VOL 2, NO 6, JUNE 1964 AIAA JOURNAL 1047

A Model for the Transition Regime in HypersonicRarefied Gasdynamics

BERNARD B HAMEL*General Electric Company, Valley Forge, Pa

A two-fluid model for the hypervelocity rarefied regime is presented in this work Incontradistinction to the near free molecular theories, which are valid for Kn > 10, the modelpresented is valid for Kn > 1 To construct the model, an a priori choice of fluids is madeThose particles reflected from the surface of the body are considered as the "wall" fluid,whereas all scattered and freestream particles comprise the "cold" fluid; the assumption isthat collisions between "wall" and "cold" particles convert "wall" particles to "cold" par-ticles Assuming hard sphere molecules, kinetic models are then constructed for both the"wall" and "cold" fluids Utilizing the kinetic models to compute the collisional moments,a moment method is then employed To close the moment equations for the "wall" fluid,the free molecular distribution function with the number density as state-variable is em-ployed For the "cold" fluid the distribution function is expressed as an expansion in thehigher derivatives of the delta function, the expansion parameter being the Mach number ofthe "cold" fluid The moment equations are then numerically analyzed for the one-dimen-sional hypersonic compression problem, and numerical results for the relevant quantitiesare presented

Nomenclature

a — diameter of hard sphere moleculesB = proportional to the inverse Knudsen number12

C = peculiar velocity (v — Vi)f = distribution functionQIJ = relative velocity between particles i and j\ = idemfactor (in Cartesian notation: 5#)k = Boltzmann s constantKn — Knudsen numberMoo = speed ratioMH i = moments of the charge distributionm = molecular massn — number densityNz = normalized number density, Eq (45)n2(0) = normalized free molecule densityn = unit vector normal to a surfacep = scalar pressureQ = heat flux tensorr = position coordinateSij i = moment of the distribution functionT = temperaturet = timev = molecular velocityV = macroscopic velocityF2

(0) = normalized free molecular velocity, Eq (4 5)x = position coordinatea. = thermal accommodation coefficientAtf = collisional transfer due to collisions between i and j

particlesAI = cosine of an angle, Eq (26c): M = v (Vi — V^)/

v\ |V, - V,|Vij = collision frequency between particle i and j11 = stress tensorp — charge distribution12 = solid angler = equation of characteristics$ = electrostatic potential\f/ = function of molecular velocity

Presented as Preprint 63-438 at the AIAA Conference onPhysics of Entry into Planetary Atmospheres, Cambridge,Mass , August 26-28, 1963; revision received April 1, 1964This paper is based on work performed under the auspices of theU S Air Force Ballistic Missile Division, Contract AF 04(694)-222 An acknowledgment is due S M Scala for stimulating theauthor's interest in this problem and for providing support andencouragement throughout the course of this work

* Research Engineer High Altitude Aerodynamics, SpaceSciences I aboratory Member AIAA

SubscriptsI2fssbw

CO

= cold fluid= wall fluid= freestream= scattered= body= wall= infinity

Superscripts

Differentials

(D)T

d/dx

gradient operator (vector notation)gradient operator (Cartesian tensor notation)

= transpose of the D matrixordinary spatial derivativepartial spatial derivativestreaming derivative: (d/d£) +

I Introduction

ONE of the fundamental problems in hypervelocityrarefied gasdynamics is a quantitative description of

the transition between free molecule and continuum flowThe problem of providing such a description entails solvingthe Boltzmann transport equation subject to initial valuesand boundary conditions This is, indeed, a difficult prob-lem, and thus far analyses have predominantly dealt withsimple linearized flows (Couette flow1 2 and Rayleigh'sproblem3)

Theoretical analysis of nonlinear flows has, in general, notproceeded as successfully as for the linear case The onlynonlinear problem that has received exhaustive treatmentis shock structure in a monatomic gas (an excellent reviewof this problem is given by Talbot4) The work on shockstructure has, for the most part, followed the ideas of Mott-Smith,5 who considered the shock layer as the mixing zonefor two Maxwellian streams These ideas have been elabo-rated by various investigators6"8 and have resulted in thepostulation of a two-fluid model6 for the shock structureproblem

The analyses of more general nonlinear flows have beeninfluenced, to a large degree, by the theoretical models em-ployed in the shock structure problem Although these

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1048 B B HAMEL AIAA JOURNAL

C COLD PARTICLEW WALL PARTICLESC SCATTERED COLD PARTICLE

C AND SC PARTICLES COMPRISE THE COLD FLUIDW PARTICLES COMPRISE THE WALL FLUID

Fig 1 Classification of particles

assumptions will become progressively invalid when themodels are based on sound physical ideas and give reasonableresults for the shock structure problem, in their application tomore complicated flows one encounters serious difficulties

Lees9 has, for example, suggested a generalization of theMott-Smith bimodal representation of the distributionfunction This approach, although successful for com-pressible Couette flow, results in a set of intractable differen-tial equations for such flows of interest as the shock structureproblem4 Several investigators,10 u have also proposedvariations of the multifluid, shock structure models for usein more general nonlinear, rarefied flows Although a multi-fluid theory is intuitively appealing for the nonlinear regime,the investigations to date do not present a consistent formula-tion nor are they applied to a really representative problemThe model proposed by Rott and Whittenbury10 assumestwo fluids, a hyperthermal, "freestream" fluid and a "scat-tered fluid " In the model, the "freestream" fluid can beconveniently represented as a molecular beam; however, theassumption that the "scattered" fluid is representable by aMaxwellian distribution function is, as pointed out by theauthors, quite arbitrary Lubonski's model,11 on the otherhand, divides the gas into thiee classes of particles: "freestream/' "reflected from the boundary/' and "scattered "Although such a classification is sensible, Ref 11 containslittle discussion on how one handles each fluid, and in factthe treatment of hypervelocity Couette flow that is givenrestricts considerations to the near free molecular regimeTo summarize, one can state that although the applicationof the ideas used in the study of wave structure to moregeneral problems is certainly worthwhile, there is as yet nosatisfactory extension of these ideas

In this work a new, two-fluid model for the hypervelocityrarefied regime is presented This work avoids many of theshoitcomings of the previous multifluid models and resultsin a set of partial differential moment equations that are of thesame order of difficulty as the conventional gasdynamicequations Although the model does not provide a descrip-tion of the entire transition between the free-molecule andcontinuum regimes, it does yield acceptable results into thetransition regime, the consistency of the simplifying assump-tions being ascertainable from the numerical results Themodel equations are analyzed for the case of one-dimensionalhypersonic compression (piston problem), which is a first-order approximation to the problem of incipient shock waveformation, during the early phase of the entry of a spacevehicle into a planetary atmosphere

II Kinetic ModelIn this section the a priori choice of fluids will be discussed,

and a kinetic model for each fluid will be formulated Itshould be pointed out here that by a kinetic model we refer

to the replacement of the collision integrals, for each classof particles, by approximate expressions

In order to make an a priori choice of fluids, it will be ofinterest to examine some salient features of hyperthermal,rarefied flows past isolated bodies When the Knudsennumber (ratio of appropriate mean free path to typical bodydimension) is much laiger than unity, collisions between gasmolecules can be neglected compared to collisions betweengas molecules and the body surface, and the body is said tobe in the free-molecule flow regime As the Knudsen numberbecomes smaller, collisions between incident freestream partides and those reflected from the surface become impor-tant (Fig 1) On the average, the scattered particlesthat result from these collisions have an ordered veloc-ity that is still larger than their thermal velocity (e g ,they are still "supersonic") Therefore, if one considersfreestream particles and "scattered" particles as an aggregatefluid, this fluid (referred to as the "cold" fluid) in the hyper-velocity rarefied regime will have a macroscopic velocitythat remains large compared to its thermal spread As thisfluid becomes populated with scattered particles, its macro-scopic velocity decreases and thermal energy increases It isthis thermalization of the freestream and scattered particles("cold" fluid) which gives rise to shock formation and thetransition to the continuum regime The particles that arereflected from the surface ("wall" fluid) will, on the otherhand, be depopulated by collisions with the "cold" fluid and,in the continuum regime, will be depleted within a shortdistance from the body (of the order of a mean free path)

The a priori choice of fluids made here is thus a "wall" fluid,encompassing those particles that are reflected from the sur-face and a "cold" fluid that includes all freestream andscattered particles It will be seen that one has a rathersimple description of the "wall" fluid, since it is the attenua-tion of the "wall" fluid density that is of most interest and notthe details of the distribution function for these particlesHowever, in describing the thermalization of the "cold"fluid, more detailed information is required because it is thisthermalization that is most important in shock formation andin the dynamics of the transition regime

One can now write the analytic form of the kinetic modelsthat are chosen For the wall fluid (subscript 2) one writes

= -^12/2 (2 la)

and for the cold fluid (subscript 1):

5>jfi/£tf = -vzifi + P2i (2 lb)

the collision frequency v^ can be written as

Vij = (a2/2) / Qij'ftfyj - QH) X

and as

P21 = (a

X [5(021 -

(2 3)where, for simplicity, the hard sphere inter molecular potentialhas been assumed in writing (2 2) and (2 3) The assump-tions implicit in writing (2 la-2 3) are that 1) all scatteredparticles become part of the "cold" fluid; and 2) "self-"collisions are neglected (e g , "cold-cold," "wall-wall"collisions)

A later discussion of the boundary conditions will make clearthat, within the present two fluid formalism, complete con-version of "cold" fluid to "wall" fluid takes place at thesurface It is expected that for hyperthermal flow these

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JUNE 1964 TRANSITION REGIME IN HYPERSONIC RAREFIED GASDYNAMICS 1049

local speed ratio of the "cold" fluid approaches 1 in the flowfield

It should be pointed out that, by combining (2 la) and(2 Ib), one can recover the complete Boltzmann equationfor / = /i + /2 Therefore, independent of the choice of /iand /2 used in the calculation of va and Pi2, the model willconserve mass, momentum, and energy; however, becauseof the neglect of self-collisions ("wall-wall" and "cold-cold") the model will not correctly predict the relaxation toa Maxwellian distribution

To calculate va and P2i, one must choose approximatingfoims foi the distribution functions of the "wall" and "cold"fluids For the "wall" fluid, with the assumption of a coldwall, Vi2(2kTw/m) ^> 1, one can characterize the "wall"paitides by a single velocity V2, where V2 can be calculatedfrom the free-molecule solution for the wall fluid:

/2 = nj (v - V2) (24)Equation (2 4) is equivalent to the treatment of the "wall"particles given by Baker and Charwat12 It should be notedthat, although the single group approximation of (2 4) isutilized in calculating the collision integrals, the thermalvelocities of the "wall" particles are accounted for in thedrift terms of the Boltzmann equation [see Eq (3 3) ] Theassumption implicit here is that the thermal velocities ofthe "wall" particles are unimportant in considering collisionsbetween "cold" and "wall" particles, but are of significancein the prediction of the density profile of the "wall" particlesin the neighborhood of the surface (e g , the dispersal of"wall" particles from the surface)

For the "cold" fluid, as has been noted earlier, the effect ofcollisions is to thermalize the freestream particles, decreasingthe macroscopic velocity Vi and increasing the temperatureTI It will, therefore, be appropriate to choose a form for/i which takes account of this thermalization lo achievethis, /i is represented by the following expansion in thederivatives of the delta function:

+ (25)

where Ci = v — V:, and the coefficients of the expansion arethe moments of the distribution function /i The expansiongiven in (2 5) was first utilized by Obermann13 in an attemptto obtain first-order thermal corrections to the problem ofcold plasma oscillations; however, its use in neutral, rarefiedgasdynamics is new

It is important to note that the use of delta functions inhyperthermal rarefied gasdynamics was begun by Enoch 14

Enoch employed the delta functions to perform Knudseniteration in the hyperthermal, near free molecular regimeReferring to the mathematical appendix of Ref 14, one canperform the integrations indicated in (2 2) and (2 3) Sub-stituting (2 4) for /2 and (2 5) for /i (retaining only the firsttwo teims) in Eqs (22) and (23), and performing theintegrations, one obtains

v2 — 1 + —————iT~2 xn\ v2 — Vi 2

(v2 — Vi)fc(v2 —

— V2

(26a)

(26b)

Pa =i — V2|-V, X

- V2)y

' f + " V i - v 2 | \ f l|Vi -V 2

[l+sgn(7i- |v!-l

- V2) - S - Vi + V2| - IS + Vi - V2

-^ ) xd2

X

(26c)

Equations (2 6a-2 6c, 2 la, and 2 Ib) constitute the kineticmodels for "wall" and "cold" fluids for a hard spheie inter-molecular intei action

It should be pointed out that the key assumption used inobtaining the model equations is that Vi/(2kTi/m)1/2 > 1,or that the "cold" fluid be "supersonic " The neglect ofself-collisions ("cold-cold," "wall-wall" collisions) and thetruncation of the moment expansion for /i become question-able assumptions when Vi/(2kTi/m)112 approaches unityAlthough it is difficult to express Vi/(2kTi/m)l/2 directly interms of the Knudsen number, some estimates are given inAppendix A of Ref 15 These seem to indicate that forKnw =: 3 the approximation is still valid Judging from theestimates given in Ref 15, one can conjecture that the modelequations derived here are valid for Knw > 1 This estimateseems to be borne out by the numerical lesults discussed inSec 4

It is important to point out that, within its range of validity,it can be expected that the kinetic model will account for both"freestream-wall" particle collisions, and scattered-wallparticle collisions This is because retention of the stresstensor in Eq (2 5) in the calculation of v\i and P2i takesaccount of the scattered particles present in the "cold"fluid; in this way, one approximately accounts for the effectof both "scattered-wall" collisions and "freestream-wall"collisions

III Moment MethodThe a piiori separation of particles into "cold" and "wall"

fluids and the development of kinetic models for each fluidprovides a framework for the development of a momentmethod for the rarefied hypervelocity regime In examiningthe dynamics of the "cold" fluid, we note that far fiom thebody the "cold" fluid is made up of hypei velocity freestreamparticles As one approaches the body, the "cold" fluidbegins to be populated with scattered particles (lesultingfrom "cold-wall" intermolecular collisions) and its pressureand temperature begin to increase, i e , it begins to thermalizeIt therefore seems appropiiate to choose a form for /i thatcan account for this thei malization and allow a self-consistenttruncation of the moment equations It will be seen herethat the delta function expansion of (2 5) has these pioper-ties This can be shown by introducing the following non-dimensionalization :

(31)

where V^ is the incident freestream velocity and Ci is thethermal velocity of the "cold" fluid With the nondimen-sionalization of (3 1), one can write (2 5) as

,o ON(32)

By examining Eq (3 2) , one can note that the expansionparameter for the "cold" fluid is the inverse speed ratio:(2kTl/m)ll2/Vm, so that when the "cold" fluid is super-sonic, one can tiuncate the "cold" fluid moment expansionby neglecting terms of highei order than the stress tensoi inEq (3 2) This tiuncation is equivalent to a neglect of thegradient of the heat flux tensor in the equation of motion forthe stress tensor compared to the other terms, thereby closingthe set of moment equations for the "cold" fluid In thislimit, one therefore obtains a hydrodynamic description ofthe "cold" fluid

For the "wall" fluid, it is expedient to adopt a much simplerdescription than for the cold fluid This is because one is, for

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1050 B B HAMEL AIAA JOURNAL

the most part, interested only in the attenuation of the "wall"fluid density as a function of distance from the body Thefluxes of energy and momentum to the body will generallybe controlled by the "cold" fluid, and so one is mainly inter-ested in the number density n2, which interacts with the"cold" fluid through collisions To this end we choose for/2

be written as:

n2(r,t)exp(" 2T0

(33)/2 = 0

where 12& is the solid angle that the body subtends at the pointr As pointed out earlier, Eq (3 3) accounts for the thermalvelocity of the "wall" particles, whereas Eq (2 4) does notBecause Eq (3 3) is the free molecular distribution functionfor /2, with the density n2 an undetermined function of timeand position, Eq (3 3) will reduce to the correct free-moleculesolution in the limit of Knudsen number -> °o The rationalefor the choice of only one free variable in (3 3) is that, forViz/(2kTw/mi) » 1, "cold-wall" collisions do not have alarge effect on the form of the distribution function of the"wall" particles One theiefore considers the main effectof "cold-wall" collisions to be the attenuation of the "wall"fluid density

The moment equations for the "cold" fluid can now bewritten by taking the moments of (2 1):

/ (»/i/»Ofc(v)d«i; = JViW [Pa - Wi]#»i (3 4)where ifr(v) = 1, v, (v - Vi)<(v - Vi),- For the wall fluid,one need only take the first moment \f/ = 1 to obtain an equa-tion forn2:

(35)

Using the approximating forms for jfi and /2 given in (3 2)and (3 3), one can easily compute the left-hand sides of Eqs(3 4) and (3 5):

j/di) + V wiVi = A12(l) (3 6a)ni(Vi V)Vi + V H = A12(v) (3 6b)

+ V (VJI) + (Hi V)Vi +(H! VVOT = A12(dd) (3 6c)

V2(0) VN2 = [AaOOlN^]-1 (36d)

where n2 = n2(0) AT2, 7?2

(0) being the free-molecule numberdensity for the "wall" fluid, and

= fJ fi = fi&

(36e)

where 72(0) corresponds to the macroscopic velocity of the

wall fluid in the free-molecule regime The computation ofthe right-hand sides of (3 4) and (3 5) will involve the use ofthe approximating forms given for jfi and /2 In Sec 4 thecalculation of the collisional moments is considered in moredetail

With the choice of fluids given here, the boundary condi-tions for isolated body flow conveniently fit into the two-fluidformalism The choice of Eq (3 3) to represent the "wall"fluid distribution function will be suitable for satisfying theboundary conditions at a surface with diffuse reflection anda thermal accommodation coefficient of unity By a suit-able alteration of Eq (3 3) one can, however, account formore realistic surface conditions; an investigation of theboundary conditions implied by a < 1 0 is currently beingpursued To complete the determination of the surfaceboundary conditions, one must impose the condition thatthere be no net mass transfer normal to the surface; this can

n) (37)

Far from the body, the distribution function must becomea Maxwellian This can easily be satisfied by the formschosen for jfi and /2 It is seen that far from body, /2 ->- 0and fi will represent the freestream distribution functionIf Fco2 ^> 2fc77oo/m, the freestream Maxwellian distribution isrepresentable by the form nmd(v — Vm) + £>co5(2)(v — Vco)Therefoie, the form chosen for/ i [Eq (32)] satisfies theboundary conditions at infinity, provided «i, Vi, and IIi are

Vl = Voo (38)

It is of interest now to discuss the assumptions and limita-tions inherent in the use of the moment approximation pio-posed here To obtain the moment approximation, we havefirst derived an approximate form of the collision integral(Sec 2) and then proposed a method for truncating themoment equations and calculating the collisional momentsThe critical assumption necessary for both the approxima-tion of the collision integrals and the truncation of the mo-ment equations is that the "cold" fluid_be supersonic (Fi/Ci > 1) and that the wall be cold (Vi/Cw » 1) From theestimates of Ref 15 and the numerical results of Sec 4, itwould seem that the assumption Vi/Ci > I breaks downwhen the lelevant Knudsen number becomes of order unityThe present theory is therefore seen to be valid into the transi-tion regime (e g , Knw of order unity) in contradistinction tothe near free molecular flow theories that break down whenthe relevant Knudsen number is of order 10 12 14 16 Itshould be pointed out that the near free molecular theoriesbreak down when the number densities of freestream andwall particles departs significantly from their free molecularvalues, whereas the present theory fails when the "cold"fluid speed ratio MI departs significantly from its freestreamvalue The important difference here is that a collision ofa freestream particle with a wall particle gives a more signifi-cant perturbation to the freestream number density than tothe "cold" fluid speed ratio This is because (see AppendixA, Ref 15) the two scattered particles that result are still"supersonic," and therefore do not produce as significanta perturbation on the "cold" fluid speed ratio as does theloss of a particle on the freestream number density

The theory, additionally, would appear to be tractable sinceit yields a set of differential moment equations that are quiteanalogous to those of continuum gasdynamics, and allows oneto simply satisfy the necessary boundary conditions

IV One-Dimensional Hypervelocity Compression

A Equations and Boundary Conditions

To illustrate the applicability of the methods of Sees 2 and3 to a realistic problem of hypervelocity rarefied gasdynamics,the one-dimensional hypervelocity compression (piston prob-lem) will be analyzed This problem is not only of value indemonstrating the applicability of the method; it also rep-resents a first-order approximation to the problem of incipientshock wave formation during the early phase of the entry ofa space vehicle into a planetary atmosphere

The problem we consider here is that of a piston that isimpulsively brought to hypersonic velocity at i = 0 In thecoordinate system of the piston, the gas molecules can be con-sidered to have a hypersonic velocity [Vm

z/(2kTm/m) ^> 1],and the plate is stationary at x = 0 It is assumed that theplate has an accommodation coefficient of unity, all incidentparticles being reflected with a Maxwellian distribution at theplate temperatures It is further postulated that the molec-ular interaction is that of hard spheres

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JUNE 1964 TRANSITION REGIME IN HYPERSONIC RAREFIED GASDYNAMICS 1051

The equations of motion of the "cold" fluid are obtainedby writing the moment expansion of Eq (31):

Hi 5(vy)5(Cx) 5(v (4

It should be noted that in Eq (4 1) only terms of the order of(2&7ym)/7o>2 are kept, this being equivalent to the neglectof the heat flux tensor compared to the stress tensor in Eq(3 6c) In addition, it should be pointed out, that in the one-dimensional compression problem (just as for the problemof shock wave structure), the nonsymmetric terms of thestress tensor need not be considered The kinetic modelfor the "cold" fluid (Sec 2) can be written to lowest order in

(42)

where

Pzi = 2

To calculate the collisional moments for the "cold" fluid,we employ Eq (3 2) for /i, and only keep the lowest-orderterms in (2kT1/m)/VJ

We now take the following moments of (42): ^ = 1,,va, (vx — 7i)2 and obtain, after calculation,

i .+

+ 7i

cte2

(43a)

(43b)

(43c)

In (4 3b) it should be noted that, to lowest order in (2kTw/m)/Vi2, collisions make no contribution to the right-handside of the "cold" fluid momentum equation; in this equa-tion, collisions manifest themselves through the stress tensorIiixx It should be noted further that r&i, 7i, and HI-,* canbe calculated independently of IIi^ and HI ; solutionsof the moment equations for II^ and IIi can then be ob-tained with the aid of the solutions for (4 3a-4 3c) Onecan therefore consider (4 3c), in a sense, as the statement ofenergy conservation It should also be pointed out thatin the calculation of the collisional moments, for simplicity,only the lowest-order terms in (2kTi/m)/Vi2 have been re-tained This assumption leads to a simplification in theanalysis and is not unreasonable in the rarefied regime

For the wall particles, as discussed in Sec 3, we assume

/2 =m

\2TrkT- )3/2' e-mv2/2kTw for vx > x/t

/2 = 0 for vx < x/t

Taking the moment ^ = 1 of Eq (2 1), we obtain

* + 7,00

(44)

(45)

where

72«» = fL

= Nz(x,t) erfcf

exp {- Vm)1'2]2}!Vm)i/2] J

Here again only the lowest order terms in kTi/Vi* are em-ployed in the calculation of the collisional moment Withthe following nondimensionalization, one writes

(46)

The governing moment equations can then be written asfollows:

I?"

2(7T)1

X

Tvr +2 7(

20)

j(7T)1/2 Mrc

where n2(°) = &Ac[(x'/t)Mm] and Mco2 =It should be re-emphasized here that, although the first-order[in(2&7Vm)/7co2] thermal correction is included in the trans-port terms, the collisional terms have been calculated usingthe lowest-order terms in 2kTi/mVm

2

In formulating the initial and boundary value problemfor the preceding equations, we assume the piston to be at aconstant temperature Tw, t > 0 For the isothermal pistonone therefore has the following boundary and initial valueproblem:

x > 0, t < 0 V 1 7i' = -1= 2

x = 0, t > 0 = 2n1'|7i'|

(48)

(49)The initial conditions are thus seen to represent a hyper-velocity streaming toward the plate with the "cold" fluidhaving ni = nm, Uixx = pm\, 7i = Vm, and the "wall" fluidbeing characterized by n2 = 0, since for t < 0, n2

(0) = 0 Atthe surface, for t > 0, Eq (4 9) represents an expression ofthe conservation of mass It should be pointed out here thatthe assumption Tw/Tm = 1 is implicit in the nondimension-alization of (4 9)

B Solutions

In examining the governing equations (4 7) one can notea striking similarity to the conventional gasdynamic equa-tions The equations of motion for the "cold" fluid (4 7) re-semble the one-dimensional, inviscid gasdynamic equationswith sources of mass, momentum, and energy and an aniso-tropic pressure [note factor of three modifying the com-pression term 3IIiXJB'(d7i'/daO] This analogy suggests theuse of the method of characteristics To use this technique,we first solve the characteristic determinant for the realcharacteristics of the system [Eq (4 7) ] The roots of thedeterminant are then used to find the differential equationsalong the characteristic curves, and a numerical scheme isformulated for solving the initial and boundary value by themethod of characteristics:

vVif - T

01/V

0 (410)

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where

OOt1 t[Tra2no0V00Mco]

-M , AT X 0

8 12 16 20 24 28 32t 1

Fig 2 "Cold" fluid speed i atio vs time at the plate (x = 0)

Equation (4 10) has four real roots, so that the system ofequations is hyperbolic These roots are

Tl =r 2 = T Y (411)

T3,4 = TV ± (Sni'/IIi**)1'2

From Eq (411), one can easily find the differential equationsalong the characteristic curves; these are

X

n/2 (412)

25

23

21

19

17

15

-H- 13n,[oo

02 04 06 08 I 12 14 16 18X 1

Fig 3 Total number density vs position

Ma dz'

' da;'

dr*

(413)

JOne can now integrate Eqs (4 12) using a characteristic netin the x-t plane Using a 7094 IBM computer, these calcu-lations have been performed for 3 < t'] for latter times, thecharacteristics begin to bunch, and the usual techniquesbecome unsuitable The results obtained are discussed inSec 4C

C Results

In Sees 4A and 4B, a discussion is given of the governingequations, boundary conditions, and method of solution forthe hypervelocity compression problem In this section,the numerical results are discussed, and some critical com-ments are made about the relation of the present work to theexisting theories for the hyperthermal rarefied regime

Before giving a general discussion of the numerical results,it is important to assess the range of Knudsen number forwhich these results are valid For the hypervelocity com-pression problem, no characteristic dimension exists whichcan be utilized in the definition of a Knudsen number; onemust, therefore, define the Knudsen number as the ratio of acharacteristic collision time to the elapsed time In most.rarefied gasdynamics problems one is faced with the taskof choosing between a number of different characteristiccollision times or mean free paths As pointed out by Prob-stein17 for hyperthermal flow, care must be exercised in choos-ing the appropriate mean free path and collision time

If one considers the particles reflected from the wall, thecollision frequency vw / for a reflected particle with a free-stream particle can be approximated as:

vw f s = 7ra2nfsVm (4 14)

Although for the incident freestream particles one approxi-mates Vf w (the collision frequency of a freestream particlewith reflected particles) as

vfsjw = ira^Foo (4 15)at the surface of the body,17 one can estimate nw as

nw = 4(700/0.)% (4 16)

Fig 4 "Cold" fluid speed ratio vs position

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JUNE 1964 TRANSITION REGIME IN HYPERSONIC RAREFIED GASDYNAMICS 1053

Therefore, in choosing a characteristic collision time for thisproblem, it would appear that [vf w}~1 would be the smallest,and hence the "critical" collision time for changes to takeplace in the freestream number density It should be noted,however, that the longer collision time [vw,f ]~l will be con-trolling time scale for the particles reflected from the surfaceand therefore foi the total fluid

With reference to the model being considered here, the"cold" fluid will experience important changes in a time ofthe order of [v/ «,]"*, whereas the "wall" fluid will changein the time [vw / ]~* Since the validity of the model de-pends on Vi/Ci > 1, from the discussion in Sec 3 it can beexpected that the model will break down when the Knudsennumber [tvf w]~l becomes of order unity For the com-pression problem, we define an effective thermal velocity as

- F M-WITH COLLISIONS

ttlxx = (417)and an effective speed ratio

Mi = Vi/Ci (4 18)In Fig 2, MI is plotted vs [fo>/ «,]"' at the plate surface(x = 0) From Fig 2 one can note that the discussion givenin Sees 3 and 4 regarding the range of validity of the modelis verified, namely, that Mi becomes of order unity when thecharacteristic Knudsen number for the "cold" fluid becomesof order one

Since the freestream number density undergoes significantchanges in a time scale of order bv >w]~"S it can be expectedthat the near free molecular flow theories for this problemwill remain valid up to times of order 0 l[v/ w}~1 There-fore, based on the Knudsen number [tv/ w]~~l, it wouldappear that the present theory remains valid into the transi-tion regime

Since the "wall" fluid changes in a time scale of order[vw,f ]~l and contains the predominate number of particles,a developed shock wave can be expected on the time scale[vw / ] ~l, and so a developed shock does not appear in thepresent result One can, however, gain insight to the phe-nomena associated with the earliest phases of shock forma-tion from the results presented heie

V Compression

One can note from the results of the compression problemthat two tendencies seem to be present First, one can ob-serve that the "cold" wall causes the total density (anddensity of reflected particles) to increase rapidly at the sur-face (x = 0) This can be seen from Fig 3 Secondly, the"cold" fluid begins to develop a shock-like structure on thetime scale [vf w]*1 (this can be noted by inspecting Figs 4and 5) The development of a shock-like structure in the"cold" fluid is anticipated from the form of the equation of

Fig 6 Normalized total number density vs position

motion of the "cold" fluid (4 7) These equations areanalogous to those of inviscid gasdynamics with sources ofmass and energy where these sources, being distributedthrough the flow field, represent the effect of the piston sur-face on the "cold" fluid and so tend to cause gradients in the"cold" fluid to steepen into a shock wave

Because the "wall" fluid density in the vicinity of the sur-face is very much larger than the density of the "cold" fluid,the shock-like development in the "cold" fluid is maskedwhen the total fluid density and velocity is plotted (see Figs6 and 7) In fact, because of the predominance of the "wall"fluid in the vicinity of the surface, the velocity of the totalfluid does not significantly depart from the free molecularsolution; however, the total density does show rather largedepartures from the free molecular case

In discussing the effects of the freestream speed ratio, onemust bear in mind that the assumptions of theory becomemost appropriate in the limit as Mm -> °° One can, there-fore, expect a rather small freestream speed ratio effect—this is borne out by the results If one considers the relevantquantities as functions of x', t", and Mm, where t" = tr(Mm) ~x,then an inspection of Fig 8 reveals that a very weak depend-ence on Mm results from a solution of the equations; allquantities depend only on x' and t"

In summaiy, the model presented here results in a set ofnonlinear differential equations The range of validity ofthe equations is shown to be, for the hypervelocity com-pression problem, Kn > 1; this can be compared to the con-ventional near free molecular flow theories which are validfor Kn > 10 Although the solutions presented here do notshow a fully developed shock, they do give insight to the

Voo

Fig 5 "Cold" fluid numbei density vs position Fig 7 Average fluid velocity vs position

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1054 B B HAMEL AIAA JOURNAL

01 03 05 07 09 II 13 15 17 19 21X 1

Fig 8 Effect of freestream speed ratio

regime of incipient shock formation In future work, withthe inclusion of self-collisions in the "cold" fluid and im-proved numerical techniques, it is hoped that solutions ofthe compiession problem can be presented for largei times,which do yield a fully developed shock

References1 Willis, D R , ' Comparison of kinetic theory analyses of

linearized Couette flow,' Phys Fluids 5, 127-135 (1962)2 Gross, E P , Jackson, E A , and Ziering, S , "Boundary

value problems in kinetic theory of gases," Ann Phys 1, no 2141-167(1957)

3 Gross, E P and Jackson, E A , "Kinetic theory of the im-pulsive motion of an infinite plate," Phys Fluids 1, 318-328(1958)

4 Talbot, L , "Survey of the shock structure problem,; ARSJ 32,1009-1015(1962)

5 Mott-Smith, H M , "The solution of the Boltzmann equa*tion for a shock wave, Phys Rev 32,885-892(1951)

6 Glansdorff, P , "Solution of the Boltzmann equations forstrong shock waves by the two-fluid model,' Phys Fluids 5, 371-379(1962)

7 Ziering, S , Ek, F , and Koch, P , "Two fluid model for thestructure of neutral shock waves, ' Phys Fluids 4, 975-987(1961)

8 Koga, T , "The structure of strong shock waves of stablemonatomic molecules " Rarefied Gas Dynamics, edited by LTalbot (Academic Press Inc , New York, 1961), pp 481-499

9 Lees, L and Liu, C Y , ' Kinetic theory description of planecompressible Couette flow ' Rarefied Gas Dynamics, edited byL Talbot (Academic Press Inc , New York, 1961), pp 391-428

10 Rott, N and Whittenbury, C G , "A flow model for hyper^sonic rarefied gas dynamics with applications to shock structureand sphere drag,' Douglas Ah craft Co Rept SM-3824 (1961)

11 Lubonski, J, "Hypersonic, plane Couette flow in rarefiedgas " Archiwum Mechaniki Stosowanej 14, no 3/4, 553-560(1962)

12 Baker R M L and Charwat, A F , "Transitional correctionto the drag of a sphere in free molecule flow ' Phys Fluids 1,73-81(1958)

13 Obermann, C , "On the correspondence between the solu-tions of the collisionless equation and the derived moment equa-tions," Project Matterhorn Rept Matt-57 Princeton Univ.(1960)

14 Enoch, J , "Kinetic model for high velocity ratio near freemolecular flow," Phys Fluids 5, 918-924 (1962)

15 Hamel, B B , "A model for the transition regime in hyper-sonic rarefied gas dynamics," General Electric Co , Missile andSpace Div , Space Sciences Lab Rept TIS R63SD85 (1963)

16 Willis, D R, "A study of some nearly free molecular flowproblems," Princeton Univ, Aeronautical Engineering ReptNo 440(1958)

17 Probstein, R , "Shock wave and flow field development inhypersonic re entry," ARS J 31, 185-193 (1961)

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