8416-13-T Report of BAMIRAC ACCELERATION OF A HYPERSONIC BOUNDARY LAYER APPROACHING A CORNER George R. Olsson Arthur F. Messiter May 1968 This document has been approved for public release and sale; its distribution is unlimited. Infrared Physics Laboratory THE INSTITUTE OF SCIENCE AND TECHNOLOGY THE UNIVERSITY OF MICHIGAN Ann Arbor, Michigan

WILLOW RUN LABORATORIES FOREWORD This report contains the work submitted by G. R. Olsson to the Graduate School of The University of Michigan in August 1967 in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The committee for the dissertation consisted of professors A. F. Messiter (chairman), T. C. Adamson, Jr., J. D. Murray, M. Sichel, and W. W. Willmarth.

WILLOW RUN LABORATORIES ABSTRACT An asymptotic description of the acceleration of a laminar hypersonic boundary layer approaching a sharp corner is obtained. The description assumes small interaction with the outer inviscid flow. Viscous forces are neglected except in a thin sublayer. The initial part of the expansion takes place over a distance O(Me6), where Me is the external Mach number, and 6 is the boundary-layer thickness. Here the transverse pressure gradient is small, and a solution can be obtained analytically. Within a distance 0(6) from the corner, the effect of streamline curvature is essential, and a numerical solution is obtained by the method of integral relations for a single strip. The solution for surface pressure is compared with experimental results for a particular case, and an approximate velocity profile at the corner is calculated. Possibilities for improving the accuracy of the calculation, both by refining the numerical procedure and by including higher order effects, are considered.

WILLOW RUN LABORATORIES CONTENTS Abstract........................... ii List of Symbols................................. vii 1. Introduction.................................. 1 2. Physical Description of the Flow....................... 5 3. Asymptotic Representations.......................... 8 3.1. Limit Processes 8 3.2. First Outer Limit 11 3.3. Second Outer Limit 15 3.4. Composite Expansions of Solutions in the First and Second Outer Limits 18 3.5. Sublayer Limit 19 4. Solution in the First Outer Limit: (x/L) - 1 O(M.R /2). 21 4.1. Transformation of the Equations 21 4.2. Asymptotic Expansion for s - -o 24 4.3. Expansion at the Critical Point 25 4.4. Numerical Results 26 5. Solution in the Second Outer Limit: (x/L) - 1 = O(R 1/2) 31 5.1. Formulation by the Method of Integral Relations 31 5.2. Solution by the Method of Integral Relations for One Strip 42 5.3. Solutions by the Method of Integral Relations for Two Strips 45 5.4. Generalization to an Arbitrary Number of Strips 54 5.5. Evaluation of Results and Comparison with Experimental Data 57 6. Solution in the Sublayer.......................... 63 6.1. Dorodnitsyn Transformation 63 6.2. Solution by the Method of Integral Relations 65 6.3. Displacement Thickness 69 7. Conclusions..................................70 Appendix I: Compressible Laminar Boundary Layer at Constant Pressure.........................73 Appendix II: Approximation by the Method of Integral Relations Compared with the Exact Blasius Solution............79 Appendix III: Reduction of Hama's Wall-Pressure Data.............81 References.....................................83 Distribution List................................. 85 V

WILLOW RUN LABORATORIES SYMBOLS A Parameter defined in equation 5.35 A. i = 0, 1. Constants defined in equation II-11 a Speed of sound a (Sec. 4) 1- 02 a (x) Coefficients in representation for pu in equation 5.18 "a( MM) Coefficients in representation for O in equation 6.13 B Function in equation 5.194 Bi i = 0, 1. Functions in equation II-7 Bij i, j = 1, 2. Functions defined in equations 5.138 through 5.141 b (Sec. 4) 02 b (x) Coefficients in representation for puiv in equation 5.19 b M() Coefficients in representation for Q-1 in equation 6.13 C Function in equation 5.197 C. i = 1, 2. Functions defined in equations 5.142 and 5.143 c (Sec. 4) 1- p2(1-_ a22) cn () Coefficients in representation for pui, equation 5.34 D Function defined in equation 5.146 D.ij i, j = 1,..., 4. Constants appearing in equations 5.171, 5.174, 5.177, 5.182, etc. d (i) Coefficients in representation for puv, equation 5.48 E Entropy function in the isentropic relation F. i = 0, 1, 2, 3. Functions defined in equations 5.110, 5.120, 5.121, and 5.155 f(M ) Scale factor in intermediate limit, equation 3.54 fi' fk Weighting functions in the method of integral relations G Function defined in equation 5.151 g Transformed stream function defined by equations I-27 and I-28 H Function defined in equation 5.157 H) Nonhomogeneous terms in equations 5.177, 5.182, and 5.189 through 5.192 Vll

WILLOW RUN LABORATORIES h Enthalpy h (Sec. 5) Function defined in equation 5.45 I(:) Function defined following equation 4.20 I. i = 1, 2. Integrals defined in equations 4.22 i Index J(J) Constants defined in equations 5.163 through 5.166 1 j Index K. i = 1, 2. Integrals defined in equations 4.20 k Heat conductivity k 0, 1 for a wedge or a cone, respectively k Index L Length of the body measured along the surface L. i = 1,..., n. Arbitrary functions introduced in equation 5.1 M Mach number, q/a M Number of strips in the calculation by the method of integral relations for the initially subsonic part of boundary layer m Index N Number of strips in the calculation by the method of integral relations for the initially supersonic part of boundary layer n Index P. i = 1,..., n. Arbitrary functions introduced in equation 5.1 Pr Prandtl number p Pressure Qi i = 1,..., n. Arbitrary functions introduced in equation 5.2 q Magnitude of the velocity R Reynolds number, puL//i r Radial coordinate r Body radius s Coordinate (equal to x) in the direction of the free stream T Temperature U Nondimensional velocity at the outer edge of the sublayer u Velocity component parallel to the wall "1 vl/q1 * *.

WILLOW RUN LABORATORIES v Velocity component normal to the wall w Transformed normal velocity component defined in equation 6.7 X Transformed x coordinate in equation 5.80 x Coordinate measured parallel to the wall y Coordinate measured normal to the wall Yi Dependent variables defined in equations 4.43 and 1-38 Z. i = 1,..., 4. Functions defined in equation 4.46 a g(O) = 0.4696 a Yi1/6c in integral-relations analysis ak Yk/6c /3 Similarity variable defined in equation I-25 /3 Function defined in equation 5.38 y Ratio of specific heats A(x) Layer thickness introduced in material following equation 5.1 6 Boundary-layer thickness 6* Boundary-layer displacement thickness e 1- Transformed coordinate defined in equation 6.21 (App. I) / - 1.21678 71 Transformed normal coordinate defined in equation 6.1 o (aut/Oar)-l 0 Flow deflection angle X Constant in equations II- 11 and II- 12 y/ Viscosity coefficient Transformed x coordinate defined in equation 6.1 p Density a Shock wave inclination angle T Typical flow deflection Transformed s coordinate defined in equation 4.9 'I' Transformed stream function defined in equation I-23 4/ Stream function co Exponent in power-law viscosity-temperature relation, equation 1-6 ix

WILLOW RUN LABORATORIES Subscripts b Condition in the base region c Critical condition e Condition at the outer edge of the undisturbed boundary layer i (App. III and fig. 4) Value at the surface which would be predicted by inviscid-flow theory i, j, k, m, n Indices max Condition at infinite Mach number M Condition on the strip boundary y = YM M + N Condition on the strip boundary y = 6 o Condition as y - 0 SL Sublayer condition t Stagnation condition w Wall condition 6 Condition at the outer edge of the disturbed layer co Condition in the undisturbed free stream 1 Condition in the undisturbed boundary layer for x/L = 1 0, 1, 2 Indices which indicate conditions on the strip boundaries y = yk, k = 0, 1, 2 (bar over letter) Dimensional quantity Condition in the first outer limit Condition in the second outer limit rt Condition in the sublayer limit (dot over letter) a/a3 or a/a 4 d/dx

WILLOW RUN LABORATORIES ACCELERATION OF A HYPERSONIC BOUNDARY LAYER APPROACHING A CORNER 1 INTRODUCTION The expansion of an inviscid supersonic flow at a sharp corner takes place through a centered Prandtl-Meyer expansion fan. Of course, this description neglects any effect of the viscous boundary layer at the solid surface. For nonzero viscosity, the expansion actually begins somewhat upstream from the corner and is completed somewhat downstream from the corner. At any given Mach number, the acceleration of the boundary layer, here assumed to be laminar, takes place over a distance which decreases as the Reynolds number increases. The details of the boundary-layer expansion are of interest, for example, in relation to the calculation of base flows. The present investigation was motivated by the need for proper initial conditions for the study of the hypersonic near wake. This work is concerned specifically with the portion of the expansion which occurs just upstream from the corner, for the case of hypersonic flow over a slender wedge or cone. The inviscid hypersonic flow past a slender wedge or cone is described by hypersonic small-disturbance theory [1]. Although the shock wave is rather close to the surface, there is a significant range of Mach numbers and Reynolds numbers for which the boundary-layer thickness is small compared with the shock-layer thickness, while the boundary layer remains laminar. For this range, the approximate boundary-layer velocity and temperature profiles can be obtained by neglecting interaction with the outer flow. At the base of the wedge or cone, the boundary layer expands rather rapidly and in a complicated manner. Since the acceleration and the pressure gradient are quite large in a relatively small region near the corner, the flow is approximately an inviscid rotational flow. Since, in this approximation, the no-slip condition is violated for a short distance upstream from the corner, a viscous sublayer, with a thickness that is small compared to the boundarylayer thickness, must also exist. As in conventional boundary-layer theory, and for the same reasons, the sublayer is considered to have only a small effect on the expansion of the outer part of the boundary layer. It has been experimentally determined that the base pressure, for the range of parameters of present interest, is considerably smaller than the pressure in the region between the shock wave and the body surface. Except for the sublayer, the entire boundary layer, therefore, ac

WILLOW RUN LABORATORIES celerates to supersonic speed. Downstream from the corner, when the complicated interaction has been completed, the inviscid-flow approximation predicts a highly rotational outer shear layer and a velocity discontinuity corresponding to the sublayer described above. Below the sublayer is a region of recirculating flow at lower velocity and more nearly constant pressure. Existing solutions for the near-wake region have been obtained without complete knowledge of the initial shear-layer profiles. For example, the integral method used by Reeves and Lees [2] to analyze the near wake of a circular cylinder is extended byGolik, Webb, and Lees [3] in a study of the wake behind a wedge. In each of these papers, the importance of the boundarylayer expansion at a corner is pointed out, but the flow near a sharp corner is not studied in detail. Some recent attempts have been made to study the portion of the expansion just downstream from a corner. The model proposed by Weinbaum [4] assumes a highly rotational outer shear layer with velocity discontinuity across a sublayer located essentially at the zero streamline. In this model, flow near the zero streamline expands to the base pressure, but an overexpansion occurs elsewhere in the boundary layer. A curved, centered expansion fan from the corner leads to reflected waves which are predominantly expansions; these, in turn, are reflected as compressions from the constant-pressure zero streamline and ultimately coalesce to form the lip-shock wave. Weiss [5] incorporates this model in a calculation of the near wake of a wedge. He uses the full Navier-Stokes equations in the recirculation region, boundary-layer equations in the sublayer, and a rotational-characteristics calculation in the remainder of the shear layer; an iteration procedure is used in obtaining a solution. However, detailed pressure measurements by Hama [6] suggest that even the flow close to the surface overexpands, and, in fact, completely turns the corner. The lip shock occurs as a separation shock, originating very close to the base at a point just below the corner. A more detailed theoretical investigation of the lip-shock wave formation would require accurate information concerning profiles of the flow properties at the corner. It is evident from Hama's [6] data for laminar flow that a significant portion of the pressure drop does indeed occur upstream from the corner. In fact, since the base pressure is sufficiently low, the air very close to the surface (i.e., just outside the thin sublayer) is expected to accelerate at least to sonic speed, and details downstream from the corner will not influence the upstream flow. Therefore, the flow upstream can be studied without further knowledge of the lip- shock wave. The upstream influence of the corner is associated with the propagation of disturbances through the subsonic part of the boundary layer. A basic assumption in this study is that the extent of upstream influence is small compared with the length of the body. In the present ap

WILLOW RUN LABORATORIES proximation, it is found that this requirement is satisfied provided that the boundary-layer thickness is small compared with the shock-layer thickness. Because the changes take place in a small region, the problem is approximately two-dimensional for a cone as well as for a wedge. The idea of describing abrupt changes in a boundary layer by the inviscid-flow equations has appeared in the literature in several other contexts. For example, Morkovin [7] has observed experimentally the effect of an expansion wave impinging upon a boundary layer on the wall of a supersonic wind tunnel and has successfully predicted the post-interaction velocity profile by an inviscid-flow calculation. That is, given the initial conditions in the boundary layer, he calculated the final velocity profile from the Bernoulli equation, the entropy equation, and the measured value of the final pressure. Except for the effect of a viscous sublayer, his prediction agrees well with the experimentally-determined velocity profile. Lighthill [8] uses a similar concept in analyzing the interaction of a supersonic boundary layer with a disturbance sufficiently weak that separation does not occur. He introduces small perturbations on a parallel shear flow and neglects viscous shear forces except in a sublayer. Zakkay and Tani [9] consider a problem of boundary-layer acceleration at a sharp corner without considering separation. Their interest is primarily in the boundary-layer development downstream from the corner, and they assume that the changes close to the corner are described by inviscid-flow equations. The concept of a sublayer again appears. For the same case, a calculation describing changes close to the corner is given by Hunt and Sibulkin [10]. They use a momentum integral and assume that pressure is constant along radial lines. After the present study had been completed it was brought to the authors' attention that closely related work has been carried out by Neiland and Sychev [11] and by Matveeva and Neiland [12]. Neiland and Sychev consider compressible boundary-layer flow at a rounded corner having radius of curvature 0(6). For a distance 0(6) in the stream direction, they obtain inviscid-flow equations, except in a viscous sublayer of thickness O(R1/46), where the boundary-layer equations are required. Matveeva and Neiland use a similar approximation to formulate a description of a supersonic boundary layer approaching a sharp corner. They carry out a one-strip calculation by the method of integral relations. The numerical integration is started by use of an asymptotic solution valid upstream where the perturbation in pressure is characterized by a small nondimensional parameter, A. In this region, nonlinear inviscid-flow equations are required at a distance 0(6A1/2) from the wall, and the disturbances extend over a distance O(6A-1/2) in the stream direction. In the present study, the case of zero wall heat transfer and unity Prandtl number is considered, primarily because of simplifications in the equations. The initial boundary-layer

WILLOW RUN LABORATORIES profile is then obtained quite easily (app. 1). As the boundary layer approaches the corner, the pressure drop causes the boundary layer to become thinner because the changes in streamtube area for the subsonic portion of the layer are dominant. Flow deflections at the outer edge of the boundary layer remain small, even if relative pressure changes are of the order one, because the flow is hypersonic. In the early stages of the expansion, the flow deflection is also small throughout the boundary layer. Since the profile eventually becomes entirely supersonic, this behavior cannot persist all the way to the corner; eventually the spreading of streamlines in the supersonic region must dominate. In a second region close to the corner, the streamline deflection remains small at the outer edge but can become quite large inside the layer because the fluid is free to turn inward when it reaches the corner. The boundary layer in these two regions might be called subcritical and supercritical. This distinction is discussed by Lees and Reeves [13]. A numerical solution of the problem to be discussed here has been obtained by Baum [14], who used a finite-difference method to solve the boundary-layer equations. For the initially supersonic part of the flow, an acceleration term was retained in the transverse momentum equation. Consequently, the normal pressure gradient was nonzero. Weiss and Nelson [15] have obtained an approximate solution by using a stream-tube calculation (zero normal pressure gradient) for the fluid which is initially at subsonic speed and a Prandtl-Meyer expansion for the initially supersonic part. In the present investigation, approximate equations are derived which are expected to be correct in an asymptotic sense for the case of a sufficiently thin hypersonic boundary layer, a simple method is shown for obtaining approximate numerical results, and the procedures for studying the largest neglected terms are considered. A more detailed physical description of the flow is given in section 2. In section 3 the asymptotic nature of the approximation is discussed, order estimates are given for the two regions of inviscid flow and for the sublayer, approximate differential equations are obtained for each of these regions, and the appropriate matching conditions are given. An analytical solution is derived in section 4 for the upstream region in which the normal pressure gradient is negligible. In section 5, the full inviscid-flow equations for the region closer to the corner are studied by using the method of integral relations. Numerical results are obtained for a one-strip calculation, and a procedure for carrying out a two-strip analysis is described. An attempt at a generalization to an arbitrary number of strips is also discussed. A composite expansion of the solutions for the wall-pressure ratio is obtained and compared with an experiment [6] for a particular case, and an approximate velocity profile at the corner is calculated. In section 6, an approximate formulation of the sublayer problem is derived by using the method of integral relations. Results and conclusions are summarized in section 7.

WILLOW RUN LABORATORIES 2 PHYSICAL DESCRIPTION OF THE FLOW The inviscid flow over a slender pointed body at high Mach number (see fig. 1) can be described by using the approximations of hypersonic small-disturbance theory. The parameter (M T)1 is taken to be of order one, where T is some typical value of the flow deflection, for example, the body thickness ratio. Order estimates for the flow variables can be obtained from the shock-wave relations [1]. The shock is inclined at a small angle of order T, and, therefore, the velocity component in the direction of the free stream remains approximately unchanged. The velocity ue just outside the boundary layer is approximately equal to the freestream velocity u (and, in fact, to the maximum velocity qmax ) Relative changes in pressure and temperature at the shock wave are of order one or larger. At the outer edge of the boundary layer, the pressure pe is of order po u T, the temperature T is of order M c T, and the density p is of order p. It follows that the Mach number Me is of order T e \ x Moo -, R -0, 7TM T Held Fixed FIGURE 1. SLENDER BODY IN A HYPERSONIC FLOW If the boundary-layer thickness is small compared with the distance from the body surface to the shock wave, then the inviscid-flow equations remain approximately correct in the region between the boundary layer and the shock wave. For a boundary layer with zero wall -1/2 heat transfer, the thickness is proportional to R, where R is a Reynolds number based w w on u and on thermodynamic properties evaluated at the surface. Then the assumption of a thin boundary layer requires that a viscous interaction parameter M R/ be small. If the body is a wedge or cone, the pressure is constant in the boundary layer. The temperature in the

WILLOW RUN LABORATORIES 2 -boundary layer is large, of order Me Te and the density is therefore small. Since the mass flow is small, the boundary-layer thickness 6 can be taken equal to the displacement thickness 6* [16]. It is known from experiment that for a wide variety of body shapes in high-speed flow, the pressure Pb in a neighborhood of the base of the body is considerably smaller than P e' Thus, the nondimensional pressure drop (pe - Pb)/Pe at the base of the body is of order unity, and the boundary layer, in separating from the body, will undergo a significant acceleration. Outside the boundary layer, this acceleration influences the flow through a change in the boundary-layer displacement thickness. The pressure varies with the flow deflection angle 06 at the outer edge of the boundary layer according to the Prandtl-Meyer formula for a simple-wave expansion. Inside the boundary layer, the large pressure drop at the corner causes the pressure and velocity gradients, ix and lia, to increase greatly over their values in the upstream, undisturbed boundary layer, while the viscous shear stress remains of the same order as farther upstream. These remarks suggest that the accelerating flow in the boundary layer (see fig. 2) can be described approximately by inviscid-flow equations. Propagation of disturbances upstream through the subsonic portion of the boundary layer will cause a significant portion of the acceleration to occur near the surface of the body upstream from the corner. However, the no-slip condition at the body surface cannot be satisfied by a solution to inviscid-flow equations. Therefore, a viscous sublayer must exist in which the viscous shear stress is of the same order as the streamwise pressure gradient jL and the inertia term puui_. The balance of viscous and inertia terms in the streamwise momentum equation provides the estimate of the order of magnitude of the thickness 6SL of the sublayer, which is found to be considerably M,P ~ e e Y, Y 1 Sonic Line Sublayer_................-.,. e w e w Dividing / Streamline FIGURE 2. ACCELERATION OF A HYPERSONIC BOUNDARY LAYER APPROACHING A CORNER 6~~~~~~~~~~~~~~~~~~~~~~~~~.............

WILLOW RUN LABORATORIES smaller than the boundary-layer thickness, provided that the interaction parameter M R1/2 e w is small. The sublayer flow is described by the boundary-layer equations with a pressure gradient. In a first approximation (as in conventional boundary-layer theory), the sublayer is ignored and an inviscid-flow calculation is made with the normal velocity component v set equal to zero at the wall. Then, from a knowledge of the pressure and velocity distributions along the wall, calculated from the inviscid equations, the sublayer equations can be solved, giving the variation in sublayer displacement thickness along the wall. In a second approximation (not carried out in this work), the normal velocity component v at the wall would be related to the rate of change in the sublayer displacement thickness d SL*/dx while, otherwise, the inviscid-flow equations would still apply. Thus, for most of the boundary layer, the primary effect of viscosity is the variation in sublayer displacement thickness. If the base pressure is sufficiently low, a streamline at the surface (i.e., just outside the viscous sublayer) will accelerate to the sonic condition at the corner. The sonic line is not expected to intersect the surface upstream from the corner because streamlines near the surface would have to bend away from the surface as the pressure continues to decrease. If the pressure is at least as low as the base pressure immediately downstream from the corner, as expected from experiment [6], the flow at the surface must have reached supersonic speed at the corner. A similar situation occurs for the flow in a convergent nozzle exhausting to a low pressure. The sonic condition must occur at the nozzle exit to permit the flow to adjust to the ambient pressure. Thus, in the present problem, the sonic line will intersect the corner in a first approximation, and the portion of the acceleration of the boundary layer which takes place upstream from the corner can be analyzed, independent of a knowledge of the base pressure. The flow deflection angle 0 will be equal to zero at the wall and comparable in magnitude with M at the outer edge of the accelerating boundary layer. Two possibilities arise for the e order of magnitude of 0 in the layer. To match 0 to the value 6 = O(M at the outer edge, one might anticipate a region where the flow deflection angle is comparable in magnitude with -1 M throughout the boundary layer. In such a region, the streamline curvature is small. e Therefore, the normal pressure gradient can be neglected, and the flow is described by inviscidboundary-layer equations. (For another application of these equations, see reference 17.) A second region might occur in which 0 is of order unity throughout the layer. The boundary condition at the outer edge, in the first approximation, would then require that 0 be equal to zero. In this region, the normal and streamwise pressure gradients would be comparable in magnitude because of significant streamline curvature. A considerable divergence of streamlines will actually occur because, in the immediate neighborhood of the corner, a streamline just ouside the viscous sublayer turns rather sharply around the corner.

WILLOW RUN LABORATORIES The two possible choices for the order estimate of 0 correspond to two rather different physical effects. A decrease in the pressure will cause the subsonic portion of the layer to contract and the supersonic portion to widen. Initially, the subsonic portion of the layer is dominant, and d6*/d-p > 0. This is designated as the subcritical condition [13] and is analogous to subsonic flow in a convergent nozzle. The pressure change generated is communicated smoothly through the subsonic portion of the layer a considerable distance upstream. Lighthill [8] finds the inverse logarithmic decrement of upstream influence for small disturbances in a subcritical shear layer to be M21 - M-1/26 -2 (2.1) M2(1 M2) X M 2()[1- M ]d (2.1) where M(O) is the Mach number at a point just outside of the viscous sublayer. Clearly, in an accelerating boundary layer, a point may be reached at which the integral in equation 2.1 vanishes. At such a "critical point," some average Mach number in the layer is sonic, and small disturbances can propagate upstream only through a distance of the same order as the boundary-layer thickness. When the expression in equation 2.1 is set equal to zero, the condition -- M-2 0)dy <M-> = 1 (2.2) -2 results. Therefore, <M-2> is the appropriate function for determining whether or not the layer is subcritical. In a subcritical flow,KM 2 > 1, while, in a supercritical flow, we have <M2 < 1. In the supercritical region, we would expect d6*/dp < 0 because the widening of streamlines in the supersonic region is dominant as the pressure continues to decrease. Since the flow deflection at the outer edge is not expected to change sign, the boundary layer in the present problem can become supercritical only in a region where the stream tubes near the wall can be displaced inward. It is shown in the next section that this effect can occur within a distance of order 6 upstream from the corner. 3 ASYMPTOTIC REPRESENTATIONS 3.1. LIMIT PROCESSES The equations of hypersonic small-disturbance theory for flow past slender bodies are obtained from the full inviscid-flow equations by taking the limit M - cO, T-0 x/L, Yr/rL, M T held fixed (3.1) 8~~~~~~~~~~~~c

WILLOW RUN LABORATORIES Here x and r are dimensional coordinates measured from the front of the body (see fig. 1), M is the free-stream Mach number, T is the body thickness ratio, and L is the length of the body. For a sufficiently high Reynolds number, the shape of the shock wave for a cone or a wedge is given in the form Y/TX = constant (3.2) In this approximation, the Mach number behind the shock wave is of order T~ (see app. I). The approximate equations describing a hypersonic laminar boundary layer on a slender body are obtained in the limit M -o, R -o, M R-1/2 0 e w e w x/L, y/R 1/2L held fixed (3.3) w where Me is the Mach number just outside the boundary layer, and R is the Reynolds number based upon ue, L, and the thermodynamic properties evaluated at the wall. The condition M R - O arises from the requirement that the ratio of boundary-layer thickness to shocke w layer thickness vanish in the limit 3.3 (see app. I). In the present study, the nondimensional parameters, in the case of a chemically inert, laminar, continuum flow, may be chosen as Me, Rw, A, Pr (3.4) where y is the ratio of specific heats, and Pr is the Prandtl number. We will consider a limit M -co, R - o, M R-1/2 0 e w ew y, Pr held fixed (3.5) Since we will be concerned with a small region near the corner at the base of the body, both the x and y coordinates will be stretched in some manner. The equations of motion for two-dimensional planar, nonreacting, laminar, continuum flow are (iu)- + (V)- = 0 (3.6) puui-x+vUi) = -F + (M Y)- +... (3.7) P(gu v_ + VV-) = -i- + (3.8) X y ~6Tx- + vT-) - ix + v = (k T-)- +... yy 7Tu +...(39~)

WILLOW RUN LABORATORIES For simplicity, the terms shown for viscous and heat conduction effects are only those important in the boundary-layer equations. In the case of axisymmetric flow past a cone, it will be shown that the effect of body curvature on the acceleration of the boundary layer at the corner of the base of the body is negligible in a first approximation. We now consider the boundary conditions to be imposed at the wall and at the outer edge of the accelerating boundary layer. At y = 0 we have u(x, O) = v(x, O) = 0 (3.10) As the boundary layer undergoes rapid acceleration, it will initially become thinner if it is in the subcritical condition in which d6*/dp > 0, where 6* is the displacement thickness of the boundary layer (see sec. 2). Then the pressure at the outer edge of the layer will be related to the flow deflection by the simple-wave relation in hypersonic small-disturbance theory [1]. p(k, 6) - 1 1j2y/(-) -_2 -M 2 M2eo + (x, 6) (3.11) Peue Y e Notice that the boundary-layer thickness 6 in the limit 3.5 is equal to the displacement thickness 6* (see eqs. 1-34, 1-35, and ref. 16), so that O(x, 6) = d6*/dx[1+ O(M02)] (3.12) with v/u = tan O ~0 when 0 is small. From the limit 3.1 and when Ap/p = 0(1), we have 0(x, 6) = (M) (3.13) If in addition to the limit 3.5, y/R 1/2L and some stretched x coordinate are held fixed in a manner such that (x/L) - 1 - 0 (3.14) the equations obtained in a first approximation will not contain terms representing viscous effects. Regardless of the limit chosen, the order of magnitude of the flow deflection should be such that a first approximation to the continuity equation shows a balance between streamline divergence and change of mass flux. If Ap/p = 0(1), it follows that 0 = O(A y/Ax) (3.15) where Ax and Ay are taken to be of the same order as the relevant lengths, respectively, in the x and y directions. 10

WILLOW RUN LABORATORIES Ify -- -/ L is held fixed, any of four possible sets of equations might be obtained, consistent with the limits 3.5 and 3.14. For x — [(x/L) - 1]/MeR /2 held fixed, equation 3.15 gives = e (3.16) throughout the layer; the y-momentum equation simplifies to p- 0, and no further approximation is made in the boundary condition (3.11). For x - [(x/L) - 1)]/Rw /held fixed, equation 3.15 gives 0 = O(1) (3.17) in the boundary layer; terms for both pressure and inertia appear in the first approximation to the y-momentum equation. Since 0 = o(Me ) at the outer edge, the leading term in 0 is required to approach zero as y - 6, and equation 3.11 is replaced by the requirement that p remain bounded as y - 6. For the class of limits x - O but x - -co, we obtain the approximate form both for the ymomentum equation and for the boundary condition at y = 6. This case turns out to be important for obtaining higher approximations. Finally, for the class of limits x - -co with (x/L) - 1 - 0, we find 0 =o(M-1 (3.18) throughout the boundary layer. Here the flow properties are only slightly perturbed from their undisturbed values, a situation related to that studied by Lighthill [8]. Following the physical arguments of the previous section, we will assume that limits for x or x held fixed mustboth be considered, with x fixed to correspond to the subcritical condition and x fixed to correspond to the supercritical condition. The choices of x and x show the same dependence on Reynolds number as those given in references 11 and 12. Since in each case the no-slip condition (3.10) is lost, we will refer to those as the first and second outer limits, respectively, and we will later introduce a sublayer limit in which y - 0 at a prescribed rate. Specification of the sublayer limit is determined, as in ordinary boundary-layer theory, by the requirement that terms for viscosity and inertia in the x-momentum equation (3.7) be of the same order of magnitude. The notation to be introduced for the boundary-layer limit, the first outer limit, the second outer limit, and the sublayer limit are summarized in table I. 3.2. FIRST OUTER LIMIT The first outer limit is described by the conditions e -o, Rw-,o, M RwM/2 -0 ~~~~~~~e w e ~~w ~(3.19) x, y, y, Pr held fixed 11

WILLOW RUN LABORATORIES TABLE I. COORDINATE NOTATION AND LEADING TERMS IN ASYMPTOTIC EXPANSIONS Original Boundary First Second Variables Layer Outer Limit Outer Limit Sublayer x A x/L-1 x/L - 1 x/L - 1 x = x = -1/2 x x - xM R L R1/2 R-1/2 e w w w y/L y/L /L yt = y/L R-121/2 12-3/4 ~R R R R w w w w u/u ~ uy(x, y) u(x, y) u(X, Y) U(x)u (x, yt) v/u R /2 v(x, y) M vl(x, y) v(x, y) R 1/4U(x)v (x, y) e w e w p/Pwue Pe 2y P ) p(x y) pt(x) P/P p(x, y) p(x, y) p(x, y) pt(x, yt) where - [(x/L) - 1]/MR and y - /R L. We wish to obtain the appropriate asymptotic representations for u, v, p, and p and the system of equations and boundary or initial conditions applicable in a first approximation. We note that in a boundary layer in the limit 3.3 Ou/u = O(1) (3.20) e= O(2Me) (3.21) These relations are also applicable in the limit 3.19. Thus from equations 3.16 and 3.20 we have vu = O(M ) (3.22) and if Ap/p is assumed to be of order one, equation 3.21 gives A (w1)ue = O(1) (3.23) From the x-momentum equation (3.7), we see that = 0(1) (3.24) 12

WILLOW RUN LABORATORIES In the y-momentum equation (3.8), we have PuvX = O(2M R1/2 L_) (3.25) — 2 \ e w w e Since Ay is of the order R L, in a first approximation, p- z 0, and p is only a function of x. w y From equations 3.20 through 3.25, we see that the asymptotic representations for u, v, p, and p in the limit 3.19 are given by u (X, y) +... Ue At. (3.26) p ue p(X) +. p P A(x, y) +... Substitution of equations 3.26 into equations 3.6 through 3.9 provides the relations for the first approximation in the limit 3.19 [puA+ u + o(1) = O (3.27) x y pUUA + VU =-pA +(M /2) (3.28) X PA X 0= A + O(M ) (3.29) p + h = UP + (M + 0(Me (3.30) x. ewhere h - h/ u. Since the equations in the first outer limit are inviscid-flow equations, the noslip condition u(x, 0) = 0 must be dropped, and only the requirement v(, 0) =0 (3.31) is retained. For convenience we introduce a nondimensional, stretched boundary-layer thickness 6= 6/R /2 L (3.32) which remains finite in the limit 3.19. Then the pressure is related to the flow deflection at the outer edge of the layer from equations 3.11 and 3.13 by A vX, 6y 2y/(y-l ) p(x) Y - 1 - 1v(x, ))j (3 33) 13

WILLOW RUN LABORATORIES Upstream we have the initial condition u (- o, Y) = U1(y) (3.34) where u1(y) is the velocity profile found from the solution to the boundary-layer equations with x/L = 1 (see app. I). For a large and negative x, it follows from equations 3.11 and 3.18 that the pressure disturbances are small, and the flow problem becomes similar to the one studied by Lighthill [8]. Lighthill found that the region of disturbed flow extends a distance of order R 3/8L in the stream w direction. The variation in the displacement thickness of a viscous sublayer has a first-order effect on the perturbations in the outer, inviscid portion of the layer and is of order R 5/8L in w magnitude: x= O(w /) -_~~~~~~~~~~~~* ~(3.35) SL = OQR) Depending upon the order of magnitude of the pressure disturbance, there are two possible order estimates for the change Au in the velocity of any fluid element in the sublayer, namely O(u) Au = (3.36) o(u) In the first case, the sublayer equations are nonlinear, and a balance of the orders of magnitude of p u u-, p-, and (Mi u-)_ in equation 3.7 with y Ax =O (R 3/8L) Ay = (Rw5/8L) provides the order estimates U/Ue= o(,R/8) (3.37) Ap (- U2) 0 (R-1/4) (3.38) In the outer, inviscid region, the orders of magnitude of puu- and p in equation 3.7 are equated, giving, along a streamline, 14

WILLOW RUN LABORATORIES -/- O( -1/4) (3.39) In the second case in equation 3.36, the sublayer equations may be linearized, and ApT u ) 0= (rrl/4) (Note that Lighthill's order estimates may be reconstructed by a systematic consideration of orders of magnitude in the limit R - coo.) The solution for u (x y) is obtained in section 4. It is shown in equation 4.29 that as x - -co, u (X, y) has the form ul(y) + O(x2), y > 0 A A u (x, y) = (3.40) Thus, if x = O(R ), the order of magnitude of u (x, y), as found from equation 3.40, agrees with the order estimates given in equations 3.37 and 3.39. That is, the velocity profile obtained in the first outer limit has the possibility of matching upstream with a solution to the problem described by Lighthill. 3.3. SECOND OUTER LIMIT The second outer limit is described by the conditions M -oo R -c, M R1/2- 0 e ' w ew (3.41) x, y, y, Pr held fixed where x [(x/L)- 1]/Rw/2, and - y/(Rw/2 L). We wish to obtain the appropriate asymptotic representations for u, v, p, and p, and the system of equations and boundary conditions applicable in a first approximation. The order estimates 3.20 and 3.21 are also valid in the limit 3.41, and, thus, from equations 3.15 and 3.20, and 3.41, we have v/ue = O(1) (3.42) In order for the inertia and pressure gradient terms to balance in the x- and y-momentum equations 3.7 and 3.8, we then must have A (wue) = 0(1) (3.43) (3.44) 15

WILLOW RUN LABORATORIES From equations 3.20, 3.21, and 3.42 through 3.44,we see that the asymptotic representations for u, v, p, and p in the limit 3.41 are u/u U(, y) +... v/ue V(X, ) +... (3.45) wUe P/Pw ~ p(', y) +... Substitution of equations 3.45 into equations 3.6 through 3.9 provides the relations for the first approximation in the limit 3.41. (P)~ + (P) + o(1) = 0 (3.46) P( x + vu) = + (R1/2)(3.47) P(uvx + vV) = - + O(R1/2) (3.48) Ur + i-i;) = -) += _ p+ - (1'/2 (3.49) uh X h Y) Xvp+ + o(%1/) x y w where h - h/ Because these relations for the second outer limit are the inviscid-flow equae tions, the no-slip condition u(x, 0) = 0 must be abandoned, and only the requirement v(x, 0) = 0 (3.50) is retained at the wall. At the outer edge of the layer, v/u is of the order Me so that the boundary condition on v at y = 6 becomes v(x, 6) = 0 (3.51) while p(x, 6) must remain bounded. The equations 3.46 through 3.49 are elliptic when the local Mach number M = [(2 ]/Yp (3.52) is less than one. Thus, it is appropriate in this limit to specify a downstream condition. As discussed in section 2, for a sufficiently low base pressure, the sonic line will intersect the corner, x = y = 0, in a first approximation where the effect of the viscous sublayer is neglected. Therefore, we have the condition 16

WILLOW RUN LABORATORIES M(O, 0) = 1 (3.53) To obtain initial conditions upstream, we assume that for any given flow quantity the solutions in the first and second outer limits can be matched. That is, we assume that both solutions are valid for some class of intermediate limits such that — oo0, x/M - 0 (3.54) x/f(M ) fixed where 1 << f(M ) << Me as M - Co. In a first approximation the matching appears straightforward, and it is expected that the matching conditions become simply u(-oo, y) = (O, y) (3.55) p(-co, y) = p(O) (3.56) v(-cc, y) =0 (3.57) See reference 18 for a further discussion of matching. For higher approximations, the situation is quite different. As x - -co, the asymptotic expansion of the normal velocity component v for the second outer limit (3.41) gives ~-3 v/ue (constant) x as x -cco. See the representation for v described by equations 5.48 and 5.93, for example. In the first outer limit, the asymptotic expansion of v, for x = x/Me - 0, is found from the solutions given by equations 4.17, 4.18, and 4.33 to be A -1/2 V /e (constant) M le x -1 (constant) M xe M 1e/2 as x/M - 0. Thus, these two expansions for v do not have the same functional form, and the e higher order matching cannot be carried out. Since these expressions are of the same order of magnitude when x = O(Me/ ), it is expected that approximate differential equations must also be derived for f(Me)= (M e/5 )in equation 3.54, i.e., for the distinguished limit in which 1/ 5 is held fixed. It can be shown that in this limit the order estimates for the perturbae tions in the dependent variables along a streamline are ~AU/Ue= O(Me2/5) 17

WILLOW RUN LABORATORIES V/U =O(M /) e e -(p2) o(M-2/5) Ap/p= O(Me2/5) In the governing equations in this limit, both the approximation p- 0 employed in the first outer limit (3.19) and the approximation v(x, 6) 0 applied in the second outer limit are valid. A solution to these equations would be required as part of a higher order approximation. In equations 3.6 through 3.9, the effect of body curvature has been neglected. The extent of upstream influence of the corner in which the disturbances are still a first-order effect is found to be (x/L) -1= o( R_/2) ( e w ) Since the radius of a slender cone is equal to TX, the relative variation in the radial coordinate r when x [(x/L) - 1] M R12 is held fixed is ew1 ( e w where r1 = TL is the radius of the base of the cone. Thus, the variation in r is relatively small, and may be classed as a second-order effect. 3.4. COMPOSITE EXPANSIONS OF SOLUTIONS IN THE FIRST AND SECOND OUTER LIMITS As long as the asymptotic expansions obtained in various limits in a solution to a singular perturbation problem have a common region of validity, a single uniformly valid expansion can be constructed from them. One method of constructing composite expansions is by additive composition. The sum of the expansions is corrected by subtracting the part they have in common so that it is not counted twice [18]. In the present problem, we will construct a composite expansion of the solutions in the first and second outer limits to provide the necessary boundary values in the sublayer problem. Following the rule for additive composition we have, to the first order, ~u,M + U(X, )) - U(0, y) +. (3.58) u e e where x/Me has been substituted for x and where the common part, to the first order, is found from equation 3.55. Since only M appears in the expansion, it is necessary,in this approach,to 18

WILLOW RUN LABORATORIES specify Me but not R in carrying out the solution to the viscous sublayer equations. When the sublayer solution has been obtained, all three expansions may be combined to give, in a first approximation, u y;M, U + u(x, Y) - u (0, y + U(x) u(w Ue~~ e (x, ) + u (o, 0) +... (359) where u tue, y-/ord W n). in the sublayer (eq. 3.64). Composite expansions for the other flow properties are found in a similar manner. In particular, the first-order, uniformly valid representation for v is simply v x, y; Mel Rw) (x )+ uu 0(, (360,e 3.5. SUBLAYER LIMIT The conditions applicable in the sublayer limit are found when an approximate stretching factor for the y-coordinate in the sublayer is determined. Again, the order estimates for u and p given in equations 3.20 and 3.21 are applicable in the viscous sublayer also. Because the pressure gradient in the sublayer is equal to the pressure gradient given by the outer inviscid solution for y = 0, equation 3.23 also provides the order estimate for ap in the sublayer. Thus, for e If we choose to consider a limit in which Me - oo, it is again necessary to introduce the two x coordinates, x and x. It seems, however, that to take the hypersonic limit in studying the sublayer does not lead to simplification, but rather to increased complication. A different approach is first to obtain the composite solutions for the first and second outer limits (e.g., equation 3.58) and then to use these results evaluated at y = O as the boundary conditions at the outer edge of the sublayer. To accomplish this, of course, it is necessary to specify the value of M. Then x = [(x/L) - 1]/ R1/2 is an appropriate x coordinate in the sublayer, and 19

WILLOW RUN LABORATORIES puu- -= o(R/2 L-1) PwUe w e (3.61) -(ju- = O(Rw l SL L) Pu w e Thus, 5SL= O(R3/4L), and the conditions in the sublayer limit are R -co w (3.62) x, yt, Me, y, Pr held fixed where x [(x/L) - 1] and yt y/Rw L. This choice is in agreement with the result given in references 11 and 12. The order of magnitude of the flow deflection in the sublayer is found from equations 3.15 and 3.62 to be v/u = O(R- /4) (3.63) From equations 3.20, 3.21, 3.23, 3.60, and 3.63, we see that the asymptotic representations for u, v, p, and p in the limit 3.62 are u/u ~ U(x)ut(x, yt) +... v/ue Rw /4U(i)vt(xt) (3.64) P/PwUe) p )+ ioPWsp (x, y)+.. where U(x) is the nondimensional x component of velocity at the outer edge of the sublayer and is found from equation 3.58 with y = 0. The introduction of U(x) into equations 3.64 simplifies the boundary conditions on ut. Substitution of equations 3.64 into equations 3.6 through 3.9 provides the relations for the first approximation in the limit 3.62. (p Uut)x + (ptUv ) + o(1) = 0 (3.65) y yt y PtUut(Uut)+vtUutt] = - (t )+ U utt + o(R 1/2) (3.66) 0=_t + (R/) (3.67) 20

WILLOW RUN LABORATORIES p ~ +vh = +0 ) (3.68) P tU th-t +VY Uut Pi tTu + 1/2 (3.68) x x yy y t wh kt k/k hf h2 where - w, h h/u. The boundary conditions (3.10) become ut(, 0) = vt(x, 0) = 0 (3.69) while, at the outer edge of the sublayer, yt o, and ut(x, oo) = 1 (3.70) 4 SOLUTION IN THE FIRST OUTER LIMIT: (R/L)-l = O(MeRw-1/2) 4.1. TRANSFORMATION OF THE EQUATIONS The approximate equations 3.27 through 3.30, obtained by taking a limit of the exact equations for x fixed, can be integrated directly by the following procedure. First, von Mises variables s, 0 are introduced. A S=X 5A= -- (4.1) A AA. = pu Then in the case of adiabatic flow of a thermally and calorically perfect gas over an insulated body, the set of equations (4.1) becomes (v/u)~ = (1/p)s (4.2) A A2 2yP u+ 2 =1 (4.3) y - A A AA (4.4) p = E(4i)p (4.4) A A A A A where u, v, p, and p are now functions of s and r4. The boundary and initial conditions (eqs. 3.31, 3.33, and 3.34) are u(-oo, 4) = g(3) (4.5) 21

WILLOW RUN LABORATORIES v(s, O) = 0 (4.6) 2y P(s) = 2j [ 1js +x 1(S )] (4.7) where g(3) = u1, the Blasius boundary-layer solution u(x, y) evaluated at x — x/L = 1 (see app. I). Notice that any streamline can be identified by specifying either the appropriate value A of 14 or the corresponding value of I. From equations 4.3, 4.4, and 4.5, E($) =2 1(1 g2) (4.8) 2y Now let us introduce a new coordinate f = 1 + 2 v(s, 0o) (4.9) Thus, from equations 4.3, 4.4, 4.7, 4.8, and 4.9 we obtain A 2.2 1/2 u =- (1 - g)2> (4.10) A y- 1 2y/(y- 1) (4.11) p - 2y 0 (4.11) P =[1 g2 -1 2/(y-1) (4.12) The normal velocity component v and the transformation s = s(o) are found from integration of equation 4.2. A A /AA A v = u(ds/do) I (1/Pu) 0 d4/ (4.13) A 2 A A Since v -- _ (1 - ) as - co, letting ' - x in equation 4.13 yields ds/d 2 (1 - ) 1/u),d (4.14) The normal coordinate y is found by integrating the third of equations 4.1. A Y =fA (1/su)d4 (4.15) 22

WILLOW RUN LABORATORIES The integrals in equations 4.13 through 4.15 can be evaluated by substituting for p, u from equations 4.10 and 4.12 and carrying out the differentiation with respect to ~. Then the A variable of integration is changed from 4 to 3. Since x1 = r1 = 1, equations I-26 and I-31 yield diq = 2k+ 1) gdo (4.16) The resulting integrals are ds/d (2 )/ l 1(- P)- K (4, i)do (4.17) ds/de= k + 72k'-1 (1 1/2 K1(, j)d A1 [1 - 0(1 g)] / (1 - ~) (4.18) [K1(~, f)do Y = ( 2k ~ ); K2f(~K(, )do3 (4.19) where K1 and K2 are given by K1, 0) =(l _ ) )[1 (2+ _2)]1 _2 (1 2- )]-3/2 I 2- " < (1 - g1 (4.20) K2(q1 i) _ - 2[ 12 2.2 -1/2:2(~,' )= g(1 - g )[1 - ~2(1 - g2)]-1/2 The function I(f3) will be defined by I(o) 2k + 1 -(1 )-1 K d I() = (2k- K(, 1 )d3 and will be used in the calculation of ds/de. Equation 4.17 now could be written as ds/de = I(oo). 1 The kernel K1 can be written in terms of the Mach number M = (u /yp) with M2 (1 - M) = -1 +102(1 - ]2)][1 - 02(1 _.2 -1 A A 23

WILLOW RUN LABORATORIES ds/dp = = -0(1 - )dy (4.21) Thus ds/do is proportional to Lighthill's result, equation 2.1, for the inverse logarithmic decrement of upstream influence for small disturbances. The derivative ds/dp clearly vanishes at some point where the Mach number just outside the sublayer is still subsonic. This "critical point" represents the downstream limit of validity of the solutions obtained in terms of x and y. As the pressure continues to drop, the spreading of streamlines in the supersonic portion of the layer will be dominant. This is expected to occur only when the flow becomes free to turn inward within a distance Ax = O/(R/ L) upstream from the corner. Since x = O(1) corresponds to a distance Ax = O(MeR 1/2L) upstream from the corner, the critical point is located at x = 0. 4.2. ASYMPTOTIC EXPANSION FOR s - -oo To obtain asymptotic expansions for the dependent variables as s - -oo, we introduce e = 1 -q so that e - 0 when - 1. The integral in equation 4.17 can be split into two parts by introducing 00 I1= J K 1(' f)do (4.22) 12 = K1(~, f)do 0 where i0 is defined to satisfy E << 30 << 1. Then in I1 the approximation (see eq. I-41) g(3) ~- o as: - 0 can be used, while in I2 the approximation E << g2(/) (4.23) is valid. Thus, we obtain 0 22 2 -3/2 2 2 y-l 1 ~t'-~1 i7, 1 (2c + a22) d(a2! 2) + 0(1) = + 01 (4.24) 24

WILLOW RUN LABORATORIES - 2y y + 1 2 (4.25) -2 1 -- 2 dfi2 Y~y Because the integral is not bounded for 30 - 0, we add and subtract a term to remove the singularity. Y2 2 i (2 23) y - 21d Y - + (4.26) When we recombine the integrals I1 and I2, the terms in cancel out so that ~2_ -1 1 1/2 ds/d21 2k +1 ) - 3/2 (12 + 1- ) s - -( - 1) 1(2k + 1)-1/2(1 - )-1/2 (4.27) or 1 (1 1)2 2 -2(2k 1) 1s-2 (4.28) From equations 4.10 through 4.12 and 4.28, as s - -o, =0 (4.29) i+ 0(s-2), > 0 P = -1 + O(s-2) (4.30) 2y p =(1- ) +O(s ) (4.31) while from equations 4.18 and 4.28 we have v (constant)s-2 for I > 0 (4.32) as s -co. Thus, in the upstream limit s - -oo, the flow properties decay algebraically as anticipated by the discussion of equations 3.40. 4.3. EXPANSION AT THE CRITICAL POINT Although ds/dq vanishes at the critical point, d s/d9 there is nonzero, so that a Taylor series expansion of the integral of equation 4.17 yields 25

WILLOW RUN LABORATORIES (4.33) -s (c (( - qbc)2 (433) as - - c where (c is found from P K1(c, I )d = 0 (4.34) Since A\ 2 d6/ds = v(01, oo) = - (1 - 0) (4.35) we have that d26/ds2 2 (ds/d)- 1 (4.36) Thus, in a neighborhood of the critical point, d6/ds oc constant + O[(-s) 1/2] (4.37) d26/ds2 oc (-s)-1/2 as s - 0. Thus, although the streamline slope at the outer edge of the boundary layer remains bounded, the curvature, in terms of the stretched coordinate s, becomes large at the critical point. 4.4. NUMERICAL RESULTS %AA A fA To obtain u, v, p, and p as functions of x and y, it is necessary to integrate equations 4.17 and 4.20 numerically. These integrations have been carried out using the IBM 7090 digital computer at The University of Michigan Computing Center. The algorithms employed are programmed in the MAD language [ 19]. The numerical technique employed is first to integrate equations 4.20 and I-39 on 3, using the Runge-Kutta fourth-order method, which is a standard computer library subroutine [20, 21]. Thus, the integration on: consists of solving the following set of first-order, ordinary differential equations: dy 1/d = 2 (4.38) dy2/do = y3 (4.39) dy3/do = -y 1Y3 (4.40) dy4/d3 (2k+ 1) K2 (4.41) dy5/d3 = (2k2+ )/2 (( 1 - -) K1 (4.42) 26

WILLOW RUN LABORATORIES where Y1 = g(P) Y2 = g(p) Y3 =g(f) (4.43) Y4 = y =2 (-1 K -1(1,- d Equations 4.38 through 4.40 are introduced to generate the function g(o), which appears in the expressions for K1 and K2. To start the numerical integrations it is necessary to utilize asymptotic expansions for Y1,..., y5 as - 0. The expansions for Y1, Y2, and y3 are given by equations I-41, while results for expansions of y and y5 are y ( 2k + 1)/ -- 2a(Z3 - Zq) + (4.44) (4.46) Y5 2k -1(1 - 0)-i 1 1 ~ 2 Z + [(Y + I)0 2 11Z +. (4.45) as/3 - o, where a = 0.4696 and where Z 2b-l(c1- a-1/2) 1 = 2c 1/2 _2a + ac-1/2 Z3=2b (c -1/2 Z4 = 2b-2 3 ac 1/2 + 2a3/2) 2 2 2 22 Here a 1 -,b, and c 1 - (1-a ). Since we find1- 2 to be a numerically small quantity over its possible range of values, 12 has not been neglected in comparison with 1 - 2 as - 0, in equations 4.44 and 4.45. With the initial values of the dependent variables given, the numerical integration proceeds, step by step, up to a value of 1 sufficiently large for the asymptotic expansion for g as 3 - oo, given in equation 1-30, to be applicable. In terms of the values calculated in the last step in the numerical integration, the values for - cc are, ~ y4(~, 13) + 0.662(2k + 1) 1- 2 exp (-2) (4.47) 27

WILLOW RUN LABORATORIES 1/2 Y+1 ds/d- y 5(', 3) + 0.662(2k2 1) -1(1 )-1-2 exp (2) (4.48) where ( = - - 1.21678. Now ds/do can be calculated for any given value of p from equation 4.48. The critical point is located when ds/d4 = 0. The critical values are found by a simple iteration scheme to be M = 0.4482 O,c c/Pe = 0.8712 (4.49) oc = 0.980495 when y = 1.4. We retain six significant figures in c because of the factor 1 - p which appears in the equations. Although the value of /c is quite close to one, the drop in pressure is more A 7 pronounced because p oc q when y = 1.4. Then s = 0 when, = c, and the second step in the numerical solution is to compute the integral s = f (ds/dp)do (4.50) c where ds/dp is found in equation 4.48. This is carried out by Gaussian quadratures [20], another standard computer library subroutine. It should be noted that when the results are considered as functions of x/ 1 and Y/61, where 61 is given in equation I-37, they are applicable to both wedge-shaped and conical con1/2 figurations. This is because not only 61 but also s and y contain the factor [2/(2k + 1)] (see eqs. 4.17 and 4.19). The velocity u/u and the function I(3) are tabulated against y/6 and j for several values of / in table II. In table III, the quantities ~, Mo, P/pe, Uo/Ue, do/ds, and 6/ 61 are tabulated against s/6 1 Figure 3 shows M, uo/e, P/Pe 6/61' and q plotted as functions A of x/651. The flow properties are seen to change rather rapidly as x - 0, a result expected from the expansions in equations 4.28 through 4.32 for s - 0. The results in the solution for the first outer limit will be utilized further in constructing composite expansions. This topic is discussed in section 5 after the solution in the second outer limit has been obtained. 28

WILLOW RUN LABORATORIES TABLE II. SOLUTION IN FIRST OUTER LIMIT, PART ONE = 1.000000 ~ = 0.990000 Y/6 1 u/ue I(3) _ y/1 u/ue I(3) 0.000 0.0000 0.0000 0.OOOE + 00 0.000 0.0000 0.1411 0.OOOE + 00 0.010 0.0059 0.0047 0.OOOE + 00 0.010 0.0001 0.1411 -0. 128E + 00 0.500 0.2910 0.2342 -0.OOOE + 00 0.500 0.1738 0.2714 -0.960E + 02 1.000 0.5502 0.4606 -0.OOOE + 00 1.000 0.4264 0.4773 -0. 109E + 03 1.500 0.7514 0.6614 -0.OOOE + 00 1.500 0.6332 0.6698 -0.923E + 02 2.000 0.8834 0.8166 -G0.OOE + 00 2.000 0.7708 0.8206 -0.764E + 02 2.500 0.9545 0.9167 -0.OOOE + 00 2.500 0.8454 0.9184 -0.667E + 02 3.000 0.9854 0.9688 -0.OOOE + 00 3.000 0.8779 0.9695 -0.623E + 02 3.500 0.9962 0.9904 -0.OOOE + 00 3.500 0.8892 0.9906 -0.607E + 02 4.000 0.9992 0.9975 -0.OOOE + 00 4.000 0.8923 0.9975 -0.603E + 02 4.500 0.9999 0.9994 -0.OOOE + 00 4.500 0.8931 0.9994 -0.602E + 02 0.000 1.0009 1.0000 0.OOOE + 00 0.000 0.8942 1.0000 -0.600E + 02, = 0.999000 q = 0.985000 ___ P~y/6u I(e ___) 1 u/u I(/f) 0.000 0.0000 0.0447 0.OOOE + 00 0.000 0.0000 0.1726 0.OOOE + 00 0.010 0.0003 0.0450 -0.423E + 02 0.010 0.0001 0.1726 -0.447E - 01 0.500 0.2305 0.2382 -0.361E + 04 0.500 0.1596 0.2881 -0.424E + 02 1.000 0.4888 0.4623 -0.385E + 04 1.000 0.4103 0.4854 -0.481E + 02 1.500 0.6905 0.6623 -0.370E + 04 1.500 0.6202 0.6740 -0.367E + 02 2.000 0.8231 0.8170 -0.355E + 04 2.000 0.7607 0.8226 -0.257E + 02 2.500 0.8946 0.9168 -0.345E + 04 2.500 0.8371 0.9193 -0. 190E + 02 3.000 0.9256 0.9689 -0.341E + 04 3.000 0.8704 0.9698 -0. 160E + 02 3.500 0.9364 0.9904 -0.340E + 04 3.500 0.8819 0.9907 -0.149E + 02 4.000 0.9394 0.9975 -0.339E + 04 4.000 0.8852 0.9976 -0.146E + 02 4.500 0.9402 0.9994 -0.339E + 04 4.500 0.8859 0.9994 -0.146E + 02 0.000 0.9412 1.0000 -0.339E + 04 0.000 0.8871 1.0000 -0. 145E + 02 ~ = 0.995000 q = 0.980495 _ Y/1 UP/ue I(j3) __ 1 u/ue I(/1) 0.000 0.0000 0.0999 0.OOOE + 00 0.000 0.0000 0.1965 0.OOOE + 00 0.010 0.0001 0.1000 -0.745E + 00 0.010 0.0001 0.1966 -0.224E - 01 0.500 0.1966 0.2536 -0.331E + 03 0.500 0.1507 0.3023 -0.239E + 02 1.000 0.4519 0.4691 -0.367E + 03 1.000 0.4002 0.4925 -0.267E + 02 1.500 0.6559 0.6656 -0.335E + 03 1.500 0.6130 0.6776 -0.175E + 02 2.000 0.7906 0.8186 -0.304E + 03 2.000 0.7563 0.8244 -0.887E + 01 2.500 0.8635 0.9175 -0.286E + 03 2.500 0.8343 0.9200 -0.361E + 01 3.000 0.8952 0.9691 -0.277E + 03 3.000 0.8683 0.9701 -0.122E + 01 3.500 0.9062 0.9905 -0.274E + 03 3.500 0.8801 0.9908 -0.378E + 00 4.000 0.9092 0.9975 -0.273E + 03 4.000 0.8835 0.9976 -0.140E + 00 4.500 0.9100 0.9994 -0.273E + 03 4.500 0.8842 0.9994 -0.833E - 01 0.000 0.9111 1.0000 -0.273E + 03 0.000 0.8854 1.0000 0.459E - 03 29

WILLOW RUN LABORATORIES TABLE III. SOLUTION IN THE FIRST OUTER LIMIT, PART TWO M P u /U d /ds / 61 s/d 1.000000 0.0000 1.0000 1.0000 0.000E + 00 1.0000 -0.0000 0.999000 0.1001 0.9930 0.0447 -0.339E + 04 0.9412 -3.3437 0.998000 0.1416 0.9861 0.0632 -0.122E + 04 0.9309 -1.8937 0.997000 0.1736 0.9792 0.0774 -0.655E + 03 0.9229 -1.2456 0.996000 0.2006 0.9723 0.0894 -0.406E + 03 0.9165 -0.8711 0.995000 0.2244 0.9655 0.0999 -0.273E + 03 0.9111 -0.6291 0.994000 0.2461 0.9587 0.1094 -0.192E + 03 0.9065 -0.4625 0.993000 0.2660 0.9520 0.1181 -0.140E + 03 0.9026 -0.3431 0.992000 0.2846 0.9453 0.1262 -0.104E + 03 0.8993 -0.2552 0.991000 0.3020 0.9387 0.1339 -0.788E + 02 0.8965 -0.1893 0.990000 0.3186 0.9321 0.1411 -0.600E + 02 0.8942 -0.1392 0.989000 0.3344 0.9255 0.1479 -0.459E + 02 0.8922 -0.1010 0.988000 0.3496 0.9190 0.1545 -0.351E + 02 0.8905 -0.0717 0.987000 0.3641 0.9125 0.1607 -0.266E + 02 0.8891 -0.0495 0.986000 0.3781 0.9060 0.1667 -0.199E + 02 0.8880 -0.0327 0.985000 0.3917 0.8996 0.1726 -0.145E + 02 0.8871 -0.0203 0.984000 0.4049 0.8932 0.1782 -0.101E + 02 0.8864 -0.0114 0.983000 0.4177 0.8869 0.1836 -0.651E + 01 0.8859 -0.0055 0.982000 0.4301' 0.8806 0.1889 -0.355E + 01 0.8856 -0.0018 0.981000 0.4422 0.8743 0.1940 -0.108E + 01 0.8854 -0.0002 0.980495 0.4482 0.8712 0.1965 0.459E - 03 0.8854 0.0000 1.0 -2_-10 0.9 0.8 0.5 0.4 0.3 0.2 0.1 30

WILLOW RUN LABORATORIES 5 SOLUTION IN THE SECOND OUTER LIMIT: (/L)-1 = O(RW -1/2) 5.1. FORMULATION BY THE METHOD OF INTEGRAL RELATIONS The equations 3.46 through 3.49 in the variables x = [(x/L) - 1]R1/2 and y = y/R 1/2L, describing the flow in the second outer limit for the first approximation, are the full inviscidrotational-flow relations, and the flow contains both subsonic and supersonic regions. We formulate a numerical procedure for obtaining the solution to this problem by an application of Dorodnitsyn's method of integral relations [22]. In this method, the differential equations are integrated across horizontal strips bounded by the lines yj = yj(x). A system of first-order, ordinary differential equations is obtained in which the dependent variables are the flow properties on the strip boundaries. These equations, then, are in a form well suited to numerical integration using high-speed electronic digital computing machinery. In the present case, numerical results are obtained for a one-strip calculation, and a procedure is described for carrying out a two-strip analysis. An attempt to generalize the calculation to an arbitrary number of strips is also discussed. In an application of the method of integral relations, there may be n equations of the form aPi/ax + aQi/ay = Li (5.1) where i = 1,..., n with a ' x ' b and 0 _ y ' A(x) and where Pi, Qi, and Li are known functions of the independent and dependent variables. We integrate equation 5.1 on y from the lower boundary Yj 1to the upper boundary yj of each of N strips, where Yj = (j/N)Ax (5.2) and obtain n first-order, quasi-linear, ordinary differential equations of the form Yj Yj dd r jP.idy = J L.dy + P.(X, Yj)Y - P.(x, YJ-1)Y-1 + Qi(x' Y)- i(' Y (5.3) j-1 j-1 where j = 1, 2,..., N. In other applications of the method, equation 5.1 is first multiplied by weighting functions fj(y) and then integrated from y = 0 to y = A(x). (See sec. 6 for an application to boundary layers.) Equation 5.3 would be obtained using weighting functions Y < Yj1 fj(y) = 1, yj_. - y - y, y>yj 31

WILLOW RUN LABORATORIES If a total of n boundary conditions are given at y = O and y = A(x), we have n(N + 1) equations in terms of x. Presumably, n suitable initial and/or boundary conditions are given at x = a and x = b. We represent Pi, Qi, and Li by interpolation formulas containing simple functions of y with unknown functions of x as coefficients. Then the integrations over y may be carried out analytically. The new functions in x can be expressed in terms of the dependent variables evaluated at the edges of the strips. Thus, we obtain a system of nN first-order, ordinary, quasi-linear differential equations plus n boundary conditions at y = O and A in n(N + 1) unknown functions. In reference 22 there are comparisons of the results of calculations using the method of integral relations for various numbers of strips with exact solutions and experimental results. It is found that a two-strip calculation provides a reasonable degree of accuracy in a number of different applications. In appendix II, it is shown that a two-strip calculation by the method of integral relations for a boundary layer on a flat plate is comparable in accuracy with a Pohlhausen calculation. In our present problem, equations 3.46 through 3.49 in the case of an adiabatic flow of a thermally and calorically perfect gas over an insulated body can be written in the form (pu)x + (pv)Y = 0 (5.4) — 2 (5.5) uv)- + (p + pv )= (5.5) x y -2 -2 u + v + [2y/(y - l)]p/p = 1 (5.6) p = Ep' (5.7) where E = [(y - 1)/2](1 2) and where g = u(1, y) (see app. II). Thus, in this problem there are two differential equations (5.4 and 5.5) and two integrated expressions (5.6 and 5.7). The latter two relations are valid along streamlines, whose locations are found from dy/dx = v/u (5.8) The initial conditions are prescribed by the first-order matching with the solution in the first outer limit, equations 3.55 and 3.56. As noted in the discussion following equation 4.21, the critical point in our approximation is expected to occur at x = (x - L)/R 1/2L = 0. Further w decrease in pressure requires a significant turning inward of the streamlines, which can take place only within a distance Ax = O(R1/2L) upstream from the corner. For convenience, we introduce the notation 32

WILLOW RUN LABORATORIES U (y) u(O, y) C P(0) Then equations 3.55 and 3.56 become u(-oo, y) = u(Y) (5.9) p(-o0, y) pC In a first approximation, the boundary condition at y = 0 is v(x, 0) = 0 (5.10) (In a second approximation, 0(x, 0) would be set equal to the flow deflection angle at the outer edge of the viscous sublayer.) Before the hypersonic limit is taken, the boundary condition at the outer edge of the layer is a Prandtl-Meyer relation between p and v. In the hypersonic limit, equations 3.12 and 3.51 give d6/dx = 0 and, therefore 6 =6 c (5.11) v(x, 6c)=0 In the subsonic region, equations 5.4 and 5.5 are elliptic so that in this region it is appropriate to impose downstream conditions. In applications of the method of integral relations, the downstream boundary condition takes the form of a requirement that the integral curves must pass through saddle-point singularities of the differential equations derived from the integral relations [22]. The corner, x = y = 0, is located by the requirement that it is a sonic point. Thus, the special downstream condition on the line y = 0 is M(0, 0) = 1 (5.12) and the corner is a singular point in the flow. In the present application of the method of integral relations, the use of equations 5.6 and 5.7 is simplified if we choose the strip boundaries to be streamlines rather than lines y. = 33

WILLOW RUN LABORATORIES th (j/N)6 as in equation 5.2. We let y. be the value of y on the upper boundary of the j strip and note that the functions yj(x) are to be found at each point from integration of dyj/dx = vj/uj (5.13) where u. - u(x, yj) and v. - v(x, yj). Because of the different nature of the boundary conditions in the supersonic region as compared with the region that is initially subsonic, we treat these regions separately in constructing the integral relations. In the region that is initially subsonic, we introduce M strips bounded by the streamlines yj = yj(x), j = 0,..., M, where YM is the normal coordinate of the streamline that is initially sonic. In the supersonic region, we introduce N strips bounded by the streamlines Yk = k = M,..., M + N, where yM+N = 6c; there is a total of M + N strips. We now consider the application of the method of integral relations in the region that is initially subsonic. It is convenient to write the integral relations, derived from integrations on y of equations 5.4 and 5.5, in the form {' (pu)Rdy = -(pvj - Pj1 lj_l) (5.14) Yj-1 fYJr i. = ~2 2 51 y (puv)xdy =-(i - j- 1 + pjvj - j-l (5.15) j-i where j = 1,..., M. Here pj m p(x, yj), etc. Equations 5.6 and 5.7, written for the strip boundaries y = yj(x), become 2 2 uj + vj + [2y/(y - 1)]pj/pj = 1 (5.16) p = E p ' (5.17) where the terms E. = [(y - 1)/2](1 -g are constants and where j = O..., M. Although p - o as y - 6 in the hypersonic limit, pu is finite and bounded everywhere in the region which is initially subsonic. Thus, an appropriate interpolation formula for pu to be used in carrying out the integration in equation 5.14 is 34

WILLOW RUN LABORATORIES M pu = am(x)(y/6c) (5.18) m=O Similarly, a representation for puv to be substituted into equation 5.15 is M puv= 2bm(x)(y/6C) (5.19) m=O Thus substitution of equations 5.18 and 5.19 into equations 5.14 and 5.15, respectively, transforms these relations into the ordinary differential equations CM ~l, 1 (+1 m+l1 -1 Zi(m + 1) ( - )a (P( - p v (5.20) aj - j-1 ) c j-vj. m=O, m - -12. 2(m + 1) 1(a+ - oaJ1)b' (x) = (p -P 1 + p.v2 - p.v1) (5.21) m=0 where aj = Yj/5 and j = 1,..., M. From equation 5.13, we obtain the equation for the streamline slope, -1 ao. = 6 v./u. (5.22) J cJJ Equations 5.18 and 5.19, when evaluated on the strip boundaries, are a system of linear algebraic equations in the a and b in terms of the values of pu and puv on the strip boundaries. M iaDrmam Ipuj. (5.23) m=O a ab = p.uvj (5.24) m=0 aM b UV (5.24) mwhere j = 0 M in these relations. where j = 0,..., M in these relations. Thus, in the application of the method of integral relations to the portion of the flow that is initially subsonic, we have 7M + 5 equations in 7M + 6 unknown functions of x: uj, v., p, p., am, bm (where j, m = 0,... M) and aj (where j = 1..., M). The 7M + 5 equations are: the boundary condition (5.10), which gives v = 0; equations 5.16, 5.17, 5.23, and 5.24 in which 35

WILLOW RUN LABORATORIES j = 0,..., M; and equations 5.20, 5.21, and 5.22 in which j = 1,..., M. The extra unknown function appears because the boundary condition at y = 6 has not yet been employed. Now we consider the application of the method of integral relations in the region that is initially supersonic. We introduce in this region N strips bounded by the streamlines Yk = Yk( k = M,..., M + N. The integral relations may be written in the form Yk, (pU;dy-d = -(PkV - k vk-) (5.25) k-1 2 2 uk + vk + [2y/(y - 1)]pk/pk =1 (5.27) Ekp Ek (5.28) Pk = EkP where the terms Ek = [(y - 1)/2y](1 - are constants and where k = M,..., M + N - 1. Let us now consider the representation for pu to be substituted into equation 5.25. Using equation 5.6 and the relation v(x, 6) = 0 we have, in the hypersonic limit, p = O[(1 - u)1] (5.29) as y - 6. From equation 4.10 it can be seen that u (y) has the same form of asymptotic behavior for y - 6c as ul(Y) in the upstream boundary layer. It follows from equation I-30 that = O{[-log (1 - )] 1/2} au/a8 = O{(1 - u)[-log (1 - u)]1/2} (5.30) as y - 6. In the boundary layer at x = 1, we see from equations 1-19, 1-25, and 1-31 that 1/2 do = [2/(2k + 1)] 1/2p dy (5.31) 36

WILLOW RUN LABORATORIES Then equations 5.29-5.31 lead us to the results ay/au = O{[-log (1 - u)] 1/2} 1- /6 = 0{(1 - u)[-log (1 - u)]1/ } (5.32) p = 0(1 - /6)-l[-log (1 - i)]-1/2} (5.33) as y - 6. Consistent with other applications of this method to boundary-layer problems [22], we approximate the integral in equation 5.25 by employing an interpolation formula for pu that omits the logarithmic factor N pu = (1 -./6 c) 1cn()(~/6c) (5.34) n=O We introduce N A = 2c(x) (5.35) n=O where A is constant, as will be shown in equation 5.43. A nondimensional stream function 4 is given by 4'4'M f pudy (5.36) which, from equation 5.34, becomes N nN7 4' 4M=6cZcn(x)J (1- )ltnd: = -6 log ) + c N, aM) (5.37) n=O 0 n=0 M where and where (, M) is given by n(a, (aM) f- (1 - ) ( - l1)d5 (5.38) M For instance, 0 = 0, /1 = a- aM, i2 = a - aM 1/) etc. Thus, we have, from equation 5.37, 37

WILLOW RUN LABORATORIES - Y/ =(1 - aM) exp K(6cA) 1(, - - A C (x) (a, aM (5.39) n=O We can write the isentropic relation 5.7 in the form p/pC = ( /cY (5.40) Then we substitute for p and Pc from equation 5.34 and get p -Y c U (Y/c) (5.41) PC n, (~c5c)I where Yc is the value of y on the same streamline at the critical point. Now we substitute for the values of 1 - Y/6C and 1 - Yc/6 from equation 5.39. c LWKA~cU(4' 4'Mc Y K~c1 ) 0/nj ) exp -6- A - o __ - A (5.42) In equation 5.42, n = 3n(ay, aM), O3C = m (aC aM C)' and cn c = Cn(-C). For the pressure PM+N at the outer edge of the layer to remain bounded and nonzero, A = A = constant (5.43) Then pM+N is given from equations 5.41 and 5.42 as PM+N = PC[(1 - aMc)/(l - aM)] exp (-yh) (5.44) where h -A [On.c c(" aMc)Cnc ~ On(1' aL)Cnc )c M=O (5.45) c (- c 2y/(y- 1) Pc =[(Y- 1)/2Y]c / c We see from equations 5.39 and 5.43 that ignoring the logarithmic factor in equation 5.32 in constructing a representation for pu is equivalent to assuming that 38

WILLOW RUN LABORATORIES u 1 - f () exp [-4t/(6cA)] (5.46) as - o. With the interpolation formula 5.34 for pu substituted in the integral, equation 5.25 becomes N -1 (5.47) Z3n(k' k- l)Cn(x) c (Pkvk Pk-l k- 1) n=O where k=M + 1,..., M + N. From equation 5.47 we see that pv is bounded and nonzero in the limit y - 5. Thus, an appropriate interpolation formula for puv in this case is N puv =~ dn(x)(y/6c)n (5.48) n=O and substitution of this relation into equation 5.26 gives us (n/+ 1( - k - =k-1 + Pk k - Pk-1k-1) (5.49) n=O where k = M + 1,..., M + N. Equations 5.34 and 5.48, evaluated on each strip in the supersonic region, give N Za c (1 -ak)pkuk (5.50) kCn (1- ak)PkUk n=O N 2and n = kuvk (5.51) n=O where k = M,..., M + N - 1. To utilize equation 5.47 for k = M + N, it is necessary to substitute for p M+NvM+N from equation 5.51. We note that uM+N = 1, and obtain lim =Zd (5.52) n=O 39

WILLOW RUN LABORATORIES From equation 5.8 we get ak(X) = 6c Vk/Uk (5.53) where k = M,..., M+ N- 1. This application of the method of integral relations for the region that is initially supersonic also can be applied across the entire layer in the case of a one-strip calculation. In the application of the method of integral relations to the supersonic portion of the flow, we end up with 7N + 3 equations in 7N + 4 unknown functions. The 7N + 4 functions are Uk, Vk, PkPk k ak (where k = M,, M + N - 1); cn, dn (where n = O,..., N); PM+N; and vM+N. The 7N + 3 equations are the condition 5.11, which gives vM+N = 0; equations 5.27, 5.28, 5.50, 5.51, and 5.53 for k = M,..., M + N - 1; equations 5.47 and 5.49 for k = M + 1,..., M + N; and equations 5.35 and 5.44. Here the extra unknown function appears because the boundary condition at y = 0 has not been employed. When we compare the equations obtained for the supersonic region with the equations obtained for the region that is initially subsonic, we see that the five dependent variables uM, vM' PM' PM' and acM have been counted twice. Thus, there are 7(M + N) + 5 equations in 7(M + N) + 5 unknown functions. In the preceding discussion we did not obtain a relation for evaluating the constant A, defined in equation 5.35. When we expand po and -P for M - M and form the difference 0 P6 0 Oc P6 - P, we see that there are two possibilities for the selection of A. O(M - M ) P6 - Po =o(M Mc ) (5.54) o oc(5.54) In the first case, the perturbations in uk, vk, Pk' etc. are all of the same order of magnitude, and the perturbations will decay exponentially in x as x - -oo. The second case is obtained when A is chosen so that the terms in p5 - p of the order M - M vanish identically. Thus, 6 o o,c in this instance a relation for evaluating A is obtained, while in the first case no relation determining A is found. In the present problem the boundary layer on the body is subcritical (see sec. 2), and, after an acceleration described approximately in the first outer limit, the boundary-layer profile reaches the critical state, in that <M -2> = 1. Therefore, the initial condition for the equations in the second outer limit is special in the sense that the initial profile is critical, and we can expect A to have a particular value, corresponding to the application of the second part of equation 5.54. 40

WILLOW RUN LABORATORIES In support of the above argument, we will show that the second part of equation 5.54 is a necessary condition for (d6*/dp6)c = 0 (5.55) to be satisfied. Here 6* is the boundary layer displacement thickness, and p6 is the pressure at y = 6. From equation I-35 and the definition of the stream function we have, in the hypersonic limit, oo 6* = (/u) di/ (5.56) where 6* = 6*/R 1/2L and = pu. We introduce a small perturbation in the pressure in a w y neighborhood of the critical point and obtain 6 6 c2 a6 - - M) dy -(A6/P)J (A (1 - M_) dy (5.57) as Ap -0. From equation 5.55 we see that A6* = o(AP6) (5.58) as Ap6 - 0 near the critical point. Also, from equations 4.17, 4.21, and 4.34 we have i (1 - M d = (5.59) Thus, in view of the results expressed in equations 5.58 and 5.59 and since Ap/Ap6 is monotonic in y, a necessary condition for the orders of magnitude in equation 5.57 to match is Ap - Ap6 = o(Ap6) (5.60) as Ap6 - 0. This condition is equivalent to the second part of equation 5.54. Thus, A is to be selected so that the second part of equation 5.54 is satisfied. The perturbations in the dependent variables are not all of the same order of magnitude; they will decay algebraically in x as x - -oo. We find, in this case, that along a streamline v= O[(AM)3/2] as x- -oo, which is consistent with transonic small-disturbance theory. 41

WILLOW RUN LABORATORIES In the use of the Bernoulli equation and the entropy equation in the application of the method of integral relations to this problem, it is convenient to introduce the Mach number as a new dependent variable. Then we may replace the Bernoulli and entropy relations by Pi 2y 1 2+ 2 2\l (1- 1 g2 (5.61) i= 2- + M 1 (5.62) Pi = M(1+ 2) (5.62) P/1 + i 1-gi ='2 1/M(1+y2'Ml) (5.66) q/qi = Mi 1( + YM) M (5.67) Also, for adiabatic flow, 2 2 2 p0 = [(y - 1)/2y]pi (5.66) c +c = i M' (5.67) 42o 1 in the above equations, i = 0,..., M + N -i. To proceed further it is necessary to specify the values of M and N. 5.2. SOLUTION BY THE METHOD OF INTEGRAL RELATIONS FOR ONE STRIP The simplest possible calculation to carry out is a single-strip application with M = 0, N =' 1. Then equations 5.27, 5.28, 5.50, 5.51, with k = 0; equations 5.47 and 5.49, with k = 1; and'equations 5.35 and 5.44 become, with v = ca = v = 0 and a~ = 1' u + [2?/(7 - 1)]Po/po =1 (5.68) Po = [(Y - 1)/2y]p (5.69) c +c =A (5.70) Pl = Pc exp (-Th) (5.71) 42

WILLOW RUN LABORATORIES where h =A (c1c and Pc = [(- 1)/2]c Ci =6 c (do+ d1) (5.72) 1 -c1 2-d =-6c (- P) (5.73) c =p u (5.74) d = 0 (5.75) From equations 5.70 and 5.72 through 5.75, we obtain (pu )' = -61 d (5.76) 0 0 C 1 d? =-26 (p - po) (5.77) from which we get the second-order, ordinary differential equation (P Uo)" = 26 2(p1- _p) (5.78) A form of the equations more convenient for carrying out the numerical solution is obtained by introducing the Mach number Mo as the independent variable. From equations 5.64 and 5.67, with i = 0, we have (p ' = p u M 1 - M)1 + 2 M m (5.79) Uo) 0 p o o 2 o) o Since x does not appear explicitly in the equations, the initial value of x is arbitrary, and we can introduce a new x coordinate, X, depending upon Mo such that 0 x = X(Mo) - X(1) (5.80) Then the boundary condition M(0, 0) = 1 will automatically be satisfied. The differential equations in the new variables are dX 6pu 1-M2 _ _dM coo1M -1 2 (5.81) dMo 1 1M0 M d(d~) 4pup - )(1 (5.82 odM 2 43

WILLOW RUN LABORATORIES The integration is to be carried out over the interval M <_- M - 1, and the values of x are recovered from equation 5.80. In equation 5.81 we have chosen d1 =:- d'l (5.83) so that dX/dMo > 0, which corresponds to a decreasing pressure. If dl = + 1 is chosen, we would obtain AM < 0 and Ap > 0 for AX > 0. o We now obtain the asymptotic expansions for d1 and X as Mo - Mo. From equation 5.61 with i = 0, we get PO PCL1 - yM0c(1 + 2 Mo c) AM0 +... (5.84) as AMo - 0, where AM Mo - M.o For i = 0, equations 5.62 and 5.67 yield o u pu 1 + Mo( - M ) (1 + M M + Po o oc oc 0Lo,c oc 2 oc ' 0 (5.85) so that, with equations 5.70, 5.71, 5.74, and 5.85, the expansion for P1 is Pi~-PCL1yA poc UcM0c 1 - M ) 1P - ' x AM +... ) (5.86) as AM ~0. Therefore, the expansion for p1 - P0 as AM - 0 is 0O - _ p yp M c ( + 1M c) F1 - Ap u M 2 x(1 - M2 AAM +... (5.87) Pi -Po P c0 oc 2 o,c - oc oc o,c oc o The condition expressed in the second part of equation 5.54 implies, in this case, that p1 - PO = o(AM ). This condition is satisfied when c o,c o,c o,c, A = p 01CuO~C MOI - 2,3 (5.88) The numerical value is A = 0.708. Thus, P1 - Po = O(AMo), and from equation 5.86 we get d(d2)/dM ~ (constant)(AMo) (5.89) Since d1- 0 as AMo - 0 because d = O(O), we have d1 (constant)(AMo)/ (5.90) 44

WILLOW RUN LABORATORIES as AMo - 0. Then 0 = O[(Mo) /as x -c. We substitute equation 5.90 into the expansion of equation 5.81 and get dX/dMo ~ (constant)(AMo) 3/2 X (constant)(AMo) 1/2 (5.91) The expansions for Mo and d1 in terms of X are M ~ Mc + (constant)X2 (5.92) O O'C d~ (constant)X3 (5.93) as X - -co. Note that, in order to obtain the numerical constants in the expansions 5.90 and 5.91 or 5.92 and 5.93, it is necessary to carry out the expansion of - pO to the order (AMo) as AM - 0. 0 When x - 0, it can be seen from equations 5.81 and 5.82 that M - 1 + (constant) lx 1/2 0 d1 ~ d1(0) + (constant)x In general, it is necessary to employ the asymptotic expansions for the dependent variables in order to get the numerical integration of the differential equations started. In the special case of one strip, however, the numerical integration of the differential equations 5.81 and 5.82 can be carried out without using the expansion for AM - 0. The reason for this is that d (d) dM0 is bounded at M0 = Mo, and is also only a function of the independent variable M. Once a value of d1 has been found at a point Mo > Mo,c' the integration of equation 5.81 may be started and carried out simultaneously with the integration of equation 5.82. The numerical results will be discussed at the end of this section. 5.3. SOLUTION BY THE METHOD OF INTEGRAL RELATIONS FOR TWO STRIPS The next step is to consider a two-strip calculation with M = N = 1. In the region that is initially subsonic, the governing equations are 5.16, 5.17, 5.23, and 5.24 for j = 0, 1; 5.20, 5.21, and 5.22 for j = 1; and v = a = 0. We also define aa- a 1. Thus, we have the equations u + [27y/(y - 1)]Po/po = 1 (5.94) 45

WILLOW RUN LABORATORIES 2 2 u2 + v + [2y/(y - 1)]p1/P11 (5.95) Po =E p' (5.96) P0 = E0 P1p E 1 (5.97) where E = ( - 1)/2y and E1 = [(y - 1)/2](1 - g), 1 2 -1 aa' + 2a a =-6 p lv (5.98) a 1 b' a -- p P- + plv 2 (5.99) o 2 1 c 1 O 1 1 a' =6 v 1/u (5.100) a = p u (5.101) a + aal p lu (5.102) 0 b =0 (5.103) ab1 = P1l1U1 (5.104) We solve for al, b1, ai, and b' from equations 5.100 through 5.104 and obtain a1 = a (plul - uo) (5.105) -1 b =-1 Pl u v (5.106) b = (p1 u v)' 1- 6 a p1v1 (5.108) Then the differential equations 5.98 and 5.99 become (po u) (P1U ) = t a p 1Vl[(PYall/p U)- 2] (5.109) 00 1 1 C 11 11 (p1uv) = 6c a [p1vl + 2a(Pl - Po F2 (5.110) 46

WILLOW RUN LABORATORIES The relations derived for the supersonic region (discounting those already noted and with V2 = 0, a2 = 1) are, in this case, c + c =A (5.111) 0 1 P2= Pc[(l - c)/(1 - a)] exp(-yh) (5.112) -1 where h =A [(1 -a c)c - (1- a)cl], (1-a)ci =bc (do + d1 - 1v) (5.113) '1 (5.113) (1- a)d' +( 1-a )dP = -6 p- p v (5.114) c + ac1 = (1 - )PlU1 (5.115) do + ad =P1U1 1 (5.116) We find c, c1, and c' from equations 5.100, 5.111, and 5.115 to be o 1~ ~'I''1 c =p u - a(1 - a) A (5.117) C =-P1Ul + (1 - )- 1A (5.118) -1 -2 =-(P1 1)' + 6A(1 - a) 2v1/u1 (5.119) Then equation 5.113 becomes f = (-1 2 -1 (p 1u1) = c (1 )-l(l_ a) Avl1 /U1 - (do + d1) + P1vlj F1 (5.120) When we substitute the result in equation 5.120 into equation 5.109, we get (p u )' a -1(1- al - a)2a v l/u + a(1 - a)(d + d1) - (1 - a)(2 - a)plv1} F (5.121) From equations 5.61, 5.62, 5.65, 5.66, and 5.67, with i = 0, 1 (these equations are equivalent to equations 5.94 through 5.97) we have y - 12( - 2o)-Y/(-l) (5.122) 47

WILLOW RUN LABORATORIES O= (1 + 2 M2) (5.123) uO2 = 2 MO( + 2 Mo (5.124) Po (Y-1 M2 - /1 \ (Y- 1) P 1 = -M1(1 2 1) M1 (5.127) 2 y -12 - 2 q =u1 + v1 (5.128) = Y M2(+Y 21)1 (5.129) - 1 2 1 1 Pq Pi =-M1 1+ q2 M1 (5.130) 1 = Ul + V F1 (5.128) y1 - 1 i (5.133) M- 11+ 2-1 (5129) v -1 q 1 + V 2V t (5.134) Thus, we wish to express (P u I, (P U and (p U V Y in terms of Mf, (l, and V. The io M, and V1 are 0 1' 1 ~= (5.135) 48

WILLOW RUN LABORATORIES B11M + B12V = C (5.136) B M' +B V'- =C (5.137) B21M + B22V1 2 (5.137) where the B.. and C. terms are given by Bll (= P(1 - M1 + 2 M) (5.138) 1M 1 1-122 11- 12 B21 = P1Ulql1M (11 + 2 (5.140) B22 = p1U1q1 (5.141) C =F1 (5.142) C2 = F2 -q1V1 F1 (5.143) We solve equations 5.136 and 5.137 for M' and V' and obtain Ml =D (CB22 - C2B) (5.144) V1 = D (C2B11 C1B21) (5.145) where D is given from D = B11B22 - B12B21 (5.146) Substitution of equations 5.138 through 5.141 into equation 5.146 gives 243 -1 2/-11 ( 2/21 D = 1Piq qM1(1 + 2 1M)[1 - u )] (5. 147) Here al is the local speed of sound on the strip y = c. Since this line is initially sonic (q 1c = al c) and since the initial value of v1 is zero, then q1,c = u1,c' and Dc =0 (5.148) The implication of equation 5. 148 will be discussed further when the asymptotic expansions for x - -cO are considered. 49

WILLOW RUN LABORATORIES There is a fourth differential equation, deriving from equation 5.114. If we pick Fo, defined in equation 5.121,as the fourth dependent variable, a relation equivalent to equation 5.82 will be obtained when Mo is chosen to be the independent variable. From equations 5.116 and 5.121 we get d = (1 - a) Plllv1 - aG (5.149) d1 = -(1 - a) 1 + G (5.150) where G=6 + P1qV1{ (P1U1) 1[a - (1 - a)-2A + a-1(1 - )-1(2 - )} (5.151) By differentiating equations 5.149 and 5.150, we obtain ' = -cG' + (1 - a) 1F2 + [(1 - a) pu1v - Gcvl/u (5.152) d' = G'- (1- a) F2 (1- a) 26 1 plv2 (5.153) 1'~ 2 c P 1 ( so that equation 5.114 becomes G' = -(1 - a) 1F2 + 2(1 - a) 261 [(1 - a)Gvl/U1 - (P2 P) + 2PVl (5.154) We obtain the differential equation for F by differentiating equation 5.151 and solving for F': FO C= G'6 Pq1 V 1 (P lq)- (Plql)'+ V1 V1 + (Pu1)-2Flal - (1 - a)2A] (PlU)1 x [l(F1 -F F - a61a1v1/u)- 2(1 a)-3 1Av/1 -2 22 1 2 - a (1 - a)- ( - 4a + 2)5 v /u1 F (5.155) In equation 5.155, Vi and G' are given by equations 5.145 and 5.154, respectively. Also, from equations 5.132 and 5.135 we have =1 1 (Pq M (1 - M' + Y 2l1 (5.156) l (l501 2 1 1

WILLOW RUN LABORATORIES Thus, with M as the independent variable, the differential equations, obtained from equations 5.135, 5.144, 5.145, and 5.155, are dX pu 1 M2) d. 0.0 0 H (5.157) o M 1+ o 2M dM1/dM =HD- B22 C2B12) (5.158) 1'o = HD(C1 B22 2 12 dV /dM = HD (C2B - C B2 ) (5.159) lo (C211 1 21 dF2 2p u (1 - M2)F3 o 00\ 0 3(5.160) dM: M o 2 o dr/dM= =- Y1H(11 2 )-1/2 da/dMo = iC1 HV (1 - V) (5.161) where X is defined in equation 5.80. In this case it is necessary to choose F = + F (5.162) in order to guarantee that dX/dMo is positive. To begin the numerical integration of equations 5.157 through 5.161, it is necessary to utilize the asymptotic expansions for M - M. This is because dF o/Mo, for example, is a function of M1 and V1 as well as Mo. Also, since p - po = O [(AM0)2 as AMo - 0 and since dV1/dM and dF /dM are, in part, dependent on P1 - o, it is necessary to carry out the expansions to second order in AMo as Mo - M c From the expansions 5.90 and 5.91 obtained in a single-strip calculation, we expect the expansions to be of the form M 1-M J AM +J 2)(AM )2 (5.163) 1 1,c 1 o 1 (0o) (5.163) J(1)(AM)3/2 + (2) (AM)5/2 (5.164) 1- J ()(AM 3/2 + J 2)(AM ) / +. * * (5.165) o 3 0(Mo) 3 0(Mo) +*** (5.165) Co + j1)AM + J 2)(AMo) +... (5.166) 51

WILLOW RUN LABORATORIES Because the initial value of X is arbitrary, it is not required to obtain an expansion for X(Mo). Since M1 c = 1, we have Dc = 0 (see eqs. 5.147 and 5.148), and D = O(AMo) (5.167) as AMo - 0. Then, to the order AMo, the expansions of equations 5.158 through 5.161 are F1 = [(M )2 (5.168) 1- Po O[(=M()A ] (5.169) 2- P= 0[(AMo)2] (5.170) (1) (1) D42J2 + D43J =0 (5.171) where D42= PU01 - Mo)M0 (1 + 2 M oc (5.172) 43 cJ4 (5.173) We get from equation 5.168 the result D12J2 1) + D13 ) =0 (5.174) where 2D c1 (a2 + 2c 2) cu Uc + (1- a)a (5.175) 12 = c c c 1,c c c D13 =-(1- a )c (5.176) From equation 5.169, to order AMo, we obtain D J(1) = H) (5.177) where D1 = Mic(1 + Yt-~1Mc) (5.178) 52

WILLOW RUN LABORATORIES H1) 2 MI- H M (c1 + 2M (5.179) Now we require the expansion for P2. From equation 5.112, we find P - P1 + 1 - + YA Pi cuA c]J) A Mo - yA (1 - a%)A(P1U1)} (5.180) while the expansion for plul as AMo - 0 gives A(1ul) = O[(AM0) (5.181) so that equation 5.170 becomes D J = H(1) (5.182) where EH = H(1) (5.183) 3 2 =34 (1 - ) + A Plculc (5.184) When J() is calculated from equation 5.182, equations 5.171 and 5.174 become a system of two linear homogeneous equations in J( ) and J3 ). For a nontrivial solution of these equations to exist, the determinant of the coefficients must vanish. D12D43 - D13D42 - 0 (5.185) Now D43, through equations 5.173 and 5.184, contains A. Thus, equation 5.185 provides a relation for determining A, and we get D43 =DD DD13D4 (5.186) D =-6 D 1 (1 (5. 187) 34 c 43 2 A= PlcUl c[D34 7y (1 - ac) (5.188) However, now equations 5.171 and 5.174 are linearly dependent, and either J() or J3 will remain undetermined. An additional relationship involving the first-order perturbations is found from the second-order relations. In the second-order expansions of equations 5.158 through 5.161, we obtain 53

WILLOW RUN LABORATORIES ~ (2) + J(2) (2) (5.189) 12 +2 133 -1 (2) (2) D21J(2) = HI2 (5.190) D34J4 ) = H( ) (5.191) D J (2) 1(2) (5.191) 34 4 3 D422 + D3 J2 ) H (5.192) The coefficient matrix D.. is the same for all orders of magnitude. The nonhomogeneous terms, H!i ) are functions of the first-order perturbation constants, J() But A has been chosen so that the determinant of the coefficient matrix in equations 5.189 and 5.192 vanishes. Thus, we must have H1(2) D 1 12) 3 0 (5.193) H(2) D H 43 and it appears that the last of the first-order perturbations is found from this equation. Equations 5.189 and 5.192 now will contain only one independent relationship, and it appears that one second-order coefficient, J 2), for instance, will be determined by a third-order relationship equivalent to equation 5.198. In utilizing the asymptotic expansions 5.163 through 5.166 to start the numerical integraonly to the second order. Hence it would be permissible to choose the value of J 2), for instance, arbitrarily, as this only incurs a relative error of the order AM or smaller in the initial estimates of the derivatives. 5.4. GENERALIZATION TO AN ARBITRARY NUMBER OF STRIPS If an arbitrary number of strips were utilized in the application of the integral relations 5.20, 5.21, 5.47, and 5.49, we would expect to obtain differential equations of the form dX/dM =1 - M)2B/F (5.194) dMi/dM~= (1_- Mo)1i/F (5.195) dVi/dMo = (1- Mo) A2,i/AF (5.196) 54

WILLOW RUN LABORATORIES dF2/dM = (1 - M2)C (5.197) dai./dM= (1 - M2O)Bvi/uFi (5.198) where F =+ F and where i = 1,..., M + N - 1. In general, A has a simple zero at each of the points where 1 - ui/ai = 0. When M + N = 2, A D is given in equations 5.146 and 5.147. In the case of strip boundaries that are initially subsonic, downstream boundary conditions are obtained from the requirement that the solution be regular at the points where A = 0 [22]. This condition takes the form A,i = 0 (5.199) at the points where u. = a.. In the asymptotic expansions of Mi, Vi, and F, one perturbation should be undetermined on each strip boundary. The solution is found by guessing values of the initial perturbations in the Vi, for example. Trial integrations are carried out until equation 5.199 is satisfied, and then the integral curves will pass through the saddle-point singularities at the points where u. = a.. 1 1 As an example, we will consider a three-strip calculation (M = 2, N = 1) where M1,c < 1, and M2c = 1. The asymptotic expansions of the dependent variables, to first order, are of the form AM 1 A(1)M M1~J1 0 AM ~ J1) AM 2 2 O V j(1)(AM)3/2 V2 J( )(AM) / (5.200) V25 ~(Mo)3 F j(1)(AMo)3/2 Au1 (1) AM 2 7 o (1) Then the J) terms are related by the equations 55

WILLOW RUN LABORATORIES (1) (1) (1) (1) (1) D J J +D ( +D J +D J 0 (5.201 -11 5 1 133 144 155 (5.201) D J(1) + D J(1) +D J-(1) =0 (5.202) 233) 244 255 D 1J(1) = H( (5.203) 31 - 3 D2J (1) = H(1) (5.204) 42J2 4 D1) (1) D J(l) = H(1) (5.205) 57 7 5 (1) (1) (1) D J) + D65( )J () - 0 (5.206) 63+3 65J6 5 D?4J ) + D75J ) 1) = 0 (5.207) where the Dij and the H(1) terms are known constants, except for D57, which will depend upon 1J 1 57' (1) A. Since there is one strip boundary that is initially subsonic, we expect one of the Ji terms '1),(1) to remain arbitrary in the solution of equations 5.201 through 5.207. When we substitute for J() and J from equations 5.203 and 5.205, equations 5.201, 5.202, and 5.207 become three 1 7 (1) (1) (1) linear homogeneous equations in J 3, J 4, and J5. The determinant of the coefficients in these equations must vanish, providing a relation for A equivalent to equation 5.185, and one of the coefficients, J), for instance, remains undetermined. In the second-order equations, the same coefficient matrix appears since the terms in J 2) and J(2) may be grouped with the nonhomogeneous terms. Thus there appears to be a second-order equation in J ), equivalent to equation 5.193, and we do not seem to have the proper initial conditions in the subsonic region. Now let us consider a three-strip calculation (M = 1, N = 2) where M1 = 1 and M2 > 1. In this case, the relations for the J) are the same as in equations 5.201 through 5.207 when the subscripts 1 and 2, and 6 and 7 are interchanged except in that D22 now has changed sign. Thus, further investigation is required to distinguish between the initial conditions on strip boundaries that are initially subsonic and the initial conditions on strip boundaries that are initially supersonic. The difficulty may be related to the fact that the approximation p- 0 as x — oo causes the characteristics to degenerate to the single family of lines x = constant. A possible alternative approach for obtaining numerical solutions to this problem would be to apply the method of integral relations only to the portion of the flow that is initially sub56

WILLOW RUN LABORATORIES sonic. Then one function remains undetermined, the normal velocity component on the streamline that is initially sonic. A numerical method of characteristics or a finite difference technique such as that employed by Baum [14] then would be applied in the supersonic region. Since the subsonic portion of the layer is relatively thin, comprising less than 20 percent of the initial thickness of the layer, a single-strip calculation by the method of integral relations for the subsonic portion of the layer together with another numerical technique applied in the supersonic region is likely to provide a relatively high degree of accuracy. Also, it would not be necessary to carry out trial integrations of the equations in this instance. However, further investigation to ascertain the proper treatment of the initial conditions in the subsonic region would be required in this case also. 5.5. EVALUATION OF RESULTS AND COMPARISON WITH EXPERIMENTAL DATA The numerical integration of equations 5.81 and 5.82 applicable in a single-strip calculation (M = 0, N = 1) has been carried out by using the IBM 7090 digital computer at The University of Michigan Computing Center. The algorithms employed are programmed in the MAD language [19]. The numerical technique applied is the Runge-Kutta fourth-order method, which is a standard computer library subroutine [20, 21]. First, d(d1)/dMo, which is only a function of the independent variable M, is integrated from the initial point, M = Moc (where d1 = 0) to a point where AMo = Mo - Mo c is small. Now the integration of dX/dM in equation 5.81 can be started since a finite value of d1 has been calculated. An asymptotic expansion for X(Mo) as M - M is not required since the initial value of X is arbitrary. Then, integration of o OC equations 5.81 and 5.82 proceeds step by step up to the corner where Mo = 1. X(1) is determined, and the value of x at each point is recovered from equation 5.80. The numerical results for y = 1.4 are presented in table IV. When the results of our calculations for the first approximation are plotted as a function of R/6- (x - L)/61, they are independent of the Reynolds number R (see, for example, eq. 3.58, the composite solution for u/ue in a first approximation). Also, the results in a first approximation are the same for a wedge as for a cone. The only conical effect is a factor of 1/V3 that appears in the formula for the boundary-layer thickness 61 (see eq. I-37). These premises can be tested by replotting Hama's [6] data as a function of (x - L)/61. In figure 4, some of Hama's wall-pressure-ratio data for laminar flow (which appears as fig. 4 of ref. 14) are presented. The ratio Po/Pi is plotted against x for three different Mach numbers, where p is the measured surface pressure and pi is the surface pressure predicted by 0 ei inviscid-flow theory. These data are replotted in figure 5, with abscissa x/61 ( - )/1. The necessary calculations for the reduction of Hama's data are presented in appendix III. The 57

x/61 N o P o Pe Pl/e dl X/1 Mo Uo/Ue,P 1, 0.0000 0.4482 0.1965 0.8712 0.8712 0.0000 -0.4353 0.7390 0.3138 0.6957 0.7676 -0.0273 -171.4514 0.4490 0.1969 0.8708 0.8708 -0.0000 -0.3935 0.7490 0.3176 0.6892 0.7656 -0.0283 -16.7019 0.4590 0.2011 0.8655 0.8656 -0.0003 -0.3550 0.7590 0.3214 0.6827 0.7637 -0.0293 - 10.7671 0.4690 0.2053 0.8601 0.8606 -0.0007 -0.3193 0.7690 0.3252 0.6762 0.7619 -0.0302 -8.0810 0.4790 0.2 09 5 0.8547 0.8556 -0.0013 -0.2865 0.7790 0.3290 0.6697 0.7602 -0.0312_ -6.4751 0.4890 0.2136 0.8492 0.8508 -0.0019 -0.2561 0.7890 0.3327 0.6632 0.7586 r —.3 r-5.3831 0.4990 0.2178 0.8436 0.8461 -0.0027 -0.2281 0.7990 0.3365 0.6567 0.7570 -0.0330 - 4.5 82 3 0.5090 0.2220 0.8379 0.8416 - 0.0034 - 0.2024 0.8090 0.3402 0.6502 0.7556 - 0.0338 -3.9652 0.5190 0.2261 0.8322 0.8371 -0.0043 -0.1786 0.8190 0.3439 0.6437 0.7542 -0.0346 - 3.472 5 0.5290 0.2302 0.82 65 0.8328 - 0.00 52 - 0.1568 0.8290 0.3476 0.6371 0.7529 - 0. 0354 -3.0689 0.5390 0.2343 0.8206 0.8286 -0.0061, -0.1369 0.8390 0.3513 0.6306 0.7517 -0.0362 -2.7313 0.5490 0.2384 0.8148 0.8245 -0.0071 -0.1186 0.8490 0.3550 0.6242 0.7506 -0.0369 -2.4444 0.5590 0.2425 0.8088 0.8205 -0.0080 -0.1019 0.8590 0.3586 0.6177 0.7496 -0.0376 Z -2.1974 0.5 690 0.2466 0.802 8 0.8167 -0.0091 -0.0868 0.8690 0.3622 0.6112 0.7486 -0.0383 r-1.9825 0.5790 0.2507 0.7968 0.8129 -0.0101 -0.0730 0.8790 0.3658 0.6048 0.7477 -0.0389 b1 - 1.79 37 0.5890 0.2547 0.7907 0.8093 -0.0112 -0.0607 0.8890 0.3694 0.5983 0.7469 - 0. 0395 O~p - 1.6267 0.5990 0.2588 0.7846 0.8058 -0.0122 -0.0496 0.8990 0.3730 0.5919 0.7462 -0.0400 (~ -1.4779 0.6090 0.2628 0.7785 0.8024 -0.0133 -0.0398 0.9090 0.3766 0.5855 0.7455 -0.0405;~ -1.3446 0.6190 0.2668 0.7723 0.7991 -0.0144 -0.0312 0.9190 0.3801 0.5791 0.7450 -0.0410 b] - 1.22 47 0.6290 0.2708 0.7660 0.7959 -0.0155 - 0.02 37 0.9290 0.3837 0.5727 0.7444 - 0.0414 — 4 -1.1163 0.6390 0.2748 0.7598 0.7928 -0.0166 -0.0174 0.9390 0.3872 0.5664 0.7440 -0.0418 (~ -1.0180 0.6490 0.2787 0.7535 0.7899 -0.0177 -0.0120 0.9490 0.3907 0.5601 0.7436 -0.0421 ~o -0.9285 0.6590 0.2 827 0.7471~ 0.7870 - 0.01 88 -0.0077 0.9590 0.3942 0.5538 0.7433 - 0.042 3 — - 0.8468 0.6690 0.2866 0.7408 0.7842 -0.0199 -0.0044 0.9 690 0.3976 0.5 47 5 0.7431 - 0.042 5 rn -0.7721 0.6790 0.2906 0.7344 0.7815 - 0.0210 - 0.0020 0.9790 0.40~11 0.5413 0.7429 -0.0427 v -0.7036 0.6 890 0.2945 0.72 80 0.7790 -0.0221 -0.0005 0.9890 0.4045 0.5351, 0.742 8 - 0.042 8 - 0.6408 0.6990 0.29 84 0.7216 0.7765 - 0.02 31 -0.0000 0.9990 0.4079 0.52 89 0.7 42 8 - 0.042 8 -0.5829 0.7090 0.3022 0.7151 0.7741 - 0.02 42 0.0000 1.0000 0.4082 0. 5283 0.742 8 - 0.0428 -0. 5297 0.7190 0.3061 0.7087 0.7719 -0.0252 -0.4806 0.7290 0.3100 0.7022 0.7697 -0.0263

WILLOW RUN LABORATORIES A 10 o A ~r rl = 0.500 in. 0.9 e 4 o A 4.02 1.2 x 10 o 3.15 1.5 x 10O o 2.35 4.4 x 104 8 0.8 -1.5 -1.0 -0.5 0 x - L (in.) FIGURE 4. HAMA'S DATA ON THE WALLPRESSURE RATIO FOR LAMINAR FLOW [ 14, fig. 4] Solution in the First 10Outer Limit for M = 4.02 0.9 Composit Solution for o o 2.35 4.402 x 10 0.8, PRESSURE RATIO AMA Sec.02 1.2 x 0Lit L = 4.783 In. o 3.1 5.5 x...... = 6. 0. - M 2.3 4.4 x R 4 =.4 -4.0 -3.0 -2.0 ll FIGURE 5. COMPARISON OF THE COMPOSITE SOLUTION FORTHE WALL-PRESSURE RATIO AND HAMATS DATA

WILLOW RUN LABORATORIES reference values Me and pe are chosen to correspond to the measurements at the staticpressure orifice farthest upstream from the corner. This orifice is located at (x - L) = -1.5 in., where L = 4.783 in. The largest experimental Mach number presented here corresponds to Me z 4.02, Rw 1.2 x 10, and 61 0.10 in. Although even this value of Me might be rather low for use of a hypersonic theory, the other sources of error to be considered seem to be at least as important. We see in figure 5 that the wall-pressure-ratio measurements are independent of the Reynolds number R within a discrepancy of Ap /p- 0.02 over the range 1.2 x 104 < R < 4.4 x 10. Thus, the theoretical prediction of Reynolds number independence appears to be confirmed by the experimental data. Also presented in figure 5 is the wall-pressure-ratio solution in the first outer limit for Me = 4.02, the solution in the second outer limit (which is independent of both Mach and Reynolds numbers) for the one-strip calculation by the method of integral relations, and the composite solution for M = 4.02. The discrepancy between the calculated solution and the experimental e data is observed to be IAP /PO I - 0.06 or less. This is a fairly good result, as compared with the sort of accuracy usually obtained in a one-strip calculation by the method of integral relations (see ref. 22 and app. II). This may be because the calculation by the method of integral relations comprises only part of the solution; the equations in the first outer limit have an exact solution. The changes in velocity profile for the accelerating boundary layer are shown in figure 6 by plots of u/ue vs. Qi. The two solid curves represent the initial velocity profile u/ue = u(-o, y) = g(j) and the profile u/ue = u (0, y) given by the upstream solution 4.10 at x = 0. It is evident that the initial acceleration of fluid particles along streamlines is significant primarily near the wall, but plots of u/ue vs. y would show that the resulting displacement of streamlines is significant all across the layer. For the solution in terms of x and y, it is probably consistent with a one-strip calculation by the method of integral relations to choose only a linear variation of u with y. u = u + (1 - )Y/6 c (5.208) If equation 5.37 is specialized for a one-strip calculation, we have = -Sc[A log (1 - /6 c) + c1( Y/ (5.209) These two equations can be combined to give plots of u/ue vs. iq for x - -co and x = 0. The plots are shown as dotted curves in figure 6. Rather good agreement is obtained between the approximate form for u as x - -oo and the solution 4.10 for u at x = 0. 60

WILLOW RUN LABORATORIES 3 Solution in the First Outer Limit.......... Solutions in the Second Outer Limit: One-Strip Calculation ~cx01 0 I I I 0 0.2 0.4 0.6 0.8 1.0 u/ue FIGURE 6. VELOCITY PROFILE AT THREE STATIONS: A A x - -oo, x - O(x - -oo), AND x = 0 We shall now consider the implications of some higher order effects. The chief correction required because of finite Reynolds number probably involves the sublayer displacement thickness effect. The relative sublayer displacement thickness, 6SL/1 = O(R /4) (see eq. 6.35), also causes a pressure disturbance, AP/Pe = O(R ). In Hama's experiment, R1/4 0.10 for M = 4.02. Since p- < 0, the sublayer thickness 6SL will decrease monotonically, and, since e x IpJ increases monotonically as (x - L)/1 - 0, the sharpest decrease in 6SL will occur near the corner. This will tend to cause a further increase in Ip-T near the corner. The overall drop in the pressure, [p(0, 0) - Pe]/Pe' is fixed because of the sonic condition at the corner. Thus, an increase in Ip-I near the corner will be balanced by a decrease in P-xI farther upstream, and a calculated wall-pressure-ratio distribution that includes the correction for the sublayer effect will lie above the solution for the first-order theory shown in figure 5. Thus, the sublayer effect may account for a substantial part of the discrepancy between the predicted and measured values in figure 5. Disturbances in the pressure of the order Rw1/4 also arise when (x/L) - 1 = O(RW /8) (see eq. 3.38). It would be consistent to treat this effect in conjunction with an analysis of the sublayer. Another Reynolds number effect appears through the displacement effect of the boundary layer on the outer inviscid flow. The order of magnitude of this effect is characterized by the interaction parameter, M R1l/2. In Hama's experiment, M R/2 0.04, and it is pointed out in reference 14 that viscous interaction has a noticeable effect is raising the initial pressure 61

WILLOW RUN LABORATORIES Pe' The parameter MeR / also is a measure of the three-dimensional effect in flow over a cone (see sec. 3). The most significant Mach number effect is a result of the solution in the first outer limit. An increase in Me should result in a greater upstream influence and thus a lower value of pi/Pe for a given value of (x - L)/ 1 (see fig. 5). However, the experimental results in figure 5 do not seem to bear this out. This predicted Mach number effect may be obscured by the sublayer influence previously discussed. Since in the experiment, the test conditions for larger values of M also have smaller values of R, the two effects tend to offset each other. e w Another Mach number effect appears if we attempt to calculate higher-order terms. It is then also necessary to consider a limit 3.54 in which x/M1/ is held fixed, and, in this limit, e we find Ap/p =O(M )e In principle, it would be possible to generalize the present theory by the following procedures: (1) Include the viscous sublayer effect by carrying out the calculations suggested in section 4. (2) Obtain a solution for the limit 3.54 with x/Me/5 fixed (3) Generalize the equations to include nonadiabatic and real gas effects We shall now examine certain aspects of the application of the method of integral relations to this problem. The system of nonlinear partial differential equations is reduced to a system of quasi-linear, first-order, ordinary differential equations. The character of the downstream boundary condition in the subsonic region where the equations are elliptic is simplified to consist of one downstream condition on each strip boundary that is initially subsonic (see eq. 5.199). Also, the procedure for obtaining the numerical solution of the equations is well adapted to the use of high-speed electronic digital computers. Increased accuracy in the numerical solution is achieved, in principle, by increasing the number of strips used in the application, rather than by increasing the number of iterations in a relaxation technique, for example. However, in the application of any integral technique, certain properties of the full equations may be only approximated or obscured. Although accurate representations for the flow properties along the strip boundaries might be attained with the method of integral relations, profiles normal to the strips are found only by interpolation of the values on the strip boundaries. In the present study, the property profiles at x = L are of particular interest since they would provide the initial conditions for a calculation of the near wake of the body. 62

WILLOW RUN LABORATORIES In the present application of the method, we have found that Mo 1 + (constant)xl 1/2 (5.210) 0 as x- 0. This behavior for x - 0 is also found by Gold and Holt [23] in a one-strip integralrelations calculation of supersonic flow past a flat-faced cylinder. However, results obtained by Vaglio-Laurin [24] and by Fal'kovich and Chernov [25], for example would seem to suggest that the correct behavior is M - 1 + (constant) Il2/5 (5.211) 0 as x 0 instead of equation 5.210. A possible explanation of this discrepancy is that the integral relations resemble a description of a generalized one-dimensional flow as discussed by Shapiro [26] where, for instance, dM2/dx = G(x)(1 - M2)-1 (5.212) (see eq. 8.71a in ref. 26). When G(x) f 0 at the point where M = 1, the expansion for M near the sonic point is similar to that obtained in equation 5.210. In general, there is a requirement that G(x) = 0 when M = 1, and equation 5.212 has properties similar to those of equations 5.195 and 5.196. Belotserkovskii, Sedova, and Shugaev [27] avoid this difficulty for the related problem of inviscid supersonic flow over a blunt axisymmetric body with a corner. They obtain a solution by the method of integral relations, with strip boundaries equally spaced between the axis of symmetry and the limiting characteristic, and join their result with Vaglio-Laurin's solution near the corner. 6 SOLUTION IN THE SUBLAYER 6.1. DORODNITSYN TRANSFORMATION The sublayer equations describe a compressible boundary layer which extends upstream to infinity. Problems of this type are discussed by Neiland [28], and a numerical solution for the boundary layer approaching a corner is given by Matveeva and Neiland. In the present work, we employ Dorodnitsyn's method [22] to derive the integral relations for N = 1 and 2. The appropriate form of the Dorodnitsyn transformation in this case is (see app. I) Pt e (6.1) 63

WILLOW RUN LABORATORIES yt =uf ptdyt where Pe = (y - 1)/2y and U = -o/% e; -Uo is the velocity just outside the sublayer. This velocity is found by setting y = O in the composite solution for the outer part of the boundary layer. a/ai = u- a/at + - a/an P x a/ayt = upt a/av Since yt7 = (1/pt U) we have yt =I (l/pt U) d Differentiating 7 (x, yt) with respect to 5, '7 = - 7 ytY d /di Hence 77= _pt U2 A (l/pt/U) dn For adiabatic flow of a thermally and calorically perfect gas over an insulated body, the density can be obtained from the integrated energy equation, (utU)2+ 2y p 1 (6.2) Also, the x-momentum equation, 3.66, evaluated at the edge of the sublayer, yields (since ut = 1 here) -dpt/dx = 2 pt (1 - U2)- 1UdU/d (6.3) while, in terms of the Mach number M at the outer edge of the sublayer, 2 y - 2 12o) 2 U = OMki + MJ (1 - U) U dU/dx= M- 1dM /dx 64

WILLOW RUN LABORATORIES Thus, the transformed equations are u +w =0 (6.5) 771 utu +wuK = 1 _u Mo/Mo+u (6.6) where M denotes dM /d,and where O7 O ~ P w = -p u UJ (l1/p U)a d+ + V6.7) The boundary conditions are ut(5, 0) = w(, 0)= = O, u(, c) = 1 (6.8) 6.2. SOLUTION BY THE METHOD OF INTEGRAL RELATIONS To obtain the integral relations, we follow reference 22 and introduce a set of smoothing functions fk(u ) where k = 1, 2,.., N. The fk terms are defined to have the properties fk(O)= 1 lim fk(uT 0 (6.9) u - 1 k= 1, 2,..., N We multiply equation 6.5 by fk and equation 6.6 by fk and add the results. Then we integrate on 71 from 0 to co, obtaining 0 0 ct~~~~o ~o~~u Introducing O = (aut/al)- 1 changing the variable of integration from V7 to ut, and integrating by parts, we get 1 1 d- =2 M' K- ut2)fdu f Id(0) 0 d fkudut M - J Odu -dut (6.11) 0 0 When ut - 1, O becomes large. In the case of an adiabatic constant-pressure boundary layer, equation I-30 gives the velocity g(3) for: - o. It follows that, as j - o, 65

WILLOW RUN LABORATORIES g (constant) exp (-/ /2) - (constant) 0(1 - g) and ~ (constant) [log (1 -g)1/2 g (constant) (1 - g)[-log (1 - )]1/2 The same kind of behavior is expected when there is a pressure gradient,with 77 playing a similar role to 3. Since yt oc n7 as 77 - co in the present notation, aut/a -7 (constant) (1 - ut)[-log (1 - ut)]1/2 as t7 =co. The procedure in the present method [21] is to approximate the singularity in 0 by omitting the logarithmic factor. We assume instead that o = 0[(1 - ut] as ut - 1. To guarantee that the integrals exist, we let fk(ut) = (1 - u)k (6.12) where k = 1, 2,..., N, and we represent O by N-1 m o = (1 - ut)-l a()u m=O (6.13) N- tm o (1-u)jbm(f)u$ m=O where the am(t) and the b (M) terms are related via N-1 m N-1 m b M f=~~ - a Mut ~~~(6.14) m= Om=O where uk = k/N, and k = 0, 1,..., N - 1. That is, the expressions for O and 1/0 are required to agree at N equally spaced values of u When N = 1, we have 0= (1 - ut)-l o(4) (6.15) 66

WILLOW RUN LABORATORIES fl= 1 -u f= -1 (6.16) f=' 0 and equation 6.11 becomes 1 d M1 o utO0 dut = - M (1 + utO0 du + or dO M2 +6 0 =4 (6.17) d5 M 0 Upon multiplication by the factor M, equation 6.17 can be integrated directly, giving 0 2m3 (i 6 do) (6.18) It follows from equation 4.29 and the definition of U that U = O(1/x) as x — oo. If the velocity on any streamline in the sublayer decreases in this manner, then the distance of the streamline from the wall must increase linearly with x. The constant of integration in equation 6.18 has been chosen so that yt obtained below in equation 6.22 has the required form for x - -o. As pointed out following equation 6.35, this behavior appears to permit an upstream matching with Lighthill's [8] results. Then from equations 6.2, 6.15, 6.18 and the relation 0 = (aut/a77)1 we obtain (1- u )M3 aut/ai = 1/2 (6.19) 2 (iM d Integration on i7 and r fixed yields 1 - exp (-Mo3n/2 1/) (6.20) where 67

WILLOW RUN LABORATORIES ~f M d (6.21) = JUx dx The normal coordinate is found from ut 2 ~p+U 1 [1 - (utU) 2]Odu 1/2 (2M6 2 t 1 d2 = _____t 3 [-(1- U2) log (1 - Ut) + U(u + U)] (6.22) 0 in the case N = 1. Since d~ = Up dx and both U and Mo are of the order 1/x asx - -, it fol'e lows that yt O(x) as x - -oo for ut = constant. Further evaluation of the properties in the sublayer requires numerical integration of the integrals for specified M (x). When N = 2, the procedure for obtaining the integral relations is as follows. Equations 6.12 and 6.13 become fl = (1 - ut) = -1 (6.23) f (1 - ut)2 f -2(1- uu) f= 2 o = (1 - ut) l[0(1 - 2ut) + Olut 1 ~~~= (1 - u.)K - 2R+(6.24) and equation 6.11 becomes 68

WILLOW RUN LABORATORIES d |u [0 0(1 - 2u) + G lut dut = —M~ (1 + u) [0(1- 2uT) + 01ut du + (6.25) 1 Jo oJo d u(1 ut) 0(1 - 2ut) + Olu dut - (1 - t [O( - 2ut) + 01u]d 0 o2 l- 2u 4u + + et ( - u (1- E) ' + 0-du (6.26) so that one gets the differential equations il0 34 32 Eo + M (9o0 + 701): O- Ol (6.27) o 0 1 0 +2(20 16 +M (400 + 601) - (6.28) O 0 1 For a specified Mo (), equations 6.27 and 6.28 can be integrated numerically to find O0(a) and 01(t). The first of equations 6.24 gives, for ~ fixed, drq= (E1 -200)( 1 _lu + Oo( lut2 dut (6.29) 77 = (01 - o) log (1 - ut) - ( - 20)u Again, the normal coordinate, yt, is recovered from equations 6.2 and the solution for 0. 6.3. DISPLACEMENT THICKNESS A displacement thickness of the sublayer may be defined by 6 6SL,=___ 6SL*0 I) dy (6.30) where po and u are the density and velocity just outside the sublayer. Let 6SL = S L. Then, in terms of the nondimensional stretched variables in the sublayer,,5* 1-41/4 _(_1 d t 6=R (631) SL o wdy 69

WILLOW RUN LABORATORIES Changing the integration variable to ut, with dyt = (l/ptU) di7 and dl7 = Odut, and using equation 6.2 and the equation of state to eliminate pt, we obtain pt/p= (1 - U 2)[1- (u tU)2 1 (6.32) 1p = 7 - U [1 - (u U)2] (6.33) 2yp and 6 R_ (1/4 > (1 - ut)( + u)U odut (6.34) SL w 2rptuI-o when N = 1, 0 = O0/(1 - uT) and where O0(t) is given by equation 6.18. Thus, the integration in equation 6.34 yields 1/2 6p U d 6 =R 3 (6.35) SL w p}M The N = 2 result can be obtained in a similar manner. Since U = O(x ) and M = O(1/x) as x - -oo, it follows from equation 6.35 that 6SL 0 '(R / i)4 as — co. Lighthill's [8] estimate of upstream influence is (-x/L) - 1 = OR 3), i.e., x = o(Rw 8) For x of this order, the above result for displacement thickness may be expressed by 6SL/ w) (6.36) which is consistent with the sublayer thickness given by Lighthill. 7 CONCLUSIONS In the present study we have developed a description of the acceleration of a laminar boundary layer approaching a sharp corner in the limit of large Reynolds number R and large external Mach number Me, with MR-1/2 tending to zero. Because of the large pressure gradient near the corner, the viscous effects are found to be confined to a sublayer thinner than 70

WILLOW RUN LABORATORIES the boundary-layer by a factor of the order R l/ In a first approximation, the effects of the w viscous sublayer are neglected, and the inviscid rotational equations govern the flow. Numerical results are obtained for hypersonic, laminar, adiabatic flow of a perfect gas over a slender wedge or cone. The approach can be generalized to apply to nonadiabatic flows of real gases. The outer inviscid flow in the accelerating layer is characterized by two distinguished limits. In a first outer limit, x = [/L) -1)]/MeR /2 and y = y/R L are held fixed, and the flow deflection angle is of order M throughout the layer. The normal pressure gradient e can be neglected in this case, and the governing equations in this limit are inviscid boundarylayer equations, which may be integrated directly. In a second outer limit, x = [(i/L) - 1)]/R- /2 and y =-y/R 1/2 L are held fixed, and the W W flow deflection angle 0 is of the order unity in the layer. The full inviscid equations govern the flow in this case. Now 0 remains of the order Me at the outer edge of the layer, and in e this limit the boundary condition 0(x, 6) = 0 is imposed. In the first outer limit, a decrease in the pressure is associated with a corresponding decrease in the layer thickness since the changes in stream-tube area for the subsonic portion of the layer are dominant. This is designated as the subcritical condition. Eventually a further decrease in the pressure would cause the layer to become thicker instead of thinner. Further acceleration of the layer is accomplished by the sharp turning of the streamlines near the corner. This turning corresponds to the flow description in the second outer limit. This is designated as the supercritical condition, with d6*/dp < 0. The critical point, where d6*/dp = 0, occurs at a distance o(M R1/2 L) upstream from the corner. This clear distinction between subcritical and supercritical flows arises as a result of taking the hypersonic limit, M - o0. Another result of the limit Me - co is that the boundarylayer thickness is clearly defined and is equal to the displacement thickness. This is a useful simplification in our application of the method of integral relations to the system of equations in the second outer limit. A composite expansion for the wall-pressure ratio, formed from the solutions in the first and second outer limits, compares reasonably well with Hama's experimental data for a wedge with M = 4.02. Hama's wall-pressure ratio data at three Reynolds numbers are correlated when plotted as a function of (x - L)/61, in agreement with the theory. The most important second-order correction appears to be the sublayer effect, since R -/4 0.10 in Hama's experiment. The governing equations in the sublayer are the boundary71

WILLOW RUN LABORATORIES layer equations with a pressure gradient. We are able to derive integral relations for the sublayer velocity and displacement thickness. Other higher order effects may, in principle, be considered in the analysis by studying a limit where x- 0, x - —, and x/M /5is held fixed, e and a limit where x - -o with x/R 8 held fixed. w In addition to the numerical solution in the second outer limit found with a single-strip application of the method of integral relations, the integral relations for a two-strip application are derived. In principle, any number of strips can be considered. However, it is necessary to carry out asymptotic expansions of the equations to the second order as x - -co in order to start the numerical integration technique, and the procedure for applying the initial conditions is not yet clear. Another approach to solving the equations in the second outer limit might be to use the method of characteristics or a finite-difference method in the supersonic region, and a onestrip application of the method of integral relations for the portion of the flow that is initially subsonic. 72

WILLOW RUN LABORATORIES Appendix I COMPRESSIBLE LAMINAR BOUNDARY LAYER AT CONSTANT PRESSURE As a preliminary to the analysis of boundary-layer acceleration at a corner, it is necessary to calculate the development of the boundary layer upstream from the interaction region. The velocity profile evaluated at the corner then provides the upstream boundary condition for the interaction calculation. For this reason and for the convenience of having the results available in notation consistent with other parts of this work, the solution of the constant-pressure laminar boundary layer on a wedge or cone, in the limit of large Mach and Reynolds numbers, is given here. In this boundary-layer calculation, the following idealizations are made: (1) Thermally and calorically perfect gas (2) Unity Prandtl number (3) Linear viscosity-temperature relation (4) Adiabatic wall However, these simplifications are not essential to the approach employed to analyze boundarylayer acceleration at a corner. To establish order estimates for flow properties just outside the boundary layer on a wedge or cone, we shall consider the oblique-shock relations for high-speed flow past a slender wedge (see fig. 1). We are concerned with the limit M -Coo 0 R - o (I-1) T- 0 If also T = 0(1) (I-2) M T o0 then it can be shown (for example, see Hayes and Probstein [1]) that 1/2 Pe (7 i + 1 (+ 1 1 (-3) and also 73

WILLOW RUN LABORATORIES (a /T)M M2 (I- (1-4) e (o/iT- l)[l + yM T2(iT)] 0(T-2) (I- 5) Here P and M are the pressure and Mach number at the outer edge of the boundary layer, T Pe e is the wedge half-angle, and a is the shock-wave angle. Although equations I-3 and I-4 are applicable only for a wedge, equation I- 5 is valid also for a slender cone. A power-law viscosity-temperature relation of the form M / =M (T /T ) (I-6) is assumed. Since TeTo = 0(M2oT) and Twe = 0(T2), we have the order estimates R O[(M T) -2CR] (I-7) Rw 0[T2(M )0 RJ (I- 8) where R = P u L/i and R = eL/ii are the Reynolds numbers based on the thermodye ee e w we namic properties at the outer edge of the boundary layer and at the surface of the body, respectively. Later in the calculation, the exponent w will be set equal to one. Since T/T = 0(1) for any point inside the boundary layer, T is a proper reference temperature. Then the order estimate for the boundary-layer thickness at the trailing edge of the body is ~1/O(0 w) (I-9) Since from equations I-2 and I-3 we have a/T = 0(1), the requirement that the boundary-layer thickness be much smaller than the shock layer thickness becomes R-12.0 (I- 10) w The limit expressed by equations I-1, I-2, and I-10 can be written in terms of local boundarylayer properties as M -oo e R -c (I- 11) w e w 74

WILLOW RUN LABORATORIES It is appropriate, in view of the order estimate for the boundary-layer thickness given in equation I-9, to introduce the stretched coordinates x= x/L Y =/R 1/2L (I- 12) r = r/L' Then, in the limit expressed in equations I-11, the leading terms in asymptotic representations for the x- and y-velocity components, pressure, and density are /-e -U(X, y)+... T /- R_ 1/2v(x,. +. _/te w ~R-1/2 (I- 13) 2 pp Ue2 - (i - 1)/2y +... w e P/jPw p(x, ) +... In the first approximation for the hypersonic limit, the equations describing laminar boundarylayer flow become (pur ) + (pvrk)= 0 (I- 14) x p(uux + vu-) = (guy ) (I1-1 5) p= (1 - u2)1 (I-16) where k = 0 for a wedge, and k = 1 for a cone. For a linear viscosity-temperature dependence, = T p-1 (I-17) where /t = i/i and T = Tr. Note that q2 e22 -1 (M2) -2 2 = 1 + 1 + O (1-18) e/ max ( - 1 e e in the hypersonic limit. The differential equations I-14 and I-15 can be converted to the form for plane incompressible flow by the Dorodnitsyn transformation (see Belotserkovskii and Chushkin [22]), which, in this case, simplifies to x jr2kx 75

WILLOW RUN LABORATORIES 7l = r pdy (I- 19) a/ax = r2k a/a + 7x a/a7 7X (Y/Y7)fx 3k f7k k 2 = pr pr (prk) 2 d Integration of the continuity equation gives pvr' [- J [(purk)r + 77x (purk) ] (prk)i d Then the resulting differential equations are u +w = 0 (I-20) uu +wu =U77 (1-21) where -k kC77k k -2 w =pvr + pur (prk)(prk) d7 The boundary conditions are u(O, ) = 1 u(4, 0) = 0 (-22) (I- 22) w(4, 0) = 0 u(g, o) = 1 The system of equations 1-20, 1-21, and I-22 has a similarity solution originally found by Blasius (see Rosenhead [29]). Equation I-20 can be satisfied identically if we define a stream function,,I, by E =-w (I- 23) ' =u 76

WILLOW RUN LABORATORIES Then equation I-21 becomes *4 ~ I * 1 7= -4 4TI' (I-24) In terms of a similarity variable: (2 )-1/2t7 (I-25) the stream function has the form - = (2 ) 1/2g() (I- 26) and equation I-24 transforms to the ordinary differential equation g + gg 0 (1-27) where the dot denotes differentiation with respect to 3. The boundary conditions are g(0)= 0 (0) = 0 (1-28) g(oo)= 1 Rosenhead [29] tabulates the solution to equations I-27 and I-28 and gives g(0) = 0.4696 (I-29) In the limit of large /, k(/) 1 - 0.331(C-1 - 3 3(5 - ) exp (1(2) (I-30) where = / - 1.21678. The values of our original variables are found from 1 x = [(2k + 1) ]2k (1-31) = (2 ) 1/r~i (1 *2d: (I- 32) Integrating by parts and using equation 1-27, equation I-32 becomes = (24)1/2r-k(- g_ - g- + 0.4696) (I-33) 2 -1 Also, r = x, u = g, and p = (1 - ), while the boundary-layer thickness is given by 77

WILLOW RUN LABORATORIES 6/R-/ 2L (2 )1/2r k lim (3 - g - " + 0.4696) = 1.68638(2n)1/2rk (I-34) /-co In the hypersonic limit, the boundary-layer thickness is finite and equal to the displacement thickness 6*, where '= (Pu'd'y='' Nowo O(M)u in thedy = 5I(p/pPe)udy Now e: O(M -)2 in the hypersonic limit so that 6/6* - 1 = o(1) (I-35) At the trailing edge, x = r = 1, and hence 12 (I- 36) 1 2k + 1 75/R 1/2 L= 1.686382k + 1/ (I-37) In the analysis of boundary-layer acceleration at a corner, it is necessary to generate the function g(O). This is carried out by transforming equation I-27 to a system of first-order equations and integrating numerically by the Runge-Kutta fourth-order method [21]. We introduce new dependent variables Y =g Y =g (I- 38) Y3=g and obtain the differential equations Yl = Y2 Y= Y3 (I-39) Y= -Y1Y3 with the initial conditions Yl(0) = 0 Y2(0) = 0 (I-40) 78

WILLOW RUN LABORATORIES Y3(0) = 0.4696 To start the integration process, we utilize the expansions of g, g, and g for small [: 2 2 5 11 3 8 g ~ 2 /35 -[ 20a[ + 4032a 2 4 11 3-7 (1-41) g - ao - --- V P + 5040 0-... "/ 5040 12.3 11 36 g 6a[3 +720a where a = 0.4696. Appendix II APPROXIMATION BY THE METHOD OF INTEGRAL RELATIONS COMPARED WITH THE EXACT BLASIUS SOLUTION The transformed relations for a boundary layer without a pressure gradient are equations 1-20, 1-21, and I-22. Since, in this case U = 1 and Ml = 0 (see sec. 6), the integral relations, equations 6.11, become d f fkdu - -du = dOet (II- 1) where k= 1, 2,..., N. When N 1, we have fl= 1 -u f = -1 fA'= 0 Assuming a similarity solution for u, O (au/an)1 = 2 41/2 (1 u1 (- -2) and u = 1- exp (-/ 21/2) (11-3) u_(0) = 0.5-1/2 (11-4) 79

WILLOW RUN LABORATORIES A nondimensional displacement thickness is 6' *=i(1 - u)di 1 - u)Odu= 2.01/2 (II-5) 0 0 When N = 2, fl =1-u f = -1 fV'= 0 f2= (1 - u)2 f* =-2(1 - u) 2 f" = 2 2 o = (1 - u)1 [ + ( 1 - 200)u] (11-6) = (1 - u)[B0 + (B1 - 2BO)u] (II-7) Thus, from equation 6.14, B0 = O;1, B1 = 40, and equation II-1 becomes d00/d = 340O1 - 3201 (II-8) dOl/d = 200O - 1601 (11-9) When we assume a form of solution similar to equation 11-2, 0. = A. 1/ (II- 10) where i = 0, 1. When substituted into equations II-8 and II-9, equation II-10 yields A 2(17 - 16-1 )1/2 = 3.1555 (II-11) A 2X(17 - 16X 1)1/2 3.4793 (II-12) where X = (13 + XF-)/17. When N = 2, the result for the nondimensional wall shear stress is u (0) = 0.3174 80

WILLOW RUN LABORATORIES while the displacement thickness is given by 6* = 1.74 1/2 Table V is a comparison of the results of the method of integral relations for N = 1 and 2 with the Blasius solution and the Pohlhausen approximation. It can be seen that when N = 2 the method of integral relations is comparable in accuracy with a Pohlhausen calculation. For a more complete assessment of the accuracy of the method of integral relations in a variety of problems, see reference 22. TABLE V. COMPARISON OF SOLUTION BY THE INTEGRAL RELATIONS METHOD WITH THE BLASIUS BOUNDARY-LAYER SOLUTION AND A POHLHAUSEN CALCULATION Method of Pohlhausen Blasius Result Integral Relations Calculation [30] N=1 N = 1 1/2u (0) 0.332 0.5 0.317 0.343 - 1/26* 1.729 2.0 1.74 1.752 Appendix III REDUCTION OF HAMA'S WALL-PRESSURE DATA In this appendix, the calculations carried out in reducing Hama's wall-pressure ratio data [6] (shown in fig. 4) to the form given in figure 5 are explained. The Mach number given is Me i' and the wall pressure p is divided by Pe i' where M,i and Pi are the values of Mach number and pressure at the surface which would be predicted by inviscid-flow theory. Also, the value of the Reynolds number Re is specified. There are three sets of experimental conditions: M R e,i e 4.02 2.16 x 105 3.15 1.34 x 105 2.35 1.97 x 105 VWe wish to express the results in terms of M, R and to calculate 61. We choose the static pressure orifice farthest upstream (x - L = -1.5 in.) as the reference point for measur81

WILLOW RUN LABORATORIES ing Pe' Then -o /e can be calculated from the relation o/-e = Po/Pe,i )/e /-e,i) (III-1) Since IM - M i/M i is considerably smaller than IPe - Pe i /Pe i' we will take M = M e e,i The Reynolds number based upon the thermodynamic properties at the wall is found from R p-1"- R + M R (III- 2) ~w w e)J e w e ( 2 e e and, in our calculation, w 1 and y = 1.4. The boundary-layer thickness is calculated from 61 = x(1.68638)R /L (III- 3) (see app. I), with L = 4.783 in. The results are as follows: M j R 61 Me Pe/Pe,i Rw (in.) 4.02 1.070 1.2 x 104 0.104 3.15 1.048 1.5 x 104 0.093 2.35 1.020 4.4 x 104 0.054 82

WILLOW RUN LABORATORIES REFERENCES 1. W. D. Hayes and R. F. Probstein, Hypersonic Flow Theory, Vol. I: Inviscid Flows, 2nd ed., Academic Press, 1966. 2. B. L. Reeves and L. Lees, "Theory of the Laminar Near Wake of Blunt Bodies in Hypersonic Flow," AIAA J., Vol. 3, 1965, pp. 2061-2074. 3. R. J. Golik, W. H. Webb, and L. Lees, Further Results of Viscous Interaction Theory for the Laminar Supersonic Near Wake, AIAA Paper No. 67-61, New York, 1967. 4. S. Weinbaum, "Rapid Expansion of a Supersonic Boundary Layer and Its Application to the Near Wake," AIAA J., Vol. 4, 1966, pp. 217-226. 5. R. F. Weiss, "A New Theoretical Solution of the Laminar Hypersonic Near Wake," AIAA J., Vol. 5, 1967, pp. 2142-2149. 6. F. R. Hama, Experimental Investigations of Wedge Base Pressure and Lip Shock, Technical Report No. 32-1033, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, 1966. 7. M. V. Morkovin, "Effects of High Accelerations on a Turbulent Supersonic Shear Layer," Proceedings of 1955 Heat Transfer and Fluid Mechanics Institute, Stanford University Press, 1955, pp. IV-1 to IV-17. 8. M. J. Lighthill, "On Boundary Layers and Upstream Influence, II: Supersonic Flows without Separation," Proc. Roy. Soc. (London), Ser. A, Vol. 217, 1953, pp. 473-507. 9. V. Zakkay and T. Tani, Theoretical and Experimental Investigation of the Laminar Heat Transfer Downstream of a Sharp Corner, PIBAL Report No. 708, Department of Aerospace Engineering and Applied Mechanics, Polytechnic Institute of Brooklyn, Brooklyn, N. Y., 1961. 10. B. L. Hunt and M. Sibulkin, An Estimate of Compressible Boundary Layer Development Around a Convex Corner in Supersonic Flow, Report No. Nonr (562) 35/6, Division of Engineering, Brown University, Providence, R. I., 1964. 11. V. I. Neiland and V. V. Sychev, "Asymptotic Solutions of the Navier-Stokes Equations in Regions with Large Local Perturbations," (Russian) Mekh. Zhid. i Gaza, 1966, pp. 43-49; Translation TR-40, Research Department, Grumman Aircraft Engineering Corporation, Bethpage, N. Y., 1967. 12. N. S. Matveeva and V. I. Neiland, "Laminar Boundary Layer near a Corner Point of a Body," (Russian) Mekh. Zhid. i Gaza, 1967, pp. 64-70. 13. L. Lees and B. L. Reeves, "Supersonic Separated and Reattaching Laminar Flows, I: General Theory and Application to Adiabatic Shock-Wave/BoundaryLayer Interactions," AIAA J., Vol. 2, 1964, pp. 1907-1924. 14. E. Baum, An Interaction Model of a Supersonic Laminar Boundary Layer Near a Sharp Backward Facing Step, Report No. BSD TR 67-81, TRW Systems Group, Redondo Beach, Calif., December 1966. 15. R. F. Weiss and W. Nelson, On the Upstream Influence of the Base Pressure, Research Report No. 264, Avco-Everett Research Laboratory, Everett, Mass., January 1967. 16. F. K. Moore, "Hypersonic Boundary Layer Theory," Theory of Laminar Flows, ed. by F. K. Moore, Princeton University Press, 1964, pp. 491-492. 83

WILLOW RUN LABORATORIES 17. J. D. Cole and J. Aroesty, The Blowhard Problem-Inviscid Flows with Surface Injection, Research Memorandum RM 5196, The RAND Corporation, Santa Monica, Calif., March 1967. 18. M. D. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press, 1964. 19. E. Organick, The Michigan Algorithm Decoder, rev. ed., University of Michigan Computing Center, The University of Michigan, Ann Arbor, 1966. 20. University of Michigan Executive System for the IBM 7090 Computer, Vol. I, University of Michigan Computing Center, The University of Michigan, Ann Arbor, September 1966. 21. B. Arden, An Introduction to Digital Computing, Addison-Wesley, 1963, pp. 274-275 22. 0. M. Belotserkovskii and P. I. Chushkin, "The Numerical Solution of Problems in Gas Dynamics," Basic Developments in Fluid Dynamics, ed. by M. Holt, Academic Press, 1965. 23. R. Gold and M. Holt, Calculation of Supersonic Flow Past a Flat-Headed Cylinder by Belotserkovskii's Method, AFOSR Technical Note No. 59-199, Division of Applied Mathematics, Brown University, Providence, R. I., 1959. 24. R. Vaglio-Laurin, "Transonic Rotational Flow over a Corner," J. Fluid Mech., Vol. 9, 1960, pp. 81-103. 25. S. V. Fal'kovich and I. A. Chernov, "Flow of a Sonic Gas Stream Past a Body of Revolution," (Russian) PMM, Vol. 28, 1964, pp. 280-284; J. Appl. Math. and Mech., Vol. 28, 1964, pp. 342-347. 26. A. H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow (Vol. I), Ronald Press, 1953. 27. 0. M. Belotserkovskii, E. S. Sedova, and F. V. Shugaev, "Supersonic Flow about Blunt Bodies of Revolution with a Corner," (Russian) Zh. Vychisl. Matem. i Matem. Fiz., 1966, pp. 930-934; Translation TR-42, Research Department, Grumman Aircraft Engineering Corporation, Bethpage, N. Y., 1967. 28. V. I. Neiland, "Solving the Equations of a Laminar Boundary Layer under Arbitrary Initial Conditions," (Russian) PMM, Vol. 30, 1966, pp. 674-678; J. Appl. Math. and Mech., Vol. 30, 1966, pp. 807-811. 29. Laminar Boundary Layers, ed. by L. Rosenhead, Oxford University Press, Oxford, England, 1963. 30. H. Schlichting, Boundary Layer Theory, 4th ed., McGraw-Hill, 1960. 84

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UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA- R & D (Security classification of title, body of abstract and indexing annotation must he entered when the overall report is elassified) 1. ORIGINATING ACTIVITY (Corporate author) Ia. REPORT SECURITY CLASSIFICATION Willow Run Laboratories, Institute of Science and Technology, Unclassified The University of Michigan, Ann Arbor 2b. GROUP 3. REPORT TITLE ACCELERATION OF A HYPERSONIC BOUNDARY LAYER APPROACHING A CORNER 4. DESCRIPTIVE NOTES (Type of report and incluaive datea) 5. AU THOR(S) (First name, middle initial, last name) George R. Olsson and Arthur F. Messiter 6. REPORT DATE 7a. TOTAL NO. OF PAGES.7b. NO. OF REFS May 1968 x + 89 30 Sa. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) DAHC15 67 C 0062 b. PROJECT NO. 8146-13-T c. 9b. OTHER REPORT NO(S) (Any other numbers that may be asaained this report) 10. DISTRIBUTION STATEMENT This document has been approved for public release and sale; its distribution is unlimited. II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Department of the Army, Defense Supply Service- Washington 13. ABSTRACT An asymptotic description of the acceleration of a laminar hypersonic boundary layer approaching a sharp corner is obtained. The description assumes small interaction with the outer inviscid flow. Viscous forces are neglected except in a thin sublayer. The initial part of the expansion takes place over a distance O(Me6), where Me is the external Mach number, and 6 is the boundary-layer thickness. Here the transverse pressure gradient is small, and a solution can be obtained analytically. Within a distance 0(6) from the corner, the effect of streamline curvature is essential, and a numerical solution is obtained by the method of integral relations for a single strip. The solution for surface pressure is compared with experimental results for a particular case, and an approximate velocity profile at the corner is calculated. Possibilities for improving the accuracy of the calculation, both by refining the numerical procedure and by including higher order effects, are considered. D D.n~ov'..14 73 UNCLASSIFIED Security Classification

UNIVERSITY OF MICHIGAN I_ t _ l _ _l__ _ll__ __ i __ l _ l l h U llI UNCLASSIFIED 3 9015 03527 5992 Security Classification 14. LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT Boundary layer Hypersonic Corners Acceleration Pressure Inviscid flow Viscous force Streamlines Integral relations UNC LASSIFIED Security Classification