ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Technical Report No. 1 A STUDY OF THE HYPERSONIC LAMINAR BOUNDARY LAYER WITH DISSOCIATION Part I T, C. Adamson, Jr. J. A. Nicholls P. M. Sherman Project 2606 BELL AIRCRAFT CORPORATION BUFFALO, NEW YORK May 1957

The University of Michigan * Engineering Research Institute TABLE OF CONTENTS Page LIST OF SYMBOLS' ii ABSTRACT OBJECTIVE v INTRODUCTION1 CONSERVATION EQUATIONS1 EQUATIONS FOR A TWO-COMPONENT MIXTURE5 QUASI TWO-COMPONENT SYTEM 9 CHEMICAL REACTION-RATE TERM 14 FLAT-PLATE EQUATIONS IN PHYSICAL PLANE 16 TRANSFORMATION OF EQUATIONS 17 METHODS OF SOLUTION 22 PROPOSED FUTURE WORK 25 PHYSICAL PROPERTIES OF AIR 26 EXPERIMENTAL TECHNIQUES FOR THE STUDY OF HYPERSONIC FLOWS 27 Hypersonic Wind Tunnels 29 Shock Tubes and Shock Tunnels 29 Arc Discharge Tunnel 30 Hyperveloc ity Pellets 30 Other Methods 30 Instrumentation 31 APPENDIX. VISCOSITY OF THE MIXTUE 33 REFERENCES 35 BIBLIOGRAPHY 38 ii

The University of Michigan * Engineering Research Institute LIST-OF SYMBOLS ( ):== vector quantity C PpL/Pelle Co = mass average velocity of gas mixture at any point C.= diffusion velocity of molecule of ith kind referred to Co Cpi = specific heat at constant pressure per unit mass, of ith species Cp = specific heat of mixture Cp S (ps/p)Cps Dij = multicomponent diffusion coefficient ij b = binary diffusion coefficient DiT = multicomponent thermal diffusion coefficient ei = internal energy of ith species per unit mass e = internal energy of-mixture, e = Z(Pi/p)ei f = Blasius function hi = enthalpy per unit.mass of i.th. species h = enthalpy of mixture h = Z(pi/p)hi k Boltzmann's constant kf r = forward or reverse reaction-rate constant X(c = equilibrium coefficient in terms of concentration p'= equilibrium coefficient in terms of partial pressures xi = rate at which molecules of:th species are formed per unit volume due to chemical reaction.mi -= mass of-molecule of ith kind Mi = molecular weight of i)th species M = molecular weight of mixture ni -number density = number of molecules of ith species per unit volume n = number density of mixture, n-= Z ni Ni = molar density = number of moles of ith species per unit volume iii

The University of Michigan * Engineering Research Institute LIST OF SYMBOLS (Concl.) N = molar density of mixture, N Z= Z-N 1 P = local pressure Pr = Prandtl number, Cpl/q = heat-flux vector Ro = universal gas constant Sc.= Schmidt number, I/pDi2 T = local temperature uo = component of mass average velocity in. X direction vo = component of mass average velocity in Y direction i.= force on ith molecules due to external field Xo <= coordinate in physical plane Yo = coordinate in physical plane ~Y.Ie~n" = dimensionless coordinate, i e-= /1i = dimensionless mass concentration of ith species, KI = Pi/P X = coefficient of thermal conductivity.= ~ coefficient of dynamic viscosity y =. k5inematic *viscosity., YP L/p Pi = density:.of ith -species ~p = density,of mixture, p P= pi Subscripts e = Outer -edge of boundary layer w s= urface of f)lat plate iv

The University of Michigan * Engineering Research Institute ABSTRACT The general conservation equations for a multicoomponent gas mixture with chemical reaction are considered. They are reduced to the boundry-%ayer equations by the usual order of magnitude analysis, for a twocomponent and a three-component system. It is shown that a two-component system can be used strictly only for a system following chemical equilibrium For the case of a nonequilibrium reaction, it is shown that the three-com"X ponent system can be reduced to a quasi-two-component system by the use of only mildly restrictive assumptions concerning the similarity of the properties of oxygen and nitrogen. The studies indicate that the boundary-layer similarity variable,'q is the sole independent variable only if chemical equilibrium is assumed. Otherwise the variables also depend on the axial coordinate, X. Arguments for the existence of a buffer layer where no reaction occurs are presented for the case where there is no chemical equilibrium. The thickness of the buffer layer is shown to depend on the wall temperature and the altitude through the pressure. An outline of the various sources of gas properties at high temperatures and their usefulness is made. A comparative study of the various methods of experimentally studying the hypersonic boundary layer is also carried out Sme consideration is given to the types of instrumentation which have been and might be used. OBJECTIVE The purpose of this study is to demonstrate the effects of dissociation on the velocity and temperature distributions in the hypersonic laminar boundary layer, thusindicating the effect of the dissociation on skin friction and heat transfer at the surfaces

The University of Michigan * Engineering Research Institute INTRODUCTION The analytical problem considered in this report has been simplified considerably from the general problem of the boundary layer in very high speed flow. To emphasize the dissociation effects, extraneous phenomena have been neglected. For example, since there will be shocks in front of or attached to any body traveling at hypersonic Mach numbers, the air entering the boundary layer will have been dissociated to some extent already, and will have begun to recombine to some intermediate concentration of atomic species. Hence, in general, the air entering the boundary layer will have an atomic concentration which varies along the boundary layer (assuming a curved shock). Furthermore, it is known that there is a pressure gradient in the flat-plate hypersonic boundary layer, which is self-induced by the change in thickness of the boundary layer. Ignoring the above effects for the present, then, one can consider a problem which consists of a typical laminar boundary layer with an external flow of normal air at a very high velocity, and with dissociation occurring within the boundary layer alone. If this flow can be solved, then an attempt can be made to add the above complicating factors, to study their interaction with the dissociation effects. In conjunction with the analytical studies, a comparative study of real-gas properties has been made, and a chart listing the various properties found in each reference is included in the body of the report. Consideration has been given to the various experimental facilities being used at the present time in the study of hypersonic flow. It was desired to find what data could be obtained from existing techniques, and to compare these facilities with a shock-detonation tube, where a detonation wave. is used to give the gas a high total temperature. CONSERVATION EQUATIONS The general equations which hold for a multicomponent gas mixture, either with or without chemical reaction, have been stated in detail by Hirschfelder, et al.1 through a generalization of the work of Chapman and Cowling.2 In particular, the equations modified for the presence of chemical reaction and polyatomic molecules are employed in this report. While the ------------------------- 1 -----------------

The University of Michigan * Engineering Research Institute coefficients of diffusion and viscosity are not changed by considering molecules with internal degrees of freedom, the coefficient of thermal conductivity and the heat-flux vector are affected. Following the notation as given in the list of symbols, the conservation.equations may be written in the following form..CONSEVATION OF MASS The continuity equation for each -.species is C) tpVni(c, t S) =7 e I~ i' Eva (1) If this equation is multiplied by mi, and summed over i,. the total continuity equation is obtained. Thus,.?~o~ =o d - 7 C (2) since n nimi =-p by definition, and Z miKi 0, and i nimi _i = 0 from mass conservation- considerations Finally, if equation (1) is multiplied by mi and the relative mass concentrationKi = (nimi)/p is substituted for nimi, then, with the aid of equation (2), it can be shown that equation.(1) becomes (/ t *CO-74)= L -V — K (,.3) This latter form of the species continuity equation will be used throughout the remainder of this report. It should be noted that for an r coponent system, only ril separate species continuity equations (3) need be used, along with the overall continuity equation, (2), since Ii ki = 1, and.i = Pi/P Hence,, in a two-component system, knowing /t and p) for example, would allow one to calculate pi and P2a CONSERVATION OF ENERGY Summation of the separate energy equations for each species gives the energy equation for the mixture In terms of the enthalpy, this equation is. —---------. n --—. —.. —---—. —. —. —

The University of Michigan * Engineering Research Institute ( Co VL ) = _P Ca. P +l P V. - 7V -: VCo -ZhX -, 3t / cde~ - - - - - z (4) where i = - 7vT i- 27^ 47 2 (,, C ) - ~.. (5)o is the heat flux vector, and h Z= X ihi is the enthalpy of the nmixture. CONSERVATION OF MOMENTUM Again> siumnation of the separate momentum equations for each species gives the overall momlentu equation for the mixture: P(dCo (Co.Co) = -V P e / S Z (6) 3'at /' P Z DIFFUSION VELOCITY The diffusion velocity for each -compnent is defined as follows:?2 5 - //, n Z r h. d. -' DTViT, (7) *where = V7/() iv ~ j^l) P- J /I ZJr" -2')._ t/ ( / -m / (8)D x e EQUATION OF STATE The equation of state of each species is that of a perfect gas: Pi = nikT, so that the overall equation of state is.P = nkT or p-~ 1 e-yi (9) where i )? t i /y Lft (10) ------— 5 —----- P ~ zW

The University of Michigan * Engineering Research Institute is the molecular weight of the mixture at any given point and Ro is the universal gas constant. The above equations are simplified by the assumptions usually made in a boundary-layer study, as well as some assumptions which are necessary only in a multicomponent flow. The assumptions necessary for a general laminar boundary-layer study are as follows: a) The flow is a two-dimensional steady flow. b) No external forces exist. c) Thermal diffusion is negligible. d) The usual boundary-layer order of magnitude assumptions apply. Assumptions c) and d) must be clarified at this point. It is not clear at the present time what role thermal diffusion would play in the case of relatively high temperature gradients such as exist in the hypersonic boundary layer. Some authors3 have included the thermal diffusion term, while others have disregarded it in view of the fact that not enough is known about the coefficients. Chapman and Cowling5 show that for mixtures of hydrogen and nitrogen one can expect the ratio of the thermal diffusion coefficient to the binary diffusion coefficient to be of the order of 0.10 at the most. Furthermore, in most combustion studies where the temperature and concentration gradients are comparable to the hypersonic boundary layer, the thermal diffusion is neglected on the grounds that the mass transferred is negligible compared to that transferred by convection or diffusion due to concentration gradients. The reasoning is that the dimensionless temperature and concentraw tion gradients are of the same order of magnitude while the coefficients are not. In this study, the latter reasons are assumed valid, and thermal diffusion is neglected. In stating that the usual boundary-layer order of magnitude assumptions hold, one is making a tacit assumption about the reaction rate, which must be clarified. The boundary-layer approximations can be made because, in the boundary layer, a change in any variable in the flow direction is very small compared to the change in this variable in a direction perpendicular to the flow direction. This physical fact is used in stating that the derivative </8x is of order 1/L while 6/6y is of order 1/8 where L and 8 are the length and thickness of the boundary layer, respectively, and 8 < L. In making the same assumption in the species continuity equation, one is really assuming that not only the rate of diffusion of species but,also the rate of creation of species by reaction is small in the flow direction compared to their values in the direction vertical to the flow. While this assumption is very restrictive in combustion problems, it should not introduce any new difficulties in the present problem because the reaction to be considered is a dissociation reaction which is self-damping, ice., if the temperature is increased, the increased rate of dissociation then lowers the temperature. Hence, any zone of dissociation in the boundary layer will tend to grow with the boundary layer ---------------------— 4 ------------

The University of Michigan * Engineering Research Institute and become more diffuse, rather than tend to become a thin self-propagating reaction as would occur in an exothermic reaction. Therefore, except at the initiation of the boundary layer where a singularity exists in all the usual boundary-layer solutions, the boundary-layer assumption should hold for the concentration, and thus the changes in x or flow direction should be small compared to the changes in the y or radial direction. With the above assumptions, equations (2) to (8) become C) _=ao y-, w4 c o C) n th _o eua U o a n an- A. co th a paeFr atwo ps ii) (11) gradient P/Xo - = 0, of course.. where t = ) because P O In the above equations uo, vo, and xo, yVo are the velocities and coordinates in the physical plane. For the flat plate with no pressure gradient aP/6Xo = O0 of course. EQUATIONS FOR A TWO*COMPOBNENT MIXTURE The assumption that a dissociating flow consists of two components is often made to simplify the equations. In this case, the two components are molecular and atomic."air." Actually, this assumption can be used only if (a) chemical equilibrium exists at every point or the reaction rate is very slow compared to diffusion across the streamlines (Lees' two cases6), or if (b) ------------------- 5 —------------

The University of Michigan * Engineering Research Institute enough is known about the rate constants in the dissociation of 02 and N2 to provide an average rate which would hold for the dissociation of air. The last condition does not seem feasible since N2 does not begin to dissociate until most of the 02 has dissociated, indicating a large difference in the activation energies for each of the reactions. However, when no chemical rate is involved, such as in the case of chemical equilibrium or rates slow compared to diffusion processes, then a two-component system seems adequate since the molecular weights of N2 and 02 and thus N and 0 are almost equal, and since the transport and thermodynamic properties are also very similar. To write the specific equations for a two-component system, one need only compute the diffusion velocities. The last of equations (11) is the relation for the diffusion velocity For a two-component mixture where Di2 =C12 for a binary mixture. Now n + n n n and _ as, nfr = AK,,, % H,.______....___ = v since /l + K2 = 1. Therefore, /) = - )= -/ z,; and,- - 2,~ ~__^ / <^ <^~ ~(12) Likewise, since Z Pvi 0 > i= -n t9 = 3 L, so that in general, for a binary system, ---------------------- 6 —---------

The University of Michigan * Engineering Research Institute k- y Thus, the condition that Picks' law be used for the diffusion velocity, as stated by Lees7, is unnecessary for a two-component flow. Actually, Ficks' law is written in terms of mole fractions for a system at constant temperature and pressure so that n, the total number density, is a constant. However, it may be used for systems with varying T or P if the con. centration of one component is very small compared to the other, or if the molecular weights are equal. These assumptions, made tacitly when Ficks' law is used, are unnecessary for this case. The equations for a binary system are as follows: /^ ~A $W0 == aA~~~~~ a ~ ~ ~ 2- ay written in the form given by Lees by notingta ffo y7 p^,eZ^ tpt^ 2^ = -^ y- t /^ ^\, /7^= 43al~ /Y /)^' /^/ where Mi and M2 are the molecular weights of species one and two. %I^ may be wri~tten in -the form given by ~Le~es by noting~ tia-t

The University of Michigan * Engineering Research Institute A= 27.t4 and _ /^ s - A= / /T *.j^ where i(o) is the heat of formation at zero degrees absolute. Then, since each component is a perfect gas, Cpi = Cp (T), and and 27/< I = _ so, e < ~ cI't -' i e7IU The above equations are essentially those considered by Lees;8 As was mentioned previously-, they should be used only when chemical equilibrium is assumed (no equation necessary for K1), or when!nican be considered small compared to the rest of the terms in the equation forAil, since it would be very difficult to evaluateri,~ the rate term, for the decomposition of air rather than its separate components. One might wonder why a two-component system could not be used where the temperatures stay low enough so that only the decomposition of 02 need be considered. After all, N2 and 02 have transport properties which are very,similar, and since only one dissociation reaction is being considered, there should be no confusion in what react'iondmrate parameters to use. The reason that this cannot be done is that the rate at which 02 decomposes depends on the local concentration.of 02,not on the concentration of 02 and N2. In a two-component system,:one can tell only what fraction-of the whole concentration is made up of molecular species, One cannot tell the difference between 02 and N2 molecules. From the standpbint of diffusion, heat capacity, etc., this information never becomes necessary. However, to know the local reaction rate., one must know the 02 concentration; that is, one must be able to distinguish between 02 and N2 molecules. For this reason, a three-component system should be used,.In general, the use of such a system involves more than simply adding another equation to the set. The transport properties become extremely complex, and terms are added to the energy equation and heatflux vectoz. However, it is possible to simplify this system of equations in8

The University of Michigan * Engineering Research Institute to the so-called quasi-two-component system used for the remainder of this study. QUASI-TWO-COMPONENT SYSTEM Consider a three-component gas mixture made up of 02 (subscript 1), 0 (subscript 2), and N2 (subscript 5). This means that the external flow Mach number must be low enough so that.only a negligible amount of N2 dissociates into N. For this range of temperatures, then, there is no reaction involving N2. The assumption is also made that the molecular weights of 02 and N2 are the same, and that the thermodynamic properties are the same. That is: / _ - z (16) and C (17) The assumption is also made that c3 =o31 = 0; i.e., that 02 and N2, having essentially the same properties, do not diffuse into each other. The gas mixture, then, consists:of N2 and 02 molecules and 0 atoms. Within this.mixture, one can distinguish between 02 and N2 only insofar as calculating the concentration of 02 for the reaction rate, For all other intents and purposes, 02 and N2 are the same gas. With these assumptions, then, |/^ ^ opt - shy =(18) Using Hirschfelder's9form for the multicomponent diffusion coefficient for a three-component mixture, one can write the following general relation between the multicoimponent and binary coefficients*' 4*j a j J 4 i'>(19) where 4,. =. / /-...3 ( / 33 /t /3 ) A' = - 6 ^'-44),,. with the.'others being obtained by cyclical permutation of the indices. Next, -------------------- 9 —------— I

The University of Michigan * Engineering Research Institute.if the assumptions involving the oeij and Mi are employedy.one obtains the following set -of relations: t/^= /? 3 = / 2, (20) ancd az s 3 = /,.'I37/3 _ _ _ _ Note that the concentration ratios may be written in terms of moles per unit volume or imolecules per unit volume The last of Equations (ll) may now-be used to calculate the diffusion velocities. Thus,. = h - 2 ^ o6(,/ja ) )~i L 4- YL r~ /*#3 dt03 (21) oz e ( ta ^ K = nt1 2^ ~ (A.;L ) a(2/2) 3 P ^/,? 2'I To convert the number or nmole fractions into relative mass fractions, the following relations may be usedo h, _ ______n,m, I__ / /1,m, * 2z/8ZX 9/ t' 33 )y2t kt6 47 3 4 z (22) 10 =....______ /t ^- )//^/lO_ (2

The University of Michigan * Engineering Research Institute since Kt2 = 1 - K1 -/;3. Likwise /7? = to^- ^ t j /= /-. /63 and _ - The derivatives of the sum of equations,(22) and (23) is taken nexty and making use of the assption that hl =t 3, one way write it as, g- ue the'as- s =_ t I 03)| gl n/ n J t ( n,t3)(_ ^0e 2 mz 197O 9ago (24)::, j?'' n'' where [( ^)(/-t-, m_ 7_Z /2 = n< c/9^ *) again frca equations (22) and (23). Finally,,to write the diffusion velocities in terms of the desired, variable^s one ast write ( n)/(n.: + xn) and (cs)/(n, + na) in terms of the /K. This can be done as follows:./, = m/_ k, ~,Q~~~~fn~,wf43M (25) 97o ^/^ < 7 /+H since mi is ass ed equal to.ms Substitution of equations (24) and (25) into equations (21) and a slight rearrangement gives the desired fora for the diffiusio terfs in the continuity equations.,,, = -, = -,z,- ~ (/& ) (26) L-, -- 26)11 -

The University of Michigan * Engineering Research Institute These relations m'ay be checked by noting that the relation for the conservation of mass by diffusion is satis fied, That is: With the above relations for the diffusional mass transport, the species continuity equations [equation (llb) ] may be writen. Also equation (lld)may be written in terms of -the concentration gradients. Thus, = = - A ar *' (Ida) But 2? ^0 = -4 / _ ) /d — t-i —anad sinee by asispton'i = A3, 27^/^ -A [, S ) &- 1 = -/7/Z< I -4 a 7 ed g. a(27) Therefore with -the atumption f made concerning the 02 and N2 the heat.iflux vector can be written in exactly the saame form as for the binary.. system;" 70 -'- / / A)2 1S 4 ~ I (28 ) where aT/oy0 has been written in terms of ah/ayQ and ai4Y/yo, as before0.Se complete set of equations for this ternary mixture can:be written now" As was- metioned before,however, it is necessary to consider only two ofthe t three possible species cotitnuity equations, and the overall continuity equatipon Since there is no reaction rate-considered for the N2, the equation for.s, being the simplest of the three, s one choidee. Th' equation in (0)) iB chosen as the other,'since K2 = 1 - (;|L +'a) and K>L + /( is a natural:. grouping in the equations, and since, as will be discussed later, the bou.nda.e 12 -------------- 1 -------------

The University of Michigan * Engineering Research Institute where (2 - 0 (limit of dissociation) are very important, The complete set of equations for the three-component, quasi-two-component system are as follows: a)?- y- /^T = ae, S I /7- Z/ Equati) o ) tten or te specific c as where M3. M, and I) ct> ^? siy c/ s 6 = /^^>) -^ ) e/ ) /2/ -S )In general,? (29) I aJ /Y / <A/ |M). /Mi) =:1/2. In ge ral, I7 = L' Z/, but fpr the above conditions., this reduces to j3 j= -P~, / - _~~ t' / and substitution of equat;ions (22) and (25),.again with the above eonditions on molecular weights, gives: the desired results. --------------- 15 —3

The University of Michigan * Engineering Research Institute A- / 2, -/ / For the special case of a flat plate with a uniform exterior flow, the main problem with which this study is concerned, (6P)/('Xo) = O. Hence for the remainder of this rport, the pressure term will be neglected,| CHEMICAL E REACTION-RATE TERM Before continuing with any revisions or transformations of the boundary-layer equations, the chemical rate term appearing in eqution (29b) should be carefully examined..This termy m~2z, expresses the rate at which mass of species 2 (atomic oxygen) is created by chemical reaction, per unit volumeo. To write this rate, something:must be known about the proper chemical equation for the reaction.According to Hirschfelder,10 the reaction for the zrecmbination of atomic oxygen depends on a three-body collision. and hence' the reaction should be rit.ten as follows AA 0 * 0 + X 0g^:O f oY ) (30) where X is any third body (ie,,e,iother 0 atom,,.or an 02 or N2 molecule) and kr and kf are the reverae and forward reaction;rate constantS, respectively, Since the reaction is usually written as a dissociation jreaction, the kr and kf appear revrsed in equation (30). Actually, this equation holds for both the diss'eiatiohn and reeombinat ionof oxygen, depending on the directioan in which the reaction happens to be proceeding. "From equation (30), one can find the rate at which atomic oxygen is being formed making uWe of the law 1f mass reaction rates. Thus, the rate at which moles of 0 are being formed per unit volume, i, using Penners' form of the rate lawy,1 aJ= -z^itz^ i-^ ^z44, (1) where N2, N.N andd x are the moles per unit volume of Oy 027 and any third bo.dy, respectively. Sinee X can be any of the eonsti.tuents of the mixture, Nx - N, | 14

- The Uhiversity of Michigan * Engineering Research Institute and, I^^-^4'A~/^r/-~ ^ j (32) where f( is the equilibriurn constant written in terms of concentration. That is, /e t= rf^ (33) The equilibrium constant is usually written in tens of partial pressipres. Then, /C t= /y=t / (34) Also, it'is deired to -calculate m. which is MP.Sd), Threrefre, d.t = -Z 7 where p, = M. Next, the equation may be rearranged as follows' ^/ /S/ ^ A j The term multiplyring the bracket can be revised, u'sing the equation of state. Thu-s /AZ9 Af3 / (! P/) % 67) 4, ot 4ir~,sopL' C8y.t it i i. = _M P. Ale 7and since M1/M2 2, and M/M1 = l/(;,2 + 1)., the equation for m2K2 can be written,. finally as D h: = /: -pA) / L- /^l - _ - 1@-,'Q c'44)jp. 7P (35) 15

The University of Michigan * Engineering Research Institute so that W is defined as Xfr - f^/ Z,?- / j=) (4 P)Z jj/4 /~A4- -/ (36) Aceording:to- Hirschfelder kr may be approximated by,conSidering the reaction H. + 21I His calculations give the following results: ~24,.. * —.,.2 ~ja'~f..ZT ~~ ~, (37) Ioo ~*'"o~in/e) (sec w) Aleso, using the values of Fricke for the equilibriuzm constant, Hirschfelder elaims that,the following expression gives good acecacy for temperatures up to 5-,0K, p r e,.^> (s8e - 00o ) (38) Using the above values, an approximation of the reaction rate may be -btained.: Note that in the definitions of w, kr, and K:, w is a function only of the tem perature and concentrations. p does not enter into the definition explic.itly FLAT:-PLATE EQUATIONS I' INPHYSICAL PLAME At this point, the egquations covered in (29) can be rewritten for the case of a flat plate with uniform external flow, (fP/xXo) = 0, and usg the definition;of the rate law, In -writing these equationa, one new variable is defi.nd to make thf eq.uati:ons look mare similarx Thus, = 1 - /2 is used. b) /p,'a' /Cs^ f =; //~eg> S ) - / ions ) be ome, thR n, 16

The University of Michigan * Engineering Research Institute d),^oA d o f/ e _ d/Z) e) > a/ f?/P a^ = ^ /j/ - (- P^ )^ Cp)}t] asr s; <,4, c. kt~ A e-12YI- ),>p = (5-9 ),Z < Con5NT (39) ftoM;^~~~~~ ((Cont ) It should be noted that them energy equation (59e) may be written in -other forfns. If the product of uo times the momentun equation is added to the energy equati, o one obtains the well-known form where the total enthalpy J' =A + (lu2/2') is involved. If the'equation is written in terms of the temperature rather than the enthalpy, then, since the equation-of state connects the density with temperature, there is a better connection between the energy,and sate -equa tio; however, the energy equation becoes more complicated through the appearance of the reactiOn,*rate term..These two forms are listed at' this t'ime for future reference. 443 S.) ^$ / 7 * - 9 | [ -//4 ) 2/ -/ ) Here (, - ) is the heat released due to reaction. In combustion prob.lems this is considered to be a constant, given by some average value, Hoo er, in this case the assmption depends on the temperature range involved, 17 pr m thi~s is ~con ~rld to;be a costants gie by 8 average vale Ho ea xIt m~y x'b )sw b3 to *n^^ a -estant value. I —------------------— 17 ------------

The University of Michigan * Engineering Research Institute Io3rodn.y1 3f i withot dissoiation ths transaormaati.on:.effectively deo:uples thfe mamentu and energy quati since z p, m c-nstant with dissociation, since p -,constant, the equations are not,?ompletel eco/pledla, and the new plane is pot.completely iCompres:ib e Eo H ew as other aut;hors have pointed out, l 5 th effects:de to thee variatlons are sall: and. can be taen carte of by an iterative aolutin.of the mm.etum equatiMon and eergy: e qat If:x, y u,u ad v ae th co0ordinates and velicities in theo t orr e-t ipo-di:qLuasin:aressibIe ft lowthen they are cecte to t- ptysical variables with the: following relatino c) t= d |e..) a = eo1 SJ t ^ ~ e46( ^ # Jl 7/- ) Here p. is the density along. the -ege of the boudary. layer. The | -:elocity relat ions ae foud by defining the stream fu-ction in th~~ usual way: (42) anld performing the give n transf;o rationt of variables. That is<or 20= iai | Also)6 ) ^~ - -/~ g' S; ). i8

The University of Michigan * Engineering Research Institute Now, defining =, and in the transfoieed plaes, one obtainS the relations given in equati.n (41e) and (41f). With the definiti'oxa given in equatio.iS (',l), and defining oec can write equatio:ns (39) in the following manner 2) a _ L-) 6 f Q^ ~ v - - Yo C }- <sI Ia ag 4 04 &3 = 04 / t?) c/) &sz = Ye(c) =) i St f ^ v^J SC?^': ^l7 (^ sc)* sJ |e is the kinmatic viscosity alon the edge of the bounday layer, an'd Pr and S, are te Prandtl n:iumber and Schmidt number, repectitvely.They are defined as'and, l6) whre all quantities are for the mixture at a given point. Thbe fors- of the -energy equation given in — euations (40) can also be writte in term..f the new variablea. Thus, L —--------- -— 19

The University of Michigan * Engineering Research Institute a) 6LS ^ ^-M =t;/e/^j -^_ )^ -.C )f, v a i ^c( p>/ @ c^; =/;' ~ y ^ At th-is point, it is tstructive to write the equatin nons e.more diferent fo. This form ivolves the use of the socalled bondary-layer variable,^ i. HR, instead of assuwing immediately all variables are fune tions of Ir.' as. iSf usually. done, one can transform the equations instead of fror x'and y, to s an.d:3, say, where sX,a X)and (8) Also, a dimersai'elesss stream futtion is defined in the usual way, exept that f is now a fu:ction of x and.no;;^ = N^ie ae t) 7t872) (49) With thiese definitions, a straightforward change of variables leads to the following set of e lquationaS, from equations ('45) Te baory c fte a s are, inenra 6/ 4/74 e^ 5^ = 2o^[~ 4~ —12 C) 9/~-)y /^^^=,/^if )r - _fa' 97 c^27, 9 2 / t <9t a d) d s~5 _4r)<e&<t) S&-O Xve where the Kergy e:quation has beeBn written kii terns of the1 to;tal enthalpy | Th'bounfd-eor4tit ias orp the above eqiuatiogn ae, in geaeral. I ------------- - --------------

The University of Michigan ~ Engineering Research Institute ='. ~e=?' —o Of course, if the incoming air has no dissociation, then Ae, A3e, andAf are not functions of x, The important thing to notice about equations (50) is the dependence of the dependent variables on x.o In the usual boundary layer analysis, these variables are ass umed to be functions of'I aloneo In terms of the momentum,equation, this means, for instance, that rf/<X = c2f/'x/s'= Oo Now, the resulting equation.must be independent of x, or the assumption -cannot be made. Looking at equation (50a), then, it is clear that K is not independent of x unless chemical equilibrium exists. This is true since even if one assumed H = f{('q) and f = f(T) so that 7/6x = K(/6x = 0, there would be a term kxw/7e on the right-hand side of the equation, so that x would appear explicitly ir4 the equation, If chemical equilibrium exists, however, the net rate of production of O, by chemical reaction is zero at every point, and w = 0. Then, however, equations involving K and /3 are unnecessary, since all cocentration may be calculated from equilibrium considerations. Equation (50b) is the only equation which has a term multiplied by x, which is not of the form where an assumption about the dependence of the variables on'q would remove the explicit Xo However, if AC depends on x as well as il. then s, As, and even f, thrugh c, depend implicitly on x since/C appears in the equation. Actually, C depends onAs andX, but, as Probstein sh.owed^l6 the dependence of f on C is not very pronounced9 so the assumption that f' depen~ds on alone should not be too inaccurate. However, as mentioned.above, the assumption is strictly true only in the case of chemical equilibrium or no reaction at all, of course, 21

The University of Michigan ~ Engineering Research Institute METHODS OF SOLUTION Before considering any actual methods of solution, it is instructive to picture the temperature distribution through a high external velocity boundary layer with heat transfer into the wall, with no dissociation. The temperature profile is shown in Fig, 1. Fig. 1, Temperature distribution with no dissociation. Next, if one allows dissociation to occur, it is evident that as the temperature increases (coming from Te toward the wall), a temperature will finally be reached where dissociation becomes important. From this point on toward the wall, the temperature will decrease from the value it would have with no dissociation, In Fig. 2, the dotted line represents the actual temperature distribution with dissociation; the full line, the temperature without dissociation: as in Fig. 1. /e7 0 —-s- / Fig - 2, Effects of dissociation on temperature profile.. The profiles are drawn with the maximum temperature much greater than the wall temperature so that there is heat transfer into the wals, because 22

The University of Michigan * Engineering Research Institute in the practical case, the wall temperature must not rise over approximately |1000~K, which is much less than the stagnation temperature at the flight Mach numbers envisaged. Hence in any practical problem, there will be heat transfer into the surface. The effect of dissociation, as can be seen by the sketch in Fig. 2, is to reduce the heat transfer by reducing the temperature gradient at the body surface, It is evident if chemical equilibrium is assumed, that the temperature at which dissociation begins to have an appreciable effect, as the temperature is increasing, is also the temperature at which dissociation ceases to have any effect as the temperature is decreasing. That is, there is a buffer layer next to the plate where no dissociation occurs. This situation is pictured in Fig. 5.!'' —'. /ssoc......r. be...s': IT EZone of I,//~ 1\8~ I dIssoc1a o \ v ~/ Ij;~es i ^ye C o_ _.. / I Fig.o 5 Buffer layer for case of chemical equilibrium. Now, this situation is sure to be altered when an actual chemical rate existso Chemical equilibrium will probably not exist. Since, however, according to equilibrium figures, the temperature at which the relative weight concentration of atomic oxygen is 1% is 2500~K, and since the wall temperature is only about 1000~K, it seems evident that there still should be a layer at the plate surface where no dissociation or recombination takes place. This is because at these temperatures the recombination rate is extremely fast, there being a zero activation energy, and because the fluid is moving slowly compared to the external velocity. Hence the relaxation time must be very small compared to the residence time. This ratio could be expressed, for example, by (t7-Z.z recombncfiton that is, by the ratio of the nitrogen convected to the atomic oxygen which disappears due to chemical reaction,* This ratio may be written in terms of *Note that the diffusion terms, in the boundary-layer equations, are of the same order as the convection terms,'so that this ratio should hold for the time associated with the cross stream diffusion of mass as well as the time associated with the convection of mass. 23

The University of Michigan * Engineering Research Institute Hirschfelderts values (equations 36, 37) and an order of magnitude.ound.'9 _ __________s *to about 2000~K, and P is assumed to be approximately 1/10 of an atmosphere, then C rf/ / or Rt 2 x Icr6 U/L, and for Rt< 1 no significant reaction occurs. where s except at or very near the leading edge This ratio maye singularitie'to exist in all the functionsT U/L can be of the order l04 before the ratio is significantly affected' Of course, all the values used are approximatations,. but they inallydicate aeneral order of magnitude and show that there should be about 2000~K, and P is assumed to be approximately 1/10 of an -atmosphere, then or RIC Z 6 /O"0 U/L, and for R << 1 no significant reaction occurs. Thus;, except at or very near the leading-edge, where singularities exist in all the functions, V/L can be of the order 104I before the ratio is significantly affected. Of course, all the values used are approximatations,. -but they indicate a general order of magnitude, and show that there should be a buffer layer even when a reaction rate is considered. The concentration profiles would then look something like those pictured in Fig. 4. I. -- rNhcoess of 0l L — f ~. — /. l |offew Za er Figs 4,, Coneentration profiles in boundary layer.: I —---------- -24

The University of Michigan * Engineering Research Institute PROPOSE FUTURE WORK With the above arguents in:mia the following method of attack i prpseaL First, with the aSsumption of chemical equilibriumt the bodaries of the ze of reactiion may be found. They are arbitrarily defined as those liles along which the relative mas coentration of atmit oyge is ti,i.e*., K 00,iOl.-. Nexty to see what the effects of a reaction rate are a perturba tion ynaly'is will be made with the net rate of raction, w being the sma1 pamaeter. This- will indiclde t changes in both the boudaries of the reaction zone, ad the profile of the teperature to be expected from a.hemical reaction rate, Finally, a numerical analysis a be e t be t best aail able, remionrate da.ta'This shouldbe perford with the ide.f fnding the effectS of a 10% or )20% hange in any of the chemi al rameters, so as to take ac t of any laer more a turate determination of these paraeter*.. The end result of these t CLulations will be to calCulate the heat t isfer and skin frictio,. at the plate sufae. For this first problemy the paraters are MaEh nmber and exte rnl flw desity and teperature (altitude) It is interesting to note that, for the case of chemical equilibrium, the bouidaries of the reaction ne are lines of j = const.. *-j say; Hence, once these lines are foundj i e., the reaction zone solved, and the temperature and slope of the teperature profile are known, the problem resolves itself in to finding the heat trasfr in a fictitious boundary layer with no reaction, which has the given alue of temperature and temperature derivative, at the given Iq* closest to the plate surface. This means, physically, that the new problem consists of aregular boundarylayer problem with a new external stream temperature id a new thermal boundarylayr thicesse these being set by the temerature ad temperature derivative at the r = r* closest tohe surface of the plate. Since the thickness of the thermal boundary layer is assoc iated with the Mach number or the Prandtl number, this. indicates that the solution of the boundary layer with dissociation following.hemical equilibrium can. be found by tosidering a boundary layer with no dissociation with a new stream temperature and with either a new stream Mach numbery or pe'rhaps a new Pra.ndtl number (i,, different fluid)* This iirplies that a type of similarity exists between high speed flows with dissociation, and those without dissociation, which involves chsnging the external conditions to tae accot of the dissociation. It may be that a change of working fluid can be acunted.for; this would mean that testing could be done with a nondiss$ciating medium which would mean a great saving in time and effort. This should be investigated more Finally, the above studies can be carried out for the case of a dissociated free stream, where the dissociation is caused by a.nOrmal shock in front of the flat plate ad also for the stagnationpoint region of a blunt body, where there must be some surface reaction. In the latter case, an at25

The University of Michigan * Engineering Research Institute tempt can be made to calculate the extent of such surface reaction, the end of such a region being the beginning of the buffer-layer region as described above PHYSICAL PROPERTIES OF AIR Information about the physical properties of air is necessary for the solution of aerodynaic problems These properties will change at high temperatures, The oxygen in air at normal pressure begins to dissociate (^1l atomic oxygen) at 2500~EK and is completely dissociated at 10,000~K. Nitrogen starts dissociating at ^-35000K. Ionization of NO starts at about 6000~K, and ^50% of air at 25,000~K is made up of ionized particles, The earliest computations of thermodynamic properties of air in thermal equilibrium were presented by Bethe.l8 He obtained the properties of air from the quantum mechanical partition functions based on the rigid rotatorharmonic oscillator with a correction for nonrigidity and anharmonicity, Kreiger and Whitey20 Gilmore,32 and Logan55 employed essentially the same approach, Combining constituents into a mixture based on the ideal-gas law, Bethe neglected the effects of NO formation and re r and White neglected ionization. Logan used: spetroscopic data instead of the rigid rotator. harmonic oscillator model for the energies A different approach is that of Hirschf elder and Curtiss,19 who em-* ployed the Beattie-Bridgeman equations of state for high pressures and the Virial equation of state for lower pressures The second Virial coefficient was computed from thne Lennard-Jones 6-12 potential. Hall and Ibell also used the Virial equation of state and computed the second., third and fourth Virial coefficients from the Lennard-jones 6-12 potential, but did not consider dissociation. Hilsenrathy et alS.r employed a similar method also making use of PVT and other experimental data ilsenratn h and Beckett57 computed.thermodynmic properties from ideal-gas thermal functions of constituents and. quilibrium concentrations. Their work is being continued to include more than just a mixture of nitrogen and oxygen. They assumed an ideal-gas mixture of oxygen and nitrogen increased proportionately to compensate for other constituents notincluded at the time., Only References 31 through 59 include the presently accepted -energy of dissociation for nitrogen (9.759 ev). All earlier references employed the lower energy of dissociation (7.37 ev). Tabulations of transport properties of air are rather limited. Viscosities, conductivities, and Prandtl numbers have been computed based on empirical expressions (References 30, 335) or the Lennard-Jones 6-12 potential and the Eucken relationship (References 19, 21, 235, 26). 26

The University of Michigan * Engineering Research Institute For the most part transport properties are unknown at high temperatures due to the lack of experimental data and the unknown aspects of atommolecule interactions. A discussion of interaction parameters for atom-molecule mixtures is presented in Reference 38. Some predictions have been made (Reference 39) which indicate that the Ptandtl number and viscosity do not vary much from the predicted by simple models at temperatures below 90000K. Computations based on the Lennard-Jones 6-12 potential have given relatively (at moderate temperatures) usable results. Many of the data, such as force constants and collision integrals, necessary for such computations, are tabulated in Reference 1. It would seem, then, that the use of the Lennard-Jones potential would at least be a consistent and reasonable approach to the problem of computing physical properties. Table I presents a summary of results of computations available. The basis of the method employed is indicated. EXPERIMENTAL TECHNIQUES FOR THE STUDY OF HYPERSONIC FLOWS A great deal has been written about the difficulty encountered in attempts to simulate hypersonic flow problems in the laboratory. The wide array of similarity parameters to be matched (ratio of specific heats, Mach number, Reynolds number, Nusselt number, Schmidt number, etc*) presents a formidable task. In addition, it now becomes imperative to consider the effects of dissociation, vibrational excitation, electronic excitation, and ionization along with the attendant relaxation times These effects are not scalable but rather depend on local temperatures and pressures. Instrumentation that might be utilized in assessing-these effects is greatly limited and obtains but questionable accuracy. In view of the complicated fluid-mechanic and chemical interrelationships, it is not surprising that different experimental techniques lend themselves more readily to specific areas of experimental interest. Accordingly, it is intended to review briefly those techniques currently being used in the study of hypersonic flow as well as some of the newer techniques suggested by various scientists. Some consideration will also be given to the instrumentation available for these studies, limited though it is. As of this time the authors are not prepared to specify one particular research tool as the optimum for studies of the hypersonic laminar boundary layer. Pertinent references to the above area are included in the bibliography. 27

TABLE I Reference No.: 18 19 20 21 22 23 24* 25 26 27 28 29* 30 31 32 33 34 35 36 37 38.3 5000'K 273'K 500'K 1500~K 40~C 0 100~R 0~K 300~K 0OR 300'K 298~K 100-1500 0 1000~K 50~K to 1000 2000 2000 2000 250 0 Temperature to to to to to to 100- to to 3000~K + to to. 25,000~K 5000~K 80000K 300~K 2000~K 5000'R 6000~K 3000~K 14,000~R 7000'K 5000~K 20000K 5000~K 24,000'K 5000K 12,000~K 10,000'K 8000~K 15,000~K 3360~K 9000K 1 to 20.008 to.02-9.0 10 -10 X 10-106 106 —10 104-102 Densityx st. 25x std. lb/fts x std.. x s td. x s td. x Air Air 02,N2, C02,H20, Air Air Air 10 of N, 02,N2,NO, Air + 29 Air,A,C02 OQases 02 Air N2,02 NO,C02 C02,H2 45 N 42 + Air /A-M\ 0,H,C HCO,C2, H2io2 constit- CO,Ha,N2, Air Air Air N2+2 N2 NO,N,O,A 02-N2 mix H2,H20 gases 02 gases H20 \mix/ combi- H20,He,A N2 uents 02,H20,0, mix nations N lO1'-102.2-30 1-104 10-x 1-200 1'o8-100.01-100 X pre88ure x atm. 150 ~H6 ^^ X arm. atm. X X X Pressure t. x atm. atm. atm. atm. x atm. atm. 0 Enthalpy XX X X X X X X X X Internal Energy X X X X X Entropy X X X X X X X Composition XX X X X X Compressibility F. X X X X X K Specific Heat X X X XX Specific Heat Ratio X Z Velocity of Sound X X Viscosity X X X X X X Thermal Conductivity X X X Free Energy X Prandtl No. X X X 3 Dissoc. EnergyX X K Ionization Energy X X Fraction of Diatomic Molecules in Each Electronic State Diffusion Coefficient X X Equilibrium Constant X X X Schmidt No. (r/pD) X X lewis No. (x/pCpD) X X Survey Ideal as X X X X X Ref. 32 X X Partition Functions X X Ref. 18 Rcf. 20 X X Virial Eq. of State X X Ref. 33 X Lennard-Jones (6-12) X X X X X Wrong energy of dissociation for N2 used before Reference 31. No dissociation.

The University of Michigan * Engineering Research Institute HYPERSONIC WIND TUNNELS Some work has been done toward extending the Mach number range of conventional blowdown supersonic wind tunnels by increasing the reservoir conditions of pressure and temperature. Noteworthy among these types are the installations at NACA Langley Field and at Brooklyn Polytechnic Institute. The former utilizes a zirconium pebble-type heat exchanger that has anticipated capabilities in the order of 4500~K stagnation temperature, The Brooklyn facility utilizes a pebble-type heat exchanger to charge an adiabatic compression tower that drives a hypersonic tunnel at anticipated temperatures up to 8000~K and pressures of 3500 psi. Even so, such schemes do not simulate flight conditions in the high Mach number range (15-20). A modification of the methods above is to use a monatomic gas, rather than air, as the working medium. In this way much lower test conditions of temperature and pressure may be utilized without the danger of component liquefaction. The experimental results are also more readily evaluated as there are no dissociation effects to assess. The latter consideration presents a., strong objection to this technique. However, many tests can be made at high Mach numbers to study the Mach number effect without the complication of dissociation. Toward this end, Bogdanoff and co-workers at Princeton University have developed a hypersonic tunnel which utilizes helium as the working gas. SHOCK TUBES AND SHOCK TUNNELS Probably the greatest source of experimental information in the hypersonic range to date has been the shock tube and its modifications. In essence the technique is simple and inexpensive and lends itself to a wide range i of experimental investigations. Tn the simple shock tube (a low-pressure test gas separated by a single diaphragm from a high-pressure, high-temperature reservoir gas), the usable testing region is that uniform zone immediately behind the shock wave and preceding the contact surface which separates the original test-section gas from the reservoir gas. Although high stagnation temperatures may be realized in this zone (and hence some meaningful experiments may be conducted), the maximum obtainable Mach number is in the low supersonic range. Furthermore, the time duration of this uniform flow condition is extremely short, about 100 microseconds being the greatest realized to date. To circumvent these limitations, Cornell Aeronautical Laboratory and the Naval Ordnance Laboratory have modified the simple tube by using an-expansion nozzle on the downstream end of the tube which serves to accelerate the low supersonic gases to extremely high Mach numbers. Further performance increases are achieved by using the hot combustion products of hydrogen and oxygen as the reservoir gas. 29

The University of Michigan * Engineering Research Institute ARC DISCHARGE TUNNEL A novel hypersonic tunnel, now in operation at the Arnold Engineering Development Center, consists of a conical-shaped nozzle wherein extremely, high stagnation conditions are realized by discharging stored electrical energy across a gap in an air-filled container. Using such techniques, stagnation temperatures of approximately 15,000~K and stagnation pressures of 20,000 psi have been obtained so that test Mach numbers of 15 still yield essentially the correct static temperature desired for simulation.'Furthermore, the run time is greatly extended over that of shock tubes. The difficulty lies in the fact that the test-section conditions of pressure and temperature are continually changing with time due to the blowdown-type operation. Furthermore, Mach number changes somewhat with time as the throat of the nozzle is gradually enlarged due to vaporization. A further disadvantage is incurred by vaporization of the discharge terminals, which serves to contaminate the air stream. Preliminary results that are available to date lead the associated tunnel personnel to believe that chemical equilibrium is attained throughout the nozzle and that the flow is approximately isentropic. It appears that a larger volume of reservoir gas and a prolonged spark duration could make this method very promising. HYPERVELOCITY PELLETS Considerable progress has been made toward firing bullets or pellets at extremely high velocities. Such a technique is especially attractive in that the free stream conditions for high speed flight can be duplicated quite accurately and thus lend credence to the results obtained. Shaped charges, hydrogen guns, helium guns, nylon pellets, in addition to the innovations, have been utilized (chiefly by personnel of the Naval Ordnance Laboratory at White Oak, Maryland, and of the Naval Ordnance Test Station at Inyokern) to attain these extremely high velocities. The chief disadvantage to these techniques is the limited instrumentation that can be applied profitably. To be sure, the pelletst, path can be photographed at a number of stations and the velocity and shock patterns determined. Different gases can be used to eliminate or vary the dissociation effects as evidenced in changes in the shock-wave stand-off distance and the characteristics of the wake. However, any measurements on the skin of the pellet or on the boundary-layer characteristics would appear to be extremely difficult, if not impossible. The shape of the pellet would also be quite limited OTHER METHODS In addition to the above outlined techniques that can be applied to the study of hypersonic flows, several variations and new methods have been - ----------- 30

The University of Michigan ~ Engineering Research Institute suggested and are in the process of experimentation. Among these are the application of an electromagnetic field to the acceleration of an ionized gas and electromagnetic fields applied to the acceleration of small missiles in a tube. Some of these methods show great promise although the equipment becomes quite prohibitive in cost. No authentic flight data have been obtained in such devices to date as far as the authors know. A possible variation of the shock tube for purposes of hypersonic research and which to the authors' knowledge has not been tried as yet, is to use a gaseous detonation wave as the reservoir gas and to test in the detonation products immediately behind the contact surface. Preliminary idealized calculations indicate that this zone is typified by high Mach numbers and stagnation temperatures (the latter because of the detonation process as well as the increase in stagnation temperature due to the unsteady expansion). Even if the predicted high Mach numbers could not be obtained (as is the case in conventional shock tubes for the gases in this zone), the stagnation temperature should be extremely high and hence an expansion nozzle could be utilized on the end of the tube. Admittedly, the test gases are not air but combustion products of water vapor, carbon dioxide, monatomic and atomic oxygen, hydroxyl radical, etc. This would, of course, limit the area of application of this technique. Howeveritay find unique applications for specific experimental studies. Several other modifications of the shock tube have been suggested. One method is the use of the nonsteady expansion region behind the contact surface as mentioned above in the description of the detonation tube arrangement. The difference here is the use of a second diaphragm in an additional driver section (air in the section between diaphragms)so that the stagnation temperature in back of the second diaphragm would be increased. There is evidence, however, that the theoretical conditions behind this interface may be difficult to achieve experimentally. -The use of a shock tube with a contracted low pressure region to make use of reflected shock waves for increasing stagnation temperature is another possibility. In general, there are many possibilities of employing expanded or contracted sections at various initial conditions combined with one or more diaphragms to obtain a variety of test conditions. Strong shocks may also be produced by means of other drivers such as spark discharges or high explosive charges. Of course, simplicity is advantageous, but may have to be sacrificed to obtain desired Reynolds numbers and static temperatures at high Mach numbers. INSTRUMENTATION The tests of interest in hypersonic flow studies involve extreme temperatures and complicated chemical changes in the composition of the air, This consideration, along with the extremely short running times available in hypersonic facilities, seriously restricts the type of transducers that can be emL ployed. In shock-tube applications the thin film resistance thermometer has 51

The University of Michigan * Engineering Research Institute been of some value and response times in the order of one microsecond are reported. One limitation for its use in the shock tub is its fragility. As a result Rose and Stark at Avco have been developing a calorimeter gage which is based on an "infinitely thick" sensing elements. Hot wires were also found to be too fragile An additional problem in the case of the hot wire is the necessity of knowing the density to determine the temperature. Optical methods for the determination of density such as interferometry and x-ray absorption techniques are affected by the composition of the gas employed, Dissociation would therefore be a problem. Determinations of shock-wave positions by schlieren or shadowgraph as a measure of real-gas effects is a possibility. Spectroscopy also offers some hope. High response pressure transducers are also available but again are subject to many doubts. Usually these transducers are subject to temperature or a temperature gradient and it becomes difficult to obtain a valid dynamic calibration at the high temperatures. 52

The University of Michigan * Engineering Research Institute APPENDIX VISCOSITY OF THE MIXTURE The viscosity of a mixture has been formulated analytically by Bromley and Wilke.17 Following their development, one defines the mixture viscosity as follows: Zc^ = 2;7 -— ^ - -- 4 (A-l) where ~ Zx - M/ = /e Zrcf/bz (A-2) i' /^ and f (jV For the present case, n = 3 with M1 Ms3 and l = s3j with M2/M1 = M2/Ms = 1/2. With these simplifications, (^~~~~~ ~~~~~~(A-5) ^ ^-^'^/y ^-^/^^r~y-^

The University of Michigan * Engineering Research Institute Hence, all the cij can be written in terms of ~12 or a constant. Substituting for the (ij, one can find the viscosity of the given mixture in terms of the viscosity of the molecular oxygen or nitrogen, pi, and the atomic oxygen, p2.l ~ =,H, 2, - 1'f-, --—' ^-A- 1 (A-4) In terms of mass concentrations, (A-4) becomes + * ct/ 3 7S1/?6LL ) 7 (A-5) -^t - e /t - e) 2 A- ) *( 4x)3)0zhj And finally, in terms of the variables used in the text, where / = 1 - = /.i +3^(, equation (A-4) is as follows: Az=^/,' h X^/^ f/ / ) (A-6)

The University of Michigan * Engineering Research Institute REFERENCES 1. Hirschfelder, J. O,, CurtisS, C. F., and Bird, R. B. Molecular Theory of Gases and Liquids. New York: John Wiley and Sons, Inc., 1954 2. Chapman, S., and Cowling, T. G. The Mathematical Theory of Non-Uniform Gaseso 2nd ed. New York: Cambridge University Press, 1952. 3. Shen S. F. The Steady Laminar Boundary Layer Over a Flat Plate with Injeetion of a Different Gas. NAVORD Report 4235, May 9, 1956, 4. Lees, L., "Laminar Heat Transfer Over Blunt Nosed Bodies at Hypersonic Flight Speeds," Jet Propulsion, 26 (April, 1956), pp. 259-269. 5. Chapman and Cowling, p. 255. 6. Lees, loc. cit. 7. Ibid. 8. Ibid. 9 Hirschfelder, J. 0. Curtiss, Co F,, Hamnel, H., Adams, E.; Henkel, M. J., *and Spaulding, W. "Theory of Propagation of Flames, Parts I, II, III." Third Symposium on Combustion Flame, and Explosion Phenomena (Baltimore: Williams and Wilkins Co., 1949), p. 124. 10. Hirschfelder, J. O. Heat Transfer in a Chemically Reacting Gas Mixture. University of Wisconsin, WIS-ONR-18, February 6, 1956, p. 15. 11. Penner, S. S, Chemical Reactions in Flow Systems. AGARDOGraph 7, Butterworths Scientific Publications, 1935, p. 8. 12. Hirschfelder, Heat Transfer in a Chemicly Reacting Gas Mixture, pp 15-16 13. Dorodnitzyn, A., "Laminar Boundary Layer in Compressible Fluids," C. Ro Academy of Sciences (U.RS.So,), 34 (1942), ppo 213-219. 14. Lees, loc. cit. 15. Probstein, R. F. The Effect of Variable Fluid Properties on the Equilibrium Laminar Boundary Layer Surface Heat Transfer Rate at Iypersonic Flight Speeds. WADC Tech, Note 56-2, December, 1955. 35

The University of Michigan Engineering Research Institute REFERENCES (Continued) 16. Ibid. 17. Bromley, L. A., and Wilke, C. R., "Viscosity Behavior of Gases," Industrial and Engineering Chemistry 43 (July, 1951), p. 1641. 18. Bethe, H. A. The Specific Heat of Air up to 25000~C OSRD Report No. 569, February, 1942. 19. Hirschfelder, J. 0., and Curtiss, C. F. Thermodynamic Properties of Air, IIo University of Wisconsin Naval Research Laboratory Department of Chemistry Report CM 518, December, 1948. 20. Kreiger, F. J., and White, B. W. The Composition and Thermodynamic Propertie of Air at Temperatures from 5000 to 8000~K and Pressures from 10 to 102 Atmospheres. RAND Corp. Report R-149, April, 1949. 21. Hirschfelder, J. 0., Bird, R. B., and Spotz, E. L o, "Viscosity and Other Physical Properties of Gases and Gas Mixtures," Trans. ASME, 71 (November, 1949), p. 921. 22. Wilke, C. Ro, "Diffusion Properties of Multicomponent Gas," Chem. Eng. Prog., 46 (1950) p. 95. 23. Bromley and Wilke, loc. cit. 24. Hall, N. A., and Ibele, W. E. Thermodynamic Properties of Air, Nitrogen, and Oxygen as Imperfect Gases University of Minnesota Technical Paper No. 85, December, 1951. 25. Huff, V. N., Gordon, S., and Morrill, V. E. General Method and Thermodynamic Tables for Computation of Equilibrium Composition and Temperature of Chemical Reactions. NACA TR-1037, 1951. 26. Knuth, E. L., "Note on Calculation of Transport Properties of Gas Mixtures. J, Aeronaut. ci., 19 (September, 1952), p. 644, 27. Moore, L, L,, "A Solution of Laminar Boundary Layer Equations for Compressible Fluid with Variable Properties Including Dissociation," J. Aeronaut. Se,, ~1 (August, 1952), p.505. 28. Hansen, C, F., "Note on the Prandtl Number for Dissociated Air," J. Aeronaut. Sci., 20 (November, 1953), Po 789. 29. Noeggerath, W. C, General Enthalpy-Temperature -Entropy Diagrams for Ideal Gaes and Gas Mixtures up to 5000~K. NAVORD Report 3544, June, 1954. 36

The University of Michigan * Engineering Research Institute REFERENCES (Concluded) 30. Hilsenrath, J., and Touloukean, Y. S., "Viscosity, Thermal Conductivity, and Prandtl Number for Air, 02, N2, NO, H2, CO, C02 H0O, He, and A," Trans. ASME, 76 (August, 1954), p. 967. 31. Wooley, H. W. Effect of Dissociation on Thermodynamic Properties of Pure Ditatomic Gases. NACA TN-3270, April, 1955. 32. Gilmore, F. R. Equilibrium Composition and Thermodynamic Properties of Air to 24,000~K. RAND Report RM 1543, August, 1955. 33. Hilsenrath, J., et al. Tables of Thermal Properties of Gases. NBS Circular 564, November, 1955. 34. Bond, J. W., Jr., and Dyer, J. N. Equilibrium Comosition of Air at Various Densities and Temperatures. Lockheed Aircraft Corp. Report MSD 1487, December, 1955, 55a. Logan, J, G. The Calculation of the Thermodynamic Properties of Air at High Temperatures. Cornell Aero. Lab. Report AD-1052-A-1, May, 1956 b. Treanor and Logan, J. G. Tables of Thermodyamic Properties of Air from 30000~.l0,O00K. Cornel Aero. Lab. Report AD-1052-A-2, June, 1956. c. Logan, J. G. Thermodynamic Charts for High Temperature Air Calculations (2000~-9000~K). Cornell Aero, Lab. Report AD-1052-A-5, July, 1956. 36. Dommett, R. L. Thermodynamic Properties of Air at High Temperatures. Royal Aircraft Establishment, Tech. Note No. GW429, August, 1956. 37. Hilsenrath, J., and Beckett, C. W. Tables of Thermodynamic Properties of Argon-Free Air to i O5000~K AEDC TN 56-12, September, 1956. 38. Lees, loc. cit. 59. Fay, Jo A., and Riddell, F. R. Stagnation Point Heat Transfer in Dissociated Air. Avco Research Note 18, June, 1956, p. 13537

The University of Michigan * Engineering Research Institute BIBLIOGRAPHY 1.. Smelt, R., "Test Facilities for Ultra-High Speed Aerodynamics," Proceedings of the Conference in ih Speed Aeronautics, Polytechnic Institute;of Brooklyn, January 20-22, 19550 2. Libby, P.Ao., and Bloom, M. H,, "Facilities for Experimental High Speed Aerodynamic Research at the Polytechnic Institute of Brooklyn," Proceedings of the Conference in High Speed Aeronautics, Polytechnic Institute of Brooklyn, January:20*.22, 1955. 3. 3.Bloxsom, D., E. Production of HighTemperature Moderate Pressure Gases Means of Electrical Spark Discharge. AEDC, TN 5617, November, 1956 4, Bloxsom, D, E. Gas Purity of HighTeaperature Hi gh Pressure Electrical Dis.charges in Air. AEDC TN 56-15, ASTIA AD:98977, November, 1956o 5.-, Dodge, J. A. traHigh Temperature Aerodynamic Testing Facilities. AEDC.TN54-61-955. 6. Bloxsom, D. E., and Perry, R. Wo,'"Use of Electric Arc Heating in Ultra Temperature Aerodynamics," Phys. Rev.. 100 (November, 1955), p 968. |7. Smith, H. Le, and Early, H, C. Investigatton of Heatin an Air Stream in a Winad Tunnel,] Electrical Discharge The University of Michigan, Engi.neering Research -Institute, 2154-35F, October, 1954o 8,.8Glass, I. I, and Patterson, Go -N,, "A Theoretical and Experimental Study of Shock Tube Flows," Jo Aeronaut. Sci, 22 (February, 1955), p. 73 9o.Hertzberg, A,, "The Application of the Shock Tube to the Study of High Tem-| perature Phenomena in Gases," Ale.o Mecho Rev., 9 (December, 1956), p. 505. 10. Reslu, Lo, and Kantrowitz, A,, "The Production of High Temperature Gases in Shock Tubest J..A o Ph ys, 23 (December, 1952), po 1.90,;1. HEertzberg, A., "The Application of the Shock Tube to the Study of the Problems of Hypersonic Flight," Jet Propulsion, 26, (July, 1956), p. 549. 12. Hertzberg, A,, and Smith, Wo Eo, "Methods for Generating Strong Shock Waves' J al Phys.., 25 (January, 1954), p. 1350 -------- 58 -

The University of Michigan * Engineering Research Institute BIBLIOGRAPE. (Continued) 13. Fowler, Re.G., Goldstein, Jo S., and Clotfelter, B. E., i"Luninous Fronts in Pulsed Gas Discharge," Phys. Rev., 82 (June, 1951), p-. 879. 14. Fowler, Ro G., Atkinson, W. R., Compton, Wo D., and Lee, B. J.., "Shock WaVe in Low Pressure: Spark Discharges,"Phys. Rev, 88 (October, 1952), p..137e.15. Shreffler, R.. G., and Christian, Ro H., "Boundary Disturbances in High Explosive Shock Tubes," Jo Appl Phy, 2 (March, 1954), po 324 16. GlickH,H.S,. Hertzberg, A*, and Smith, W. Eo Flow Phenomena in Starting Bypersonic Shock Tunnel. Cornell Aero. Laba Report AD 789-A-3, AEDC-TN-5516, March, 1955. 17. Squire, Wo Hertzberg, A., and Smith, Wo E. Real Gas Effects in a eyr;-; so.nic Shock Tunel. AEDC-TN-55-14, March, 1955. 18. Hertzberg, A, Glick, H.o S, Squire, W,, and Smith, W. E,.Modifications of the Shock Tube for the Generation of Hypersonic Flow. AEDC-TN*55-15, March, 1955. 19. Bromberg,,,R, "Use of the Shock Tube Wall Boundary Layer in Heat Transfer Studies," Jet Propulsion, 26, (September, 1956), po 737..20. Trimpi, R L.,, and Cohen, R, B..A Theory for Predictig the Flow of Real Gases in Shock Tubes with Experimental Verificatin o L.angley, NACA-TN 3375 March, 1955. 21. Bird, Go A.A Note on. Multiple Diphrag Shock Tubes. Royal Aircraft Establishment Tech Note No. Aero. 2469 October, 1956o 22,.Mirels, Ho Attenuation in a Shock Tube due to Unsteady Bounda Layer.Action. NACA TN-3278, August, 1956o 235.,Lundquist, Go A. The NOL 8 x 8 Inch Shock Tube Instrumentation and..'.......... ~;' —... -.-.- - Operation. NAVORD Report No 2449, 19520 24.. Kurzweg, H, Ho, and Wilson, Ro E.,'Experballisticst.Aeronaut En. Rev., 15 (December, 1956), p. 532.25. Allen, W.-A,, Rinehart, J. S., and White, Wo Co, "Phenomena Associated with the Flight of Ultra Speed Pelletso Part I, Ballistics,n Jo Ap3.p Phys, 23 (January, 1952), p. 152. 39

The University of Michigan * Engineering Research Institute BIBLIOGRAPHY (Continued) 26. Allen, W. A., Rinehart, J. S., and White, W. C., "Phenomena AssOciated with the Flight of Ultra Speed Pellets. Part II, Spectral Character of Luminosity," J. App hys., 23 (February, 1952) p.. 198. 27. Bennett, S. D., Carter, W. C. and Bergdolt, V. E., "Interoermeter Analysis of Airflow About Projectiles in Free Flight, J..Ap. Phy 2 (April, 1952), p. 453..28. Schwartz, R. N. and Eckermann, J. "Shock Location in Front of a Sphere as a Measure of Real Gas Effects," J. App. Phys., 27 (February, 1956), p. 169. 29. Rabinowicz, J., Jesse, M. E.o, and Bartach, C. A..Resistance Thermometer for Heat Tanser Meeasureants ina Shock Tube, ypersonic Research Project, GAI IT, AD 103315, July, 1956..30, o HOse P., H and Stark, W. I. Stagnation Point Heat Transfer Measurements in Air at High Teperature. AVCO Research Note Noo 24, December, 195 6 31. Bendersky, D,. "A Special Thermcouple for Measuring Transient Temperatur'es,. Mech... 7..5. (F5ebruary, 1953), P..117.. 32,.Marlow, D. G., Nisewenger, C, R., and Cady, We.M., "A Method for the Instantaneous Measureent of Velocity and Temperature in High Speed Air Flow J. A ppl sPhyD, 20 (August, 1949), p. 771. 33. Yolar, Y. A., and Nagamatsu, H. T. A StU4y of Piezoelectric Elements for the Measurement of Transient Forces. GALCIT Hypersoni Wind Tunnel, Memorandu No. 23, February, 1955..54 Shunk, R. A., Dranetz, Ao. I, and Budenstein, P.P., "RingS*Shaped Piezolectric Gauge for Shock Tubes," Rev.. Sci-, Inst., 24 (Novet.er, 1953), p. 1069. 35. Blue, Ro o. Inteferf eter Correcti-ons ad Measureents of Laminar Bo Layers in a Suersonic Stre NACA T -2110, June, 1950..36.Ashkenas, H, I,, and Bryson, Ao E., "Design and Performance of a Simple Interferometer for Wind-Tunnel Measurements, J Aeronaut.S ci., 1 (February, 1951) p. 82.,37.Rosa, R. J., "Shock Wave Spectroscopy^,".. R.ev.,,(July, 1955), po 635 40

The University of Michigan * Engineering Research Institute BIBLIOGRAPHY (Concluded) 358. Baloga, P. E., and Nagamatsu, H. T. Instrumentat-ion of Galcit Hypersonic Wind Tnnels. GALCIT, AD 70405, Memorandu Noo 29, July, 1955. 39. Dhawan, S. Dircot Measurement of Skin Friction, NACA TN-2567, January, 1952.

The University of Michigan * Engineering Research Institute DIorodnit yn1 ransfora3:ition transfmatio W. wt i iatr hi tran ration:effectively deoquples the momentum and energy euatio since'p. cm- nstant with -dissociatio, since -p, constant, the u s ar not.cpleel eco0upled< and the new plpne isot c3. plete i sib lesb o Hwer, as ot|her authors have pointed out, h1 the ffects due to the:e variat4ls are small- an d can e taken care of by an ite-ative aolution. Of the entum equ tion and e:ergy e:quation If Xt, y, u, a.d v are the cordinwates and velocities in the cor evariables;wth.the following relationso |z)j Pez b) 2C =o I e) 61 = jf 1 |ere pD iis.th desity along the of the b dary layer The |~l,@city relati'ons aa.re fod by defining the stream f.uXction in the: usual way: %, = dg d) t(42) and performing the given transftoration of variables. That is: /24 \= aO A A r ^ = a z Also ) 1i8 I ) /aC_ ~fedS f 4J --------------------- IQ ----------------