conditions in rarefied gas flows

Nội dung text: Effect of viscosity on slip boundary conditions in rarefied gas flows

1. Vietnam Journal of Mechanics, VAST, Vol.41, No. 3 (2019), pp. 203 – 215 DOI: EFFECT OF VISCOSITY ON SLIP BOUNDARY CONDITIONS IN RAREFIED GAS FLOWS Nam T. P. Le1,2,∗ 1Industrial University of Ho Chi Minh City, Vietnam 2Ton Duc Thang University, Ho Chi Minh City, Vietnam ∗E-mail: letuanphuongnam@tdtu.edu.vn Received: 17 January 2019 / Published online: 25 July 2019 Abstract. The viscosity of gases plays an important role in the kinetic theory of gases and in the continuum-fluid modeling of the rarefied gas flows. In this paper we inves- tigate the effect of the gas viscosity on the surface properties as surface gas temperature and slip velocity in rarefied gas simulations. Three various viscosity models in the liter- ature such as the Maxwell, Power Law and Sutherland models are evaluated. They are implemented into OpenFOAM to work with the solver “rhoCentralFoam” that solves the Navier-Stokes-Fourier equations. Four test cases such as the pressure driven backward facing step nanochannel, lid-driven micro-cavity, hypersonic gas flows past the sharp 25- 55-deg. biconic and the circular cylinder in cross-flow cases are considered for evaluating three viscosity models. The simulation results show that, whichever the first-order or second-order slip and jump conditions are adopted, the simulation results of the surface temperature and slip velocity using the Maxwell viscosity model give good agreement with DSMC data for all cases studied. Keywords: Sutherland; Power Law; Maxwell viscosity models; rarefied gas flows; slip ve- locity; surface gas temperature. 1. INTRODUCTION The accuracy of the Navier–Stokes–Fourier (N–S–F) simulations for rarefied and mi- croscale gas flows depends on the slip velocity and temperature jump conditions, and also the constitutive relations supplied, such as the viscosity-temperature relation, ther- mal conductivity and heat capacity. We did an investigation for the slip and jump con- ditions in [1] to find the most suitable choice of slip velocity and temperature jump con- ditions for rarefied gas simulations. Flow regimes in rarefied gas dynamics are charac- terized by the Knudsen number, Kn, defined as the ratio of gas mean free path (i.e. the average distance a molecule moves between successive intermolecular collisions) to a characteristic length of the vehicle body, as free molecular (Kn ≥ 10), transition regime (0.1 ≤ Kn ≤ 10), slip regime (0.001 ≤ Kn ≤ 0.13), and continuum regime (Kn ≤ 0.001). c 2019 Vietnam Academy of Science and Technology
2. 204 Nam T. P. Le The CFD method, which solves the Navier–Stokes–Fourier (N–S–F) equations with ap- propriate slip and jump conditions, may simulate successfully rarefied gas flows in the slip regime, up to a Knudsen number of 0.1. The Direct Simulation Monte Carlo (DSMC) method is a commonly used to investigate the rarefied gas flows. But this method is also very expensive both in computational time and memory requirements. The viscosity affects to the accuracy of the N–S–F simulation results through the shear stress, heat transfer and the Maxwellian mean free path presented in the slip ve- locity and temperature jump conditions. In gas microflows, the mean free path of the gas molecules becomes significant relative to the characteristic dimension of the micro- devices. The action of viscosity can be achieved from a consideration of the transfer of molecular momentum between two contiguous layers of the mass flow. Momentum is carried by the molecules from one layer to the other both by direct translation and by in- termolecular collisions. If this transfer process is undergone then viscous flow occurs [2]. So the viscosity of gases played an important role in the kinetic theory of gases and rar- efied gas simulations. Various viscosity models such as the constant viscosity, Power Law and Maxwell viscosity models were investigated for one-dimensional (1D) shock structure by the CFD and DSMC methods [3,4]. The Maxwell viscosity model gave good simulation results of the shock structure in comparing with experimental data [5]. The Sutherland and Power Law viscosity models have been commonly using in CFD simu- lations. The viscosity of real gases can be matched by a power law over a small temper- ature range only, because the long-range attractive forces (the van der Waals forces) are ignored. More realistic is the Sutherland potential which combines a short-range hard sphere repulsion with a long-range inverse 6th power attractive potential [6]. So far there is not yet any comparison between these viscosity models in two-dimensional (2D) rar- efied gas simulations. In this paper three various viscosity models found in the literature such as Sutherland, Power Law and Maxwell viscosity models are numerically investi- gated to evaluate their performance in rarefied gas flows in the slip regime (Kn ≤ 0.1). Four cases such as the pressure driven backward facing step nanochannel [7], lid driven micro-cavity, [8], hypersonic gas flow past the sharp 25-55-deg. biconic [9] and a circular cylinder in cross-flows [10] are considered to investigate the effects of viscosity on the slip velocity and surface gas temperature. The first-order and second-order slip conditions in [11–13] are adopted to simulate four cases within the OpenFOAM frame- work [14]. The simulation results of the surface gas temperature and slip velocity are compared with the DSMC data published in [11–14] to find out which viscosity model should be used for predicting the surface quantities in rarefied gas flow simulations. 2. VISCOSITY MODELS In 2D simulations, the Maxwell viscosity model employed for 1D simulation in [3], p 2 µ = 2 mkBT/π/3πd , is slightly corrected that would be presented below; where m is mass of a molecule; kB is the Boltzmann constant, d is the molecular diameter and T is temperature. Whichever model for viscosity, µ, is adopted, the coefficient of thermal conductivity, k, may be determined from the formula k = cpµ/ Pr where the Prandtl number, Pr, is assumed to be constant and cp is the constant pressure specific heat.
3. Effect of viscosity on slip boundary conditions in rarefied gas flows 205 When two molecules collide with each other, energy, momentum and mass are all conserved. If we examine the transport of momentum it means we have been studying viscosity of a gas [15]. The phenomena of viscosity occur in a gas when it undergoes a shearing motion. It is found experimentally that the stress acts in the gas across any plane perpendicular to the direction of the velocity gradient is not only the nature of a simple pressure normal to the plane but also contains a tangential or shearing component. The net transfer of momentum of molecules crossing the plane appears as the effect of viscosity for a two-dimensional gas and is computed by [15] r mk 1 µ = B T0.5. (1) π πd2 This equation of gas viscosity was inspired by Maxwell, so-called the Maxwell vis- cosity model. In comparison with the Maxwell viscosity model mentioned-above in 1D simulation, the factor (2/3) vanishes in the 2D Maxwell viscosity model. Observing that according to the kinetic theory of gases, µ is proportional to T0.5, and molecular diameter. In the other approach, the viscosity also depends on the intermolecular force that determines how molecules interact in collision with each other. The Power Law viscosity model is simple and expressed in the well-known relation, 1 2 µ = A Ts, where s = + , (2) P 2 v − 1 where AP is a constant of proportionality and depends on the reference temperature. The accuracy of the Power Law model depends on the exponent s over the range of temperature. The values v and s for the intermolecular force law can be determined from the limiting theoretical cases [15, 16]. The values s and v for the intermolecular force law for hard-sphere molecules are v = ∞, s = 0.5, and v = 5, s = 1 corresponding to Maxwellian molecules. Real molecules generally have v ranging from 5 to 15 [15]. Moreover, the values s is suitably chosen to satisfy experimental data [5]. However, the viscosity can match by a power law over a small temperature range only, because the attractive forces are ignored. It is seen that the Maxwell viscosity model above (Eq. (1)) s p  2 can be re-written in the Power Law form µ = AMT , with AM = mkB/π /πd and s = 0.5. The Sutherland viscosity model is more complicated than Power Law viscosity model. It adds a weak attractive force to the intermolecular force which is more realistic. This law is valid only if the attractive force of the intermolecular force is small. The Sutherland model is expressed as. T1.5 µ = As , (3) T + Ts where AS and TS are constant. The coefficient AS depends on the reference temperature, and TS is a measure of strength of the attractive force [6]. These constants are interpreted from experimental data and taken in [3,5,6] to fit the viscosity as accurate as possible. The values AS and TS for different gases in the range of gas temperature from 58 to 1000 K are −6 −1/2 given in [3,17]: for argon AS = 1.93 × 10 (Pa.s/K ) and TS = 142 K, and for nitrogen
4. 206 Nam T. P. Le −6 −1/2 AS = 1.41 × 10 (Pa.s/K ) and TS = 111 K. Finally, the macroscopic viscosity model using for DSMC simulations [10] is expressed as follows, √  T ω 15 πmk T = = B ref µ µref , where µref 2 , (4) Tref 2πdref(5 − 2ω)(7 − 2ω) where ω is the variable-hard-sphere temperature exponent. This model requires a refer- ence temperature, Tref, reference diameter, dref and the exponent, ω. Eq. (4) can be written s −ω in the power-law form µ = AT if we set the constant A = µref/Tref and s = ω. The open source CFD software, OpenFOAM [11], is used in the present work. It uses finite volume numeric to solve systems of partial differential equations ascribed on any 3-dimensional unstructured mesh of polygonal cells. The Maxwell viscosity model s presented in the form of µ = AMT , the Power Law and the Sutherland viscosity models are implemented into OpenFOAM to work with the CFD solver “rhoCentralFoam” that solves the N–S–F equations. 3. SLIP VELOCITY AND TEMPERATURE JUMP CONDITIONS In this paper, we focus on the numerical evaluation of viscosity models in rarefied gas flows in slip regime (Kn ≤ 0.1). So the simple slip and jump conditions are selected in the present work. The first-order conventional Maxwell slip boundary condition can be expressed in vector form as [11]     2 − σu 2 − σu λ 3 µ S · ∇T u + λ∇n(S · u) = uw − S · (n · Πmc) − , (5) σu σu µ 4 ρ T  2  where Π = µ (∇u)T − I tr(∇u) . The right hand side of Eq. (5) contains 3 mc 3 terms that are associated with (in order): the surface velocity, the so-called curvature effect, and thermal creep; p is the gas pressure; u and uw is the velocity and the wall velocity, respectively; n is the unit outward normal vector; S = I − nn where I is the identity tensor, removes normal components of any non-scalar field; T is the transpose and tr is the trace. The tangential momentum accommodation coefficient, (0 ≤ σu ≤ 1), determines the proportion of molecules reflected from the surface specularly (equal to 1 − σu) or diffusely (equal to σu). The Maxwellian mean free path is calculated by [15] µ r π λ = . (6) ρ 2RT Experimental observations show that the temperature of a rarefied gas at a surface is not equal to the wall temperature, Tw. This difference is called the “temperature jump” and is driven by the heat flux normal to the surface. The Smoluchowski boundary con- dition can be written [12] 2 − σT 2γ T + λ∇nT = Tw, (7) σT (γ + 1) Pr where γ is the specific heat ratio; σT is thermal accommodation coefficient (0 ≤ σT ≤ 1). Perfect energy exchange between the gas and the solid surface corresponds to σT = 1,
5. Effect of viscosity on slip boundary conditions in rarefied gas flows 207 and no energy exchange to σT = 0. The second order velocity slip boundary condition for a planar surface can be expressed as follows [13] 2 2 u = −A1λ∇n(S · u) − A2λ ∇n(S · u) + uw, (8) where A1 and A2 are the first and second order coefficients. It was assumed there is no more heat flux along the surface. The values A1 and A2 are proposed either from theory or from experiment. Recently we suggested the second order jump condition in a new form as follows [13] 2γ 1 T = − C λ∇ T + C λ2∇2 T
   + T , (9) γ + 1 Pr 1 n 2 n w where C1 and C2 are the first and second order coefficients. The first-order and the second-order slip and jump conditions were also implemented into OpenFOAM presented in our previous work [1, 14, 17] to employ with the solver “rhoCentralFoam” for running all CFD simulations. In this solver, the laminar N–S–F equations are numerically solved using a finite volume discretization and high-resolution central schemes to simulate high-speed viscous flows, and a calorically perfect gas for which p = ρRT is assumed. 4. NUMERICAL RESULTS AND DISCUSSIONS Four cases such as the pressure driven backward facing step nanochannel, Kn = 0.025 [7], lid driven micro-cavity, Kn = 0.05 [8], hypersonic gas flows past the sharp 25-55- deg. biconic with Mach number Ma = 15.6 [9], and past a circular cylinder in cross-flow, Ma = 10, Kn = 0.01 [10] are considered in the present work. The characterized lengths to calculate the Kn numbers for cases are 1) the height of the channel, H, 2) the length of cav- ity, L, 3) diameter of the biconic base, 2R, and 4) the diameter of cylinder, D. Their values are found in Tab.1. In all CFD simulations at the walls, the slip and jump boundary con- ditions are applied for (T, u), and zero normal gradient condition is set for p. For the step nanochannel case, pin and Tin are set at the entrance, and pout is set at the outlet. The gas flow is driven by the pressure gradient, and the velocity of gas flow depends on the pres- sure gradient. The velocity is then calculated explicitly, and the Neumann type is used for both inlet and outlet for velocity. Zero normal gradient condition is applied for u at the entrance and exit, and for T at the exit of channel, seen in Fig. 1(a). For the lid-driven micro-cavity case, initial pressure and temperature are set as initial values in the com- putational domain, shown in Fig. 1(b). For the two-dimensional axisymmetric biconic Table 1. Gas properties and characterized lengths of all cases Cases ω Tref (K) dref (m) m (kg) Gas Characterized lengths Step nanochannel 0.74 273 4.17 × 10−10 46.5 × 10−27 Nitrogen H = 17.09 nm Micro-cavity 0.81 273 4.17 × 10−10 66.3 × 10−27 Argon L = 1µm Biconic 0.74 273 4.17 × 10−10 46.5 × 10−27 Nitrogen 2R = 261.8 mm Cylinder 0.734 1000 3.595 × 10−10 66.3 × 10−27 Argon D = 304.8 mm
6. Nam T. P. Le present work. The characterized lengths to calculate the Kn numbers for cases are 1) the height of the channel, H, 2) the length of cavity, L, 3) diameterNam of T.the P. Lebiconic base, 2R, and 4) the diameter of cylinder, D. Their values are found in Tab. 1. In all CFD simulations at the walls, the slip and jump boundary conditions are applied for (T, u), and zero normal gradient condition is set for p. For the step nanochannel case, pin and Tin are set at the entrance, and pout is set at the outlet. The gas flow is driven by the pressure gradient, andpresent the wovelocityrk. The characof gasterized flow lengths depends to calculate on the the pressure Kn numbers gradient for cases. The are velocity 1) the he ightis th ofen the calculated explicitly, andchanne thel, H , Neumann 2) the length type of cavity is , used L, 3) diameter for both of inthelet biconic and base outlet, 2R, forand 4) velocity the diameter. Zero of cylinder normal, gradient condition isD applie. Their dvalues for u are at ftheound entrance in Tab. 1 .and In allexit, CFD and simulation T at thes atexit the ofwalls, channel, the slip seen and jump in Figures boundary 1a. For the conditions are applied for (T, u), and zero normal gradient condition is set for p. For the step nanochannel lid-driven micro-cavity case, initial pressure and temperature are set as initial values in the computational case, pin and Tin are set at the entrance, and pout is set at the outlet. The gas flow is driven by the pressure domain, showngradient, in and Figure the velocity 1b. Fofor gas the flow two depends-dim onens theional pressure axisymmetric gradient. The biconicvelocity is case,then calculated the geometry is specified asexplicitly a wedge, and of theone Neumann cell thickness type is usedrunning for both along inlet the and plane outlet offor geometry. velocity. Zero The normal axisymmetric gradient wedge planes mustconditio be specifiedn is applie asd forseparated u at the entrance patches and of exit, type and “wedge”, T at the exit seen of channel, in Figure seen 1c in. FiguresFor the 1a. sharp For the 25 -55-deg. lid-driven micro-cavity case, initial pressure and temperature are set as initial values in the computational biconic anddoma crossin, -flow shown cylinder in Figure cases, 1b. For a t the the two inflow-dimens boundary,ional axisymmetric the freestream biconic case, (p, theT, geometryu) conditions is were maintained specifiedthroughout as a wedge the computational of one cell thickness process. running At along outflow the plane boundary of geometry. for Thethese axisymmetric both cases, wedge zero normal gradient conditionplanes must are be specified applied as for separated (p, T patches, u). Atof type the “wedge”, bottom seen boundary in Figure 1c of. For the the biconic sharp 25 -55 and-de g. cylinder, a symmetry boundarybiconic and condition cross-flow cylinderis applied cases, to aallt the flow inflow variables, boundary, shown the freestream in Figu (p,res1c T, u and) conditions 1d. were 208maintained throughout the computational Nam T. process. P. Le At outflow boundary for these both cases, zero normal The geometrygrad ientdimensions, condition arenumber applieds forof (cellp, Ts, foru). At blocks the b ottom in computational boundary of the domain, biconic and input cylinder, parameters a and symmetry boundary condition is applied to all flow variables, shown in Figures1c and 1d. workingcase, the ga geometryses of all is cases specified are given as a wedge in Figures of one 1a, cell 1b, thickness 1c and 1d. running Number alongs of cells the plane are 60x60, 140x60 and 140of geometry. x60 forThe blocks The geometry axisymmetric of the dimensions, backward wedge number facing planess of stepcell musts nanochannel for blocks be specified in computational case as, seen separated domain,in Figure patches input 1a parameters. Those of are and 120 x 120 fortype the “wedge”, cavityworking case seen ,ga andse ins ofFig.256 all cases1(c)x 256. are For for given thethe in sharpbiconic Figures 25-55-deg. 1a, case 1b, (i.e.1c and biconic256 1d. cellsNumber and ins cross-flowtheof cells axial, are 60x60,streamwise cylin- 140x60 directionand and 256 cells in140 thex60 for radial, blocks surface of the bac normalkward facing direction). step nanochannel For the case circular, seen in Figurecylinder 1a. Those case, are t120he x computational 120 der cases, atfor the the cavity inflow case boundary,, and 256 x the256 for freestream the biconic (casep, T (i.e., u )256 conditions cells in the wereaxial, streamwise maintained direction and structuredthroughout mesh256 the cells computationalis constructed in the radial, process.to surface wrap normal around At outflow direction). the leading boundary For thebow circular for shock these cylinder with both thecase, cases, smallest the zero computational cell sizes grading nearnormal the gradient surfstructuredace ∆x condition mesh= 0.1 is mm, constructed are applied∆y = 1.1to forwrap96 (mm.p around, T, u). the At leading the bottom bow shock boundary with the ofsmallest the biconic cell sizes grading and cylinder,near a the symmetry surface ∆x boundary= 0.1 mm, ∆y condition = 1.196 mm. is applied to all flow variables, shown in Figs. 1(c) and 1(d). Effect of viscosity on slip boundary conditions in rarefied gas flows Effect of viscosity on slip boundary conditions in rarefied gas flows (a) Backward facing step nanochannel (b) Lid-driven micro-cavity a) b) a) b) 5 5 c) d) (c) Sharp c) 25-55-deg. biconic (d) Circular d) cylinder Fig. 1.Fig. Numerical 1. Numerical setups, inp setups,ut parameters input parametersand geometry anddimensions geometry of four dimensions cases a) backward of four facing cases step Fig. 1. Numericalnanochannel, setups, b inp) lidut-driven parameters micro- cavity,and geometry c) sharp 25dimensions-55-deg. biconic, of four and cases d) circular a) backward cylinder. facing step nanochannel, b) lid-driven micro-cavity, c) sharp 25-55-deg. biconic, and d) circular cylinder. The second-order slip and jump conditions obtained good results for simulating rarefied gas microfThe lows. secondThe So geometry- orderthey are slip dimensions,adopted and jumpfor simulating numbersconditions two of obtainednano/micro cells for good-flow blocks resultscases in in computational forthe pr simulatingesent work domain, rarefwithied the gas microflows.coefficientinput So parameters theyvalues are A 1adopted = andC1 = working 1.3for and simulating A gases2 = C2 of =tw 0.23 allo nano/micro cases proposed are givenin-flow our inprevious cases Fig. 1in. work Numbersthe pr[13].esent The of work cellsfirst- order arewith the Maxwell/Smoluchowski60 × 60, 140 × 60 and conditions 140 × 60are for select blocksed for of simulat the backwarding hypersonic facing cases step with nanochannel the coefficients case, σT = coefficient values A1 = C1 = 1.3 and A2 = C2 = 0.23 proposed in our previous work [13]. The first-order σu seen= 1. In in the Fig. present 1(a) .work Those the CFD are 120 results× 120 wou forld be the compared cavity with case, DSMC and 256 data× using256 the for values the biconic σT = σu Maxwell/Smoluchowski conditions are selected for simulating hypersonic cases with the coefficients σT = = 1.case For (i.e. a fair 256 comparison, cells in thethe axial,viscosity streamwise should be treated direction as equivalent and 256 as cells possible in the between radial, the surface DSMC σu = 1. In the present work the CFD results would be compared with DSMC data using the values σT = σu andnormal CFD simulations direction) This For means the circularthe parameters cylinder (m, ω case,, dref, theTref),computational that are chosen to structured calculate the mesh constant is = 1. For a fair comparison, of the DSMC the viscosity macroscopic should viscosity, be treated will be asadopted equivalent for viscosity as possible models between in CFD asthe 1) DSMCs = and CFD simulations. This means the parameters (m, ω, d , T ), that are chosen to calculate the constant ω for the Power Law viscosity model, and 2) the constantref refA for the Maxwell of the DSMC macroscopic viscosity, will be adoptedM for viscosity models in CFD as 1) s = viscosity model. These parameters of gas properties are shown and characterized lengths in Tab. 1. ω for the Power Law viscosity model, and 2) the constant A for the Maxwell M viscosity model. These parameters of gas properties are shown and characterized lengths in Tab. 1. Table 1: Gas properties and characterized lengths of all cases. Cases ω T (K) d (m) m(kg) Gas Characterized Table 1: Gasref propertiesref and characterized lengths of all cases. lengths -10 -27 CasesStep nanochannel ω 0.74T ref (K)273 d4.17ref (m) x 10 46.5m( xkg) 10 NitrogenGas H C=harac 17.09nmterized Micro-cavity 0.81 273 4.17 x 10-10 66.3 x 10-27 Argon L =lengths 1µm Step nanochannelBiconic 0.740.74 273273 4.174.17 x x10 10-10-10 46.546.5 xx 10 2727 NitrogenNitrogen 2R H= 261.8mm= 17.09nm Micro-Cylindercavity 0.810.734 2731000 4.173.59 x5 10x 10-10- 10 66.366.3 xx 10 2727 ArgonArgon D = 304.8L = 1 mmµm Biconic 0.74 273 4.17 x 10-10 46.5 x 10-27 Nitrogen 2R = 261.8mm Cylinder4.1. Pressure driven0.734 backward 1000 facing3.59 step5 x nanocha10-10 66.3nnel x case.10-27 Argon D = 304.8 mm In the pressure driven backward facing step nanochannel, Kn = 0.025 [7], we present the simulation results on the wall-3 of the step channel only in the streamwise direction because the separation 4.1. Pressure driven backward facing step nanochannel case. In the pressure driven backward facing step nanochannel, Kn = 0.025 [7], we present the simulation results on the wall-3 of the step channel only in the streamwise direction because the separation
7. Effect of viscosity on slip boundary conditions in rarefied gas flows 209 constructed to wrap around the leading bow shock with the smallest cell sizes grading near the surface ∆x = 0.1 mm, ∆y = 1.196 mm. The second-order slip and jump conditions obtained good results for simulating rar- efied gas microflows. So they are adopted for simulating two nano/micro-flow cases in the present work with the coefficient values A1 = C1 = 1.3 and A2 = C2 = 0.23 pro- posed in our previous work [13]. The first-order Maxwell/Smoluchowski conditions are selected for simulating hypersonic cases with the coefficients σT = σu = 1. In the present work the CFD results would be compared with DSMC data using the values σT = σu = 1. For a fair comparison, the viscosity should be treated as equivalent as possible between the DSMC and CFD simulations. This means the parameters (m, ω, dref, Tref), that are −ω chosen to calculate the constant A = µref/Tref of the DSMC macroscopic viscosity, will be adopted for viscosity models in CFD as 1) s = ω for the Power Law viscosity model, p  2 and 2) the constant AM = mkB/π /πdref for the Maxwell viscosity model. These parameters of gas properties are shown and characterized lengths in Tab.1. 4.1. Pressure driven backward facing step nanochannel case Nam T. P. Le In the pressure driven backward facingNam T. P. step Le nanochannel, Kn = 0.025 [7], we present the simulation results on the wall-3 of the step channel only in the streamwise direction because the separation zone is located over this wall. The surface gas temperatures in- crease to the peak temperature and then gradually decrease along the wall-3, seen in Fig.2. The prediction of the Maxwell viscosity model for the gas surface temperature gives good agreement with the DSMC data [7] while the CFD other results do not. Slip zonezone is is located located over over this this wall. wall. The The surface surface gas gas temp temperatureseratures inc increaserease to to the the peak peak temperature temperature and and then then velocities on the wall-3 consist of negative and positive components shown in Fig.3. Neg- gradualgradually lydecrease decrease along along the the wall wall-3,- 3,seen seen in inFigure Figure 2. 2.The The prediction prediction of ofthe the Maxwell Maxwell viscosity viscosity model model for for theativethe gas gas surface ones surface represent temperature temperature the gives gives separation good good agreement agreement zone, with andwith the thethe DSMC DSMC distance, data data [7] where[7] while while the indicates the CFD CFD other other the res negativeresultsults do do not.slipnot. Slip velocities,Slip velocities velocities ison definedon the the wall wall-3 as -con3 theconsist lengthsist of ofnega nega oftive thetive and and separation positive positive components components zone. It isshown shown seen in that inFigure Figure the 3. prediction 3.Negative Negative onesusingones represent represent the Maxwell the the separation separation viscosity zone, zone, and model and the the distance, givesdistance, betterwhere where indicates slip indicates velocity the the negative negative than slip the slip velocities, CFD velocities, other is isdefined resultsdefined asin asthe comparingthe length length of ofthe withthe separatio separatio DSMCn zonn zon datae. e.It It [is7 is].see seen thatn that the the prediction prediction using using the the Maxwell Maxwell viscos viscosityity model model give give betterbetter slip slip velocity velocity than than the the CFD CFD other other results results in incomparing comparing with with DSMC DSMC data data [7]. [7]. Fig. 2. Surface gas temperature along the wall- Fig. 3. Slip velocity along the wall-3, Fig.Fig. 2. 2Surface. Surface gas3, gas tKnemperature temperature= 0.025 along [ 7along] the the wall wall-3,- 3Kn, Kn Fig.Fig. 3 . 3Slip. Slip velo velocitycityKn along along= 0.025 the the wall [7 wall]-3,- 3, Kn Kn = = 0.025 0.025 = 0.025= 0.025 [16]. [16]. [17].[17]. 4.24 2 Lid. Lid driven driven micro micro-cavity-cavity case case. . ForFor the the lid lid driven driven micro micro-cavity-cavity case, case, Kn Kn = 0.05= 0.05 [8] [8], the, the gas gas flow flow ex expapandsnds at atthe the location location x/L x/L = 0= as0 as it isit isdr ivendriven by by the the moving moving lid, lid, and and it itis iscompressed compressed at atthe the location location x/L x/L = =1. 1.Considering Considering the the surface surface gas gas temperaturetemperature along along the the lid lid wall, wall, the the Power Power Law Law and and the the Sutherland Sutherland viscosity viscosity models models underpredicts underpredicts the the tempetemperaturerature in inth eth rangee range x/L x/L >T wT =w =300K. 300K. It Itmeans means there there is isviscous viscous heatheat generation generation which which results results in inthe the heat heat transfer transfer from from the the gas gas to tothe the wall wall toward toward the the location location x/L x/L = =1 of1 of thethe cavit cavity casey case. . ForFor the the slip slip velocity velocity alo alongng the the lid lid w allwall in iFiguren Figure 5, 5,all all simulations simulations showed showed that that the the slip slip velocities velocities areare very very slow slow at atthe the locations locations x/L x/L = 0= and0 and x/L x/L = 1,= 1,and and obtained obtained the the peak peak value value around around the the location location x/L x/L = = 0.5.0.5. The The Power Power Law Law and and the the Suthe Sutherlandrland vi sc viosityscosity models models underpredict underpredict the the slip slip velociti velocitieses al ong along the the lid lid surfacesurface in incomparing comparing DSMC DSMC data data [8]. [8]. The The simulation simulation result result using using the the Maxwell Maxwell viscosity viscosity model model is isclose close to to DSMCDSMC data data while while those those of ofthe the Power Power Law Law and and Sutherland Sutherland viscosity viscosity models models ar ear not.e not. 7 7
8. 210 Nam T. P. Le 4.2. Lid driven micro-cavity case For the lid driven micro-cavity case, Kn = 0.05 [8], the gas flow expands at the lo- cation x/L = 0 as it is driven by the moving lid, and it is compressed at the location x/L = 1. Considering the surface gas temperature along the lid wall, the Power Law and the Sutherland viscosity models underpredicts the temperature in the range x/L Tw = 300 K. It means there is viscous heat generation which results in the heat x L = transfer from the gas to the wall toward the location / 1 of the cavity case. Fig. 4. Surface gas temperature along the lid Fig. 5. Slip velocity along the lid wall, Fig.Fig. 55 Slip velocity along along the the lid lid wall, wall, Kn Kn = 0.05= 0.05 Fig. 4Fig Su 4rface. Surface gaswall, gastemperature temperatureKn = 0.05 along along [8 the] the lid lid wall wall, , Kn = 0.05 [8] Kn =Kn 0.05 = 0.05[17] .[1 7]. [17].[17]. For the slip velocity along the lid wall in Fig. 5, all simulations showed that the slip velocities are very slow at the locations x/L = 0 and x/L = 1, and obtained the 4.3. 4Sharp.3. Sharp 25-55 25 deg.55-deg. biconic biconic case case. . peak value around the location x/L = 0.5. The Power Law and the Sutherland viscosity modelsAn oblAn underpredict iqueoblique shock shock forms forms the from slip from the velocities the tip tip of of the the along first first cone theconelid andand surface locateslocates inalongalong comparing towardstowards near near DSMC the the end end data of of [ 8]. this cone,this cone, and andthen then separates separates creating creating a shock.a shock. Latter Latter one one interacts interacts withwith thethe obliqueoblique shock shock and and meets meets the the detachedetacheThed bow simulationd bowshock shock being result being formed formed using over over the the theMaxwell second second cone. cone. viscosity A A low low model speedspeed recirculationrecirculation is close to zone zoneDSMC forms forms data at at the whilethe junctionjunctionthose between of between the the Power thefirst first and Law and the andthe second second Sutherland cones cones in in the viscositythe range range 0.0754m0.0754m models ≤ arex ≤≤ 0.1021m not.0.1021m where where presents presents the the negativenegative4.3. slip Sharp slipvelocity, velocity, 25-55-deg. seen seen in Figurein biconic Figure 7. 7. case Figures 6 compares the CFD surface gas temperatures with those of DSMC data [9]. The surface AnFigures oblique 6 compares shock th formse CFD from surface the gas tip temperatures of the first with cone those and of DSMC locates dat alonga [9]. The towards surface near gas temperaturegas temperature with with the theMaxwell Maxwell viscosity viscosity model model is is close close to to thethe DSMCDSMC data [9][9] near near the the tip tip of of biconic. biconic. the end of this cone, and then separates creating a shock. Latter one interacts with the The surfaceThe surface gas gastemperatures temperatures obtain obtain the the peak peak values values at at the the biconicbiconic tip, and therthereaftereafter rapidly rapidly decrease decrease in in the rangetheoblique range x ≤ 0.754m.x shock ≤ 0.754m. andIn tInhis meets t hisrange range the the the detachedsurface surface gas gas bowtemperature temperature shock predicted beingpredicted formed by thethe MaxwellMaxwell over the viscosity viscosity second model model cone. A give givelowgood good speedagreement agreement recirculation with with the the DSMC zoneDSMC data. forms data. There There at the is is a junction adrop drop ofof temperature betweentemperature the inin the firstthe recirculationrecirculation and the second zone. zone. All conesAll CFDCFD intemperatures the temperatures range and 0.0754 and DSMC DSMC m data≤ datax are≤ are close0.1021 close together together m where in in 0.0754m 0.0754m presents ≤≤ xx ≤ the 0.02m. negative slip velocity, seen in Fig.Figur6. Figures 7es compares 7 compares the the CFD CFD and and DSMC DSMC [9] [9] slip slip velocities velocities along the the biconic biconic surface. surface. Slip Slip velocitiesvelocities Fig.on the on7 comparesbiconicthe biconic surface surface the consist CFD consist of surface of negative negative gas and and temperatures positive positive components.components. with NegativeNegative those of ones ones DSMC represent represent data the the [9 ]. recirculationrTheecirculation surface zone, zone, and gas and the temperature thedistance, distance, wher wher withe indicatese indicates the Maxwellthe the n negativeegative viscosity slipslip velocities, model isis defined defined is close as as the tothe length thelength DSMCof of the recirculationthe recirculation zone. zone. The The slip slip velocities velocities obtain obtain the the peak peak values values at the biconic biconic tip tip and and then then quickly quickly decreasedata along [9] nearthe forecone the tip until of biconic. the locations The x surface= 0.075m. gas The temperatures CFD results using obtain the Maxwell the peak viscos valuesity at decrease along the forecone until the locations x = 0.075m. The CFD resultsx ≤using the Maxwell viscosity modelmodthe areel biconic closeare close to tip, the to and the DSMC DSMC thereafter data. data. Past rapidly Past this this zone decrease zone the the slip slip in velocities thevelocities range increase0.754 and and oscillate m. oscillate In this along along range the the the secondsecond 55- deg. 55-deg. cone, cone, and and there there is isgood good agreement agreement between between all all CFD CFD results and and the the DSMC DSMC data data in in the the rangerange 0.105m 0.105m ≤ x ≤≤ x0.02m. ≤ 0.02m. Overall, Overall, the the Maxwell Maxwell visco viscositysity momodeldel predicts betterbetter slipslip velo velocitycity than than the the SutherlandSutherland and andthe Powerthe Power Law Law models models in incomparing comparing with with DSMC DSMC data.data.
9. Effect of viscosity on slip boundary conditions in rarefied gas flowsNam 211 T. P. Le Nam T. P. Le surface gas temperature predicted by the Maxwell viscosity model give good agreement with the DSMC data. There is a drop of temperature in the recirculation zone. All CFD temperatures and DSMC data are close together in 0.0754 m ≤ x ≤ 0.02 m. Fig. 6. Surface gas temperature distribution over the Fig. 7. Slip velocity distribution over the biconic Fig. 6. Surface gas temperature distribution over the Fig. 7. Slip velocity distribution over the biconic biconic surface [18]. Fig.surface 6. Slip [18]. velocity distribution over the bi- Fig. 7biconic. Surface surface gas [18]. temperature distribution surface [18]. conic surface [9] over the biconic surface [9] 4.4. Cross-flow circular cylinder cases. Fig.6 compares the CFD and DSMC [9] slip4 velocities.4. Cross-flow along circular the biconic cylinder surface. cases. Slip In the cylinder cases, various values of accommodation coefficients σ = σ = 1, σ = σ = 0.8, σ u T u T u In the cylinder cases, various values of accommodation coefficients σ = σ = 1, σ = σ = 0.8, σ = σ = 0.6 and σ = σ = 0.4 are conducted for allvelocities simulations on. The the surface biconic gas temperatures surface consist and sli ofp negative and positive components. Negative u T u T u T u T = σ = 0.6 and σ = σ = 0.4 are conducted for all simulations. The surface gas temperatures and slip velocities are plotted against with the cylinder angle.ones All representCFD simulations the recirculationpredict a higher zone,slip velocity and the distance,T whereu indicatesT the negative slip velocities are plotted against with the cylinder angle. All CFD simulations predict a higher slip velocity than the DSMC data [10], as seen in Figures 8, 10, 12velocities, and 14 for the is cas definedes σu = σ asT = the 1, σu length = σT = 0.8 of, σ theu = recirculation zone. The slip velocities obtain than the DSMC data [10], as seen in Figures 8, 10, 12 and 14 for the cases σu = σT = 1, σu = σT = 0.8, σu = σT = 0.6 and σu = σT = 0.4, respectively. The DSMC theand CFD peak slip values velocities at theincrease biconic gradually tip from and 0 then ≤ θ quickly decrease along the forecone until the ≤ 130-deg., reaching peak normalized values around the location θ = 130-deg., and then gradually σT = 0.6 and σu = σT = 0.4, respectively. The DSMC and CFD slip velocities increase gradually from 0 ≤ θ decrease in 130-deg. ≤ θ ≤ 180-deg. The slip velocitylocations using thex = Maxwell 0.075 m. viscosity The model CFD obtains results the using the≤ 13 Maxwell0-deg., reaching viscosity peak model normalized are values close around to the location θ = 130-deg., and then gradually lowest values, and are relatively close to the DSMC thedata [10]. DSMC Cons data.idering Past the surface this zonegas temperature, the slip velocitiesall decrease increase in 13 and0-deg. oscillate ≤ θ ≤ 180 along-deg. theThe secondslip velocity using the Maxwell viscosity model obtains the the CFD and DSMC results are shown in Figures 9, 55-deg.11, 13 and cone, 15 for andthe cas therees σu is= σ goodT = 1, σ agreementu = σT = 0.8, betweenlowest all values, CFD and results are relatively and the close DSMC to the data DSMC in data [10]. Considering the surface gas temperature, all the CFD and DSMC results are shown in Figures 9, 11, 13 and 15 for the cases σ = σ = 1, σ = σ = 0.8, σu = σT = 0.6 and σu = σT = 0.4, respectively, in whichthe the range one using 0.105 the mMaxwell≤ x ≤ viscosity0.02 m. model Overall, is close the Maxwell viscosity model predicts better slip u T u T σ = σ = 0.6 and σ = σ = 0.4, respectively, in which the one using the Maxwell viscosity model is close to the DSMC data. There are differences between thevelocity CFD and than DSMC the temperatures Sutherland along and the the cylinder Power Lawu modelsT in comparingu T with DSMC data. surface. These differences may be explained by the calculation of the translational surface gas temperature to the DSMC data. There are differences between the CFD and DSMC temperatures along the cylinder in DSMC depending on the components of gas velocity4.4. and Cross-flow the slip velocity circular only. cylinderWhile that casesin CFD is surface. These differences may be explained by the calculation of the translational surface gas temperature calculated by the normal gradient of gas temperature, and is independent of the gas velocity. This leads to in DSMC depending on the components of gas velocity and the slip velocity only. While that in CFD is the profile of the DSMC temperature being very similar toIn that the of the cylinder DSMC slip cases, velocity various. values of accommodationcalculated by the normal coefficients gradient oσfu gas= temperature,σT = 1, and is independent of the gas velocity. This leads to σ = σ = 0.8, σ = σ = 0.6 and σ = σ = 0.4 are conducted for all simulations. The Finally, the average errors between all CFD uand DSMCT simulationsu areT shown in Tableu 2. The T the profile of the DSMC temperature being very similar to that of the DSMC slip velocity. solver “dsmcFoam” is used to run the DSMC simulations, and generates the DSMC data. CFD simulations using the Maxwell viscosity model obtain the smallest average errors in comparing with Finally, the average errors between all CFD and DSMC simulations are shown in Table 2. The those of the CFD simulations with the Power LawThe and surface Sutherland gas viscosity temperatures models. The and reduction slip of velocities CFD are simulations plotted using against the Maxwell with theviscosity cylinder model obtain the smallest average errors in comparing with thermal accommodation coefficient affects the factorangle. (2 - σT All)/σT CFDin the simulationsjump temperature predict condition a higherthat slipthose velocity of the CFD than simulations the DSMC with data, the Power as seen Law and Sutherland viscosity models. The reduction of results in the increases of the surface gas temperatures. It is also seen that the reduction of the surface in Figs.8–11 for the cases σu = σT = 1, σu = σTthermal= 0.8, accommodationσu = σT = 0.6 coefficien and σtu affects= σT the= 0.4,factor (2 - σT)/σT in the jump temperature condition that accommodation effectively decreases the effect of viscosity on the flow field, and leads to the increases of results in the increases of the surface gas temperatures. It is also seen that the reduction of the surface the slip velocity. respectively. The DSMC and CFD slip velocities increase gradually from 0 ≤ θ ≤ 13-deg., reaching peak normalized values around the locationaccommodationθ = 13-deg.,effectivelyand decreas thenes the gradually effect of viscosity on the flow field, and leads to the increases of the slip velocity. decrease in 13-deg. ≤ θ ≤ 180-deg. The slip velocity using the Maxwell viscosity model obtains the lowest values, and are relatively close to the DSMC data. Considering the surface gas temperature, all the CFD and DSMC results are shown in Figs. 12–15 for the cases σu = σT = 1, σu = σT = 0.8, σu = σT = 0.6 and σu = σT = 0.4, respectively, in which the one using the Maxwell viscosity model is close to the DSMC data. There are differences between9 the CFD and DSMC temperatures along the cylinder surface. These differences 9
10. Effect of viscosity on slip boundary conditions in rarefied gas flows 212 Nam T. P. Le Effect of viscosity mayon slip beboundary explained conditions by in the rarefied calculation gas flows of the translational surface gas temperature in DSMC depending on the components of gasNam velocity T. P. Le and the slip velocity only. While that in CFD is calculated by the normal gradient of gas temperature, and is independent of the gas velocity. This leads to the profile of the DSMC temperature being very similar to that Fig. 8. Temperature jump distribution around the Fig. 9. Slip velocity distribution around the cylinder of the DSMC slip velocity. surface, σ = σ = 1. cylinder surface, σu = σT = 1. u T Nam T. P. Le Fig. 8.Fig. Slip 10. velocityTemperature distribution jump distribution around around the the Fig. 9Fig Slip 11. velocity Slip velocity distribution distribution around around the the Fig.cylinder 9. Slip surface velocity, σ distributionu = σT = 0.8 arou. nd the cylinder cylinder surface, σu = σT = 0.8. Fig. 8. Temperature jump distribution around the cylinder surface, σu = σT = 1 cylinder surface, σu = σT = 0.8 surface, σ = σ = 1. cylinder surface, σu = σT = 1. Fig. 12 . Temperatureu T jump distribution around the Fig. 13. Slip velocity distribution around the forebody cylinder surface, σu = σT = 0.6. cylinder surface, σu = σT = 0.6. Fig.Fig. 1012 Temperature jump distributiondistribution around the Fig.Fig.Fig. 1 3 .114. 1Slip. TemperatureSlip velocity velocity distribution jump distribution distribution around arou aroundnd the the the Fig. 15. Slip velocity distribution around the u T cylinder surface, σu = σT = 0.4. cylinderforebody surface cylinder, σ surface,u = σT = σ 0.8u = .σ T = 0.6. Fig.cylinder 10cyli. for Slipndebody serurface, s velocityurface cylinder σu, =σ uσ surface,=T distribution =σ 0.T =6. 0.8 σ .= σ = around 0.4. the Fig. 11. Slip velocity distribution around the cylinder surface, σu = σT = 0.6 cylinder surface, σu = σT = 0.4 Table 2: Average errors between the CFD and DSMC simulations of the cylinder cases. Finally, the averageCases errors betweenMaxwell all CFDSutherla and DSMCnd simulationsPower Law are shown in viscosity model viscosity model viscosity model Tab.2. The CFD simulations using the Maxwell viscosity model obtain the smallest av- erageerrors in comparing withT those ofu the CFDT simulationsu withT the Poweru Law and Sutherland viscosity models. The reduction of thermal accommodation coefficient affects σu = σT = 1 15.12% 16.84% 28.98% 34.01% 35.65% 33.81% the factor (2 − σT)/σT in the jump temperature condition that results in the increases of σu = σT = 0.8 2.84% 13.56% 20.90% 34.82% 55.34% 61.81% the surface gas temperatures. It is also seen that the reduction of the surface accommoda- σu = σT = 0.6 1.15% 16.44% 9.56% 29.50% 53.39% 58.78% tion effectively decreases the effect of viscosity on the flow field, and leads to the increases of the slip velocity. 11 Fig. 14. Temperature jump distribution around the Fig. 15. Slip velocity distribution around the forebody cylinder surface, σu = σT = 0.4. cylinder surface, σu = σT = 0.4. Table 2: Average errors between the CFD and DSMC simulations of the cylinder cases. Cases Maxwell Sutherland Power Law viscosity model viscosity model viscosity model T u T u T u σu = σT = 1 15.12% 16.84% 28.98% 34.01% 35.65% 33.81% σu = σT = 0.8 2.84% 13.56% 20.90% 34.82% 55.34% 61.81% σu = σT = 0.6 1.15% 16.44% 9.56% 29.50% 53.39% 58.78% 11
11. Effect of viscosity on slip boundary conditions in rarefied gas flows Effect of viscosity on slip boundary conditions in rarefied gas flows Nam T. P. Le Effect of viscosity on slip boundary conditions in rarefied gas flows 213 Fig. 8. Temperature jump distribution around the Fig. 9. Slip velocity distribution around the cylinder surface, σ = σ = 1. cylinder surface, σu = σT = 1. u T Nam T. P. Le Fig. 12. Temperature jump distribution Fig. 13. Temperature jump distribution Fig. 11. Slip velocity distribution around the Fig. Fig.9. Slip 10 .velocity Temperature distribution jump distributionaround the cyli aroundnder the aroundFig. 8 the. Temperature cylinder jumpsurface, distributionσu = σ aroundT = 1 the around the cylinder surface, σu = σT = 0.8 Fig.cylinder 12. surfaceTemperature, σu = σ jumpT = 0.8 distri. bution around the Fig.cyli nd13er. surfaceSlip , velocity σu = σT = distribution 0.8. around the cylinder surface, σu = σT = 1. surface, σu = σT = 1. for ebody cylinder surface, σu = σT = 0.6. cylinder surface, σu = σT = 0.6. Fig.Fig. 1102 TemperatureTemperature jump jump distri distributionbution around around the the Fig. Fig.11. 14.Slip Temperature velocity dis jumptribution distribution around around the the Fig. 15. Slip velocity distribution around the Fig. 13. Slip velocity distribution around the Fig.cylinder 14. surface Temperature, σu = σT = jump 0.8. distribution cyliFig.nd 15er .surface Temperature, σu = σT = 0.8 jump. distribution cylinder surface, σu = σT = 0.4. forebody cylinder surface, σu = σT = 0.6. cylinder fsorurface,ebody σ cylinderu = σT = 0.surface,6. σu = σT = 0.4. around the forebody cylinder surface, around the forebody cylinder surface, Table 2: Average errors between the CFD and DSMC simulations of the cylinder cases. σu = σT = 0.6 σu = σT = 0.4 Cases Maxwell Sutherland Power Law Table 2. Average errors between the CFD and DSMC simulations of the cylinderviscosity cases model viscosity model viscosity model T u T u T u Maxwell Sutherland σu = σT = 1 Power15.12% Law 16.84% 28.98% 34.01% 35.65% 33.81% viscosity model viscosity modelσu = σT = 0.8 viscosity2.84% model 13.56% 20.90% 34.82% 55.34% 61.81% Cases T u T uσu = σT = 0.6 T 1.15% 16.44%u 9.56% 29.50% 53.39% 58.78% σu = σT = 1 15.12% 16.84% 28.98% 34.01% 35.65% 33.81% σu = σT = 0.8 2.84% 13.56% 20.90% 34.82% 55.34% 61.81% 11 σu =Fig.σT 14.= 0.6Temperature 1.15% jump distribution 16.44% around the 9.56%Fig. 15. 29.50%Slip velocity distribution 53.39% around 58.78% the σu = forσebodyT = 0.4 cylinder surface, 8.87% σu = σT = 0.4 6.70%. 16.43%cylinder surface, 17.55% σu = σT = 0. 49.36%4. 36.53% Table 2: Average errors between the CFD and DSMC simulations of the cylinder cases. Cases Maxwell Sutherland Power Law viscosity model viscosity model viscosity model T u T u T u σu = σT = 1 15.12% 16.84% 28.98% 34.01% 35.65% 33.81% σu = σT = 0.8 2.84% 13.56% 20.90% 34.82% 55.34% 61.81% σu = σT = 0.6 1.15% 16.44% 9.56% 29.50% 53.39% 58.78% 11
12. 214 Nam T. P. Le 4.5. Discussion Although the Sutherland viscosity model has been currently using mostly in the CFD rarefied gas simulations but the simulation results show that the Maxwell viscosity model give the good agreement with DSMC data for both the first-order and second-order slip velocity and temperature jump conditions, and with various accommodation coefficients in all cases considered. This may be explained that the Maxwell viscosity model was de- rived based on the net transfer of momentum since the gas molecules across any plane perpendicular in direction of velocity gradient resulting in the fixed coefficient s = 0.5, and did not depend on the reference temperature. While the Power Law and the Suther- land viscosity models were developed based on the intermolecular force law and attrac- tive force, in which the exponent, s, and constants (AS, TS, AP) are determined from the limiting theoretical cases or the limited ranges of temperatures in experiments. Com- paring Eqs. (1) and (4), both of the DSMC and Maxwell viscosity models depend on the molecular mass and diameter leading to the simulation results of the Maxwell viscosity model are close to those of DSMC data while two other viscosity models do not. 5. CONCLUSIONS From the simulation results obtained, whichever the slip and jump boundary con- ditions are adopted, the viscosity models effect the accuracy of the simulation results of surface gas temperature and slip velocity. The simulation results show that the Maxwell viscosity model provides better predictions of the surface gas temperature and slip veloc- ity than the Sutherland and Power Law viscosity models in comparing with the DSMC data, and pointed out the importance of the viscosity in rarefied gas flow simulations. A good viscosity model will increase the accuracy of the N–S–F simulations for rarefied gas flows, and gives better prediction the peak surface gas temperature to design the thermal protection system in hypersonic vehicles. ACKNOWLEDGEMENTS This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED), under grant number 107.03-2018.27. REFERENCES [1] N. T. P. Le, C. J. Greenshields, and J. M. Reese. Evaluation of nonequilibrium boundary con- ditions for hypersonic rarefied gas flows. Progress in Flight Physics, 3, (2012), pp. 217–230. [2] G. N. Patterson. Molecular flow of gases. John Wiley and Sons, (1956). [3] C. R. Lilley and M. N. Macrossan. DSMC calculations of shock structure with various viscos- ity laws. In The twenty-third Proceeding International Symposium Rarefied Gas Dynamics, (2003), pp. 663–670. [4] K. B. Jordan. Direct numeric simulation of shock wave structures without the use of artificial viscos- ity. PhD thesis, Marquette University, (2011). [5] H. Alsmeyer. Density profiles in argon and nitrogen shock waves measured by the ab- sorption of an electron beam. Journal of Fluid Mechanics, 74, (3), (1976), pp. 497–513.
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