Effect of viscosity on slip boundary conditions in rarefied gas flows

Vietnam Journal of Mechanics, VAST, Vol 41, No (2019), pp 203 – 215 DOI: https://doi.org/10.15625/0866-7136/13564 EFFECT OF VISCOSITY ON SLIP BOUNDARY CONDITIONS IN RAREFIED GAS FLOWS Nam T P Le1,2,∗ Industrial University of Ho Chi Minh City, Vietnam Ton 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 investigate 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 literature 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 2555-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 velocity; surface gas temperature INTRODUCTION The accuracy of the Navier–Stokes–Fourier (N–S–F) simulations for rarefied and microscale gas flows depends on the slip velocity and temperature jump conditions, and also the constitutive relations supplied, such as the viscosity-temperature relation, thermal conductivity and heat capacity We did an investigation for the slip and jump conditions in [1] to find the most suitable choice of slip velocity and temperature jump conditions for rarefied gas simulations Flow regimes in rarefied gas dynamics are characterized 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 204 Nam T P Le The CFD method, which solves the Navier–Stokes–Fourier (N–S–F) equations with appropriate 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 velocity and temperature jump conditions In gas microflows, the mean free path of the gas molecules becomes significant relative to the characteristic dimension of the microdevices 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 intermolecular 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 rarefied 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 simulations The viscosity of real gases can be matched by a power law over a small temperature 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) rarefied gas simulations In this paper three various viscosity models found in the literature such as Sutherland, Power Law and Maxwell viscosity models are numerically investigated 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 framework [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 VISCOSITY MODELS In 2D simulations, the Maxwell viscosity model employed for 1D simulation in [3], µ = mk B T/π/3πd2 , is slightly corrected that would be presented below; where m is mass of a molecule; k B 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 = c p µ/ Pr where the Prandtl number, Pr, is assumed to be constant and c p is the constant pressure specific heat 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] µ= mk B 0.5 T π πd2 (1) This equation of gas viscosity was inspired by Maxwell, so-called the Maxwell viscosity 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 T 0.5 , 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, µ = AP Ts , where s= + , v−1 (2) where A P 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 = corresponding to Maxwellian molecules Real molecules generally have v ranging from 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)) mk B /π /πd2 and can be re-written in the Power Law form µ = A M T s , with A M = 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 T 1.5 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 given in [3,17]: for argon AS = 1.93 × 10−6 (Pa.s/K−1/2 ) and TS = 142 K, and for nitrogen 206 Nam T P Le AS = 1.41 × 10−6 (Pa.s/K−1/2 ) and TS = 111 K Finally, the macroscopic viscosity model using for DSMC simulations [10] is expressed as follows, √ T ω 15 πmk B Tref µ = µref , where µref = , (4) Tref 2πd2ref (5 − 2ω )(7 − 2ω ) where ω is the variable-hard-sphere temperature exponent This model requires a reference temperature, Tref , reference diameter, dref and the exponent, ω Eq (4) can be written −ω in the power-law form µ = AT s 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 presented in the form of µ = A M T s , 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 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] u+ − σu σu − σu σu λ ∇ n (S · u ) = uw − λ µ S · ∇T S · (n · Πmc ) − , µ 4ρ T (5) I tr (∇u) The right hand side of Eq (5) contains 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 − σu ) or diffusely (equal to σu ) The Maxwellian mean free path is calculated by [15] where Πmc = µ (∇u)T − λ= µ ρ π 2RT (6) 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 condition can be written [12] − σT 2γ T+ λ∇n T = 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, Effect of viscosity on slip boundary conditions in rarefied gas flows 207 and no energy exchange to σT = The second order velocity slip boundary condition for a planar surface can be expressed as follows [13] u = − A1 λ∇n (S · u) − A2 λ2 ∇2n (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 C1 λ∇n T + C2 λ2 ∇2n T + Tw , (9) T=− γ + Pr 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 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 25-55deg 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 cavity, L, 3) diameter of the biconic base, 2R, and 4) the diameter of cylinder, D Their values are found in Tab 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, and the velocity of gas flow depends on the pressure 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 computational domain, shown in Fig 1(b) For the two-dimensional axisymmetric biconic Table Gas properties and characterized lengths of all cases Cases Step nanochannel Micro-cavity Biconic Cylinder ω Tref (K) 0.74 273 0.81 273 0.74 273 0.734 1000 dref (m) −10 4.17 × 10 4.17 × 10−10 4.17 × 10−10 3.595 × 10−10 m (kg) −27 Gas 46.5 × 10 Nitrogen −27 66.3 × 10 Argon 46.5 × 10−27 Nitrogen 66.3 × 10−27 Argon Characterized lengths H = 17.09 nm L = 1µm 2R = 261.8 mm D = 304.8 mm domain, shown Figure 1b For axisymmetric the geometry is gradient,inand the velocity of gasthe flowtwo-dimensional depends on the pressure gradient The biconic velocity iscase, then calculated type is used for both inletthe andplane outlet of for geometry velocity Zero normal gradient specified asexplicitly, a wedgeand of the oneNeumann cell thickness running along The axisymmetric wedge condition is applied for u at the entrance and exit, and T at the exit of channel, seen in Figures 1a For the planes must be specified as separated patches of type “wedge”, seen in Figure 1c For the sharp 25-55-deg lid-driven micro-cavity case, initial pressure and temperature are set as initial values in the computational biconic anddomain, cross-flow cylinder cases, at the inflow boundary, the freestream (p, T, u) conditions were shown in Figure 1b For the two-dimensional axisymmetric biconic case, the geometry is maintained specified throughout the computational process outflow boundary for The these both cases, zero normal as a wedge of one cell thickness runningAt along the plane of geometry axisymmetric wedge planes mustare be specified separated of type in Figure 1c sharp 25-55-deg gradient condition appliedasfor (p, T,patches u) At the“wedge”, bottomseen boundary ofFor thethebiconic and cylinder, a biconic and cross-flow cylinder cases, at the inflow boundary, the freestream (p, T, u) conditions symmetry boundary condition is applied to all flow variables, shown in Figures1c and 1d were 208 maintained throughout the computational At outflow boundary for these both cases, zero normal Nam T process P Le gradient condition are applied for (p, T, u) At the bottom boundary of thedomain, biconic and cylinder, a geometry dimensions, numbers of cells for blocks in computational input parameters symmetry boundary condition is applied to all flow variables, shown in Figures1c and 1d The and working gases of all cases are given in Figures 1a, 1b, 1c and 1d Numbers of cells are 60x60, 140x60 and case, the geometry is specified as a wedge of one cell thickness running along the plane The geometry dimensions, numbers of cells for blocks in computational domain, input parameters and 140 x60 for blocks of the backward facing step nanochannel case, seen in Figure 1a Those are 120 x 120 of geometry The axisymmetric wedge planes must be specified as separated patches of working gases all cases are given Figures25-55-deg 1a, 1b, (i.e 1c and 1d cells Numbers of cells are 60x60, and for cavity case, and 256 x 256 for theinsharp biconic case 256 in cross-flow the axial, streamwise and typethe “wedge”, seen in of Fig 1(c) For the biconic and cylin-140x60 direction 140 x60 for blocks of the backward facing step nanochannel case, seen in Figure 1a Those are 120 x 120 256 cells in the radial, surface normal direction) For the circular cylinder case, the computational der cases, at inflow boundary, freestream u)256 conditions maintained forthe the cavity case, and 256 x the 256 for the biconic(p, caseT,(i.e cells in thewere axial, streamwise direction and throughoutmesh thecells computational At outflow boundary for these boththe cases, structured is constructed to wrapnormal around the leading bow shock with smallest cell sizes grading 256 in the radial, process surface direction) For the circular cylinder case, thezero computational normal condition are applied T, u).the Atleading the bottom boundary the biconic structured mesh constructed to for wrap(p, around bow shock with the of smallest cell sizes grading near thegradient surface ∆x = 0.1ismm, ∆y = 1.196 mm neara the surface ∆xboundary = 0.1 mm, ∆y = 1.196 mm and cylinder, symmetry condition 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 a) (b) Lid-driven micro-cavity b) a) b) 5 c) (c) Sharp c) 25-55-deg biconic d) (d) Circular d) cylinder Fig Numerical setups, input parameters and geometryand dimensions of four cases a) backward Fig Numerical setups, input parameters geometry dimensions of fourfacing casesstep Fig Numerical setups,b)input parameters and geometry dimensionsbiconic, of fourand cases a) backward facing step nanochannel, lid-driven micro-cavity, c) sharp 25-55-deg d) circular cylinder 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 The So geometry dimensions, numbers cells forgood blocks in in computational domain, microflows they are adopted for simulating twoofobtained nano/micro-flow cases the present work with the gas The second-order slip and jump conditions results for simulating rarefied input parameters and working gases of all cases are given in Fig Numbers of cells are the coefficient values A = C = 1.3 and A = C = 0.23 proposed in our previous work [13] The first-order 2 two nano/micro-flow cases in the present work microflows So they are 1adopted for simulating with 60 × 60, 140 × 60 and 140 × 60 for blocks of the backward facing step nanochannel case, Maxwell/Smoluchowski conditions are selected for simulating hypersonic cases with the coefficients σ T = coefficient values A1 = C1 = 1.3 and A2 = C2 = 0.23 proposed in our previous work [13] The first-order σu seen = Ininthe present the CFD results would compared data×using σT = σu Fig 1(a).work Those are 120 × 120 forbethe cavity with case,DSMC and 256 256 the for values the biconic Maxwell/Smoluchowski conditions are selected for simulating hypersonic cases with the coefficients σT = = For(i.e a fair256 comparison, the axial, viscosity should be treated as equivalent possible between DSMC cells in the streamwise direction and 256 as cells in the radial,thesurface σu = 1.and Incase the present work the CFD results would be compared with DSMC data using the the values σT = σu CFD simulations This means the parameters (m, ω, d , T ), that are chosen to calculate constant ref the ref computational structured mesh is normal direction) For the circular cylinder case, = For a fair comparison, the viscosity should be treated as equivalent as possible between the of the DSMC macroscopic viscosity, will be adopted for viscosity models in CFD as 1) DSMC s= and CFD simulations This means the parameters (m, ω, d ref, Tref), that are chosen to calculate the constant ω for the Power Law viscosity model, and 2) the constant AM for the Maxwell of the DSMC macroscopic viscosity, will be adopted for viscosity models in CFD as 1) s = viscosity model These parameters of gas properties are shown and characterized lengths in Tab ω for the Power Law viscosity model, and 2) the constant AM for the Maxwell viscosity model These parameters of gas properties are shown and characterized lengths in Tab Table 1: Gas properties and characterized lengths of all cases Cases ω Tref (K) dref (m) m(kg) Gas Characterized 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 rarefied 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 proposed in our previous work [13] The first-order Maxwell/Smoluchowski conditions are selected for simulating hypersonic cases with the coefficients σT = σu = In the present work the CFD results would be compared with DSMC data using the values σT = σu = 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, mk B /π /πd2ref for the Maxwell viscosity model These and 2) the constant A M = parameters of gas properties are shown and characterized lengths in Tab 4.1 Pressure driven backward facing step nanochannel case Nam T.step P Lenanochannel, Kn = 0.025 [7], we present In the pressure driven backward facing Nam T P Le 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 increase to the peak temperature and then gradually decrease along the wall-3, seen in Fig The prediction of the Maxwell viscosity model for the gas surface temperature gives good agreement withThe thesurface DSMCgas data [7] whileincrease the CFD other results not Slip zone located overthis thiswall wall temperatures peaktemperature temperature andthen then zone is is located over The surface gas temperatures increase to tothethepeak and velocities on the wall-3 consist ofseen negative and positive components shown in Fig.model Neggradually decrease along the wall-3, in Figure The prediction of the Maxwell viscosity gradually decrease along the wall-3, seen in Figure The prediction of the Maxwell viscosity model forfor ative ones represent the separation zone,with and the distance, indicates theresults negative the surface temperature gives good agreement with the DSMC data [7] while the CFD other results the gasgas surface temperature gives good agreement the DSMC data [7]where while the CFD other dodo not Slip velocities on the wall-3 consist of negative and positive components shown in Figure Negative slip velocities, is defined as the length of the separation zone It is seen that the prediction not Slip velocities on the wall-3 consist of negative and positive components shown in Figure Negative ones represent separation zone, and distance, where indicates negative slip velocities, defined using the Maxwell viscosity model gives better slip velocity than slip the CFD other results ones represent thethe separation zone, and thethe distance, where indicates thethe negative velocities, is is defined as the length the separation zone seen that prediction using Maxwell viscosity model give asin the length of of thewith separation zone It It is is seen that thethe prediction using thethe Maxwell viscosity model give comparing DSMC data [7] better slip velocity than CFD other results comparing with DSMC data [7] better slip velocity than thethe CFD other results in in comparing with DSMC data [7] Fig Surface gas temperature along the wall3,gas Kntemperature = 0.025along [7] Fig Surface along wall-3, Fig 2 Surface gas temperature thethe wall-3, KnKn = 0.025 [16] = 0.025 [16] Fig Slip velocity along the wall-3, Kn = 0.025 [7] Fig.3 3.Slip Slipvelocity velocity along wall-3,KnKn= =0.025 0.025 Fig along thethewall-3, [17] [17] 4.2 Lid driven micro-cavity case 4.2 Lid driven micro-cavity case ForFor thethe lidlid driven micro-cavity case, KnKn = 0.05 [8], thethe gasgas flow expands at at thethe location x/L =0 driven micro-cavity case, = 0.05 [8], flow expands location x/L = as as it is driven byby thethe moving lid,lid, and it it is is compressed at at thethe location x/L it is driven moving and compressed location x/L= =1 1.Considering Consideringthethesurface surfacegas gas temperature temperaturealong alongthethelidlidwall, wall,thethePower PowerLaw Lawand andthetheSutherland Sutherlandviscosity viscositymodels modelsunderpredicts underpredictsthethe temperature in in thethe range x/L< 0.10.1 in in comparing with DSMC data [8][8] and that with thethe Maxwell viscosity temperature range x/L< comparing with DSMC data and that with Maxwell viscosity model, model,seen seenin inFigure Figure4 4.The Thesimulation simulationresult resultobtained obtainedwith withthetheMaxwell Maxwellviscosity viscositymodel modelgive givegood good agreement with DSMC data along thethe lidlid surface AtAt thethe location x/L = 1= the gasgas flow is is reattachment, and agreement with DSMC data along surface location x/L the flow reattachment, and 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 location x/L = as it is driven by the moving lid, and it is compressed at the location x/L = 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 < 0.1 in comparing with DSMC data [8] and that with the Maxwell viscosity model, Effect viscosity onslip slip boundary conditions inin rarefied flows Effect ofof viscosity on boundary conditions rarefiedgas gas flows seen in Fig The simulation result obtained with the Maxwell viscosity model give good agreement with DSMC data along the lid surface At the location x/L = the gas flow is reattachment, and all simulation results show that the temperature increasing with T > Tw = 300 K It means there is viscous heat generation which results in the heat transfer from the gas to the wall toward the location x/L = of the cavity case Fig Surface gas temperature along the lid Fig Surface gaswall, temperature along lidlid wall, Fig Surface gas temperature along wall, Kn = 0.05 [8]thethe Kn =Kn 0.05 [17].[17] = 0.05 Fig Slip velocity along the lid wall, Fig velocityalong alongthe thelidlidwall, wall, = 0.05 Fig 5 Slip velocity KnKn = 0.05 Kn = 0.05 [8] [17] [17] For the slip velocity along the lid wall in Fig 5, all simulations showed that the slip velocities are verycase slow at the locations x/L = and x/L = 1, and obtained the 4.3 4.3 Sharp 25-55-deg biconic Sharp 25-55-deg biconic case peak value around the location x/L = 0.5 The Power Law and the Sutherland viscosity An oblique shock forms the first and locates along towardsnear near theend end Anunderpredict oblique shock forms from thetiptipofofthe the firstcone conelid andsurface locates in along towards the of of[8] models thefrom slip velocities along the comparing DSMC data this cone, and and thenthen separates creating a shock the oblique obliqueshock shockand andmeets meets this cone, separates creating a shock.Latter Latterone oneinteracts interacts with the thethe The bow simulation result using the Maxwellcone viscosity model is close to zone DSMC data while detached shock being formed over recirculation forms detached bow shock being formed overthethesecond second cone AA low low speed speed recirculation zone forms at atthethe those of the Power Law and Sutherland viscosity models are not junction between the the firstfirst andand thethe second 0.1021mwhere wherepresents presents junction between secondcones conesininthe therange range 0.0754m 0.0754m ≤ x ≤≤ 0.1021m thethe negative velocity, seen in Figure 7.case negative slip velocity, seen inbiconic Figure 4.3.slip Sharp 25-55-deg Figures compares the CFD surface gas with those of DSMC DSMCdata [9].The Thesurface surface Figures compares CFD surface gastemperatures temperatures with thoseand of [9] An oblique shock the forms from the tip of the first cone locatesdata along towards near gas temperature with the Maxwell viscosity model is close to the DSMC data [9] near the tip of biconic gas temperature with the Maxwell viscosity model is close to the DSMC data [9] near the tip of biconic the end of this cone, and then separates creating a shock Latter one interacts with the The surface gas gas temperatures obtain thethe peak and thereafter thereafterrapidly rapidlydecrease decrease The surface temperatures obtain peakvalues valuesatatthe thebiconic biconic tip, tip, and in in oblique shock and meets the detached bow shock being formed over the second cone the range x ≤ 0.754m In this range thethe surface by the theMaxwell Maxwellviscosity viscosity model A the range x ≤ 0.754m In this range surfacegas gastemperature temperature predicted predicted by model give give goodgood agreement with the DSMC data There is a drop of temperature in the recirculation zone All low speed recirculation zone forms at the junction between the first and the second agreement with the DSMC data There is a drop of temperature in the recirculation zone Allcones CFDCFD temperatures and DSMC data are close together in 0.0754m ≤ x ≤ 0.02m temperatures and DSMC close together in 0.0754m ≤ x ≤the 0.02m in the range 0.0754 m ≤data x ≤are0.1021 m where presents negative slip velocity, seen in compares the CFD and DSMC [9] slip velocities along the biconic biconicsurface surface.Slip Slip Fig.Figures 6.Figures compares the CFD and DSMC [9] slip velocities along the velocities on the surface ofsurface negative and positive Negative ones represent Fig compares theconsist CFD gas withNegative those ones of DSMC data velocities on7biconic the biconic surface consist of negative andtemperatures positive components components represent thethe[9] recirculation zone, and the distance, where indicates the slip defined thelength length recirculation zone, and the distance, where indicates thenegative negative slip velocities, velocities, isisdefined asasthe of of The surface gas temperature with the Maxwell viscosity model is close to the DSMC the recirculation zone The slip velocities obtain the peak values at the biconic tip and then quickly the recirculation zone.tip The velocities obtain the peak at the biconic tip and data [9] near the ofslip biconic The surface gas values temperatures obtain the then peakquickly values at decrease along the the forecone until thethe locations using theMaxwell Maxwell viscosity decrease along forecone until locationsx x==0.075m 0.075m.The The CFD CFD results results using the viscosity the biconic tip, and thereafter rapidly decrease in velocities the rangeincrease x ≤ 0.754 m In this range the model are close to the DSMC data Past this zone the slip and oscillate along model are close to the DSMC data Past this zone the slip velocities increase and oscillate along thethe second 55-deg cone, and there is good agreement between all CFD results and the DSMC data in the second 55-deg cone, and there is good agreement between all CFD results and the DSMC data in the rangerange 0.105m ≤ x ≤≤ x0.02m Overall, thetheMaxwell betterslip slipvelocity velocitythan than 0.105m ≤ 0.02m Overall, Maxwellviscosity viscosity model model predicts predicts better thethe Sutherland and the Power Law models in comparing with DSMC data Sutherland and the Power Law models in comparing with DSMC data Effect of viscosity on slip boundary conditions in rarefied gas flows 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 perature distribution over the r cylinder cases 211T P Le Nam Fig Slip velocity distribution over the biconic Fig.surface Slip [18].velocity distribution over the biconic surface [9] Fig Surface gas temperature distribution over the Fig Slip velocity surface [18] Fig 7.biconic Surface surfacegas [18].temperature distribution over the biconic surface [9] Cross-flow circular cylindersurface cases Slip Fig compares the CFD and DSMC [9] slip4.4 velocities along the biconic es, various values of accommodation coefficients σu = σT = 1, σu = σT = 0.8, σu In the cylindercomponents cases, various values of accommodation coefficients the biconic surface consist of negative and positive Negative 0.4 are conducted for allvelocities simulations.on The surface gas temperatures and slip = σ = 0.6 and σ = σ = 0.4 are conducted T u T ones the recirculation and the distance, where indicates the negative for slipall simulations The surfa nst with the cylinder angle All represent CFD simulations predict a higher zone, slip velocity velocities are plotted against with the cylinder angle All CFD simulations as seen in Figures 8, 10, 12 and 14 for the σu = σas 1, σulength = σT = 0.8, velocities, is cases defined of σthe T =the u = recirculation zone The slip velocities obtain than the DSMC data [10], as seen in Figures 8, 10, 12 and 14 for the cases σu respectively The DSMC the and CFD slip velocities increase gradually from ≤ θ peak values at the biconic tip and then quickly the forecone The until the and CFD slip velocities σT = decrease 0.6 and σu =along σT = 0.4, respectively DSMC k normalized values around the location θ = 130-deg., and then gradually locations x = 0.075 m The CFD results using the Maxwell viscosity model are close to ≤ 130-deg., reaching peak normalized values around the location θ = ≤ 180-deg The slip velocity using the Maxwell viscosity model obtains the DSMC data Past this zone the slip velocities increase and oscillate along the decrease in 130-deg ≤ θ ≤ 180-deg The second slip velocity using the Maxwell tively close to the DSMC the data [10] Considering the surface gas temperature, all lowest are relatively to thedata DSMC there agreement allvalues, CFD and results and theclose DSMC in data [10] Considering th s are shown in Figures 9, 55-deg 11, 13 andcone, 15 for and the cases σu is = σgood T = 1, σ u = σT = 0.8, between the CFD and DSMC results are shown in Figures 9, 11, 13 and 15 for the cas = 0.4, respectively, in which one using them Maxwell is closethe Maxwell viscosity model predicts better slip thetherange 0.105 ≤ x ≤viscosity 0.02 m.model Overall, σu = σT = 0.6 and σu = σT = 0.4, respectively, in which the one using the Max are differences between the CFD and DSMC temperatures along the cylinder velocity than the Sutherland and the Power Lawtomodels in comparing with DSMC data the DSMC data There are differences between the CFD and DSMC tem may be explained by the calculation of the translational surface gas temperature surface These differences may be explained by the calculation of the translat he components of gas velocity the slip velocity only.cylinder While that cases in CFD is 4.4 and Cross-flow circular radient 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 velocit cylinder cases, various values of accommodation coefficients σugas=temperature, σT = 1, and is independent of t mperature being very similar toIn thatthe of the DSMC slip velocity calculated by the normal gradient of σu = σT = 0.8, σu = σT = 0.6 and σu = σT = 0.4theare conducted for all simulations profile of the DSMC temperature being veryThe similar to that of the DSMC ge errors between all CFD and DSMC simulations are shown in Table The “dsmcFoam” is errors used in tocomparing run the DSMC simulations, and generates the DSMC data Maxwell viscosity modelsolver obtain the smallest average with Finally, the average errors between all CFD and DSMC simulation temperatures slip of velocities plottedusing against with the cylinder ions with the Power LawThe and surface Sutherlandgas viscosity models Theand reduction CFDare simulations the Maxwell viscosity model obtain the smallest aver oefficient affects the factor (2 - σTAll )/σT CFD in the simulations jump temperature condition that angle predict a higher slip velocity thansimulations the DSMC as seen those of the CFD withdata, the Power Law and Sutherland viscosi the surface gas temperatures It is 8–11 also seen reduction surface in Figs forthat thethe cases σu =ofσthe = 0.8,accommodation σu = σT = 0.6 and σu affects = σT the = 0.4, coefficient factor (2 - σT)/σT in the jum T = 1, σu = σTthermal y decreases the effect of viscosity on the flow field, and leads to the increases of in the increases of the surface gas13-deg., temperatures It is also seen that respectively The DSMC and CFD slip velocitiesresults increase gradually from 0≤ θ ≤ accommodation effectivelyand decreases effect of viscosity on the flow field reaching peak normalized values around the location θ = 13-deg., thenthegradually 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 between the CFD and DSMC temperatures along the cylinder surface These differences 212 Nam T P Le Effect of viscosity may on slipbe boundary conditions rarefied gas flows of the translational surface gas temperature in DSMC explained byinthe calculation T P Leand the slip velocity only While that in depending on the components of gas Nam velocity CFD is calculated by the normal gradient of gas temperature, and is independent of the gas velocity This leads to the profile of the DSMCFig temperature veryaround similar to that Slip velocitybeing distribution the cylinder Fig Temperature jump distribution around the of the DSMC slip velocity surface, σ = σ = u T cylinder surface, σu = σT = Nam T P Le 10 velocity Temperature jump distribution aroundthe the Fig 8.Fig Slip distribution around cylinder surface, σ = σ = 0.8 u T Fig 9.cylinder Slip velocity distribution p distribution around the surface, σu =around σT =the cylinder surface, u = σT = Fig 12 σTemperature jump distribution around the forebody cylinder surface, σu = σT = 0.6 = distribution around the = σT = 0.6 0.8 u Fig 14 Temperature jump distribution around Slipvelocity velocitydistribution distribution around Fig.Fig 13.11 Slip around the thethe Fig cylinder 10 Slip velocity distribution around the forebody cylinder σu = σT = 0.4 cylinder surface, = σ0.6 T = 0.8 surface, σu =σuσsurface, T= cylinder surface, σu = σT = 0.6 Slip velocity distribution around the Fig 9.Fig Slip11.velocity distribution around the cylinder surface, σu = σT = 0.8 cylinder surface, σu = σT = 0.8 Fig 13 Slip velocity distribution around the cylinder surface, σu = σT = 0.6 Fig 15 Slip velocity distribution around the cylinder σu =distribution σT = 0.4 Fig 11 Slipsurface, velocity around the cylinder surface, σu = σT = 0.4 Table 2: Average errors between the CFD and DSMC simulations of the cylinder cases Finally, the average errors between and DSMC simulations Cases Maxwell all CFDSutherland Power Laware shown in model viscosity modelmodelviscosity Tab The CFD simulations viscosity using the Maxwell viscosity obtain model the smallest average errors in comparing withT those of with Law and u the CFD T simulations u T the Power u Sutherland viscosity models The reduction of thermal accommodation coefficient affects σu = σT = 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 accommodaσu = σT = 0.6 the effect 1.15% of 16.44% 29.50% tion effectively decreases viscosity9.56% on the flow field, 53.39% and leads58.78% to the increases of the slip velocity ump distribution around the e, σu = σT = 0.4 11 Fig 15 Slip velocity distribution around the cylinder surface, σu = σT = 0.4 age errors between the CFD and DSMC simulations of the cylinder cases Maxwell viscosity model Sutherland viscosity model Power Law viscosity model 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 Temperature jump distribution around the cylinder surface, σu = σT = Fig Slip velocity di surface, σu = σT = Nam T P Le Fig 12 Temperature jump distribution Fig 8.the Temperature distribution around cylinder jump surface, σu = σaround T = the Fig 13 Temperature jump distribution cylinder surface, σu = σT = Maxwell viscosity model Table Average errors between the CFD and DSMC simulations of the cylinder cases T u σu σu σu σu T u the uσu = σT = 0.6 T 1.15% 28.98% 34.01% 35.65% 20.90% 34.82% 55.34% 9.56% 29.50% 53.39% Fig 15 Slip velocity distribution cylinder surface, σu = σT = 0.4 16.43% 17.55% 49.36% Table 2: Average errors between the CFD and DSMC simulations of the cylinder cases Cases Maxwell viscosity model T u Sutherland viscosity model T 15.12% Sutherlandσu = σT = Power Law16.84% 28.98% viscosity model model σu = σT = 0.8 viscosity 2.84% 13.56% 20.90% Maxwell viscosity model = σT = 15.12% 16.84% = σT = 0.8 2.84% 13.56% =Fig σT 14 = 0.6 1.15% 16.44% Temperature jump distribution around =forebody σT = 0.4 8.87% 6.70% cylinder surface, σu = σT = 0.4 Fig 15 Slip ve cylinder surface, σu around the forebody cylinder surface, Table 2: Average σu = σT = 0.4errors between the CFD and DSMC simulations of Cases T Fig 13 Slip velo cylinder surface, σu = Fig 11 14 SlipTemperature velocity distribution aroundaround the the jump distribution Fig 13.Fig Slip velocity distribution around the Fig.forebody 15 Temperature jump distribution cylinder surface, σ = σ = 0.8 u T surface, σu = σT = 0.4 cylinder surface, σcylinder u = σT = 0.6 Fig 12 10 Temperature Temperature jump jump distribution distribution around around the the Fig Fig 14 surface, Temperature cylinder u = σT = forebody cylinderσsurface, σjump = σ T =distribution 0.6 u0.8 around the forebody cylinder surface, σu = σT = 0.6 Cases Fig 11 Slip vel cylinder surface, σu Fig 10 Temperature jump distribution around the Fig Slipthe velocity distribution around around cylinder surface, σuthe=cylinder σT = 0.8 cylinder surface, σu = σjump T = 0.8 Fig Temperature distribution around the surface, σ12 u = σT = forebody cylinder surface, σu = σT = 0.6 Sutherland viscosity model T Power Law viscosity model u T u σu = σT = 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% 29.50% 53.39% 58.78% 9.56% u 16.44% 33.81% 61.81% 11 58.78% around the 36.53% 9.56% u 34.01% 35 34.82% 55 29.50% 53 214 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 derived 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 Sutherland viscosity models were developed based on the intermolecular force law and attractive force, in which the exponent, s, and constants (AS , TS , A P ) are determined from the limiting theoretical cases or the limited ranges of temperatures in experiments Comparing 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 not CONCLUSIONS From the simulation results obtained, whichever the slip and jump boundary conditions 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 velocity 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 conditions for hypersonic rarefied gas flows Progress in Flight Physics, 3, (2012), pp 217–230 https://doi.org/10.1051/eucass/201203217 [2] G N Patterson Molecular flow of gases 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https://doi.org/10.1016/j.ijheatmasstransfer.2012.04.050 ... cylinder, symmetry condition 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. .. temperature in the range x/L < 0.1 in comparing with DSMC data [8] and that with the Maxwell viscosity model, Effect viscosity onslip slip boundary conditions inin rarefied flows Effect ofof viscosity on. .. Le Effect of viscosity may on slipbe boundary conditions rarefied gas flows of the translational surface gas temperature in DSMC explained byinthe calculation T P Leand the slip velocity only