THE UNIVERSITY OF COLLEGE OF ENGIN] Department of Aeronautical and Ast: Aircraft Propulsion L: THE EFFECT OF THE BOUNDARY LAYER UPON THE FLOW IN A CONICAL HYPERSONIC WIND TUNNEL NOZZLE by Martin Sichel ORA Project 02953 under contract with: Department of the Navy Office of Naval Research Mathematical Sciences Division Washington, D. C. Contract No. Nonr-1224(31) NR 061-108 administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR July 1963

TABLE OF CONTENTS Page LIST OF FIGURES............ iv SUMMARY............. V ACKNOWLEDGMENT.................. v NOMENCLATURE.................. vi I. INTRODUCTION................. 1 II. FLOW IN THE HYPERSONIC NOZZLE......... 3 A. Formulation of Integral Equations for Nozzle Flow 3 B. Discussion of the Nozzle Integral Equations 8 C. Approximate Hypersonic Solutions of the Nozzle Momentum Integral Equations 17 III. CALCULATION OF HYPERSONIC NOZZLE FLOW USING EMPIRICAL RELATIONS FOR THE BOUNDARY LAYER DISPLACEMENT THICKNESS............ 29 IV. DISCUSSION AND CONCLUSIONS........ 0... 37 REFERENCES.................. 42 APPENDIX A APPENDIX B APPENDIX C iii

LIST OF FIGURES Figure 1. Conical Nozzle Boundary Layer Coordinate System. 2. Approximate Nozzle Velocity Distribution for the Case 0 < 1. w 3. Arc Chamber Conditions for Equilibrium Air. 4. Variation of Test Section Boundary Layer Thickness with Stagnation Conditions. 5, The Mach Number Function 0 for Equilibrium Air. 6. Variation of Test Section Mach Number with Stagnation Conditions. 7. The Effect of the Boundary Layer upon the Test Section Mach Number, 8. Variation of Test Section Static Pressure with Stagnation Conditions. 9. The Effect of the Boundary Layer upon the Test Section Static Pressure. 10. Variation of Test Section Static Temperature with Stagnation Conditions. 11. Variation of Test Section Density and Limiting Velocity with Stagnation Conditions. 12. The Effect of the Boundary Layer upon the Test Section Density. 6* A-lo Mass Flow Correction Factor E a;C-1. Area Ratio Correction Factor fL (ho po). C-2. Mass Flow Correction Factor f2 (hop0). C-3. Ratio of Average to Ideal Gas Specific Heat, f (p, h*). C-4, Variation of the Boundary Layer Thickness with the Function P. iv

SUMMARY Integral equations for the flow in a conical hypersonic nozzle with a thick boundary layer have been formulated and some approximate solutions of these equations have been studied. A method developed by Burke for calculating boundary layer thickness using empirical correlations has been extended to a relatively broad range of stagnation conditions and has been used to compute the effect of boundary layer development upon hypersonic nozzle flow. ACKNOWLEDGMENT The work reported here was sponsored by the Office of Naval Research under contract Nonr-1224(31) NR 061-108 for the "Study of High Velocity Flow with Dissociation and Ionization. " The author would like to thank Professor P. M. Sherman for her help and encouragement, and Mr. Stanley Wu who carried out the bulk of the often tedious calculations. v

NOMENCLATURE At throat area A geometric area geom A e effective area eff a atom mass fraction Cp specific heat at constant pressure cf skin friction coefficient 6 boundary layer thickness 6* displacement thickness 62 momentum thickness 63 energy thickness 2 6* 6*- 6* /2R 5 ^ 6- 6/2R 2 62 2 2 2/2R 63 63 63 /2R A thermal boundary layer thickness e stagnation internal energy per unit mass e. initial internal energy per unit mass H 6*/62 H 68*/62 H. incompressible value of 6*/62 0 useful core angle uc w nozzle wall angle w h adiabatic recovery enthalpy aw h stagnation enthalpy h enthalpy hA atom heat of formation per unit mass Le Lewis number M Mach number rh mass flow vi

NOMENCLATURE (continued) A1 viscosity v kinematic viscosity p pressure Pr Prandtl number q heat flux Q arc chamber volume c R nozzle radius Re Reynoldst number r radius rt throat radius p density St Stanton number T temperature T stagnation temperature T shear stress u velocity u free stream velocity V radial velocity in conical flow x distance from nozzle throat y distance from wall o viscosity index Y ratio of specific heats Subscripts and Superscripts aw adiabatic recovery e free stream or core R reference value uc useful core w wall ( )* quantities evaluated at the Eckert reference temperature vii

Io INTRODUCTION In supersonic wind tunnels the boundary layers on the walls of the test section are quite thin and so have only a small effect upon the test section flow. On the other hand in hypersonic tunnels for M > 4 the boundary layer occupies an appre1) 2, 3 ciable portion of the test section. and then the performance of a wind tunnel depends strongly upon the boundary layer behavioro In the present work the effect of the boundary layer upon the test section conditions in a hypersonic wind tunnel with a nozzle of fixed geometry is investigatedo In supersonic wind tunnels with fixed test section to throat area ratio the test section Mach number is constant and the pressure and temperature are directly proportional to the stagnation chamber pressure and temperature. The effect of the boundary layer on the test section flow is essentially of second order and is taken into account by adding the boundary layer displacement thickness to the perfect fluid nozzle contouro Even if this thickness is off by a factor of 2 the effect upon the nozzle flow will be extremely small. In hypersonic tunnels on the other hand the test section Mach number varies appreciably with stagnation conditions even when the nozzle geometry is fixed, and of course the relation between test section temperature and pressure and stagnation conditions is considerably more complex than in the supersonic case. This drastic change in behavior comes about because, at the high stagnation enthalpies needed to produce hypersonic flow, variations in the molecular weight and specific heat of the test gas during the expansion appreciably affect the flow, and because at these high Mach numbers and temperatures the boundary layer may occupy 50% or more of the test section area. Both of these effects are quite sensitive to the stagnation pressure and enthalpyo 4 In the design of hypersonic nozzles the contour of the inviscid core of the nozzle can be calculated using the method of characteristicso The boundary layer displacement thickness is then added to the core in order to determine the

hypersonic nozzle contouro The situation is quite different if the variation of test section conditions with stagnation conditions is to be determined for a fixed geometry nozzle. Then because of the preponderance of the boundary layer, and because of the interaction between the boundary layer and the core, the inviscid and boundary layer flows must be computed simultaneously. Thus just as in the case of external 5 hypersonic flows there is a strong interaction between the viscous boundary layer and the external inviscid flowo The computation of hypersonic nozzle flow thus poses many difficulties. The simple analytic formulae which apply to supersonic isentropic nozzle flows are no longer applicable, and to compute the flow in the inviscid core it becomes neces6-,19 20=27 sary to use either gas tables or charts or to employ empirical formu28-35 lae for the properties of the gas in questiono Further there is the question of whether or not the gases in the core are in chemical equilibrium or whether the 36-45 flow is frozeno Usually there is equilibrium until some point beyond the nozzle throat where then over a fairly short region freezing occurs. Finally in computing nozzle flow it is necessary to determine the thickness of the boundary layer. The stagnation densities under consideration in the present report are sufficiently high that, at least from the discussion of Enkenhus, it appears reasonable to assume that the boundary layer will be turbulent throughouto A laminar boundary layer comes under consideration only in the case of low density tunnels. Boundary layer thicknesses have been calculated by either integrating the VonKarman boundary layer integral equations starting at the throat and contrinuing to the test section 5 or by using empirical formulas to 1~ 3, 52, 53 compute the boundary layer thickness at the test section directly 9 9 5 Because of a lack of a detailed understanding of turbulence either method of determining the boundary layer thickness is inherently empiricalo In the work below the integral equations for the combined core and turbulent boundary layer flow in a conical hypersonic nozzle are derivedo Assumptions and empirical relations required to solve these integral equations are discussed; -2 -

however, an exact solution of these equations is beyond the scope of the present study. Finally, empirical boundary layer thickness correlations have been used to compute the effect of boundary layer development upon the performance of a specific nozzle over a wide range of stagnation conditions. II. FLOW IN THE HYPERSONIC NOZZLE A. Formulation of Integral Equations for Nozzle Flow The problem at hand is the simultaneous solution of the conservation equations in both the inviscid core and the boundary layer of the tunnel nozzle. 54 55 Schlichting and Atkinson and Goldstein have calculated the simultaneous development of the incompressible boundary layer and the core flow in a diffuser, and in the inlet section of a pipe. Burke3 appears to be the only one to have made a similar calculation for the flow in a hypersonic nozzle. Following Schlichting the conservation equations will be simultaneously formulated for both the core and boundary layer flowso In the present case the boundary layer thickness is of the same order as the radius of the nozzle, consequently the effect of transverse curvature, which has 56 48 been discussed by Durand and Potter9 and Michel, must be taken into account. The coordinate system used in the present case is the same as that used in References 48 and 56 and is shown in Figure 1o The analysis will be restricted to a concical expansion nozzle. The flow is divided into a useful inviscid core and a boundary layer of thickness 6. The mass flow will be computed across the surface S = constant in the boundary layer and the adjoining spherical cap of radius R in the inviscid core. If V is the component of the core velocity normal to the spherical cap, then the mass flow m is given by -3 -

5 uc m= 27T rpu dy + p V - 27 R sin 0 dO. (1) 27r rpu dy+ + 27r Pee p ej j e e c 0 0 In general the velocity distribution within the core must be computed by the method of characteristics. The flow through an inviscid conical nozzle will be of a source type, and it will be assumed that even with a boundary layer the core flow behaves as a source flow so that pe and V in Equation 1 are constant. There e e are some indications that this is not too unreasonable an assumption to make. As in References 48 and 56 the displacement thickness is defined by the relation 6* 6 27Trp u dy = 27r(pu - pu) dy (2) 0 0 Here u e, the velocity at the edge of the boundary layer, is u = V cos ( - 0uc) (3) e e w uc and since r R - y cos 0 (4) it follows from Equation 2 that 6 16*2 \2 r(pu -pu) dy = 2rpeV cos (0 - R6 6 *- -cos0 w (5) 0 Combining Equations 4 and 5 with 1 the following expression is obtained for mr -4 -

2h- 2e (cos + ~ m = 27rp V LRc (1 - cos 0 cos (0 - 0 ) e e c uc 2 W W UC //*2 - R6* - 2 Ccos os (0 - uc) (6) In Appendix A it is shown that to a very close approximation Equation 6 can be replaced by the much simpler approximate expression m =rpVe (R - 6*) (7) provided that 0 < 1. This generally is the case; for example, in the University Hot Shot Tunnel 0 = 7o 50 and the error introduced by using Equation 7 is of w the order of 1% assuming 6*/R - 0. 25 and 6/6* - 2. In formulating the momentum equation it will be assumed at the outset that 0 << 1 so that the velocity distribution across the nozzle will be approximately as shown in Figure 2. In view of the calculation of mass flow above, this assumption appears reasonable. Assuming that the pressure remains constant across the boundary layer application of conservation of momentum to a control volume of length dx (see Figure 2) yields the equation R dx w dd | 2Tpu2rdrj= - s7R dp- 7T27rRdx, (8) For the case 0 < 1 the relation between r and y becomes r R-y (9) and Equation 5 defining the displacement thickness 6* becomes 6*2 R i pu dr (1) 2R = R d 0 -5 -

where the subscript e refers to the isentropic core flowo It is in the present case expedient to carry the integration in Equation 10 from the tunnel axis to the wall rather than only across the boundary layero Since it is assumed that there is a uniform core this change from Equation 5 will not affect the results. Again as in References 48 and 56 a momentum thickness 62 is defined by the relation 62 6 27r(p u 2) dy = 2Tr rp(uu - u) dy. (11) 0 0 Changing variables and extending the integral on the right side of Equation 11 across the entire nozzle there results the relation 2 R 6 2 pu ur 62 2 2 = pu (1 - — ) rdr. (12) 2 2R 2-u R Pee e After some simple algebraic manipulation it is possible to show that R 2 pur dr - pue R 62 - Rpee 6 + 2 Pe (13) Je 2 e e -2ee 0 so that the momentum Equation 8 becomes d 2 d 2 d R 2 Rdp -d(p u R 6 ) + (Rp u 6* ) - d - R Pe d 4 dx (Pe e 2 dx e e dx 2 e e 2dx w From Equation 7 for the conservation of mass it follows that d - d 2 d (p u R 6*) 1 d (P R2 (15) Combining Equations 14 and 15 the momentum equation becomes -6 du dpc 2 -i+ 2 due 1 de d R P du -x +62 L u ^+ ~dx R dx + R x- 2 dx+ e dx ~ (16) dx ~ 2 du+ peu dxIeedx L e 2PeUe -6 -

If the core flow is isentropic, which implies either frozen or equilibrium flow, then du dp d e dx -eue dx (17) so that Equation 16 becomes d6 du 2 -H +2 e d cf dx + 62 u dx +dx(lnp R) = (18) L e where H = 6*/62. Thus, with an isentropic core the integral momentum equation for the entire nozzle reduces to the same form as the boundary layer momentum 48 integral equation. This result is identical with that of Schlichting for the incompressible flow in an axially symmetric diffuser. It should be observed that since the flow in question is turbulent, the fluid velocities used in formulating Equations 18 and 7 are average values, The effect of transverse curvature enters through the fact that Equation 18 is in terms of the quantities 62 and H rather than in terms of the usual parameters 6, 6* and H = 6 2' 62 If ho is total average enthalpy then application of conservation of energy to the control volume in Figure 2 yields. dx 27 rpuhodr = 27 Rq, (19) where q is the heat flux per unit area from the wall to the fluid. Definition of an energy thickness 63 by 63 63 A 27rp ueh oedy 27rrpu (h -h ) (20) 0 0 57 as is also done in Shapiro, leads to the relation -7 -

2 R 6 h 3 pu r dr (21) 3 2R =3 p u R h 0 e e In Equation 20 A is the thickness of the thermal boundary layero Using Equation 21 and the continuity equation, and assuming an adiabatic core flow so that hoe remains constant. Equation 19 can be recast into the form d6~ q d +3 +3 d _ _w (22) dx 3 dx n PeUeR) u h(22) e e oe Equation 22, which is valid for the entire nozzle, has the same form as the bound57 ary layer energy integral equation given by Shapiro. 5 B. Discussion of the Nozzle Integral Equations Determination of the nozzle flow now requires simultaneous solution of Equations 7, 18, and 22 for the conservation of mass, momentum and energy in order to determine the displacement and momentum thicknesses 6* and 62. Equations 18 and 22 are closely related to the conventional boundary layer integral equations; however, while u e, the external velocity, is usually known the nozzle problem is complicated by the fact that the variation of u is itself dependent upon the solution of Equations 18 and 22 for ue must satisfy Equation 7 for the conservation of mass. Methods of solving the turbulent boundary layer integral equations are discussed in References 1, 46, 48, 49, 51, 57, and 58. Because the relation between the turbulent transport of momentum and energy and the average fluid velocity is not yet understood, all of these methods require a number of empirical assumptions, which are discussed below. A key assumption in many boundary layer studies is that the velocity profile at any point in the boundary layer can be described by a universal relation of the form (-) = (23) -8 -

This assumption is verified experimentally for fully developed incompressible pipe flow and for the incompressible turbulent boundary layer on a flat plate. It is found that boundary layer and pipe velocity profiles are well represented by the relation5 1 ( = n (24) e where n varies from approximately 5-12 with increasing Reynolds numbero A universal logarithmic velocity profile valid over a wide range of Reynolds number also can be derived starting from the Prandtl mixing length hypothesis or from the VonKarman similarity hypothesis 58 Because of its greater simplicity, the power law profile of Equation 24 has been used in most investigations. The fact that Equation 24 is equally valid for flat plate and pipe flows means that this relation is unaffected by the transverse curvature of axisymmetric nozzle boundary layers. Even though Equation 24 is strictly valid only for pipe or flat plate incompressible flows, many investigators have used this velocity profile in the study of 1 4 48, 57 compressible turbulent boundary layers with a favorable pressure gradiento 57. An alternative assumption is based on the application of the Stewartson Transformation to the turbulent boundary layer momentum integral equations. In that case 63 49 Cohen and Reshotko and Tucker have assumed that the velocity profile satisfies Equation 24 in the transformed variables. The validity of this approach is in part supported by Spence's study of boundary layer data. Most boundary layer studies make the assumption that favorable pressure gradients do not have a major effect upon the boundary layer velocity distribution. In incompressible flow the momentum and displacement thicknesses can be determined as a function of the boundary layer thickness 6 once the velocity profile is knowno In compressible boundary layers, on the other hand, a knowledge of the density and therefore the enthalpy variation is also required. Thus the -9 -

energy and momentum equations are coupled and must be solved simultaneously. One approach to this problem which eliminates the need to directly solve the energy equation is to use the Crocco relation h -h o w = u (25) h -h u oe w e between the total enthalpy and velocity variation across the boundary layer. The Crocco relation is strictly valid only when free stream and wall properties are constant and when the laminar Prandtl number is equal to one. Equation 25 implies that thermal and velocity boundary layer thicknesses are equal. Even when free stream pressure is variable and when Pr X 1 the Crocco relation has been used in the modified form9 9 5 2 h - h + (h - h )- (h - h ) —, (25a) w aw hw) u aw eu e e where the recovery enthalpy, haw is given by 1/3 h =h + P /3 (h -h ) (26) aw e r oe e 52 60 Burke has advanced the argument, similar to that given by Lees for laminar flow, that for h/h aw < 1 the effects of favorable pressure gradients upon boundary layer behavior will be small so that then it is not unreasonable to extend the use of the Crocco relation. An alternative approach, which was used by Bartz, is to actually solve the energy integral equation. Thus Bartz assumed a power law of the form 1 T -T 7 oe (y (27) T -T IA oe w' for the total temperature variation across the boundary layer, where A is the thickness of the thermal boundary layero Clearly Equation 27 is closely related -10 -

to the Crocco relation. In order to solve the energy Equation 22 a relation between the wall heat flux and other boundary layer parameters is required. For this purpose Bartz used the modified Reynolds analogy. qw cf/2 eUe aw w A 0. 46 uhe -a h ) ()(28Pr for the case in which thermal and velocity boundary layer thickness are not equal. Even when the Crocco relation is used it still is necessary to use some form of the 5, 59 Reynolds analogy to compute wall heat transfer. Other authors ' have found that the incompressible form of the Reynolds analogy with a correction factor to account for diffusion effects remains valid so that f- -2/3 2/3 e aw St = Pr23 +(Le/3 -1) - h. (29) + (Le - aw w) Equation 29 is based on the assumption that air may be approximated as a binary mixture of atoms and molecules. a is the mass fraction of atoms hA is the 'A heat of formation per unit mass of atoms, and Le is the Lewis number. If Le = 1 or if a = a = 0 as is to be expected in the hypersonic portion of a wind tunnel e w nozzle, then Equation 29 indicates that diffusion has no effect upon the Reynolds analogy. For the solution of the momentum Equation 18 and for the calculation of heat transfer from Equation 29 a relation between the skin friction coefficient and the parameters of the flow is required. Since the mechanisms of turbulent transport are not yet understood it is necessary to introduce an empirical skin friction law, and this fact introduces the major uncertainty into the results of turbulent boundary layer calculations. 4 Measurements of fully developed pipe flows have provided the basis for the empirical power law expressions for the friction coefficient. ' In the incompressible case it has been found that experimental results for smooth pipes could -11 -

be correlated over a limited Reynolds number range by a relation of the form T u R-m C 1 - 2 A / ' (30) f 1 2e 2P Pe e where u is the velocity at the center of the pipe and R is the pipe radius. What is significant is that these laws remain valid for boundary layers provided that R and ue are replaced by the boundary layer thickness and the free stream velocity. Prandtl has shown that Equation 30 leads to a velocity distribution of the form m 2 - m =Ue) - I (31) which of course has the same form as Equation 24. By substituting Equations 30 and 31 into the momentum integral equation for a flat plate turbulent boundary layer it is possible to express the friction coefficient in the form 1m m Am m+ (m 1)( " m + m 1 m u x+ m+ 1 "f " ~ m (2- m) L e where x is the distance from the leading edge of the plate and it is assumed that the boundary layer is turbulent over the entire plate. In the case of the well known Blasius law with A = 0466 and m = 1/4 this leads to the result 1 5 c =. 0592 0^. (33) For a flat plate it also can be shown from Equation 3 that For a flat plate it also can be shown from Equation 31 that 62 m (2 - m) 6 2 (m + 2) so that the friction coefficient can be expressed in the form -12 -

, m (2 - m) _ 4 f = A 2(m+2) Re (34) 62 where u 6 PeUe 2 Re = 62 e Since Equation 30 has been found to apply both to pipe flow and flat plate boundary layers its validity should in no way be affected by the transverse curvature effect. On the other hand the derivations of the friction law in terms of the distance x from the leading edge or the momentum thickness are based on the assumption of two dimensional boundary layer flow, and therefore may no longer be applicable when 6/R ~ O(1). In Appendix B it is shown that in this case the relation between cf and x should, to first order in 6/R, take the form m U m+ 1 cfcf l1 k4 X ) 1 (34) where the constant k4 is defined in Appendix B, and cfp is the plane value of the friction coefficient given by Equation 33. Numerous authors have used the flat 4, 5, 46 plate relations in the analysis of nozzle boundary layers 9 and of course this procedure will be valid if the boundary layers are thin compared to the radius of curvature. On the other hand Michel48 in his nozzle boundary layer study has taken transverse curvature into account by starting with a friction law based on the boundary layer thickness 6. Incompressible skin friction laws also have been derived starting from either the Prandtl mixing length hypothesis or the VonKarman similarity hypothesis. The resultant friction laws, the best known of which are perhaps the Prandtl and the VonKarman Schoenherr relations, are logarithmic in form, and are valid over a much larger Reynolds number range than the power laws. 5 -13 -

Remarkably it has been found that the incompressible friction laws can be extended to compressible flows if the density and viscosity are evaluated at an 5, 61, 66, 68 appropriate reference temperature. 'Frequently the Eckert reference temperature T*= 0. 50 (T + T ) + 0. 22 (T - T ) (35) e w aw e has been used, or in the case of variable specific heat and dissociation Hayes and 5 Probstein suggest evaluation of p and Liat a temperature corresponding to the reference enthalpy h* = 0. 50 (h + h )+ 0. 22 (h - h ). (36) e w aw e The situation is somewhat different for a highly cooled stagnation point boundary layer. In this case, since the free stream Mach numbers are small and since the major part of the temperature change is confined to the laminar sublayer Rose, et al, suggest use of the incompressible skin friction formula. Compressible skin friction formulas also have been derived using the Prandtl mixing length hypothesis. Van Driest has used this method to derive his well known skin friction formula50 for a perfect gas and others have extended his work 4, 59 59 to include the effects of dissociation. Dorrance also has computed a friction coefficient for dissociated turbulent boundary layers starting with the Prandtl hypothesis. The Van Driest and Dorrance formulae are considerably more complex than the power laws, though perhaps applicable over a wider Reynolds number range. It is clear that a large number of skin friction formulas are available. 4 Enkenhus and Maher have calculated the heat transfer at a nozzle throat using several different friction laws and found heat transfer rates ranging from 4200 2 2 BTU/ft sec to 12, 300 BTU/ft sec indicating the importance of the choice of friction coefficient. The best agreement with experimental results seems to be obtained when the reference enthalpy concept is used in conjunction with either a power law or one of the incompressible logarithmic formulas, or if the -14 -

Van Driest or Dorrance friction laws are used. 4' ' In general it is assumed that the friction laws are not affected by a favorable pressure gradient. Variation of free stream properties is taken into account by using local values of velocity, density and viscosity to evaluate the Reynolds number. Solution of the momentum integral equation also requires a knowledge of H, the ratio of displacement to momentum thickness. In boundary layers with adverse pressure gradients, the velocity profiles do not obey the power law (Equation 23) and in the case of incompressible flow Von Doenhoff and Tetervin have shown that H acts as a form parameter which determines the nature of the velocity profile. Thus one approach to a solution of the boundary layer integral equations is to consider H and 62 as the basic variables and to use an empirical equation relating H, 629 shear stress and pressure gradient in conjunction with the momentum equation in order to obtain H and the displacement thickness. With a favorable pressure gradient, such as will be the case in a wind tunnel nozzle most investigators have assumed a velocity profile of the form given by Equation 24. In incompressible flow H. then is determined by the definition of 6* and 62, its value being dependent on n in Equation 24, and hence on the Reynolds number. The subscript i here denotes the incompressible value of H. In compressible flow H also depends upon the variation of the density across the boundary layer. With increasing free stream Mach number M and recovery enthalpy there is a large increase of the displacement thickness 6* relative to the momentum thickness so that H increases drastically. Most investigators 48 62 have used the modified Crocco relation (Equation 25a) to compute p and hence the form parameter Ho Using this approach Michel68 has computed and tabulated H for a perfect gas with constant specific heats over the range /h h 0< aw -1 < o; O< w< 1.5 \ ~~~e ^aw for n = 5 (Equation 23) and H. = 1i. 4^ and for this case also gives the approximate formula -15"

h \ h -h H = 1. 4 + 2. 222 a + 1. 22 a+ (37) Hl+22 (wl+h l2h +H, (37) \e / e where 6H is a correction factor which depends upon h /h and (h /h - 1). For a perfect gas with constant specific heats Sivells gives the formula / T -T -T Vae e H=Hi t(e)+ (aT T (37a) which agrees with experiment up to Mach numbers of 5. 0. Equations 37 and 37a appear to be in reasonable agreement and with the exception of the factor 6H, are identical when T = T w e In Reference 57 values of H are tabulated for H. = 1. 286, n = 7 and up to M = 10. Corrections for real gas effects are discussed by Michel. The form factor appearing in Equation 18 is 62 rather than the usual factor H * 62 The effect of transverse curvature upon H has been investigated by Michel, 48 who found that to a reasonable degree of approximation H Ad H for turbulent boundary layers with a cold wall. In the extreme case of 6/R ~ 1 and h = h Michel found that H/H = 1. 05. w aw -16 -

C. Approximate Hypersonic Solutions of the Nozzle Momentum Integral Equation Solution of the boundary layer integral equations usually requires numerical integration. Under certain conditions, such as when the core flow is hypersonic, the equations can be sufficiently simplified to permit closed form integrations. One cannot expect the resulting solutions to agree exactly with experiment; however, the solutions do provide a guide to the interpretation of experimental data in that they show how key parameters such as the free stream Mach and Reynolds numbers influence the boundary layer behavior. Closed form solutions have, for example, been found by Cohen,3 who applied the Stewartson transformation to the 52 boundary layer integral equations, and by Burke, who utilized assumptions valid when the core flow is hypersonic. The assumptions utilized by Burke are used below to simplify the complete nozzle integral momentum equation (Equation 18). The effect of using various forms of the skin friction law are then exploredo The analysis starts from the momentum equation d62 H+ 1 du cf e du 2 6 (jH 1 e d n I _) f(18) dx 2 u dx dx pe 2 ' e and the continuity equation m p Peue (R - 6*)2 (7) It is assumed that the inviscid core flow is hypersonic so that (- 1) M 2>> 1 (38) 2 e and that T T w e (39) T, T << T w e o -17 -

A futher assumption is that the fluid behaves as a perfect gas at the low static temperature of the hypersonic free stream so that Reecets approximate flow equations (see Appendix C) can be used. The coefficients of 62 and cf can then be simplified sufficiently to make an explicit solution of Equation 18 possible, as is shown belowo Since the free stream behaves as a perfect gas it follows that o y-1 2 cT -1+ M (40) pe e Since u =VyRT M (41) e e e combination of Equations 40 and 41 yields du dM 1 e 2 e 1 + ___ 1 (42) ud dx 2 (42 e e y - 1) Me M 4 As long as the free stream behaves as a perfect gas 1 e = P F(Po, h) [1+ 2 M 9 (43) Pe Po~o^ l+2-T l (43) where F(p, h ) is a function of the stagnation conditions. Thus for fixed stagnation conditions dpe d(lnp e) edM 1 1 + e. (44) p dx dx y- 1 Me dx L ( 1 1)M 2 Pe e (y7- 1) Me If it is assumed that the diverging part of the nozzle is conical R = 0 x. Therefore d(ln R) 1 dR 1 = — --- =- 8(45 -) dx Rdx x -18 -

As shown in Appendix C, the Mach number M and the effective core area are e related by + 1 1 2 1+- 1 2 2(- 1) eff 1f (Poho) At M y+ 1 1 e where f1 (Po, h0) is a function of the stagnation conditions and is discussed in Appendix C. It now follows that for fixed At and stagnation conditions 1 d Aeff 2 1 dM e{1 + O. (46) Af dx y - 1 M dx Ml+ 2(46) Aeff de (y- 1) M If the modified Crocco relation holds it follows that, since hw ~ h, the ' w 0 maximum enthalpy within the boundary layer is only about 1/4 h. Consequently it should not be too unreasonable to use Equation 37, developed for a perfect gas with constant specific heats, for Ho Assuming h = h and P = 1, Michel's w e r expression reduces to 4+2- 1 2 H 1. 4 + Y M 1 M (47) 2 e 2 e Further as mentioned previously H ~ Ho Introducing these results the momentum equation becomes 2d -5 ( - 3) 1 deff 1 f + 6 +[ eff d (48) d- 2+ 2 Aeff dx + 2 and there remains the problem of evaluating cf and 1/Aeff d Aeff/dx. Since 2 Aeff = (R - 6*) it follows that efdAeff 1 eff _ ( w dx ) eff w -19 -

4, 46 Results of boundary layer calculations indicate that, at least over a limited distance, 6* - x. Assuming this to be the case d Aeff 1 eff= - (50) Aeff dx x so that the momentum equation becomes dM2 (2- )- Cf dT"6 0IT-^'T ~(51) dx x 2 2 The form which the solution of Equation 51 takes now depends upon cf. One possibility is to use the flat plate type of power law combined with the reference temperature concept so that T (p*u x\ =2 = A ~ (52) p*ue ue From Equation 52 it follows that ~s cf Tw p*U x f w _ ie-A (53) 2 2 p s u* p u e ue e and this is the skin friction law used by Burke. 52 Since u ~ /2h, Equation 53 e o0 can be expressed in the form -s -- — S c..... -s p \ p/2h x f A P* ( P ( Pe (% O xo (54) 2 s p e P PO 'IO Once again since h* h* h in the hypersonic part of the nozzle it appears reasonable to assume that T h en e T* h* -20 -

Now h* = 0. 50 (h + h)+ 0. 22 (h -h ), and when, h,h < h h e w aw e e w aw o it is approximately true that h* O /3h (55) where 13 is a constant. Thus it also follows that T h h e~ e e 1 y-e -e- = -e J(56) T* h* /3h /3E ( where from Equation C-3 it follows that o~Y- 1 M 2 E h 2-Th M e (57) 2 e Further for given stagnation conditions T /T* will be constant. Now it is assumed that j- T. Also since the pressure across the boundary layer is constant T p* e p T* e and, hence, using the above results 2f s T F(po0 hSo)1S ) x SE s (58) 2 —S S 1 )(- <59) T ~ ~.-A( i 2 Now once again using the assumption that 6* x so that 6* = Sx where S = a constant, Equation 59 yields the effective coreng arlatween E and x: -21-(A/A 2 e =+i -21 -

(Y-1 (o -S)2 -1) 2(y- 1) E I2 w;_ (rt (60) Y+ 1 f(y- 1) - 1 Combining the above results the momentum equation becomes dx x L where /1 is a dimensionless function given by T -Ws V2 -h L-S To -S PO 41 (po9ho) =As ) [F-PAhF f ~(o - )]-s -1 (~-1) (e-s)2y'^s-1 6) '-1+ 1 2 --- —T T?(62) 1 +1,) and where L is some characteristic dimension such as the nozzle length. Integration of Equation 61 is meaningful only over that part of the nozzle for which the hypersonic approximation is applicable. It will now be assumed that 62 = 62R at some position xR downstream of the throat, where the hypersonic assumptions first become valid. If the pressure gradient term in Equation 61 is arbitrarily neglected the equation yields the following for 62/x: -s 2 6 K 1 I 2 L() )Y_ X) ] (63))\ x = x + 2(y - l)(s - 1)+ s + 1 ( -22 -

where - = 2(y - 1)(1 - s) - s. This solution is often called the flat plate solution, and the only justification for dropping the pressure gradient term is that the form of the resulting solutions often agree with experimental results. Now if 62R 62 (R x X X the flat plate solution becomes approximately -S r __e"~~~ ^i~~~ ~ (~~-) -a(64) x 2(y - 1)(s - 1) + s + 1 rt (64) Equation 64 thus implies that the major portion of the boundary layer development has occurred in the hypersonic section of the nozzle. From Equation 47 it follows that 62 < 6* when the core flow is hypersonic. Thus even when 6*/R ~ O (1), 2. 6 2 6=62 2 - =62 2 (65) From Equations 64, 61, and 53 it follows that the flat plate solution can be expressed in the form [ - s +s- 1 2(s +ws-1) x x 2(y - l)(s - 1) + s + 1 2(y- 1)(s - 1)+s +1 s Re x and from Equation 47 it follows that I r- IS+ wSS - 1+ o/s A y - 1 - 1M+1 2s(l + w) 6* s 2 2 e(67) x 2(y- 1)(s- 1)+s+1 Re ( ) x 52 Equation 67 was originally derived by Burke; however, he did not take into account the upstream boundary conditions and the effect of the boundary layer upon the core flow. The form of Equation 67 is in agreement with the experimental -23 -

coorelation formulas developed by several authors for the hypersonic nozzle boundary layer 52, 9 Equation 67 is essentially a local similarity solution and implies that the upstream history of the boundary layer is not important. In view of the many approximations needed to derive the above solution it is surprising that it should actually have the same form as the experimental correlations, If the pressure gradient term in Equation 61 is retained the solution takes the form -S rt 62x nx~() + r (3=) [o)1~.Yj (68) x x x i -v- r ~ 1+- x x x R 1+ r X tX and it can be seen that the solution (Eqo 68) is quite different from the flat plate solution (Eqo 63). The exponent I and 1 +. - y is shown in the table below for several values of s which might be expected in the case of a turbulent boundary layer, and for y = 1o 4. s 0. 20 0. 25 0O 30 0.35 Io r 44 35 o 26 17 1+1 - y o04 -o05 14.23 In the range of interest9 say 0. 20 < s < 0O 3, (1 + Q - y) 1 so that Equation 68 may to reasonable accuracy be approximate by RR1-7 (-~tt)r~R In i(68a) x R '1 t t XR since Z z - a - 1 lim = in a o z —0 -24 -

There are now no simple arguments by which the effect of boundary layer history can be eliminated. If 6R/xR is sufficiently small it may be possible to find an empirical value of the dimensionless reference distance xR/rt such that experimental data can be correlated using the formula 6 M 2(s + ws - 1) 2- const e ln( I -n 1, (69) x Re s (t x XR where In -is now essentially an empirical constant. Whether or not Equation 69 rt will actually result in improved correlations of experimental data is certainly a worthwhile subject for further investigation. Of course, as stated in Section II-A above, the flat plate skin friction formula is not really applicable to hypersonic nozzles with thick boundary layers. Rather it is more consistent to use a power law based on Re so that c *u 5\ 2f = B P- ) (70) 2 mp e! — To find a solution of the momentum Equation 51 using this friction law it is necessary to find a relation between 6 and 62 From the definition of 62 in Equation 12 it follows that 1 6 6J p R pLr u u 0dj(Y) u (71) 0 e Now assuming that 1 h n p hn r u 2 Pe e and using the Crocco relation, Equation 25a, integration of Equation 71 yields the approximate result -25 -

1E - 2 + 2ln E) (73) Now Equation 70 for cf/2 can be expressed in the form c (r )-m -2(y - 1 - m) 6_/_r (-m '^ L r 1 -J6- (74) 1 2R where m T _W m n nL-m ^ H/B 1 -1^) -- — T* 'IO 2 -= Bmf- 1T'~-rm y'- 1 ( S)2(y - 1)7 1 )w ) - 1 F(ph)]-m Y + i /0-m -1 2-1 2 1f y+ 1 Now -6 m 6 m(m- 1) 1 6 2R 2R 12 4 R2 R and since m is usually small, for example. 25, the important result follows that the effect of transverse curvature upon cf will be quite small even when 5/R O (1). The momentum equation now becomes d6 r -m 2 - i m m 2 (2 - y) 6 rt -2(y — dx x 2 % 2 L r) r If the pressure gradient term is ignored Equation 75 has the solution -26 -

-m Am + 1 ( m + 1 (m+ 1) t | x /\ x | + 1 - 2(y - 1 m) rt X oi -(- 1 - m) -2(y- 1 - m) X x-2(- -m) J[(x m i n) ) (76) and if once again 62R is sufficiently small and 1 - 2(y - 1 - m) R) << 1 this solution becomes approximately 1 mn- 1 m+ m+ 1: (m + 1)B m(+) - 2 1 = [ 1-2( -1- m) 2 mo- 1 o m + (M 2) e ) (77) Re m+ 1 Re x If, as in the incompressible case, the coefficient s of the flat plate friction law m+ ber dependence of %6/x indicated by Equations 66 and 77 is identical. This is not surprising since we have shbwn that the effect of transverse curvature cf is small. -27 -

When the pressure gradient term is retained the solution becomes m+ 6 (2R +1 (2 - )(m+) 2R i () + 1) x x x ir^t - x - -2(y - 1) + m y(IT r~ r-.m) (78) For values of m of interest in turbulent flows and with y = 1. 4, y(l - m) - 1 will be a small number. Hence if 6R is sufficiently small the solution can be expressed in the approximate form r -m- 1) + m+ 1 x t 2 In (79) x 'P2 L r x which can also be written mw- 1 m+ 1 2(M ) M m+ = const e n XR In- (79a) x m r (Rex + and this result is very similar to Equation 69 which was obtained using the flat plate friction formula, the only difference being in the exponent of the logarithmic term. The hypersonic solutions of the momentum and continuity equations found above are relatively insensitive to the transverse curvature effect. The boundary layer thickness correlation formulas (Equations 67, 69, 77, and 79) will be valid only if the major part of the boundary layer development takes place in the hypersonic portion of the nozzle. Because of the usual values assumed by y, -28 -

m, and s, the boundary layer thickness correlation formula obtained with the full momentum equation is the same as the flat plate formula multiplied by a term involving in (x/xR). As mnetioned in Section III below, results of boundary layer measurements can be correlated using the flat plate type of formula similar to Equation 67. The logarithmic term in the more exact solution (Equation 79a) varies very slowly with (x/rt), and for nozzles with exit Mach numbers between 10 and 20 (x/rt) varies from approximately 1000-2000. Thus the logarithmic term may be buried in the empirical constants of the experimental flat plate type of correlation formula. The results above were obtained only after many simplifications and assumptions. The assumption that 6*; Sx actually gives the solutions the character of a first iteration of the momentum and continuity equations. The solutions found above show that actually 6* - x 5. Clearly a more precise and simultaneous solution of the momentum and continuity equations is a worthwhile subject for further study. The fact that the hypersonic solutions above, though they may not provide exact quantitative predictions, have the same form as the experimental results appears to indicate that most of the boundary layer development occurs in the hypersonic section of the nozzle. III. CALCULATION OF HYPERSONIC NOZZLE FLOW USING EMPIRICAL RELATIONS FOR THE BOUNDARY LAYER DISPLACEMENT THICKNESS An exact integration of the nozzle flow Equations 7, 18, and 22, is a major undertaking which is beyond the scope of the present investigation. Hence, experimental boundary layer thickness correlations have been combined with the continuity Equation 7, in order to calculate the boundary layer and core flow in a conical hypersonic nozzle. 52, 53- 74 Several investigators 5 have found that experimental measurements of boundary layer thickness at the test section of a hypersonic nozzle can be correlated by an equation of the form -29 -

Ma 6* e - = K (80) X b Re x which is the same as the flat plate formula found in Section II-C above, or of the form 6* K (81) x P*uex) K, K, a, b, and X are empirical constants, and x is the distance from the nozzle throat to the test section. In Equation 81 the effect of free stream Mach number 52 is taken into account by using the reference enthalpy concept. Burke has shown that Equation 81 correlates experimental results from several sources quite adequately over the Reynolds number range lo 5x 105 < Re < 8 x 106 X or 9 x 103 Re < 3. 5 x 105 if K and X have the values 0. 49 and 0. 3 respectively. Burke's empirical constants have been used in the present calculations; however, outside the Reynolds number range for which Equation 81 agrees with experiment, results can only be considered qualitative. It has been assumed that the flow in the nozzle is in thermodynamic equilibrium, and at relatively high stagnation pressures this assumption is not 43 too unreasonable. Using the results presented by Lukasiewicz a curve corresponding to the case in which 1% of the oxygen in the air freezes out in atomic form has been drawn on a stagnation pressure-density diagram (Figure 3), which also shows the operating domain of the University of Michigan Hot Shot -30 -

Tunnel. For stagnation conditions below the 1% freeze out curve the equilibrium assumption should certainly be valid. Since oxygen freeze out nowhere exceeds 10% within the U of M hotshot operating range, the discussion of Lukasiewicz4 indicates that the effects of nonequilibrium even above the 1% freeze out curve should not be excessive. The above considerations are qualitative; however, a full investigation of the effects of nonequilibrium is beyond the scope of the present investigation. 3 It now is possible to use the method developed by Burke and Bird to simultaneously calculate the development of the core and boundary layer flows. The details of these calculations are presented in Appendix Co In a hotshot tunnel stagnation conditions are determined from the arc chamber pressure after firing and from the charging density, which, since energy addition occurs at constant volume, is also the stagnation density. Consequently all results have been presented on stagnation pressure-density diagrams similar to that in Figure 3o It is of course desirable to put the nozzle calculation results into a universal form which is a function of the stagnation conditions only, but is independent of the detailed nozzle configuration, and to some extent this is possibleo However, in the case of the detailed boundary layer thickness calculations not only stagnation conditions but also a factor related to the conical nozzle configuration must be specified. By combining the continuity Equation 7 with the correlation Equation 81 and using Equations C-3 - C-6, Burke and Bird have derived the following equation for the ratio of boundary layer displacement thickness to test section radiusliR -I 1 - R F<lpo h 82) R R where F. is a function of stagnation pressure and enthalpy described in Appendix C and where,/4is a factor containing the conical nozzle geometry which is given by -31 -

to X 0x.A yx 2yX i geom x R rex "A- _K r 71 x 3) 0aaiA) X) ( )Y X A (83) The factor provides for some universality in that for a particular stagnation condition, ad test gas all nozzle configurations having identical values of will have the same value of 6*/R. In the present case boundary layer calculations have been made for/ = 363 (ft)0 3 corresponding to = 7o 5~ = 0, 131 radians w -3 rt = 005 ino = 416 x 10 ft x=72,0 ino = 6 00ft X = 0o3 The results of the nozzle performance calculations will now be presented in detail. The range of stagnation conditions of interest in the case of the U of M Hot Shot Tunnel is shown in Figure 3o The stagnation chamber of this tunnel has been: 73 designed for a peak pressure of 809 000 psi and this determines the upper boundary of the operating domain. The high temperature boundary is determined by the fact that as T increases contamination of the gases from vaporized arc chamber material as well as radiation losses increase above acceptable levels. Calculations have been extended to T = 10, 0000K subject to the assumption that flow contamination is negligible0 Indications are that the maximum operating tempera72 ture will be less than this value for practical hot shot tunnels. Below a stagnation temperature of 3000 K test section velocities are too low to be of interest in connection with high speed flight problemso Figure 3 also includes lines of constant stagnation temperatureo In hotshot tunnels energy is added to the test gas at a constant volume. Consequently the energy, EA, added to the test gas by the arc discharge will be e e..( R TsR %P ~e.1s (84) EA =PQc (e~ - e)- =(R) (R (PRRTR) c (1 - (84) -32 -

where Qc is the arc chamber volume and e and ei are the test gas stagnation and c 1o initial energies per unit mass. Since ei/e << 1 for the domain of interest the curves of (e /RTR) (po/PR) = const. correspond to constant values energy addition, EA. Figure 3 shows that the stagnation pressure depends almost entirely upon EA and is relatively independent of the charging density. For fixed EA an increase in charging density simply lowers the stagnation temperature. For p /pR > 100 the 0 R thermodynamic data used for Figure 3, as well as for the results to follow was obtained by extrapolating the air Mollier diagramo Results in this region must therefore be regarded with considerable caution. The results of the boundary layer calculations are presented in Figure 4 in the form of lines of constant 6*/S x = 6*/R drawn on a stagnation pressuredensity diagram. Figure 4, unfortunately, only is valid for one particular value of the geometrical factor/, It can be seen that 6*/R decreases with increasing stagnation pressure, a result which is not surprising. What is remarkable is that the curves in Figure 4 pass through a maximum. This means that for a constant stagnation pressure p, as the stagnation density p increases the boundary layer thickness at first increases and then begins to decrease. Consideration of the displacement thickness correlation formula (Equation 80) and Figure 6, which shows contours of constant test section Mach number provides an explanation for this strange resulto With p0 constant an increase in p causes the test section Mach number to increase and this tends to make 6* increase. On the other hand the static density as well as the Reynold s number also increase with increasing po, which of course tends to decrease 6*. The maximum in the boundary layer contours in Figure 4 thus corresponds to the point where these opposing effects balance. Fromt Equations 82 and 83 it can be seen that c6* (85) R 2~/ -33 -

or when y = 1. 4, = 0. 3 6* 1 R.84 rt Consequently for a nozzle with fixed length and divergence angle the boundary layer thickness varies almost inversely with the throat radius. This result suggests that it may be possible to experimentally determine X by measuring the effect of varying the throat radius rt upon the boundary layer thickness. The strong dependence of 6* upon rt has another interesting consequence. The effective area ratio of a conical nozzle of fixed length and exit area can be increased by decreasing the throat radius rt. Now because of the relation between rt and 6* there is, for a given nozzle and stagnation condition a critical value of throat radius, rtc such that for rt < rt the effective nozzle area ratio decreases with decreasing throat radius because of the rapid boundary layer growth. If 6*/R is not too great Equation 82 becomes approximately /F1 (Po9ho) = 2 rt where K is constant for fixed p0, ho,x,, y and X. Now the effective area ratio is given by Aeff = (R - 6*)2 2 1 K At 2 rt 2X + 1 At = ^rtL rt and Aeff/At, considered as a function of rt, has a maximum when 1 rt rt = [(2yX + 1)K] y and this is the critical throat radius mentioned above. For example, for the nozzle considered above when po = 4000 atm, h/RTR = 220, rtc =.0123 in. -34 -

and when p -- 200 atm, h/RTR 180, rtc =00342 in. At low stagnation pressures the critical throat radius is thus rather close to the actual values used. To obtain an overall view of nozzle performance and to determine the range of test section Reynolds numbers encompassed by the domain of stagnation conditions considered here other test section parameters have been computed. It has been possible to present the variation of test section Mach number in a semi-universal form. Equation C-11 relating Mach number to area ratio can, using Equation C-10, be written in the form - 1 AlA\ 2 M e A ff (86) e \A^tI where the function P is defined by Y+ 12 2(y - 1) fl (ho 9 In the absence of a boundary layer A ff/At will be constant so that Me is directly proportional to the function 0, which in that case could be considered a universal Mach number function. Hence contours of 0 - const. are plotted on a stagnation pressure-density diagram in Figure 5. When boundary layer thickness is appreciable 6*/R, which can be read from Figure 4, must be known in addition to 0 in order to determine M. Comparison of Figures 3 and 5 shows that M decreases with increasing stagnation temperature if the effective area ratio remains constant, This is entirely a real gas effect for in the case of an ideal gas a constant area ratio produces a constant Mach number independent of the stagnation conditions. The results shown in Figures 4 and 5 have been combined to compute the test section Mach number variation for a nozzle with~= 363 ft ' o The resultant constant M lines are shown in Figure 6. For purposes of comparison seve eral constant Mach number curves have been computed both with and without considering the nozzle boundary layer and the results are shown in Figure 7o -35 -

Figure 8 shows constant static pressure contours and in Figure 9 several of these contours are compared to pressure contours computed neglecting the boundary layer. It can be seen that in the present case the static pressure is doubled by the presence of the boundary layero Figure 10 shows the variation of test section static temperature with stagnation conditions. Lines of constant static temperature appear to almost coincide 70 with lines of constant stagnation temperature. It should be noted that Reece's7 perfect gas approximation used in the present calculations is accurate to 1% for T < 550 K therefore, it follows from Figure 10 that the perfect gas approximation remains valid over most of the stagnation conditions of interest. The variation of static density and the variation of the limiting velocity uQ given by u~ = 2h u 0 are shown in Figure 11. In hypersonic flow the free stream velocity u is almost equal to uQ. For the Mach number range shown in Figure 6 the free stream velocity ue is at most 2 1/2% less than the ultimate velocity so that uQ can, for all practical purposes, be regarded as the free stream velocity ueo Since uQ depends only on the stagnation enthalpy h the free stream velocity is essentially independent of the boundary layer development. The free stream density on the other hand will be considerably modified as can be seen from Figure 12, which shows constant density contours computed both with and without the boundary layer. The presence of the boundary layer almost doubles the value of the free stream density. A comparison of Figures 11 and 6 exposes an interesting aspect of hypersonic nozzle performance. As the free stream velocity, and hence stagnation enthalpy, increase the free stream Mach number actually decreases if the geometric area ratio remains fixed. Thus the flow actually becomes less "hypersonic" as stagna5 tion enthalpy is increased. The Mach number independence principle states that -36 -

as Me -oo the flow about bodies becomes independent of M and depends only on Ue and p e Consequently the above effect will not be important insofar as testing is concerned as long as the free stream Mach number remains sufficiently high. Using the results of the calculations described above the free stream Reynolds number in the present case has been found to lie in the range 0 4 x 105 < Re < 4 x 106 x Fortunately this range almost coincides with the range of validity of the empirical formula used to compute boundary layer thickness. IVo DISCUSSION AND CONCLUSIONS In the present work integral equations for the flow in a conical hypersonic nozzle with a thick boundary layer have been formulated, and solutions of these 3 equations valid in hypersonic flow have been studied. Burke's method of calculating boundary layer thickness using empirical correlations has been extended over a relatively broad range of stagnation conditions and has been used to compute the effect of boundary layer development upon hypersonic nozzle flow. The main effect of the nozzle boundary layer is to cause appreciable changes in static pressure, density, and temperature, and to reduce the useful core area of the test section, The free stream velocity depends mainly upon the stagnation enthalpy, and in hypersonic nozzles is unaffected by the boundary layer developmento For a fixed exit diameter and nozzle lengthy the boundary layer thickness at the nozzle exit varies almost inversely with the throat radius, rto This result has two important implications. First as mentioned in Section III, it should be possible to obtain information about the boundary layer thickness correlation formula by measuring the effect of varying throat radius upon the boundary layer thickness. The effective area ratio of a conical nozzle with a fixed length and exit area can be increased by decreasing the throat radius. Now because of -37 -

the relation between throat radius and the boundary layer, there is, for a given nozzle, and stagnation condition, a critical value of r such that for r less than this critical value the effective area ratio of the nozzle will actually decrease because of the rapid boundary layer buildup. Contours of constant boundary layer thickness drawn on a stagnation pressuredensity diagram are found to pass through a maximum. Physically this means that for a fixed nozzle geometry and stagnation pressure, as the stagnation density increases the boundary layer thickness first increases and then begins to decrease. This strange behavior has been explained in Section III by considering the form of the empirical displacement thickness correlation formula. The boundary layer calculations described in Section III are only as accurate as the boundary layer correlation formula employed and as the approximations to the real gas properties. Over most of the domain of interest the static temperature and nonequilibrium effects seem to be sufficiently small that Reece's approximations are applicable. The range of free stream Reynolds numbers for the particular nozzle considered and the range of validity of the empirical boundary layer thickness correlation formula almost coincide. Mollier charts and cross plotting have been used to compute the results described in Section III, and a certain degree of random error is inherent in this procedure. A precise determination of this error is difficult, though a reasonable estimate would seem to be that the results are valid to an accuracy of ~5% assuming that the boundary layer correlation formula and ReeceVs approximations are valido The incompressible analysis in Appendix B as well as the analysis in Section II show that the transverse curvature of the nozzle boundary layer causes an increase in boundary layer thickness and a decrease in the skin friction coefficient, as compared to a plane boundary layer. The approximate analysis of the momentum equation in Section II shows, however, that in most cases this effect will be negligible even when 6*/R ~ 0 (1), because 62 < 5* in hypersonic boundary layers and because the exponent m of the term (Re)m in the skin friction law is a relatively small number in turbulent flows0 -38 -

Using approximations valid in hypersonic flow Burke52 has shown that the boundary layer momentum equation can be reduced to a relatively simple form. In Section II it is shown that the nozzle integral equations reduce to the same form if it is assumed that 6* - x. Solutions of this equation both with and without the pressure gradient term have been investigated using power skin friction laws based on Re6 and Rex. Either friction law leads to essentially the same 6 x u result. The flat plate solutions, which are obtained when the pressure gradient term is dropped, can be reduced to the form M a M 6* e = const x Re b Re x provided the boundary layer thickness at the start of the hypersonic portion of the nozzle is sufficiently small and provided the nozzle is sufficiently long. When the pressure gradient term is retained the corresponding solution has the form M ~ xR 6* x__ x _ R x const rt - rt X L J where xR is an arbitrary distance from the throat where conditions are applied. In view of the numerous approximations which have been introduced to obtain the above approximate solutions it is remarkable that the form of the flat plate solution is in agreement with the form experimental correlations of boundary layer thickness. This agreement implies that the bulk of the boundary layer growth occurs in the hypersonic portion of the nozzle and this contention is borne 46 48 out by exact calculations and measurements of boundary layer development. 9 An interesting feature of highly cooled hypersonic boundary layers is that the maximum enthalpy within the boundary layer is only about one quarter of the stagnation enthalpy, assuming that the Crocco relation is valid. Thus real gas effects -39 -

will not be important throughout most of the boundary layer, and of course this may be another reason why the approximate solution takes the same form as experimental correlations. It appears strange that the flat plate rather than the more exact solution seems to provide the best agreement with experiment. There are several possible explanations. The logarithmic term in the exact formula varies very slowly with (x/rt), and for nozzles with exit Mach numbers between 10-20, x/rt varies over approximately the range 1000-2000. Consequently it is quite possible that the logarithmic variation is buried in the constant of the empirical correlation formula. In the approximate solution it was assumed, at least for what might be considered a first iteration, that 6* - x. Actually of course 6* increases more 1. 5 rapidly than this since, approximately, 6* x This more rapid boundary layer growth will tend to decrease the free stream pressure gradient and thus provides another possible reason for the success of the flat plate solutions. There are a number of questions which have not been considered. When the boundary layer is almost as thick as the nozzle there is some question as to whether the boundary layer approximation remains valid and whether the pressure remains constant across the boundary layero The answers to these questions should depend upon the rate of variation of the core velocity. In the present case both conditions above have been assumed to holdo Effects of chemical nonequilibrium in the core flow and the boundary layer have been neglected. The assumption of equilibrium core flow appears to be valid over most of the domain of stagnation conditions of interest hereo Whether or not the boundary layer flow remains in equilibrium is open to questiono In addition to the above the present study suggests a number of further problems which should be investigated in order to gain a fuller understanding of hypersonic nozzle flow. In view of the above results, the use of the approximate hypersonic nozzle flow integral equations appears as a promising approach to the analysis of -40 -

hypersonic nozzle flow, and should be investigated further. A simultaneous solution of the approximate momentum and the continuity equations, without assuming 6* ~ x, would certainly be desirable. Solutions also should be found using skin friction laws valid over a broader Reynolds number range than the power laws. A numerical solution of the exact nozzle equations should be obtained at least in a few cases in order to provide a means of evaluating the results of the approximate analysis. Finally an extensive study of existing experimental and theoretical data should be made in order to further check the results of the approximate theory and the experimental correlation formula used for the boundary layer calculations in the present investigation. -41 -

REFERENCES 1. Yu, U. N,, "A Summary of Design Techniques for Axisymmetric Hypersonic Wind Tunnels," AGARDograph 35, Nov. 1958. 2. Johnson, R. H, "Hypersonic Viscous Effects in Wind Tunnels, " ARS Journal, v. 31, pp. 1022-1024, July 1961, 3. Burke, A. F., Bird, K. D., "The Use of Conical and Contoured Expansion Nozzles in Hypervelocity Facilities," from, Advances in Hypervelocity Techniques, Proc. of Second Symposium on Hypervelocity Techniques, Plenum Press, N Y., pp. 373-424, Mar. 1962. 4. Enkenhus, K. R., Maher, E. F,, "The Aerodynamic Design of Axisymmetric Nozzles for High Temperature Air'" U. S. Naval Ordnance Lab., White Oak, Md., NAVWEPS Rept. 7395, 5 Feb. 1962. 5. Hayes, W. D., Probstein, R. F., Hypersonic Flow Theory, Academic Press, New York, Capt. 9, 1959. 6. Hilsenrath, J., et al, "Tables of Thermodynamic Properties of Air Including Dissociation and Ionization from 15000K to 15, 000K," Arnold Engineering Dev. Center, AEDC-TR-59-20, Dec. 1959. 7. Hilsenrath, J., et al, Tables of Thermodynamic and Transport Properties, Pergamon Press, London, 1960. 8. Humphrey, R. L, Neel, C. A., 'Tables of Thermodynamic Properties of Air from 90 to 1500~K'" Arnold Engineering Dev. Center, AEDCTN-61-103, Aug. 1961. 9. Treanor, C. E., Logan, J. G, "Thermodynamic Properties of Nitrogen from 2000~K to 8000~K, ' Cornell Aero. Lab., Rept. BE-1007-A-5, Jan. 1957. 10. Little, W. J., Neel, C. A., "Tables of the Thermodynamic Properties of Nitrogen from 100 to 15000K, " Arnold Engineering Dev. Center, AEDC-TDR-62-170, Sept. 1962. 11. Hilsenrath, et al, "Thermal Properties of Gases, " National Bureau of Standards, Circular 564 (1955). 12. Gilmore, F. R., "Equilibrium Composition and Thermodynamic Properties of Air to 24, 000~K " The Rand Corp., Santa Monica, Calif., RM-1543, Aug. 1955. 13. Hochstim, A. R., 'Gas Properties Behind Shocks at Hypersonic Velocities, III. Tables of Thermodynamical Properties of Air," Convair Div. of General Dynamics Corp., San Diego, Calif., Physics Sec., Rept. ZPh-004, Aug. 1958, -42 -

REFERENCES (continued) 14. Blackwell, F., et al, "Properties of Argon-Free Air, " Ramo-Wooldridge Corp., Los Angeles, Calif., Rept. GM-TR-76, Oct. 1956. 15. Goin, K. L., "Mach Tables for Real Gas Equilibrium Flow of Air in Hypervelocity Test Facilities with Total Temperatures to 10, 000~K," Sandia Corp., Rept. SCR-288, Mar. 1961. 16, Logan, J. G., Jr., Treanor C. E., "Tables of Thermodynamic Properties of Air from 30000K to 10, 000K at Intervals of 100~K, " Cornell Aero. Lab., Rept. BE-1007, Jan. 1957. 17. Hansen, C. F., Heims, S. P., "A Review of the Thermodynamic, Transport, and Chemical Reaction Rate Properties of High-Temperature Air, " NACA TN 4359, July 1958. 18. Gilmore, J. R., "Additional Values for the Equilibrium Composition and Thermodynamic Properties of Air," The Rand Corp., Rept. RM-2328, 30 Dec. 1959. 19. Chance Vought Research Center Staff, "Thermodynamic Properties of High Temperature Air," Rept. RE-IR-14, 28 June 1961. 20. Feldman, S., "Hypersonic Gas Dynamic Charts for Equilibrium Air," AVCO Research Lab., Research Rept. 40, Jan. 1957. 21. Humphrey, R. L., Little, W. J., Seeley, L. A., "Mollier Diagram for Nitrogen," Arnold Engineering Dev. Center, AEDC-TN-60-83, May 1960. 22. Korobkin, I., Hastings, S. M., "Mollier Charts for Air in Dissociated Equilibrium at Temperatures of 2000~K to 15, 000~K, " NAVORD Rept. 4446, May 1957. 23. Moeckel, W. E., Weston, K. E., "Composition and Thermodynamic Properties of Air in Chemical Equilibrium," NACA TN 4265," Apr. 1958. 24. Jorgensen, L. H., Baum, G. M., "Charts for Equilibrium Flow Properties of Air in Hypervelocity Nozzles," NASA TN D-1333, Sept. 1962. 25. Yoshikawa, Kenneth K., Katzen, Elliott D., "Charts for Air-Flow Properties in Equilibrium and Frozen Flows in Hypervelocity Nozzles, " NASA TN D-693, 1961. 26. Erickson, W. D., Creekmore, Helen S., "A Study of Equilibrium RealGas Effects in Hypersonic Air Nozzles, Including Charts of Thermodynamic Properties for Equilibrium Air, " NASA TN D-231, 1960. -43 -

REFERENCES (continued) 27. Gibbs, G. J., "Correlation of Real Gas and Perfect Gas Flow Parameters at Hypersonic Velocities in Computing the Free Stream Conditions between Mach 6 and 10,? STA Meeting Proc., Vol. II, Apr. 27, 1961. 28. Grabau, M., "A Method of Forming Continuous Empirical Equations for the Thermodynamic Properties of Air from Ambient Temperatures to 15, 000~K with Applications," Arnold Engineering Dev. Center, AEDCTN-59-102, Aug. 1959. 29. Grabau, M., Humphrey, R. L., Little, Wo Jo, "Determination of Test Section, After-Shock, and Stagnation Conditions in Hotshot Tunnels using Real Nitrogen at Temperatures from 3000 to 40000K, " Arnold Engineering Dev. Center, AEDC-TN-61-82, July 1961. 30. Smith, C. E., Jr., "Thermodynamic Properties of Nitrogen, " Lockheed Missiles and Space Co., Rept. 6-90-62-111, Dec. 1962. 31. Heims, Steve P., "Effects of Chemical Dissociation and Moelcular Vibrations on Steady One-Dimensional Flow, " NASA TN-D-87, Aug. 1959. 32. French, E. P., des Jardius, Po R, Radbill, J. R., Luntz, N. G., "Computer Programs for Thermodynamic Properties of Air and Homonuclear Diatomic Gases," presented at Supersonic Tunnel Association Meeting, Apr. 1962. 33. Hansen, C. F., "Approximations for the Thermodynamic and Transport Properties of High Temperature Air," NASA TR R-50, 1959. 34. Lee,..., "Empirical Relationships between Test Section and Stagnation Chamber Conditions in a High Temperature Wind Tunnel using Air, " presented at Supersonic Tunnel Association Meeting, Oct. 1962. 35. Hochstim, A. R., "Approximations to High-Temperature Thermodynamics of Air in Closed Form," Kinetics, Equilibria and Performance of High Temperature Systems, Proco of the First Conference, Butterworth and Co., Ltd., London, (1960). 36. Bray, K. N. C., "Atomic Recombination in a Hypersonic Wind Tunnel Nozzle," J. Fl. Mech., v. 6, pt. 1, pp. 1-3, July 1959. 37. Hall, J. G., Russo, A. L., "Studies of Chemical Nonequilibrium in Hypersonic Nozzle Flows," Cornell Aero. Lab., Rept. AD-1118-A-6, 1959. 38. Eschenroeder, A. Q., Boyer, D. W., Hall, J. G., "Exact Solutions for Nonequilibrium Expansions of Air with Coupled Chemical Reactions, " Cornell Aero. Lab., Repto AF-1413-A-1, 1961. -44 -

REFERENCES (continued) 39. Bray, K. N. C., Appleton, Jo P., "Atomic Recombination in Nozzles: Methods of Analysis for Flows with Complicated Chemistry," Univ. of Southampton, Rept. AASU 166, 1961. 40. Anderson, J. P., 'The Effect of Recombination Rate on the Flow of a Dissociating Diatomic Gas, " Arnold Engineering Dev, Center, AEDCTR-61-12, Sept. 1961. 41. Glowacki, W. J., "Effect of Finite Oxygen Recombination Rate on the Flow Conditions in Hypersonic Nozzles, " U. S. Naval Ordnance Lab., NOLTR 61-23, 15 Sept. 1961. 42. Bray, Ko N. Co, "A Simplified Sudden Freezing Analysis for Nonequilibrium Nozzle Flows, ' Univ. of Southampton, AASU 161, Dec. 1960. 43. Lukasiewicz, Jo, discussion in, "An Assessment of our Present Status and Further Requirements for High Temperature Hypersonic Facilities, Round Table Discussion," Training Center for Experimental Aerodynamics, Rhode Saint Genese, Belgium, TCEA TM 14, 6 Apr, 1962. 44. Emanuel, G., Vincenti, W. G., "Method for Calculation of the OneDimensional Nonequilibrium Flow of a General Gas Mixture through a Hypersonic Nozzle, " Arnold Engineering Dev. Center, AEDC-TDR62-131, June 1962. 45. Levinsky, E. S., Brainerd, J. J., "Inviscid and Viscous Hypersonic Nozzle Flow with Finite Rate Chemical Reactions, " Arnold Engineering Dev. Center, AEDC-TDR-63-18, Jan. 1963. 46. Sivells, J. Co, Payne, R. G., "A Method of Calculating Turbulent Boundary Layer Growth at Hypersonic Mach Numbers, " Arnold Engineering Dev. Center, AEDC-TR-59-3, Mar. 1959. 47. Bartz, D. R., "An Approximate Solution of Compressible Turbulent Boundary Layer Development, " Trans. ASME, Nov. 1955. 48. Michel, R., "Developpement de la Couche Limite Dans une Tuyere Hypersonique, " presented at AGARD Specialists Meeting, High Temperature Aspects of Hypersonic Flow, Rhode Saint Genese, Belgium, Apr. 1962. 49. Reshotko, E., Tucker, M., "Approximate Calculation of the Compressible Turbulent Boundary Layer with Heat Transfer and Arbitrary Pressure Gradient," NACA TN 4154, Dec. 1957. 50. VanDriest, "Turbulent Boundary Layer in Compressible Fluids;'J. Aero. Sci, Mar. 1951. =45 -

REFERENCES (continued) 51. Persh, J., Lee, R., "A Method for Calculating Turbulent Boundary Layer Development and Convective Heat Transfer in Convergent-Divergent Nozzles, " NAVORD Rept. 4200, June 1956. 52. Burke, A. F., "Turbulent Boundary Layers on Highly Cooled Surfaces at High Mach Numbers, " ASD Symposium on Aeroelasticity, WrightPatterson Air Force Base, Ohio, Oct. 30-Nov. 1, 1961. 53. Lee, J., "Axisymmetric Nozzles for Hypersonic Flow,' Ohio State Research Foundation, Rept. 459-1, July 1959. 54. Schlichting, H., "Berechnung der Stromung in rotationsymmetrischen Diffusoren mit Hilfe der Grenzschitht-theorie, z. Flugwissenschaften, v. 9, n. 4/5, (1961). 55. Goldstein, S., ed., Modern Developments in Fluid Dynamics, v. 1, Clarendon Press, Oxford, p. 304, 1938. 56. Durand, J. A., Potter, J. L., "Calculation of Thicknesses of Laminar Boundary Layers in Axisymmetric Nozzles with Low Density, Hypervelocity Flows, " Arnold Engineering Dev. Center, AEDC-TN-61-146, Dec. 1961. 57. Shapiro, A. H., The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. II, Ronald Press, New York, 1954. 58. Schlichting, H., Boundary Layer Theory, McGraw Hill, New York, 1955. 59. Dorrance, W. H., Viscous Hypersonic Flow, McGraw Hill, New York, 196' 60. Lees, Lo, "Laminar Heat Transfer over Blunt-Nosed Bodies at Hypersonic Flight Speeds, " Jet Propulsion, v. 36, pp. 259-269 (1956). 61. Schubauer, G. B., Tchen, CO M,, "Turbulent Flow, " Section B, "Turbulent Flows and Heat Transfer," v. V of Princeton Series, High Speed Aerodynamics and Jet Propulsion, C. C. Lin, ed., Princeton Univ. Press, 1959. 62. Shapiro, A. Ho, The Dynamics and Thermodynamics of Compressible Fluid Flow, v. II, Chapt. 27, Ronald Press, 1954. 63. Cohen, N. B., "A Method for Computing Turbulent Heat Transfer in the Presence of a Streamwise Pressure Gradient for Bodies in High Speed Flow," NASA Memo 1-2-59L, Mar. 1959. 64. Spence, D. A., "Distributions of Velocity, Enthalpy and Shear Stress in the Compressible Turbulent Boundary Layer on a Flat Plate," ERoyal Aircraft Establishment (Farnborough), Rept. AERO 2631, Nov. 1959. -46 -

REFERENCES (continued) 65. Persh, Jo, "A Theoretical Investigation of Turbulent Boundary Layer Flow with Heat Transfer at Supersonic and Hypersonic Speeds, " NAVORD Rept. 3854, May 1955. 66. Eckert, Eo Ro G., "Engineering Relations for Heat Transfer and Friction in High-Velocity Laminar and Turbulent Boundary Layer Flow over Surfaces with Constant Pressure and Temperature, " Trans, ASME, v. 78, noo 6, p. 1273, Augo 1956. 67. Rose, P. H., Probstein, Ro F, and Adams, M. C0, "Turbulent Heat Transfer Through a Highly Cooled Partially Dissociated Boundary Layer," JAS, v. 25, pp. 751-760, Dec. 1958. 68. Matting, J. W., Chapman, Do Ro, Nyholm, J. Ro, and Thomas, A. G., "Turbulent Skin Friction at High Mach Numbers and Reynolds Numbers in Air and Helium, " NASA TR R-82, 1961. 69. Hidalgo, Ho, "On the Application of VanDriest's Method to a Highly Cooled, Partially Dissociated Turbulent Boundary Layer, " Jet Propulsion, v. 28, no. 7, ppo 487-489, July 1958. 70. Reece, J. Wo, "Test Section Conditions Generated in the Supersonic Expansion of Real Air, " JAS, v. 29, ppo 617-618, May 1962, 71. Hilsenrath, Jo, and Beckett, C. Wo, "Tables of Thermodynamic Properties of Argon-Free Air to 15, 000K, " Arnold Engineering Dev. Center, AEDC-TN-56-12, Sept0 1956. 72. Lukasiewicz, J., Jackson, R., Whitfield, J, D. "Status of Development of Hotshot Tunnels at the AEDC, " paper presented at AGARD Meeting on High Temperature Aspects of Hypersonic Flow, Rhode-Saint-Genese, Belgium, Apr. 3-6, 1962. 73. Sherman, P. M., Early, Ho Co, Lawrence, W. N., "Design Considerations for Arc Heated Hypersonic Tunnel, " Final Rept,, ORA Project 02953, The Univ. of Mich, July 1960. 74. Whitfield, J. D., Private Communication, Arnold Engineering Development Center, AEDC, Tullahoma, Tenn,, 1961. -47 -

FIGURE IC I N FIGURE I. CONICAL NOZZLE BOUNDARY LAYER COORDINATE SYSTEM.

R Control volume FIGURE 2. APPROXIMATE NOZZLE VELOCITY DISTRIBUTION FOR THE CASE w< I.

EA = Energy added to arc chamber, Joules o= 2860 0C Q eo c,= Arc chamber volume - Ft.3 4 '04',Z/ o10% oxygen freeze out_ 1\'-0 / / 03T 0 s LO ~L~~~~~~~~~1~~~~ //Z5 — I. I 0 - I2 -__, _ __, ____/ uJ ~ 100 10 STAGNATION DENSITY, fi'// FIGURE 3. ARC CHAMBER CONDITIONS FOR EQUILIBRIUM AIR (Thermodynamic data from Ref.20). _.Z 40 I;~\0" I00 2 5 8 2 STAGNATION DENSITY, /Ion/P AIR (Thermodynomic doto from Ref. 20).

R X -A =363 Ft.)-0.3, 1.4, 0 'Q;A 0.3 Ld I 2- ________. ^ 0- -- gE //~ / -) 0.- ________- _ _____ ___ - 00 -8!~0 000 (f)R. 40 - 0 __ JI// /. / / 2 58 2 10 100 STAGNATION DENSITY, P//OP

* 81tV nIWsn1no3 8O- i NOIlONdnJ 8381/WnN HOVtiW 3H1 ' 3n91si d/~0d 'AilSN3a NOlIVN9ViS 001 01 z 8 s Z _-_. —....... - - —.-. I 001 / /, // X1 __ -____ -.1./y // — I -— / 2, I//I / /, /// m or-^7^^- oo 1/t</t/ ////1//, ~ ~00,~~~~~ ($U) U) X S 000 lRt ) aX s ~ o' ooo' 0o0~~~0 -*y01~~~ 1 00 ~ / ', O ~-t~~~~~~~~~

'SNOIIIONOO NOI1VN9V1S HIIM 83BiWnN HOVt N0113S iS31 So NOIVlWVA '9 3fn9IJ 'd/ ~/ 'AlISN3C NOI1VN9V1S 001 01 z 8 S z - 001 /'0 Il ~/? // - - -] Io I I I I o' I1 _.,_ / _,// /i, ---- 1^ ^^- --- 000'1 -o 1.01 / // / 0 X '" / l p',,,, 6 = ~.0.'~j ~9~, =

= 363 Ft.-0.3 o-' R = 9.45" Oo rt =.05" - 5 ' _ With boundary layer - ~ ----- Without boundary layer /, / / 8/ / |e / LJ / / / 'n,/ ILL / - w 10O__ _ / 2) T E / /S aI 1,000 - -- -r~y / -A -^- / -- - - u ---— 7- — ^^^-^__.-_-__-___ I)/, ' / J_ a: / / / H ' /,i''a-/ 1 s/ //"- / / / 2 58 2 10 100 STAGNATION DENSITY, /1% / FIGURE 7. THE EFFECT OF THE BOUNDARY LAYER UPON THE TEST SECTION MACH NUMBER.

*SNOIIIaNOO NOI1VN9V1S HJJM 38nSS3Jd OiVS N01193S lS31 JO NOIlVlVA '8 3uni91d c// od 'AIISN30 NOIlVN9VIS 001 01 W X, 20 4 m 001 / / / / - _ _ _ _F <-/ g ---- \ - /ll/-1001 - -- — oo ~/ / --- oool,^- ',.,,,,/ 8 0\: — ~ --- ——, --- —-.::: p~_~ o' ~9~ = ~ %-~ Jl --- '~~o~,,.o '~:1 ~9~: I

; 363 Ft.-3 _o 0 R = 9.45",~' rt-.05" _ - With boundary layer ---- ------ Without boundary layer - " --- 02 --- 1 o n. 1,,- 1,1' 1^ 1.1 IJu/.., /. a2 5 ' 0oj oo =- -: 10 100 FIGURE 9. THE EFFECT OF THE BOUNDARY LAYER UPON r% -^^- -7 — -.00' de.do ---:2n --- ^ --- - - -— 2 5 8 2

'SNOIIIONOO NOII VN9VIS HIM 38nl -Vd3dWl31 i11ViS NO0113S IS31 JO NOIlVlWVA '01 3fln91 I 8cd/~o/'AllSN30 NOI.VNOVlS 001 01 2 89~ g ~ — __- --- " 001 / / / / / --- -- / / / o ___ __ ///./,.// -.0 C /////.,'// "' /r// ///V/ rn;7- od I M 'W --- — I I0I 000'1.-t~ / /v^ OO- 00'.^ ^QC?.,t6= *^, —^,? /, 0._~o ~.n-L-J ~9~: [_

SNOlIl0NO NOI1VN9V1S HUIM A119013A 9NII -IV1I1 ONV A11SN30 N01193S IS31 JO NOlVlIVA 'II 38n91uI d/ O, ' AllSN3a NOI1VNE)V1S 001 01 Z 8 S 001 / / X / > / -, _/ / / / X / // X I/ /O~ -_01~1. ~L~d/d 2 I - -- -— _ _ _ - - /.*' i'/I - ' / / -- -~' rr '~~~~~~~~~~~~~~~-^^ffc^^^S"'-8 sO~~~ i~/ / 6 ' — ==^-/ — " -' /00 Ii 0 1!,//r -,, I/ /, / rii_ ',,'7,,//'6,o' Nbo,,/,,_ _/ 01),,6 = a 0/ 0,/OI ~9,,,/,

- = 363 Ft. -0.3 R = 9.45",0 I - rt=.05", —.4 / - ' With boundary layer '''-___ ____ ) - - layer.2 / LU 20- /.___o- 1jJ _ — 4_ 0 _ / - yi ^ / I7~/ j / ILJ /7,, / I I '~__/5, - -- _I I /,__ S DENST10 / _/_, 8-I _ _,) - _ I - -II / ) / / / r LL. ____:I / _____ /.I / z 0 /I / 100 - --------— _- -- --- U)2~~ / STAGNATION DENSITY, /P FIGURE 12. THE EFFECT OF THE BOUNDARY LAYER UPON THE TEST SECTION DENSITY.

APPENDIX A It has been shown that the mass flow through the nozzle is 2 2 \2 rfi = 27 peV Rc (1 - cos ) + (R 2 -cos 2 cos ( - 0 uc eue c ucos 2 cos (w -w RS 5*2. - (R6* - cos 0w cos ( 0 (A-1) Now let a= 0 - 0 (A-2) w UC then from Figure 1 it can be seen that R R c cos a sin 0 w (A-3) tan a 6=R sina=R t c sin 0 w It follows that 2 r 2 m R (1 - cos 0) 2 tan 1 R2tan a cos m uc' os [2 stan a c 12 w + cos a R -- R 27pV2e 2 2 sin 2 2 Pee e cos a sin 0 w sin 2 w w - cos a R6* cos (A-4) Now assuming that w' 0uc' << 1 (A-5) series expansion of the trigonometric functions yields the following for the terms in Equation A-4: A-1

R (1 - cos 0) R2 /2 2 6 a uc_ R 2a \ w 1la w ( 6 2 2 = 2 /l ---+ l+ -- -- 2 2 2 (6 cos a sin 0 w 0 2 w sn w 8 2 6 t ta2 an acos sa Lsin 0w R2 sin2 I 2a2 2 R2 1 i - - 1 (A-7) w6 t 6 6 J Lw w (R6*Etion6-2 a- cnbwrt2 2+ w a2 6* - os V cos a [ + - E2a 1 ) (A-8) and using the relations 6 6* tana a 2 w w Equation A-A can be written in the form e e 6*1 2 6 Ehe function E (an b i is tten fir =(R66*2

1.0.8 0 l I l I.26 b J. * 0.1.2.3.4.5 R FIGURE A-l. MASS FLOW CORRECTION FACTOR E( a- s). \ R

APPENDIX B In order to estimate the effect of transverse curvature upon the skin friction coefficient in incompressible flow consider the case of an axisymmetric boundary layer with R and u constant, even though this case is not entirely realistic from a physical point of view. The momentum equation becomes d62 cf d62 f (B-i) dx 2 Now cf is assumed to be given by the power law (Equation 30) -m Cf = A( e ) (30) and correspondingly the velocity distribution will be (U1 6) (B-2) where s - m From Equations 9 and 12 it now follows that 2- m 62 66 (B-3) so that 62 6 (k - k2 (B-4) where B-1 B1

Combining Equations 30, B-1, and B-4 yields the following differential equation for the boundary layer thickness 6;: d _ \ m I(B-5) dx (kI - 2k2 R Equation B-5 shows that the rate of boundary layer growth increases with increasing 6/R. Thus one reaches the tentative conclusion that in internal flows the transverse curvature results in thicker boundary layers than in the cases in which the transverse curvature effect is absent. Integrating Equation B-5 and assuming that 6 = 0 when x = 0 yields the result e m2 + 1 A Rem =+-Re. (B-6) m+l 1 kl m + 2* (Bx) It must be pointed out that the assumption 6 = 0 when x = 0 is inconsistent with the fact that 6/R O (1) and the assumption that ue and R are constant. Nevertheless the results of the integration above may be valid in the limiting case when 6/R is quite small. In the absence of transverse curvature (6/R = 0) Equation B-6 together with Equation 30 immediately leads to flat plate power law for the friction coefficient (Equation 32). Clearly as 6/R O (1), cf can no longer be expressed as a simple power of Rex, rather m — ": rm + 1 A - Re f - 2 x m++ 1, cf[ Z(e) (B-7) jr m2k and from Equation B-7 it follows that the transverse curvature effect decreases the skin friction coefficient. B-2

If 6/R is sufficiently small it can be shown from Equation B-6 that to first order in 6/R 6 = + k ( 2) (B-8) where 6p is the plane value of boundary layer thickness given by m 1 ux + 1 m+ 1 6p=x ) ( 1) (B-9) From Equations 30, B-8, and B-9 it follows that to first order in 6/R m cf F- 1 -k xRe ) (B-10) -cf - 4R x where 12mko.m + 1 kk 2mk f m i, ^____2 m+ 1 A 4 k1 (m + 2) k 2 B-3

APPENDIX C 3 Burke used the empirical displacement thickness relation either of the form M 1.311 6* e -= 0463 (C-1 X Re 0. 276 x or of the form -0. 3 6* P* = 0. 49 (C-2) x \ I* in conjunction with the continuity equation m = 7reu (R - 6*) (7) e e to calculate the combined nozzle and boundary layer flow at the test section. At the hypersonic test section conditions of interest the static temperature of the isentropic core is low so that the perfect gas assumption is applicable. In this 70 case, Reece has shown that the usual perfect gas nozzle flow equations can be used with correction factors to account for the real gas effects in the reservoir and upstream portion of the nozzle. Consequently static temperature and pressure are determined from the equations h 2 =1 + 1M 2 (C-3) h 2 e h T= ' R 7- 1 and y_ AS C-1

The ratio of nozzle to throat area is written in the form Y + 1 y+ 1 A f (h,p ) + 1 1 2 2 (C- 5) M 1 M: (C-5) A 0 '2 e 2 e and the mass flow density at the nozzle throat is related to reservoir conditions by p Ptat f2 (hosp) G1 (C-6) 1 1 ',1 + y- y+1 Equations C-1 to C-6 form the basis of Burke's calculation method. Before applying these equations it is necessary to discuss the various real gas correction factors. In Equation C-4 ASS (Po' ho) 1S R R R ideal where S(p0, h )/R is the real gas dimensionless entropy at stagnation conditions, 20 which in the present case was determined from Feldman's Mollier diagram for argon free air. 2(S/R)id in the perfect gas dimensionless entropy corresideal ponding to h and po and given by / h o R () 2' ln.n p+ (C-7) Rideal Y - 1 R R Subscript R refers to the reference condition of 273. 16 K and 1 atm pressure. hR and SR/R were taken from NBS Circular 564. Since the NBS tables are based on air including argon it has been assumed that h and S/R are the same for real and argon free air at the reference condition. The values of the gas constant R and the molecular weight used here correspond to those given in Reference 20. C-2

The correction factor f1 (ho, po) in Equation C-5 has been determined from G1onstb 5 ar20 Goin's tables and from graphical calculations using Mollier charts for air. The two calculations were in excellent agreement and the resulting curves for fl (ho,po) are shown in Figure C-1. It should be noted that Burke and Reece' 70 assumed that fl = fl (ho) and this is approximately true over the enthalpy range 25 < h /RT < 150 covered by their calculations. At higher values of stagnation enthalpy h Figure C-1 shows that f strongly depends upon the stagnation pressure p. 0 PO~ The correction factor f2 (ho P) in Equation C-6 has been computed using 15 oo Goin's tables and is plotted in Figure C-2. Once again there are considerable variations in f2 (ho p) with p at the higher values of stagnation enthalpy. In using the reference temperature form of the boundary layer correlation (Equation C-2) it becomes necessary to determine the reference temperature T* from the reference enthalpy h* as defined by Equation 36. For this purpose an average specific heat, Cp, defined by Cp T* = h* yR f (p, h) T* (C-8) P -Y -1 3 has been introduced. To compute this function it has been found convenient to use Blackwell's tables. 1The function f3 (p, h*) is plotted as a function of h*/RTR in Figure C-3. y-1 2 Using the assumption that 2 Me >> 1, letting h* 0. 235 h, assuming 2 e o that the thickness of the boundary layer at the nozzle throat is negligible, and that I* = k2 (T*)1/2 Burke has shown that Equations C-2 and 7 can be reduced to the single equation f-i. ~-^1~ I(C-9) w w C-3

where for X = 0. 3, y - 1. 4, 0. 3 0.3 0.354 L1 -0.12 -0.3 o tw w -0.45 ff wf - 1 2 or i f3 and for a boundary layer correlation of the form; = k1 I p*uex I i the constant k5 is given by 5 | yR3/2 -X + 1 --"G, -yB- 1 2(y- 1) k k1 {Y '( Y- } k5k l =1 __ k2 R(O. 22)3/2 For the correlation formula (Equation C-2), with y = 1. 4, R = 1. 724 x 103 ft2/sec2 R (argon free air), and k = 2. 01 x 10i8 slugs/ft sec R1/2 (taken 3 2 -3 2 03 from Burke ) the constant k5 has the value 0. 362 x 103 (ft /slug)' 3 C can be calculated once the stagnation conditions, and the nozzle throat radius, length, and half angle are known. This in turn makes it possible to solve Equation C-9 for 6*/0 x = 6*/R. A graphical solution obtained by reading off values of 6*/0 x from a graph of 6*/0 x vs. C determined from Equation C-9 is w w most convenient. A plot of Equat ion C-9 corresponding to the correlation formula (Equation C-2) is shown in Figure C-4. 6*/0 x determines the effective test section area and consequently all the w test section parameters. Thus A ( x0 2 C2 Ae Xw 1-___* (C-10) t ( t \ C-4

Combining Equations C-5 and C-10 it follows that when - 1 M 2 > 1 2 e Y- 1 fx 2 - 2 Xw 16* [(xO~ (1::~2 M = rt 1 (C-ll) e r- 1 + 1 f2 4 Knowing Me, p and T can be calculated using Equations C-3 and C-4. The Ke, Pe e dimensionless free stream density can be computed from PR T e(PR (C-12) where R is the density at standard or reference conditions and where P/P is RC R sometimes referred to as density in Amagat units. The procedure described above combined with extensive cross plotting has been used to compute the results described in Section III of the main report. One source of error in the calculations above is that while Goin's tables, used to compute fl and f2, are based on the most recent NBS calculations for real air, the Blackwell Tables and the Feldman Mollier Diagram, used to calculate f, and ~AS 71 are based on the older NBS data for argon free air. In view of the large uncertainty introduced in using the empirical formulas for 6*/x, the effects of the above inconsistency can be ignored. C-5

fI(hoPo) (t'A/A, real / /A.)ideal 0 > 2 'Q~~~~~~~~~~~~~,4' h0, FT. 2/SEC2,",,6 8 i /// ||/// / 5L l//// / l --- 40 80 120 160 200 240 280 hoFT.2/SEC.10o6 FIGURE C- I. AREA RATIO CORRECTION FACTOR fe(hn, pa).

1.23 -1.21 - _ _____ — 1.19 / o 1.17 -1.1 1.II 1.09 50 70 90 110 130 150 170 190 210 230 250 ho, FT. 2/SEC.2.10-6 FIGURE C-2. MASS FLOW CORRECTION FACTOR f2(ho,po) ~

(C)equil.! / 2.8- f3(P,h*) = f R10-4.8 2.6-4 10 ia-4 1- 0 2.4 -2.2 - i, 1.8 y0 60 80 100 120 140 DIMENSIONLESS REFERENCE ENTHALPY, h"/RTR FIGURE C-3. RATIO OF AVERAGE TO IDEAL GAS SPECIFIC HEAT, f3(p, h*).

I^-O.I2~ -0.3 h0\0.31/ \0.3/ \0.54 2-__- 00o I 11 I 10-' -I - - -- -- ewx () - W_ _ - _C / --- -- - --- k5.362 10(Ft2/SIug) 3 =1.4 A=0.3 - - -— ___ I- II 1-1 111111 —1 1 --- —. 2 5 8 2 5 8 2 5 103 104 FIGURE C-4. VARIATION OF THE BOUNDARY LAYER THICKNESS WITH THE FUNCTION.

UNIVERSITY OF MICHIGAN 3 9015 03525 0193