Progress in Aerospace Sciences 52 (2012) 80–87

Contents lists available at SciVerse ScienceDirect

Progress in Aerospace Sciences

0376-04

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/paerosci

Rarefied hypersonic flow simulations using the Navier–Stokes equationswith non-equilibrium boundary conditions

Christopher J. Greenshields a, Jason M. Reese b,n

a OpenCFD Ltd (SGI), 200 Berkshire Place, Wharfedale Road, Winnersh RG41 5RD, UKb Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK

a r t i c l e i n f o

Available online 17 September 2011

Keywords:

High-speed flow

Thermodynamic non-equilibrium

Boundary conditions

Compressible flows

Simulation

Rarefied gas dynamics

21/$ - see front matter & 2011 Elsevier Ltd. A

016/j.paerosci.2011.08.001

esponding author.

ail address: jason.reese@strath.ac.uk (J.M. Ree

a b s t r a c t

This paper investigates the use of Navier–Stokes–Fourier equations with non-equilibrium boundary

conditions (BCs) for simulation of rarefied hypersonic flows. It revisits a largely forgotten derivation of

velocity slip and temperature jump by Patterson, based on Grad’s moment method. Mach 10 flow around a

cylinder and Mach 12.7 flow over a flat plate are simulated using both computational fluid dynamics using

the temperature jump BCs of Patterson and Smoluchowski and the direct simulation Monte-Carlo (DSMC)

method. These flows exhibit such strongly non-equilibrium behaviour that, following Patterson’s analysis,

they are strictly beyond the range of applicability of the BCs. Nevertheless, the results using Patterson’s

temperature jump BC compare quite well with the DSMC and are consistently better than those using the

standard Smoluchowski temperature jump BC. One explanation for this better performance is that an

assumption made by Patterson, based on the flow being only slightly non-equilibrium, introduces an

additional constraint to the resulting BC model in the case of highly non-equilibrium flows.

& 2011 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2. Standard non-equilibrium BCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3. Patterson’s derivation of non-equilibrium BCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.1. Mass conservation condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.2. Momentum conservation BCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.3. Energy conservation BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.4. Patterson’s temperature BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4. Mach 10 flow around a cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1. Preliminary work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5. Mach 12.7 flow over a flat plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

1. Introduction

The aerothermodynamic design of hypersonic vehicles incorpo-rates wind tunnel testing, flight experiments, and modelling andcomputer simulation. Computer simulation is particularly attractivedue to its relatively low cost and its ability to deliver data that wecannot measure or observe, under conditions we cannot reproduce

ll rights reserved.

se).

in a laboratory. Its efficacy relies upon the quality of the underlyingnumerical methods and physical modelling for the particularapplication.

The use of computational fluid dynamics (CFD) for aerothermo-dynamics of hypersonic vehicles is complex, compared to aerody-namics of automotive vehicles, for example. Flow discontinuities,high gradients and other flow features impose far greater demandson the underlying numerical methods. Extreme temperaturesrequire complex models for thermodynamics that might includenon-equilibrium effects, radiation, and molecular dissociation andthe modelling of chemical reactions. The changing constitution of the

C.J. Greenshields, J.M. Reese / Progress in Aerospace Sciences 52 (2012) 80–87 81

gas also affects diffusive and turbulent transport, which is difficult tomodel accurately.

A significant limitation is also due to an underlying assumptionof the Navier–Stokes–Fourier (NSF) equations that the fluid is acontinuum in local thermodynamic near-equilibrium. Where the gasdensity is low and/or where high density gradients exist, thisassumption becomes less valid. To be precise, it is the modellingof momentum and energy transfer when gas molecules interact(diffusion) that is limited to near-equilibrium in the Navier–Stokesequations by the use of linear relationships between stress andstrain-rate (Newton’s viscosity law) and between heat-flux andtemperature-gradient (Fourier’s law), respectively. Furthermore,where the gas meets a solid body, boundary conditions (BCs) arerequired to model momentum and energy transfer through inter-action of gas molecules with the solid surface. The no-slip andisothermal conditions, in which the fluid velocity and temperatureare set to that of the solid surface, is similarly based on theassumption that the gas is in local thermodynamic equilibrium withthe surface and is typically used in conjunction with the NSFequations.

To indicate the extent to which a gas is in local thermody-namic equilibrium, we define the Knudsen number,

Kn¼lL�

lQ

l�rQ , ð1Þ

where the molecular mean free path

l¼ffiffiffiffiffiffiffiffiffip

2RT

rmr , ð2Þ

L is a characteristic length scale of the fluid system, Q is a flowquantity of interest, such as the gas density r, pressure p ortemperature T, and l represents a suitable spatial direction (generallythe direction of greatest variation of Q). Wherever Knt0:001 in aflow, gas molecules undergo a large number of collisions over thetypical length scale, and the conventional no-slip NSF equationsare appropriate. As Kn rises, it is generally thought that the mostsignificant departures from near-equilibrium occur at or near solidsurfaces, where the use of non-equilibrium BCs offer improvementover no-slip, isothermal conditions. Further increases in Kn renderthe standard NSF equations for the gas inappropriate. One alternativeis to turn to discrete particle methods, such as direct simulationMonte Carlo (DSMC), that provide an approximation to the solutionof the Boltzmann equation. Because it provides something closer to asolution of the Boltzmann equation itself, DSMC is the preferredmethod of solution at, for example, Kn¼ 1. However, the use ofDSMC is severely limited by computational cost because Dx,Dtplwhere Dx is the length of a computational cell and Dt is the time stepfor a given simulation. To a first approximation, a decrease in Kn of afactor of 10 requires decreases in cell volume, Dx3, and Dt of 103 and10 respectively, leading to a factor of 104 increase in simulation time.The practical use of DSMC becomes severely strained as Kn isreduced through the transition regime 0:001oKno1.

There is therefore a practical need for continuum-based CFDfor hypersonic flows within a particular range of Kn. Whether weuse the NSF equations or more complex continuum models, theBCs are a critical component of any such simulation. This is thesubject of this paper, in which we review non-equilibrium BCscommonly proposed for hypersonic flows and investigate mod-ifications to these BCs using test cases of hypersonic flow over acylinder and a flat plate.

2. Standard non-equilibrium BCs

The classical BCs for non-equilibrium flow are the Maxwellmodel for velocity slip [1] and the Smoluchowski model for

temperature jump [2]. For the velocity slip, Maxwell relies on afairly descriptive derivation, beginning with the idea that thetangential momentum brought up to unit area of surface per secondby approaching molecules is responsible for the tangential stress atthe surface. He assumes that the approaching stream of moleculescarries the same level of tangential momentum contained within theregion of gas some distance away from the boundary. That gasconsists of approaching and receding streams, each of which con-tribute to half of the stress within it. On striking the boundary, somefraction s of the tangential momentum relative to the boundarysurface is transferred from the molecules to the boundary. The gasmust then slip in order that tangential momentum is conserved.

The tangential momentum balance includes contributions dueto: (1) the shear stress s; (2) the slip at the surface, calculatedfrom the tangential slip velocity us and mean molecular speed v;(3) thermal creep. The tangential momentum due to thermalcreep is derived separately and accounts for the fact that hottermolecules have more momentum than colder molecules so loseproportionally more of it when they collide with a surface.

The Maxwell model is commonly expressed as

us ¼ A2�ss

� �l@ux

@nþ

3

4

mrT

@T

@x, ð3Þ

where A is a constant of proportionality, x is the tangentialdirection, n is in the direction normal to the surface, pointinginto the fluid, and m is the viscosity. The second term on the righthand side relates to thermal creep.

By analogy with Maxwell’s arguments for the momentum slip,the Smoluchowski model for temperature jump assumes theenergy brought up to a boundary by approaching molecules perunit area of surface per second is responsible for the heatconducted through the boundary. On striking the boundary, themolecules give it some fraction a of the heat. The resulting modelis usually expressed as

T�Tw ¼2�aa

� �2g

ðgþ1ÞPrl@T

@n, ð4Þ

where g is the ratio of specific heats and Pr is the Prandtl number.These expressions for the Maxwell and Smoluchowski BCs,

given by Eqs. (3) and (4), are typically adopted for hypersonicflows (if non-equilibrium BCs are used at all). In the authors’opinion, the equations are imprecise and misleading in thefollowing ways.

First, Eq. (3) only specifies the tangential component of velocityand does not specify the normal component, which is the normalcomponent of the surface itself and generally non-zero in the case ofa moving surface. It omits a reference surface velocity so that it isonly valid for a stationary surface. This also creates the illusion thatthe equation is fundamentally different in form to Eq. (4).

Second, Maxwell originally related us to s, which for a New-tonian fluid is

s¼ m½rUþrUT�ð2=3ÞtrðrUÞI�, ð5Þ

where U is the velocity vector and I is the identity tensor. Eq. (3)incorporates the first of the terms on the right hand side of Eq. (5),but ignores the other terms. The second term equates to zero for aflat boundary surface, but when the surface is curved its inclusion isnecessary, as demonstrated in the studies of a cylindrical Couetteflow and drag on a sphere [3]. The third term is also clearly non-zerofor compressible flows.

Finally, the way Eqs. (3) and (4) are expressed, with the slip/jumpvalue on the left hand side and the gradient and other terms on theright, gives the impression they are Dirichlet (specified value) BCswhose specified boundary values are calculated from the gradientand other terms. However they are Robin BCs, i.e. a weightedcombination of Dirichlet and Neumann (specified normal gradient)

C.J. Greenshields, J.M. Reese / Progress in Aerospace Sciences 52 (2012) 80–8782

conditions, which can be expressed as

fþan�rf¼ b, ð6Þ

where n is the unit normal vector at the surface pointing outwards ofthe domain, f is the dependent variable, and a and b are modelcoefficients. In this form, it is fairly clear that the BC represents aDirichlet condition f¼ b in the limit of a-0 and a Neumanncondition n�rf¼ 0 in the limit a-1, but in general represents acondition between these extremes.

Where f is the velocity, a vector field, the Robin conditionapplies to the tangential component only, with the normalcomponent being represented by a Dirichlet condition. The Robincondition can be reduced to a Dirichlet condition in the normalcomponent by transformation of the normal gradient with tensorS¼ I�nn, so that Eq. (6) becomes

fþaS�rnf¼ b, ð7Þ

in which we introduce the shorthand n�r �rn. Both slip andjump BCs can be expressed in the form of Eq. (7), although thetransformation by tensor S may be omitted in the case of tempera-ture jump since T is a scalar so is unaffected by the transformation.

3. Patterson’s derivation of non-equilibrium BCs

The Maxwell–Smoluchowski BCs given by Eqs. (3) and (4) arethe most commonly used non-equilibrium BCs and most devel-opment of the BCs involves determining optimal values of modelconstant A and the accommodation coefficients s and a. Thesecoefficients are either determined empirically or calculated fromkinetic theory, e.g. [4,5].

Rather than optimise coefficients for these standard BC equa-tions, in this work we have revisited an alternative derivation fornon-equilibrium BCs based on Grad’s moment method [6]. Weattribute this alternative derivation to Patterson since it seems toappear first in his book of 1956 [7]. It is based on integrals ofmass, momentum and energy for incident and reflected moleculesat a surface. It assumes diffuse reflections everywhere, so ulti-mately generates BCs for accommodation coefficients s,a¼ 1,although adaptation of the method for specular reflections isrelatively trivial and is included in a later book [8].

The non-equilibrium BCs of Patterson appear in some signifi-cant work soon after their publication, notably in a study ofhypersonic flow over a flat plate [9] in which velocity slip andtemperature jump are analysed and discussed in detail, and inwhich mention is made of the existence of pressure jump. Themethod was also extended to produce a system of boundaryconditions for a multicomponent mixture including concentrationslip boundary conditions [10]. Despite that, Patterson’s workseems to have been largely forgotten, so first we will reproducethe key elements of the derivation here using tensor notation.

Conservation equations for mass, momentum and energybetween incident and reflected molecules are expressed in termsof integrals over molecular velocity c. It is assumed that the incidentmolecules have a distribution function f that departs slightly fromthat of a Maxwellian distribution f0, and the reflected molecules areassumed to have a Maxwellian distribution at the wall temperatureTw. From c¼UþC, where C is the peculiar velocity, the generaldistribution function f is expressed as an expansion into Hermitepolynomials:

f ¼ f0½1þFðHÞ�, ð8Þ

where H¼ffiffiffiffiffiffi2b

pC is a normalised peculiar velocity and b¼

3=ð2C�CÞ ¼ 1=ð2RTÞ, where R is the gas constant. Following Grad’smethod, FðHÞ is approximated by a power series:

FðHÞ ¼ a�Hþ12 A��HHþ1

6A�3HHH, ð9Þ

where a, A, A are tensors of ranks 1, 2 and 3 respectively and ‘��’ and‘�3’ represent the double and triple inner products, respectively. Both

A and A are symmetric and two relationships between coefficientscan be deduced from constraints on mean values of H, i.e. trðAÞ ¼ 0;and 2aþA�

�I¼ 0. The coefficients a and A are determined such thatthe distribution function reproduces the corresponding basic vari-ables (averaged functions of H) as its moments:

a¼�1

pffiffiffiffiffiffiRTp q, A¼�

sp

, ð10Þ

where q is the heat-flux vector. The tensor A can be related to asuch that for the purpose of developing expressions at a boundary,the following relations are satisfied:

A�3nnn¼�6

5 a�n, S�ðA��nnÞ ¼�2

5S�a: ð11Þ

3.1. Mass conservation condition

The analysis of mass conservation arrives at the followingequation:

rw

r¼

ffiffiffiffiffiffiT

Tw

sð1þzÞ, ð12Þ

where

z¼1

2A��nn¼�

s��nn

2pð13Þ

is a ratio of the normal viscous stresses to pressure. The subscriptw denotes properties of the fluid with an equilibrium distributionat the wall temperature Tw and those with no subscript are thoseof the actual fluid at the wall surface.

Eq. (12) is not really itself a BC, but an expression for densityrw of diffusely reflected particles that is used in the derivation ofsubsequent BCs. The expression accounts for the difference inmolecular speeds of the incident and reflected particles, both interms of mean speed (due to different temperatures) and dis-tribution function.

3.2. Momentum conservation BCs

Conservation of momentum is split into tangential and normalmomentum components leading to equations for velocity slip andpressure jump respectively. The tangential momentum balanceequation produces [7,8]:

Uþ1

2ffiffiffiffiffiffi2b

pð1þzÞ

�2�ss

ffiffiffiffiffiffi2pp

S�ðA�nÞ�2

5S�a

� �¼Uw: ð14Þ

Substituting expressions for a and A from Eqs. (10), (14) becomes

Uþ

ffiffiffiffiffiffiffiffiffipRT

2

r1

pð1þzÞ2�ss

S�ðs�nÞþ1

5pð1þzÞS�q¼Uw: ð15Þ

Substituting into Eq. (15) the viscous stress tensor s¼ mrUþstr

where str represents the second and third terms (transpose andtrace respectively) on the right of Eq. (5), and applying the perfectgas law p¼ rRT, gives the slip condition for U:

UþaUS�ðn�rUÞ ¼Uw�a

mS�ðn�strÞ�

1

5pð1þzÞS�q, ð16Þ

where the coefficient

aU ¼1

1þz2�ss

l: ð17Þ

Eq. (16) follows the general form for a slip BC of Eq. (7) inwhich, in addition to the wall velocity Uw, the right hand sidecontains the term in str which is non-zero in the case of a curvedboundary (i.e. the curvature correction term discussed in Section 2

C.J. Greenshields, J.M. Reese / Progress in Aerospace Sciences 52 (2012) 80–87 83

and analysed previously [3]). The term in q is the thermal creepterm, which can be expressed in terms of T by substitutingexpressions for Fourier’s law of conduction, q¼�krT , the perfectgas law and Prandtl number Pr¼ mCp=k as follows:

�1

5pð1þzÞS�q¼

1

5ð1þzÞg

ðg�1ÞPr

mrT

S�rT : ð18Þ

The normal momentum balance results also in a jump condi-tion for pressure as follows:

p 1þ2zþ2�ss

4

5ffiffiffiffiffiffi2pp a�n

� �¼ pw: ð19Þ

Eq. (19) for pressure jump is usually ignored in rarefied gassimulations, and even when it does appear, its use is unclear.Gupta et al. [10], for example, state that ‘‘yusually, the equationfor pressure slip is not required as a boundary condition but isneeded to obtain the surface pressure’’. By ‘surface pressure’, theymust mean p rather than pw (which is the pressure of the fluidwith an equilibrium distribution at the wall temperature Tw); pw

must be calculated from Tw, an input to the temperature jump BC,and rw which can be calculated from Eq. (12).

Fig. 1. Mach 10 flow around a cylinder.

3.3. Energy conservation BC

The analysis of energy conservation arrives at the followingequation [7,8], in which it is assumed Uw ¼ 0:

bU�Uð1þzÞþ

ffiffiffib2

rU� �

2�ss

ffiffiffiffiffiffi2pp

A�nþA��nn

� �

þ2 1þ3

2zþ

1

2

ffiffiffiffip2

r2�aa a�n

� �¼ 2

bbw

� �3=2 rw

r : ð20Þ

Combining Eqs. (11), (14) and (20), and noting that U�v¼U�ðS�vÞ for any vector v since n�U¼ 0,

�bU�Uð1þzÞþ2 1þ3

2zþ

1

2

ffiffiffiffip2

r2�aa

a�n

� �¼ 2

Tw

Tð1þzÞ: ð21Þ

Substituting Eq. (12) gives

T 1þ1

2þ3z

ffiffiffiffip2

r2�aa a�n

� �¼ Tw

2þ2z2þ3z

� �1þ

1

4RTwU�U

� �: ð22Þ

Finally, introducing the expressions for Fourier’s law, perfectgas and Prandtl number, the equation can be written as

TþaTrnT ¼ Tw2þ2z2þ3z

� �1þ

1

4RTwU�U

� �, ð23Þ

where the coefficient

aT ¼1

2þ3z2�aa

gðg�1ÞPr

l: ð24Þ

3.4. Patterson’s temperature BC

Eq. (23) follows the general form for a slip condition of Eq. (7),and yet Patterson takes the analysis further, first assumingU�U54RT then rearranging Eq. (20) to give

T

Tw¼ 1�

1

2zþ

1

2

ffiffiffiffip2

r2�aa

a�n

� ��1þ

3

2zþ

1

2

ffiffiffiffip2

r2�aa

a�n

� �: ð25Þ

Patterson then assumes the denominator of the fraction on theright hand side is approximately unity, yielding

TþaTrnT ¼ Tw2�z

2

� �, ð26Þ

where the coefficient

aT ¼2�aa

g2ðg�1ÞPr

Tw

Tl: ð27Þ

4. Mach 10 flow around a cylinder

We consider the hypersonic cross-flow of argon over a two-dimensional cylinder of radius R¼152.4 mm (6 in) as shown inFig. 1. The selected test case is one used in a previous numericalstudy [11]. The freestream parameters are T1 ¼ 200 K, U1 ¼ð2624:1,0;0Þm=s, p1 ¼ 47 mPa. These flow parameters corre-spond approximately to Mach 10 and, taking the length scale asthe cylinder diameter in Eq. (1), Kn� 0:25. The cylinder wall isheld at a uniform temperature Tw ¼ 500 K.

Our simulations were undertaken using OpenFOAM [12], theopen source CFD toolbox. OpenFOAM includes solvers for con-tinuum CFD that use the finite volume method, and solvers thatuse discrete/particle methods such as DSMC, all operating on anunstructured mesh of polygonal cells of any shape. The dsmcFoamapplication [13] was used for DSMC simulations on the same testcase, using the variable hard sphere (VHS) model with tempera-ture exponent o¼ 0:734, reference temperature Tref ¼ 1000 K,molecular diameter 3.595�10�10 m and mass 66.3�10�27 kg.Particles that collide with wall surfaces are reflected diffusely, andthe velocity slip and temperature jump are calculated based onthe particles that strike the surface, according to relations used inthe DS2V code of Bird [14]. The simulation used 11.9�106 equiva-lent particles, each representing 5.3�1010 argon atoms. The meshconsisted of 36,000 cells whose smallest size was approximately2�2 mm. In this and subsequent simulations the number of cells(and the number of particles in DSMC) were set at a level whereresults fell within 1% of subsequent simulations on meshes withtwice the number of cells.

The rhoCentralFoam application [15] was used to solve theNavier–Stokes equations with non-equilibrium BCs for continuumCFD with s¼ a¼ 1, which are equivalent to the diffuse reflectionssimulated in the DSMC solver. A power law model was used forviscosity, equivalent to the VHS model [11], of the form m¼ mref

ðT=Tref Þo, with mref ¼ 50:7� 10�6 Pa s. The simulations used a mesh

of 28,000 cells that were graded from Dx,Dy� 4 mm in the far-fieldto Dx,Dy� 25 mm at the cylinder surface.

4.1. Preliminary work

Initial simulations were undertaken using the momentum BCof Eq. (16) combined with, in turn, the energy BCs of Eqs. (23) and(26). Initial simulations on this case suffered numerical instabil-ity, originating from the downstream side of the cylinder surface.The instability appears to be caused by the terms in z in the BCequations which, using the definition of n pointing into thedomain in Eq. (3), and assuming the flow is locally trace-free,

Fig. 3. Tangential velocity along line normal to cylinder surface at y¼ 901.

C.J. Greenshields, J.M. Reese / Progress in Aerospace Sciences 52 (2012) 80–8784

approximates to

z��mp

@un

@n: ð28Þ

It is clear that Eq. (12) is unphysical when zr�1 and somewhatimplausible when z41. In the previous work [7,8,10], it has beenassumed that 9z951, but in the case of rarefied, high speed flowsthis is not guaranteed. In this cylinder case, for example, 9@un=@n9is particularly high in the stagnation region and fairly high in theexpansion region downstream of the cylinder. In the stagnationregion, @un=@no0 so z40 and p is quite high so 9z9 is notexcessively high. However, in the downstream region @un=@n40and because, p is small, it is certainly possible that zr�1.

In our preliminary simulations on this case we evaluated z atthe downstream side of the cylinder surface along the geometrycentre line. Over a range of simulations, values in the range �20ozo�10 were measured. We therefore attributed the instabilityin the BCs to the unphysical nature of the equations when zr�1.To make the BCs physical, we first considered clipping z when itfell below some level. Clipping would need to occur before zreached �1, but there is no obvious choice of clipping ‘level’.Similarly, some clipping would probably be required beyond apositive level of z but, again, there is no obvious choice of level.Therefore, in the absence of any physical basis to choose clippinglevels, we decided to set z¼ 0. This setting was adopted for theremainder of the work presented in this paper. In addition, asPatterson’s temperature BC, Eq. (26), was derived using the assump-tion U�U54RT , we applied the same assumption to Eq. (23). Thisand the previous setting, z¼ 0, reduces both Eqs. (23) and (26) to

TþaTrnT ¼ Tw: ð29Þ

The ‘energy conservation BC’ of Eq. (23) thus reduces to a formof the Smoluchowski temperature BC in which

aT ¼2�aa

g2ðg�1ÞPr

l: ð30Þ

In the remainder of this paper we refer to Eq. (29) with thecoefficient aT in Eq. (30) as the ‘Smoluchowski BC’, and with thecoefficient aT in Eq. (27) (that includes an additional factor Tw=TÞ

as the ‘Patterson BC’.

4.2. Results

We ran CFD and DSMC simulations of Mach 10 flow around acylinder to steady-state, and analysed the results. Fig 2 shows thevelocity slip at the cylinder surface as a function of angle y aroundthe cylinder, starting from y¼ 01 at the stagnation point. Bearing

Fig. 2. Slip velocity around cylinder surface.

in mind that it is inevitable that 9U9-0 as y-01 and y-1801, thediscrepancy between CFD and DSMC is quite appreciable between301oyo1101, although the peak slip velocity compares quitewell. There is some difference in results using the Smoluchowskiand Patterson temperature BCs due to appreciable differences inpredicted temperature jump described later. The results with thePatterson temperature BC are marginally better.

Fig. 3 shows the tangential velocity profile at y¼ 901. Theprofile in the DSMC shows 9rU9 begin to increase when d=Ru1,reaching a level several times higher than in the CFD when d-0.Given such a difference in 9rU9 in this region, discrepancy incalculated slip between the two methods is likely.

Fig. 4 compares the fluid temperature along the cylinder wallin the DSMC and CFD simulations. The CFD with the Patterson’stemperature BC compares far better with the DSMC than with theSmoluchowski temperature jump. Between 01ZyZ1201, there isagreement in surface temperature between the results withPatterson’s BC and DSMC to within 710%. The SmoluchowskiBC predicts surface temperatures that are � 100% too high.

The inaccuracy of the Smoluchowski BC is quite surprising,particularly at the stagnation point ðy¼ 01Þ, where we estimatethe local Kn� 0:025 based on a characteristic dimension L¼2R.Fig. 5 shows the temperature profile along the geometry centreline between the stagnation point and a distance 2R upstream.Both CFD and DSMC predict similar normal temperature gradientsrnT as would probably be expected by the temperature jumpboundary condition models. However, while the Patterson BCgives good agreement near the surface, the Smoluchowski BC

Fig. 4. Temperature around cylinder surface.

Fig. 6. Temperature along line normal to cylinder surface at y¼ 901.

Fig. 8. Laminar flow over a flat plate.

Fig. 5. Temperature along line normal to cylinder surface at y¼ 01 (note: left is

upstream of cylinder; right is at the cylinder surface).

Fig. 7. Temperature along line normal to cylinder surface at y¼ 1801 (left cylinder

surface, right downstream of cylinder).

C.J. Greenshields, J.M. Reese / Progress in Aerospace Sciences 52 (2012) 80–87 85

does not. The difference in results further upstream ðd=R40:5Þ istypical, due to the inability of the NSF equations to replicate thediffuse behaviour exhibited in non-equilibrium flows.

At the highest point on the cylinder surface ðy¼ 901Þ, there isan increase in the estimated local Kn to approximately 0.25. AsFig. 6 shows, there is a larger discrepancy in rnT between theDSMC and CFD, and while the Patterson temperature BC gives

better surface temperature prediction, its superiority over theSmoluchowski BC is less strongly evident than before.

In the wake of the cylinder, along the geometry centre line, thedensity is extremely low, driving the local Kn beyond 10. The DSMCresults in Fig. 7 show a rapid change in rnT near the cylindersurface that the Navier–Stokes model clearly does not predict in theCFD. However, the Patterson temperature BC predicts a walltemperature that, combined with the local rnT gives a match in T

within 75% to the DSMC results further downstream. The Smolu-chowski BC, by comparison, produces a solution that is typically� 20% too high further downstream.

5. Mach 12.7 flow over a flat plate

We now consider a hypersonic flow of argon over a flat-plateat 01 angle of attack, as shown in Fig. 8. The selected test case is thatof a previous set of experiments [16]. The freestream parameters areT1 ¼ 64:5 K, U1 ¼ ð1893:7,0;0Þm=s, p1 ¼ 3:73 Pa. These flow para-meters correspond approximately to Mach 12.7 and we can define alocal Kn¼ l1=x where x is the distance from the plate tip andl1 ¼ 0:23 mm is the free stream mean free path for this case. Theplate surface is held uniform at Tw ¼ 292 K.

The rhoCentralFoam application was used as in the Mach 10cylinder case to solve to steady-state the Navier–Stokes equationswith non-equilibrium BCs using the models and parametersdescribed. Equivalent DSMC simulations were also conducted. TheDSMC simulation used 1.58�106 equivalent particles, each repre-senting 9.7�108 argon atoms. The mesh consisted of 75,600 squarecells, each approximately 0.17�0.17 mm. The CFD simulations useda mesh of 28,000 cells that were graded from Dy� 1:2 mm aty¼25 mm to Dy� 0:12 mm at the plate surface (y¼0 mm) andDx� 0:15 mm at the plate tip (x¼0 mm) to Dx� 1:5 mm at theoutlet (x¼55 mm).

Fig. 9 compares the fluid temperature along the plate surface.Near the plate tip, i.e. xo5 mm, the DSMC and CFD results areclearly different. In the CFD, the fluid does not see the plate until itreaches the tip, when the shear stress begins to develop, althoughfairly slowly due to high slip. The formation of a shock, and thecorresponding temperature jump across it, occurs slightly down-stream of the plate tip, yielding the result we see in Fig. 9 that T

increases from T1 to a level downstream of the shock over 1–2 mm.In the DSMC, however, the particles detect the presence of the

plate slightly before the tip because a proportion of particles,reflected diffusely from the surface close to the tip, travelupstream and collide with incoming particles. The shock isestablished � 0:2 mm ahead of the plate tip, with the resultingdeceleration in the flow causing a sharp rise in mean T of theparticles behind the shock. The temperature of particles at thesurface remains high until sufficient collisions have occurred withthe plate surface at Tw.

The discrepancy between DSMC and CFD for xo5 mm is perhapsnot surprising, given that this region corresponds to Kn40:05 as

Fig. 10. Temperature at cross section 25 mm from plate tip.Fig. 12. Velocity along surface of flat plate.

Fig. 11. Temperature at cross section 50 mm from plate tip.

Fig. 13. Tangential velocity at cross section 25 mm from plate tip.

Fig. 9. Temperature along surface of flat plate.

C.J. Greenshields, J.M. Reese / Progress in Aerospace Sciences 52 (2012) 80–8786

defined above. For x45 mm the results are better, with thePatterson temperature BC giving better predictions of fluid tem-perature at the surface than the Smoluchowski BC. Fig. 10 shows theprofiles in T at x¼25 mm, from the plate surface through the viscousboundary layer to the edge of the shock. At the surface the PattersonBC predicts T to within 7%, whereas the Smoluchowski BC over-predicts T by � 18%. The DSMC predicts a rise in T beyond 1000 K inthe region 1–2 mm from the wall, which subsequently falls back to� 600 K at a wall distance of 3 mm. Beyond that, the comparisonbetween the predicted temperatures from DSMC and CFD with thePatterson BC is very good, with DSMC exhibiting a more diffuseshock as expected. The over-prediction of surface temperature in theSmoluchowski BC propagates across the entire temperature profile.

Fig. 11 shows similar profiles in T at x¼50 mm. Here thePatterson temperature BC predicts T at the surface to within 4% ofthe DSMC results, whereas the Smoluchowski BC over-predicts T by� 16%. Again the DSMC exhibits a rapid rise in T beyond 1000 Knear the wall, but beyond a wall distance of 6 mm the comparisonbetween DSMC and CFD with the Patterson BC is excellent. Again,the over-prediction of surface temperature in Smoluchowski BCpropagates across the entire temperature profile.

Fig. 12 compares the predictions of velocity slip along the platesurface between the DSMC and CFD simulations. When the flowmeets the plate tip, gas particles begin to collide with the walland the mean velocity falls. Beyond a distance of approximately10 mm from the tip, the slip velocity stops decreasing rapidlywhen the rate of decrease of mean tangential velocity of particles

due to surface collisions is matched by a similar rate of increase ofmean tangential velocity by collisions from high-speed particlesin the bulk. The main discrepancy between results from DSMCand CFD is in the level at which the velocity levels out, the slipvelocity from DSMC is approximately 50% of that from the CFD.

Fig. 13 shows the tangential velocity profiles at x¼25 mm. Thecomparison between CFD and DSMC is reasonable, with the profile

C.J. Greenshields, J.M. Reese / Progress in Aerospace Sciences 52 (2012) 80–87 87

using the Patterson temperature BC providing a closer match to theDSMC results than that using the Smoluchowski BC. The results areconsistent with Fig. 3 for the Mach 10 cylinder case, in which theDSMC predicts a steeper velocity gradient near the wall and a lowerslip velocity at the surface.

6. Discussion

In this paper we have investigated non-equilibrium fluiddynamic BCs first by revisiting Patterson’s derivation of modelequations based on the moment method. We note first that theresulting equation for velocity slip reinforces previous arguments[3] that the conventional form of Maxwell’s equation omits a termrelating to surface curvature. Second, additional modifications tothe BCs appear due to changes in the particle distribution functionbetween incident and reflected particles, which appear in theequations through terms in z. For the Mach 10 cylinder casestudied here, the modifications are unphysical since calculatedvalues of z fall within a range that causes unboundedness (negativer or T) in the region near the cylinder surface at y¼ 1801. However,z¼ zðs,pÞ and s is calculated using the linear relationship of Eq. (5),which is inapplicable in this region of flow. It is possible that theinclusion of higher order terms for the stress will modify the flowitself and the calculation of z so that unboundedness does not occur.

In the simulations presented in this paper we ignored themodifications relating to z by setting z¼ 0 in all equations. OurCFD simulations used a single equation for the velocity slip BC buttwo forms of equation were tested for the temperature jump BC.Perhaps surprisingly, Patterson’s BC gave consistently betteragreement with results from the DSMC comparison cases thanthe Smoluchowski BC did. The difference lies in an additionalTw=T in the respective model coefficients aT of Eqs. (27) and (30).This originates from Patterson’s assumption that the denominatorof the fraction on the right hand side of Eq. (25) is approximatelyunity. This assumption relies on 9z951 and 9a�n951, which is nottrue in this case. However, Patterson’s analysis begins with theparticle distribution function in Eq. (8) which represents a smalldeviation from a Maxwellian distribution and is only valid if thecoefficients a,A,A, and consequently z, are small compared tounity. In other words, the non-equilibrium BCs become invalidbeyond a certain Kn, in the same way that the NSF equationsthemselves become invalid. Fig. 4 certainly seems to confirm this,in which the Smoluchowski BC hugely over-predicts the tem-perature jump, to the extent that using an isothermal ðT ¼ TwÞ

condition would arguably be better.One argument why Patterson’s temperature BC generally out-

performs the conventional Smoluchowski BC in the benchmarksimulations presented in this paper is that the assumption used inarriving at Patterson’s equation, based on a validity constraint ofthe underlying theory, is effectively applying an additional modelconstraint for cases where the BC is used outside its normal rangeof applicability. The simulations performed here indicate stronglythat the Smoluchoswki BC increasingly over-predicts temperaturejump with increasing Kn, so scaling the model coefficient by Tw=T

at least offers a heuristic solution that tends towards the originalSmoluchowski condition in the limit T-Tw.

Rarefied high-speed flows of practical interest exhibit differentlevels of thermodynamic non-equilibrium behaviour across theflow domain as the examples in this paper show. If these flows aresolved using continuum CFD only, then it is beneficial that thegoverning equations and BCs do not, at least, produce unphysicalor wildly inaccurate solutions within the more strongly non-equilibrium regions of flow for which the equations are notstrictly applicable. For a temperature BC, Eq. (26) seems to offeran improvement over the conventional Smoluchowski BC. Furtherwork is needed to confirm this and there is perhaps also scope forexploring similar improvements to the velocity slip BC.

Acknowledgements

This research was supported by the UK’s Engineering andPhysical Sciences Research Council under Grants EP/F014155/01and EP/F005954/1. OPENFOAM is a registered trademark of SGICorp. The authors thank Craig White at the University of Strath-clyde for providing data from his DSMC simulations of theMach 10 cylinder case, and the reviewers of this paper for theircomments.

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