-L I B HAHY OF THE U N IVER.SITY OF ILLINOIS 0.84 U6r no. 257-264 cop. 2 CENTRAL CIRCULATION AND BOOKSTACKS The person borrowing this material is re- sponsible for its renewal or return before the Latest Date stamped below. You may be charged a minimum fee of $75.00 for each non-returned or lost item. Theft, mutilation, or defacement of library materials can be cautes for student disciplinary action. All materials owned by the University of Illinois Library are the property of the State of Illinois and are protected by Article 16B of Illinois Criminal Law and Procedure. TO RENEW, CALL (217) 333-8400. University of Illinois Library at Urbana-Champaign JUN 2 8 1999 When renewing by phone, write new due date below previous due date. L162 't> Report No. 262 yruuZ% ILLIAC IV Doc. No. 188 KINETIC STUDY OF THE HYPERSONIC FLOW PAST THE LEADING EDGE OF A FLAT PLATE by Rate"b Jaber El-Assar THE LIBRARY OF THE AUG u isss WERSITY If ILIMHS May 15, 1968 Digitized by the Internet Archive in 2013 http://archive.org/details/kineticstudyofhy262elas Report No. 262 ILLIAC IV Doc. No. 188 KINETIC STUDY OF THE HYPERSONIC FLOW PAST THE LEADING EDGE OF A FLAT PLATE* by Rateb Jaber El-Assar May 15, 1968 Department of Computer Science University of Illinois Urbana, Illinois 6l801 This work was supported in part by the Department of Computer Science, University of Illinois, Urbana, Illinois, and in part by the Advanced Research Projects Agency as administered by the Rome Air Development Center, under Contract No. US AF 30(602)4lMK ACKNOWLEDGMENT The author would like to express his appreciation to his adviser, Dr. W. L. Chow, for his valuable guidance throughout the course of the study. The author is also indebted to Dr. E. A. Jackson who provided helpful criticism and suggestions in the initial stages of the analysis. In addition, the author would like to convey his most sincere gratitude to Dr. D. L. Slotnick, Director, ILLIAC IV Project, Department of Computer Science, University of Illinois, for his encouragement and sponsorship of this work under Contract No. US AF ^,0(602)klkk. Thanks are also extended to Mrs. Sharon Hardman for her cooperative assistance in preparing the final manuscript. 111 TABLE OF CONTENTS Page ACKNOWLEDGMENT iii LIST OF FIGURES v 1. INTRODUCTION . 1 1.1 General . 1 1.2 Leading Edge Problem „ 5 2. REVIEW OF LITERATURE 7 2.1 Continuum Theories and Experimental Observations. . . 8 2.2 Kinetic Theory Investigations . 11 3. THEORETICAL ANALYSIS. 13 3.1 Description of the Present Model ....... 13 3c 2 Governing Equations l6 3.3 Boundary Conditions . 19 3»^- Solution for the Distribution Function „ 23 3.5 Evaluation of the Macroscopic Flow Properties. ... 30 h. RESULTS AND DISCUSSION 33 5» CONCLUSIONS AND RECOMMENDATIONS 37 TABULATION OF SYMBOLS 39 FIGURES . ..... kl LIST OF REFERENCES . . . kQ VITA 53 IV LIST OF FIGURES Figure Page 1. Skematic diagram of the flow regimes in rarefied hypersonic flow. kl 2. Surface pressure distribution at M = 25, 5 = .24. 1+2 00 3. Surfa£e pressure distribution using the parameter p/(p x ) at M = 25, 6 = ,2k. 1+3 k. Comparison of surface pressure distribution with various experimental data. hk 5. Comparison of available theoretical results in the merged-layer and the kinetic flow regimes. 1+5 6. Comparison of the slip velocity at M^ = 8.3> 6 = .24 with the experimental data of Chuan and Waiter. 1+6 7. Comparison of the slip velocity at M^ = 10 with the experimental and theoretical results. h6 8. Comparison of skin friction at M^ = 8.3 with experimental data of Chuan and Waiter. 1+7 v KINETIC STUDY OF THE HYPERSONIC FLOW PAST THE LEADING EDGE OF A FLAT PLATE Rateb Jaber El-Assar, Ph.D. Department of Mechanical Engineering University of Illinois, 1968 An attempt is made to solve the two-dimensional Boltzmann equation. The form of the distribution function is assumed apriori as the sum of two half range Maxwellian distribution functions containing three unknown parameters. This approach is an extension of the Mott -Smith bimodal model for the normal shock wave problem. The three unknowns in the distribution function are determined by satisfying three moments of the Boltzmann equation. Order of magnitude approximations (e.g. the hypersonic approximations) are subsequently employed in the analysis to make decoupling of the moment equations possible. The resulting equations are then solved using the Maxwell Slip boundary condition- -after adjusting it to the present model--in conjunction with the boundary conditions for the half -range densities. The results are presented in analytical form and are plotted for comparison with the available theoretical and experimental data. They give the trend which is believed to exist in the kinetic flow region. 1. INTRODUCTION 1.1 General Two independent approaches to describe the "behavior of gases in motion have been developed. The first one, which was pioneered by Euler, is based on the continuum hypothesis of the matter. Euler's and Lagrange's equations were formulated for inviscid fluids and from them evolved the science of the classical theoretical hydrodynamics. In many instances the results obtained from this mathematical treatment stood in glaring contra- diction with the experimental observations. The Navier-Stokes equations were formulated to describe the motion of viscous fluids. Owing to the mathematical difficulties encountered in obtaining solution of these equa- tions, the two divergent branches of fluid dynamics, namely the classical hydrodynamics and the highly empirical science of hydraulics, prevailed. The boundary layer theory introduced at the beginning of the present century by Prandtl, laid a milestone in the history of continuum fluid dynamics. It paved the way for the mathematical analysis of viscous flows and unified the two divergent branches mentioned above. This led to the establishment of an impressive body of knowledge in the present-day science of continuum fluid dynamics. The results of this science has attained, in general, a high degree of agreement with experimental observations. The second approach was pioneered by Boltzmann and Maxwell and is based on the molecular hypothesis of the matter, i.e„ matter is not a con- tinuum but rather composed of discrete particles called molecules. This approach led to the formulation of the Boltzmann equation and the kinetic theory. It is interesting to note that we became so accustomed to the universality of the Navier-Stokes equations with the classical zero-velocity 1 2 boundary condition that the purely empirical character of the stress-rate of strain relations is often forgotten. Because of the many simplifying assumptions regarding the structure of the actual molecules and their interaction with each other on the one hand, and with solid surfaces sub- merged in the fluid on the other, the kinetic theory so far formulated is equally empirical. I Intuitively, it should be expected that the continuum hypothesis which implies smooth continuous indivisible structure of the matter, should at best be a rough approximation of the actual behavior of the gas when, roughly speaking, the density level is highly rarefied. Thus, in rarefied flow fields the coarse molecular hypothesis should offer a better descrip- tion of the flow. Experimental and theoretical investigations support this notion. Hence, rarefied flow problems are best studied by the kinetic theory. It should be mentioned that the absolute density of a gas is not an appropriate measure to define the domain of application of the continuum theory for gas dynamics problems. In other words, certain regions of high absolute density are best studied by employing the Boltzmann equation, and certain regions of low absolute density can be described by the Navier- Stokes equations. An example of the former is the strong shock wave where rapid changes of fluid properties occur within a narrow region. An example of the latter is the flow region far downstream of the leading edge of a semi-infinite plate immersed in a low density gas flow where the viscous layer is fairly thick. The Knudsen number K is traditionally employed to predict the departure from the continuum behavior provided that the mean free path and the characteristic length of the problem be chosen appropriately. The Knudsen number can be related to the Mach number and 3 the Reynolds number yielding a relation depending on the choice of which characteristic length to be most significant for the flow region under consideration [l] . Such a relation is often referred to as the rarefaction parameter. A flow regime with K > > 1 is called a kinetic flow regime which n to implies that it can only be treated by the kinetic theory. However, the Boltzmann equation is general enough that it should be applicable to all flow regimes in a rarefied gas flow problem. It is therefore believed that the kinetic theory ought to provide a general formulation such that it would cover flow cases with any Knudsen number K including the two extremes, namely the kinetic flow regime (K > > l) and the classical continuum gas dynamics regime (K < < l). In spite of this apparent generality and in spite of the fact that the Boltzmann equation was formulated about a century ago, the literature still lacks a kinetic theory solution that covers the full spectrum of flow regimes. This may be attributed to the fact that the kinetic theory of gases has been, mostly but not exclusively, confined to study cases of very slow speeds and internal flow problems associated with vacuum installations. Moreover, it has been dominated, for about sixty years, by the approach of Chapman -Enskog [2] which was primarily concerned with the transport proper- ties of gases. The Chapman-Enskog approach utilizes an iterative expansion scheme for the distribution function when K < < 1. The first approximation leads to the Navier-Stokes equations and the second to the Burnett equations. No proof of convergence of the iteration scheme is available and it intro- duces rigidity in the analysis. It should be noted that application of the Burnett equations to study the structure of the normal shock wave gave little h improvement, if any, over the Navier-Stokes equations. This may be indica- tive of slow convergence of the Chapman- Enskog iterative scheme. The slow convergence of this scheme, its inherent rigidity and the fact that it does not allow for the "two-sidedness" in the distribution function which is essential for rarefied flow problems dim the prospects of utilizing any Chapman -Enskog type expansion to study the kinetic flow regime. Lees [3] suggested that the Chapman -Enskog approach may only be applicable to regimes where K < < 1. Such regimes can however be directly studied through the Navier-Stokes equations. Within the last twenty years many other iterative expansion schemes have been explored. However, all of these schemes are still limited in scope and not flexible. Some of these approaches are specialized for the linearized Boltzmann equation or for the linearized BGK model equation. They were applied to the Couette flow and Raleigh problems. Others are specialized for the near-free-molecule flow. Work by Grad [k] , Gross et al. [5], Mott -Smith [6], Wang Chang [7] and Gross and Ziering [8] are examples of these recent approaches. It was pointed out by Koga [9] that the convergence in any known iterative scheme is either not good or very slow at high Mach numbers. Most iteration schemes are based on the search for higher order moment equations so that they will yield some improvement over the Navier-Stokes equations. Lees [3] > who gave an excel- lent assessment of the different approaches, pointed out that some of the so-called "improvements" on the Navier-Stokes equations only make matters worse. The need for approaches which are radically different from the Chapman -Enskog tradition has been well recognized for the last fifteen years. One approach that broke up the classical tradition was introduced by Mott- Smith [10] on intuitive grounds in his study of the shock wave structure. 5 The success of this "bimodal" model stimulated other researchers to suggest other new approaches which are in the same spirit as that of Mott-Smith. The two-fluid model introduced by Glansdorff [11] for the normal shock wave and later modified and extended by other researchers is an example of this type. The original bimodal model and the two-fluid model have been restricted to the normal shock wave problem. An extension of these models to problems involving solid surfaces is long overdue. Lees [3] introduced the "two-stream maxwellian" model which is based on the so-called "line of sight" principle. This model is an extension of the Mott-Smith approach in such a way that the resulting model allows to take account of the presence of solid surfaces within the flow field with its two-sided character of the distribution function. In the present analysis we will consider an extension of the Mott-Smith approach of a different nature. Instead of accounting for the presence of a solid surface through the line of sight principle we use an everywhere half -range bimodal distribution. 1.2 Leading Edge Problem The problem of hypersonic flow past a sharp leading edge of a flat plate is a two-dimensional problem. The flow region in the immediate vicin- ity of the leading edge, which is our concern in the present analysis, is basically within the realm of gas kinetics. In other words, the leading edge problem is in the domain of the Boltzmann equation. The development of rocket technology in the last fifteen years created the need for the solution of rarefied gas dynamics problems espe- cially when they are directly related with practical applications. The last decade witnessed considerable activity concerning the hypersonic flow past 6 a semi-infinite flat plate with a sharp leading edge. Although this problem might seem to be of little practical value, it has certain features which are of great interest to gas dynamicists. Some of these features are: (a) It provides the simplest model as far as geometry is concerned to examine the viscous effects on the development of the flow field; (b) It involves all flow regimes starting from the kinetic flow regime in the immediate vicinity of the leading edge to the classical continuum flow regime farther downstream. Hence, it provides an opportunity to study the departures from continuum gas dynamics; (c) The knowledge obtained from a solution of this problem can have ready application to the control of Space vehicles because the flow model can be assumed to remain valid for slender bodies with sharp leading edge; (d) It provides a means of assessing the traditional belief that true free-molecule flow exists at the leading edge. If the flat plate is mathematically thin and in the absence of viscous action the flow field will be undisturbed. It is the presence of frictional forces on the surface of the plate that create a viscous layer which induces a shock wave. Downstream of the kinetic flow region where continuum analysis with slip boundary conditions is believed to be valid, it is the merging of the shock wave with the boundary layer that makes such a continuum analysis rather complicated and controversial. If the tip of the plate is somewhat blunt so that the Knudsen number based on the leading edge thickness is small, it is believed that the flow close to the tip of the plate would resemble a flow around a blunt body and a detached shock wave forms in front of the plate. In this case the viscous effects close to the leading edge become relatively small and inviscid flow over the blunt leading edge predominates. This case is drastically different from the case of a sharp leading edge which is the main problem of the present investigation. 2. REVIEW OF LITERATURE ' Referring to Figure 1 the flow field can generally be divided into different flow regimes. Downstream, where Reynolds number based on the dis- tance from the leading edge is sufficiently large, there is the conventional continuum flow regime. In this regime the shock wave and the boundary layer are distinctly separated by an inviscid flow region. As one moves upstream, the flow regime passes from the so-called weak interaction regime where the induced changes of the flow properties within the inviscid layer is merely a perturbation on the free stream flow, to the so-called strong interaction regime where the development of viscous layer is strongly coupled to that of the inviscid flow [12], Upstream of the strong interaction regime, the shock wave merges with the boundary layer and the shock wave is no longer a Rankine-Hugoniot shock. Here, one has the so-called merged layer regime. Because of the differences between this regime and the strong interaction regime were often attributed to the effect of slip, the merged layer regime is sometimes called the slip flow regime. The merging of the shock wave and the boundary layer has been established experimentally [13, 1^, 15, l6] and theoretically [12], Sufficiently close to the leading edge where the Knudsen number based on the distance from the tip is large, the kinetic flow regime exists. Intermediate between the kinetic flow regime and the merged layer regime, a narrow transitional regime exists in which the flow is believed to have mixed kinetic and continuum characteristics. A survey of the literature of the flat plate leading edge problem can be found in a paper by Pan and Probstein [IT]. Most recently an excel- lent survey was presented by Potter [18], 8 2.1 Continuum Theories and Experimental Observations Numerous theoretical and experimental investigations have "been carried out within the last decade. Most experimental and all theoretical continuum investigations were primarily intended for flow regimes downstream of the kinetic flow regime, with the hope that an extrapolation of these results to the leading edge can at least give a qualitative description of the flow field in the immediate vicinity of the tip. Early work by Li and Nagamatsu [19]? Lees [20], Lees and Probstein [21] and Cheng et al. [22] have established the weak and strong interaction theories. Experimental work by Bertram [23] and Kendall [2U] demonstrated the validity of these theories. Upstream of the strong interaction regime where the boundary layer and the shock wave are merged, the strong-interaction theory no longer applies. A number of models have been proposed for the merged layer regime. Oguchi [25] and a number of subsequent authors proposed the wedge flow model. The shock wave was considered to be straight and negligibly thin and the states upstream and downstream of the shock wave were assumed to satisfy the Rankine-Hugoniot relations. The assumption of a straight Rankine-Hugoniot shock with no slip and jump conditions at the wall resulted in a pressure plateau which has not been observed experimentally. Experimental observa- tions demonstrated the presence of a peak in the surface pressure profile, rather than a plateau. Moreover, the assumption of a Rankine-Hugoniot shock implies the absence of shear stress immediately behind the shock wave, which is not correct if the boundary layer and the shock wave are fully merged. Oguichi's work was criticized by Street [26] and Aroesty [27] who attempted to take the slip effects into consideration. Later Oguichi [28] modified his original analysis by considering the effect of wall slip and temperature 9 jump. The resulting shock wave showed a reverse in curvature, which, however, has not been observed experimentally. Another model was proposed by Nagamatsu et al. [13, 29] where it was assumed that the formation of the shock wave is delayed, i.e. the shock wave starts somewhere downstream of the leading edge. The resulting shock wave in this model, again showed a reverse in curvature. Laurmann [30] proposed a slightly different model with a boundary layer type behavior at the leading edge which produces coa- lescing compression waves. The delay in the shock wave formation and the reverse in curvature in the last two models have not so far been observed experimentally. Pan and Probstein [17] > in a recent study, considered the effects of wall slip, shock wave curvature and introduced in the analysis a slightly modified Rankine-Hugonoit shock wave. They also introduced the assumption of local similarity which was later criticized by Garvine [31] • The results of Pan and Probstein were in poor agreement with the available experimental data. One tends to believe that the deficiency in the models cited so far is that none of them recognized the fact that the shock wave in the merged layer region is drastically different from a Rankine-Hugoniot shock wave. McCroskey et al. [32] who have been active on the experimental side suggested a major modification of the model. It consists of an outer region which is drastically different from a Rankine-Hugoniot shock wave and an inner region which is to be coupled and analyzed with the outer region. Laurmann [33] and earlier Chow [3*0 worked out two basically different theories* Laurmann' s theory, however, was based on controversial assump- tions, and its results are in poor agreement with the experimental data. Moreover, the two inner and outer expansions used by Laurmann did not join with each other and the inner and outer profiles were joined together in an 10 empirical fashion. Chow's theory does not encounter this difficulty and yields continuous profiles of the thermodynamic variables along y. His results are in good agreement with the experimental observations downstream of few mean free paths from the leading edge. The most recent theory was worked out by Butler [35] who used the particle in cell and fluid in cell methods for solving nonlinear partial differential equations. His work is quite unique and his results are in fair agreement with the experimental data downstream of the kinetic flow regime. On the experimental side, work has been carried out by Nagamatsu et al. [13, 29], Vidal and Wittlif [36], Chuan and Waiter [37], Wallace and Bruke [l^], Schaaf et_ al. [38], and most recently by McCroskey et al . [32, 39], Becker and Boylan [l6, 1+0 ], Moulic et al. [15], Vas and Allegre [4l] and Harbour and Lewis [k-2 J. Experimental data from different sources are not in fair agreement with each other which may indicate that there is no existing well established experimental technique to determine the rarefied flow properties. This should be expected, because taking reliable measurements in a rarefied flow regime is a difficult task due to the inter- action of the probes with the flow field. Also, surface pressure measure- ments depend on the configuration of the pressure tap [15 ] and the orifice correction is empirical. As for the density measurement by electron beam probes one still has to rely upon other experimental data to calibrate the probe. It can therefore be concluded that any attempt to compare the theo- retical results with the available experimental data should be made with caution. The present status of the theoretical studies and experimental observations for the fully merged layer region is still a confused one. 11 2.2 Kinetic Theory Investigations On the kinetic theory front, very few investigations are available. This might be expected because of the difficulty encountered in solving the two-dimensional Boltzmann equation and in describing gas -surface interaction in a reliable fashion. In the present study of the hypersonic leading edge problem, it is hoped that an approximate analysis may be carried out which may give at least qualitatively valid results. The kinetic flow region has been examined by Charwat [h3] who based his analysis on considering only the contribution of the first colli- sion for any given reflected molecule. His theory contained two empirical parameters which had to be evaluated by matching with the experimental data. Kogan [hk] carried out a Monte-Carlo simulation of the first collision technique. Bird [U5] using- a Monte-Carlo simulation studied the kinetic flow region. The theories of Charwat, Kogan and Bird are qualitatively in disagreement with each other. They are also in disagreement with the exper- imental observations if these theories are extended to the merged layer region. The extension of any kinetic theory analysis downstream of the leading edge may not be valid unless the analysis is designed to solve the Boltzmann equation in an exact fashion; a task that seems to be impossible at the present. One should therefore bear in mind that extending the pres- ent investigation beyond the region in the immediate vicinity of the tip can at best be of a speculative nature. Most recently Huang and Hartley [k6] solved the BGK model equation using an approximate numerical technique for the supersonic leading edge problem. Their results of the density profiles within the first ten mean free paths from the leading edge are unexpected and can be highly controversial. 12 It should be mentioned that all continuum and kinetic theory investigations cited so far rely upon the sweeping simplication that gas- surface interaction can be adequately described by the so-called accommoda- tion coefficients. Furthermore, these accommodation coefficients are assumed to be constant along the wall. The gas -surface interaction is actually a complex phenomenon. In addition recent experimental investigations on the gas-surface interaction do not support the assumption of constant accommoda- tion coefficients [U7] . It has always been believed that the flow at the nose of the plate is a true free-molecule flow. None of the kinetic theory investigations supports this belief and none of the experimental investigations carried out so far have been able to prove the existence of this free -molecule flow at the tip. 13 3- THEORETICAL ANALYSIS 3-1 Description of the Present Model The present moment method is designed to deal with strictly nonlinear aspects of rarefied gas flows. The nonlinearities become more and more significant as the Mach number increases. The conventional methods of linearizing the collision integral in the Boltzmann equation (or the BGK model equation) become inappropriate at hypersonic flow speeds. The classical Chapman- Enskog iterative scheme is equally invalid for hypersonic rarefied flow fields where K is in the neighborhood of unity or greater. The only different method available in literature that can hopefully be applied to all speeds is the one suggested by Lees [3]« Lees method is basically a moment method and differs from the present one in the manner of choosing the form of the distribution function. In addition Lees assumed constant accommodation coefficients along the wall. Lees method proved so far to be hard to handle and was only applied to one -dimensional problems amenable to linearization, i.e. small temperature differences and slow speeds [3]« Ziering [48] applied Lees method to solve the BGK model equa- tion for the subsonic leading edge problem. His iterative scheme was not proved to be convergent. Furthermore, his results exhibited unanalytic behavior as x->- oo or y->oo. Ziering, in his study, did suggest that it would be worthwhile to explore the leading edge problem on the basis of a more direct and simpler extension of the original bimodal model. He recom- mended choosing the form of the distribution function on the basis of an inspired guess in the spirit of the Mott-Smith bimodal "Anstaz". The key to the bimodal model is the mixing of the two extreme distribution functions (boundary conditions) governing processes at the two boundaries. Intuitively, Ik the two distribution functions in the leading edge formulation are the wall distribution and the upstream distribution. In the present investigation we will attempt to construct a two- dimensional, highly nonlinear bimodal model. The bimodal "Anstaz" will be chosen in the simplest form possible. In other words we will be sacrific- ing accuracy in favor of simple mathematical analysis. Because of the discontinuous or two-sided nature of the distribu- tion function at the wall, we choose the distribution function to be the sum of two half-range Maxwellian distributions with three unknowns. The unknowns are to be determined by solving an equal number of moments of the Boltzmann equation. This procedure is certainly approximate because an exact solution should satisfy an infinite number of moments. It is believed, however, that the most important moments are the low order moments which lead to the conservation principles of mass, momentum, and energy and to the expressions relating to the shear stress and heat transfer. Lees [3] suggested that any extension of the Mott-Smith model to problems other than the normal shock wave should satisfy the requirements: (a) It must be capable of providing a smooth transition from rarefied flows to the Navier-Stokes regime; (b) It must have two-sided character; (c) It should lead to the simplest possible set of partial differential equations and boundary conditions consistent with requirements (a) and (b) . We believe that with the proper choice of the boundary conditions and with no excessive approximations in the course of mathematical analysis to solve for the three unknowns, the present model ought to satisfy the above requirements. 15 The distribution function which we choose to use is: f = f + [n+(x,y), u+(x,y)] + f-[n"(x,y)] where f + = for c < and y f " = for c > 0. y In this model we arbitrarily choose to have v + = v = 0., u" = u and 00 T = T = T . Restricting the present study to the case where T = T W oo or j W oo merely simplifies the analysis. Kinetic flow problems with T = T have W oo been the subject of many investigations, some of which are reported in Refs. [18, 32, 46]. It is true that the present choice of the distribution function has no rigorous justification. However, it should be remembered that the Mott-Smith model which was very successful for the problem of the shock wave structure has no rigorous justification as well. Our distribution function has the explicit form + { m \ I -2kT- (G " 1U } d_ 00 e n "W -(c - iu ) 2kT e °° (3-D 16 3-2 Governing Equations The steady state two-dimensional Boltzmann equation can be written in the form hfi c x f a? + hfi V ^'fi^r d? < 3 - 2 > where cp is a. function of the molecular velocity c. The required three moment equations can be obtained from Eq. (3-2) by choosing three different forms for cp. We choose to use cp = m (continuity equation); cp = m c (normal momentum equation) and cp = m c c • We also restrict ourselves to Maxwellian Y x y molecules. Substitution of Eq. (3-1) in Eq. (3*2) and setting cp = m, m c , XJ m c c successively one obtains x y D ^(n-u^ + n + u+) + jL(n+ - n") = (3-3) D^(n+u+- n-uj + ||- ( n + + n-) = ( 3 A) *-.*1.-#-d--].sfe<.- CKJ - S(u ■■ u + )n + n~ (3-5) where p™ o I 2\l/a A 2 . / sin ( X )pdp, p . b^j , IT X, b and g are the deflection angle, the impact parameter and the relative speed, for a molecular encounter, respectively, a is the proportionality constant in the intermolecular potential — where (X = h for Maxwellian a r molecules. It is traditionally assumed that the minimum number of moments to satisfy should be equal to the number of conservation equations. Hence, it may be argued that the present model is deficient because it does not satisfy the energy equation and the horizontal momentum equation. This deficiency, however, is expected to be negligible because: (a) When T = T } the temperature field and the velocity field are usually weakly coupled. Splitting between the velocity and temperature fields is well known in the Navier-Stokes equations and was reported to occur for the Grad equations [3]- One can therefore expect that the energy equation in the present model to be of minor significance. (b) Momentum changes in the normal direction are expected to be faster than those along the horizontal axis. The normal moment equation is therefore more important than the hori- zontal momentum equation. Hence, replacing the horizontal momentum equation by the moment cp = m c c may not be serious. Besides this latter moment x y has the advantage of having a nonvanishing collision integral. The presence of a nonvanishing collision integral in the model introduces the effect of the intermolecular interaction in a direct fashion. A model that would satisfy all the conservation equations should include a minimum of five unknowns in the assumed bimodal distribution. This, however, complicates the analysis extremely. The solution for n + , n", u + from Eqs. (3-3)> (3«^) and (3-5) depends on the mathematical skill of the researcher and the approximations one introduces in the overall analysis. 18 kT • kT Using the hypersonic approximations — — « u and — — « u + m oo m in Eq. (3.5) one obtains D |-(n + u +2 - n-u 2 ) + § i-(n"u + n+u+) = S(u - u+)n+n" (3- 5a) O-X- oo £- Oy oo oo It can be shown that with some manipulation of Eqs. (3«3)> (3«^) and (3- 5a) one gets the following equations: t^ + dn + , _ + du+ /jl±2n dn + /jt-2\ dn~ , C \ D u ^r + D n aT ( t- } sr + (V" } aT = ° ' (3 ' 6) ■n/ + \ + dn it / + v dn + , _ + /_ + >. du + D(u - u ) u T ^ — + — ixr - u ) «r — + D n^(2u^ - u ) «r — v oo ' dx 2 v oo' dy oo y dx + I n+ ST ■ s(u ~ - u+)n+n " ' (3 ' 7) _ bn ,n + 2x dn~ /2 - jt\ dn + /„ q n D u oo 5T - ( ~T-) 5T + ( ~T-) dT = o (3-8) A solution based on decoupling can be attempted by assuming ^ — ~ ^ — • This approximation holds good in the region close to the wall which is the most important region in the flow field. Setting ^ — ~ ^ — ^ n Eqs. (3*6) and (3*8) one gets Du + |^ + Dn^ + f^= (3-9) dx dx 2 dy 19 D ^ ^-fy 1 = ° (3.io) Multiplying Eq. (3.9) by (u + - u ) and subtracting the result from Eq. (3.7)> one obtains Eqs. (3.9), (3.10) and (3.11) could have also been obtained by replacing "sr:(n + - n~) in Eq. (3.3) with - ^~(n + - n-) and manipulating the resulting equation with Eqs. (3»*0 and (3.5). We are now left with the task of solving for n + , n~ and u 4 " using Eqs. (3«9)> (3-10) and (3.11). Since Eq. (3=9) is not a completely decoupled equation for n + , one has to solve for u + first using Eq. (3. 11) with an appropriate boundary condition. 3.3 Boundary Conditions From a mathematical point of view the boundary conditions for u + , n + and n~ must be derived from the requirement that lim f 4 " = lim f" = f . This requirement gives the conditions u + = u and n~ = n + = n as 00 00 00 y -* 00. However, these boundary conditions fail to include the very basic conditions at the wall, namely the condition n + (x,0) = n~(x,0) and a con- dition describing the gas-surface interaction phenomenon. This phenomenon must be included in the analysis because it is the gas-surface interaction that produces changes in the properties of the flow from one point to another within the flow field. It should be noted that for the present problem u(x,0) is independent of n 4 " and n~ (see Eq. 3.17) > i.e. any change in u + (x,0) along x must come through the gas-surface interaction phenomenon. 20 Any realistic set of boundary conditions on u + , n + and n" must therefore include two conditions, namely n+(x, 0) = n" (x, 0) (which is an exact boundary condition) and a condition on u + at the wall, derived through the gas- surface interaction phenomenon. If one uses the boundary condition at y = oo for u + , the result can either be trivial or not unique. The fact that the boundary condition for the velocities at the wall must be employed has been recognized in the boundary layer theory and in kine- tic theory studies of the Couette flow problem. In the present analysis the wall boundary condition for u + is derived from the Maxwell slip boundary condition which has been exclu- sively used in the literature with occasional minor modifications. The classical Maxwell boundary condition, however, must first be modified to fit the present model as follows: The classical Maxwell slip boundary condition after assuming X ~ X is w — oo l r^\ 2 - o . du ' a so <3y (3-12) y=o where o can roughly be approximated by the horizontal momentum accommoda- tion coefficient given by T. - T a - -±—Z (3-13) T i where r. represents the horizontal momentum of the incident molecules while t represents that of the reflected molecules. For the present model one has T. = 77 m n~(x, 0) u 1 2 v / 00 T r = | m n + (x, 0) u + (x,0) 21 Substituting these relations in Eq. (3*13) and noting that n + (x, o) = n~(x, 0) one obtains u - u+(x,o) 00 a = — u (3.1*0 By definition one has and nu = — u + — u + (3-15) d oo d n = — + — (3-16) From Eqs. (3*15) and (3«l6) one obtains n" u + n + u + u(x ' y) = n- ro + n+ (3 * 17) Now, within one mean free path above the wall n+(x,y) is approximately c)n~ ^ dn + ... approximation is consistent with the stipulation that intermolecular col- equal to n - (x,y), i.e.; ^ — ~ ^ — i- n the region close to the wall. This lisions are negligible within a distance of one mean free path above the wall. With this approximation for the region close to the wall; Eq. (3' IT) becomes u + + u u(x,y) s—2 (3.I8) After differentiating Eq. (3«l8) one obtains + Bu ^ 1 du' §y - 2 3y~ 22 or in particular du(x,y) 3y ^ 1 du+(x,y) - 2 3y~ y=o (3-19) y=o At the wall, it should be mentioned that Eq. (3»l8), which is a rough approximation of Eq. (3- 17), is employed for the purpose of obtaining a wall boundary condition which is simple enough to make an analytical solution possible. Substituting Eqs. (3-1^) and (3«19) into Eq. (3.12) one obtains the following boundary condition: u + (x,o) - u_ = - \_ r— du + (x,y) oo d~y~ (3-20) It may be worthwhile to mention that the approximation intro- duced in decoupling and deriving the boundary condition (3*20), namely ^ — ~ ^ — } is not worse than the traditional method of assum- dy - dy ming the reflection phenomenon as either completely specular or completely diffuse. It is well known that the reflection is neither specular nor diffuse but can have mixed specular and diffuse properties. The propor- tion of specularly reflected molecules, intuitively, should depend on the Mach number, i.e. on the energy of the incident molecules. At hyper- sonic speeds, thermal motion is negligible compared with that of the mass motion, accordingly it is believed that the reflection process is dominated by specular reflection thereby yielding a large slip velocity on the surface. As the Mach number decreases along the wall, the specu- lar reflection is expected to become less and less predominant. This is, 23 indeed, a different way of stating that the accommodation coefficient varies along the wall. 3.^ Solution for the Distribution Function The distribution function f = f + + f " become fully known if one solves for u + , n + and n~. Inspection of Eqs. (3.9) > (3.10) and (3.11), and noting that only one decoupled boundary condition (Eq. (3.20)) is available, would reveal that two approximate approaches to solve for n~, n + , and u + can be explored. The first approach is to use an empirical boundary condition to solve for n~ from Eq. (3.10). After knowing n - , one can solve for u + from Eq. (3.1l) with the boundary condition given by Eq. (3.20). Eq. (3*9) and the exact boundary condition n~(x,0) = n + (x,0) would give the solution for n + . The second approach is to approx- imate n" in the RHS of Eq. (3.1l) by a constant. One can then solve for u + . After u + has been determined, a solution 1 and n~ can be sought by inspection in such a way that the condition n - (x,0) = n + (x,0) and the conditions at y = <» are satisfied. Approximating n - in the term S(u + - u )n~ by a constant, say n , is not a bad approximation because 00 00 this term is related to the molecular collision which is not dominating in a rarefied flow field. It is well known that the collision integral is usually assumed to be equal to zero in a free-molecule flow, and to be a small quantity in a near-free-molecule flow. In the present analysis we shall employ the first approach, namely using an empirical boundary condition to solve for n~. The general solution of n~ from Eq. (3. 10) is given by n " = Vl X + Du a> y) (3.21) 2^ where Y, is an arbitrary function. Using the boundary condition n"(- x , y) = n , where x^ is greater than zero, Eq. (3«2l) gives the (J 00 (J solution n (3-22) Substituting this result in Eq. (3'll) one obtains the general solution — = 1 - e u 00 ■R 2 y * 2 ( R i x - V + r) (3-23) Sn 2Sn where Y~ is an arbitrary function, R, = - — and R_ = . 2 J 1 Du 2 jt 00 For Maxwellian molecules R and R can be evaluated in terms of the free stream mean free path \ as follows: 00 From the kinetic theory (Ref. 2, h^) , one has i± htm mn A 2kT If we choose the viscosity ju to be given by the first approximation of Chapman-Enskog theory, the above relation gives kT a \| a i6A 2 r/7 (1) n ° Noting that Sn | *A Q D J^ n 2 2 Nm o. and Du = \J nm 2kT u 4^- M , one obtains \| C. 00 R, .78/2 1 M slny k 25 (3-24) R, 2X (3.25) Eq. (3-23) gives £u+ U y=o -f *<«> + \.M3r 1_ $)u+ u cVy~ y=o ^r (3-26) where 1 u For the sake of obtaining an explicit solution of ¥ > we introduce the approximation 1 2\ » 1 ^u+ u 37 in the coefficient of the derivative y=o d¥ dg This approximation is reasonable for the region close to the leading edge where — ^ — is expected to be small. Using this approximation and 00 combining Eqs. (3' 20), (3-23) and (3.26), one obtains d¥ ¥ = The solution of this ordinary differential equation is given by ¥ = 5e- (3.27) 26 where & is the constant of integration. From Eqs. (3-23) and (3«27)> one can immediately obtain the solution u+ , u R.X-2R y U = l-SeVe 1 2 (3-28) u 00 U+ The fact that the solution of — in the above equation is implicit, 00 makes the subsequent analysis to solve for n + very difficult. We, there- fore, choose to approximate Eq. (3*28) by an explicit solution of — . I u U + oo One way of doing this is by expanding u into a power series and oo e truncating the series at its third term, i.e. for + u u + 2TU ) + JAu J oo V oo / \ oo / we introduce the approximation u u + e ~ 1 + — — u 00 and Eq. (3-28) yields the explicit approximate solution R n x-2R Q y \v 1 - 5 e u R n x-2R Q y 1 + 8 e (3-29) 27 By definition the slip velocity u (x) at the surface is given by s u u + u + (x, 0) S oo u 2u 00 00 Employing Eq. (3*29) for the foregoing expression, one obtains s 1 u 2 R x 1 + X - S e p R x 1 + 5 e . (3-30) It should be mentioned that the parameter 6, which was introduced as a constant of integration, can only be determined from the detailed study of gas- surface interaction at the tip of the plate. At the present, there are no established tneoretical results or reliable experimental data for the flow properties at this point. It is, therefore, suggested that 6 value appearing in Eq. (3*30) should be determined by matching the flow properties (e.g. the surface pressure) somewhere downstream of the tip, either with the established theoretical results or experimental data. It should also be noted that for full slip at the tip of the plate, 5 must have a vanishing value. However, experimental studies cited previ- ously do not indicate the existence of full slip at the tip. One, there- fore, expects & to be greater than zero. The presence of the parameter 6 gives some flexibility to the present theory and other methods of obtain- ing 6 should not be ruled out. 28 It is interesting to note that the slip velocity given by Eq. (3«30) goes to zero as x goes to infinity. This should be expected from any theory that is capable of spanning the spectrum of the flow regimes, starting from the kinetic flow regime to the classical continuum flow regime. However, because of the nature of the many approximations used in the present analysis, we may not be justified to extend the pres- ent theory far downstream of the leading edge. The extension of the pres- ent theory to regimes downstream of the kinetic flow regime becomes more .justifiable as the free stream absolute density becomes lower and the free stream hypersonic Mach number becomes larger. For very low absolute density and for Mach numbers greater than 25, there are no wind tunnel experimental results in the literature; thus, there is no way of checking the validity of extending this theory to large distances downstream of the leading edge. To solve for n + , we substitute Eq. (3.29) into (3.9) and use the boundary condition n + (x,0) = n~(x,0). Since the expression for u + given by Eq. (3.29) is quite complicated, we prefer to simplify this rela- tionship through the expansion u+ i - 5 e R i x - 2 v / v- 2 v)( n , R i x " 2 V ,2 2R i*- u v) ~ = Rl x-2R p y = V 1 - 5 e / V- " 5 e + 6 e * ' • / 1 + 5 e Since & is expected to be much smaller than unity for hypersonic flow, one 2 3 can neglect terms containing 5,5, . . . , hence , R x-2R y — ~ 1 - 25 e (3.29a) u — 00 29 It should be remembered that the above equation provides good approxima- tion only within the region close to the leading edge. Introducing Eq. (3.29a) into Eq. (3.9) > one obtains the general solution -R x+R y / -R,x+R y -R y \ n + = e 1 2 ¥ 3 \e X + 28 e 2 j (3. 3D where ^ is an arbitrary function. With the remaining boundary condition, i.e. n + (x,0) = n _ (x J ,0) = n Eq. (3.31) yields -R x / -R x n = e Y le + 2oJ -R x -R x n^ Setting e + 25 = r\, one gets e = r\ - 25 and ¥~(t]) = pp- This immediately gives -R x+R y -R y Y U + 26 e 3\ ' -R x+R y -R y e +■ 28 e - 28 and n -R lX +R 2 y n -R x+R y -R y R x-2R y R x-R y e 1 d + 26 e - 26 1 t 25 e X - 26 e X (3.32) 30 Eqs. (3.22) and (3.32) imply that the wall density p(x,0) is a constant equal to the free stream density. Such a result has been obtained by Ziering [1+8]. Moreover, the results of Huang and Hartley [U6] for T = T w demonstrated that the wall density is approximately constant. 3.5 Evaluation of the Macroscopic Flow Properties From the definition of the macroscopic properties p, u, p, T, v, etc., in terms of the distribution function, one can easily obtain the following relations: -) n / 00/ (3.33) n 4 " u + n u 00 00 n n (3.3k) -) n / 00 / 7M , 00 / (3.35) v 2_ M n~ n 00 n- (3.36) _ 00 T -I ■yM 2 n n~ n oo oo / U n n~ ,n n V oo oo 2 U and the friction coefficient C based on local properties is T w 1 2 f ^ 2P (x?0) a (x,0) oo U u (x,0) 2 P u 00 00 or 31 2_ 3n _ III i n 00 00 n + nT n 00 00 (3.37) W T (x,0) 27 M \j Jt "oo 00 1 n^(x,0) / u^(x.O) 1 2 n \ u 00 v 00 (3.38) w f« 1 2 ^P u ^ oo oo 2_ 1_ vjrcy M n^(x,0) / x _ u^(x,0)V n I u / 00 \ 00 ' (3.39) foo p (x,0) u (x,0) L P co (3.^0) 32 The surface pressure can be obtained from equation (3.35). 00 \ 00 / In the above relations n , n~, and u + are already known from Eqs. (3.21), (3.29) and (3-32). 33 h. RESULTS AND DISCUSSION In the preceding chapter the macroscopic flow properties, based on the half-range distribution function, were determined. The results contain the parameter 5 which is still arbitrary or undetermined. The presence of this parameter should be expected in any kinetic treatment of the leading edge problem. All previous theoretical investigations that attempted to study the kinetic flow region starting from the tip of the plate, recognized the need to know precisely the conditions at the tip apriori. In this sense the parameter 5 in the present theory can only be determined if one knows the conditions at the leading edge. Owing to the lack of information at the very tip, however, we choose to vary 8 and match the res alts of same flow property (e.g. the surface pressure) with the experimental data, within certain range of x. Through this matching, one can obtain the value of 5 which gives the best agree- ment with the experimental results. The value of the chosen 5 is shown on plots in Figs. 2 to 8. It should be mentioned that there is no reason for 6 to have the same value for different free stream Mach numbers. Indeed, one should expect 5 to vary with M because it is clear that 5 is associated with the conditions at the tip and these conditions would certainly depend on M . However, from the experimental data of the sur- face pressure available so far in the literature, it was found that the value of 5 obtained through this matching process is not sensitive to changes of the free stream Mach number and assumes a value in the neigh- borhood of 0.2U. 3^ In the presentation of the results we employ two frequently C /R and the quoted parameters; the rarefaction parameter v = M O I hypersonic interaction parameter y = M C /R . The former is a \o oo \| 00/ e^^ measure of the significant Knudsen number in the low speed slip-flow regime. It is therefore the slow-speed rarefaction parameter and using it as a rarefaction parameter for hypersonic flow may seem to be question- able. Talbot [50], however, provided a justification, on the "basis of strong interaction estimates, for employing this parameter as a rarefaction parameter in the hypersonic regime. The rarefaction parameter correlation is useful for locating the strong rarefaction effects which cause depar- tures from the trends of the viscous interaction region (downstream of the kinetic and fully merged layer regions). Talbot [50 ] showed, to first order, that v is independent of the wall temperature: therefore, it can 00 be used as a rarefaction parameter for either insulated or cooled walls. Later Deskins [51] presented more data substantiating the independence of this correlation from wall temperature. The hypersonic interaction param- eter, y , tends to reduce the effect of the Mach number differences on the parameter p /(p % ) within the rarefied flow regime (upstream of the strong interaction regime). In addition, this latter variable has a constant value for the strong interaction theory for a given wall stagnation tem- perature ratio. It should be mentioned that in comparing different results, it is logical to use sets of results that have the same free stream and wall conditions. However, it has been a standard practice in literature to compare results from different sources with each other regardless of the fact that stream and wall conditions are significantly different. 35 The present result of the wall density may seem to be erroneous and may evoke criticism of the whole theory. Although we cannot defend the validity of such a result rigorously, we do believe that if the wall density is not a constant, it will vary slowly along the wall. Furthermore, it can be seen that the nondimensional surface pressure formula is domi- nated by the term containing M and u + , i.e. it is weakly dependent on the wall density. Accordingly, any error in the value of n_(x,0) will not affect the value of the nondimensional surface pressure. Fig. 2 compares the present results with the experimental data of Vas and Allegre [Ul] for the surface pressure. The agreement is excel- lent close to the leading edge, i.e. upstream of the pressure peak. The divergence between the two, downstream of the pressure peak, is not sur- prising because the present theory has no justification to be extended beyond few mean free paths downstream of the leading edge. Fig. 3 is a plot of the parameter p /(p y ) versus the rarefac- ~\/J' 00 00 tion parameter. The experimental data of Vas and Allegre [Ul] for cold wall are shown on the plot. The agreement is good within the range of v from 0.7 to 2.0. Fig. h is a comparison of the present theory with the experimental data from various sources. One can see from this figure that disagreement between the experimental data from different sources still prevails. The theory of Chow [3^] is also shown on the plot. Fig. 5 compares all the existing theories including the present theory. The agreement of the present results with the results of Bird [h^] is good. The slip velocity in the present theory is a monotonously decreas- ing function, i.e. the slip velocity decreases as x increases. The rate of decrease becomes slower and slower as Mach number increases. The fact 36 that the present slip velocity goes to zero as x -» «> does not imply that the theory can be extended downstream to the classical continuum flow region. Because of the nature of the approximations employed through the course of analysis, one has no justification for extending the application of the results beyond the region in the immediate vicinity of the leading edge. In this region other theories give either full slip (specular reflec- tion), or 50$> slip (diffuse reflection), at the tip of the plate. Neither of the two values is realistic. As for the experimental observations, there are no slip velocity measurements very close to the leading edge. Figo 6 compares the present results of the slip velocity to the experimen- tal data of Chuan and Waiter [37] . Fig. 7 shows another comparison between the theory of Chow [3^]> Oguchi [28], the present theory and experimental data reported in Ref. [l6]. The results of the skin friction are shown in Fig. 8. The exper- imental data of Chuan and Waiter [37] are shown on the same plot. Their results were evaluated by measuring the slope of the experimentally deter- mined velocity profiles. The experimental data as presented in Fig. 8 allow for scatter due to inaccuracy in slop determination. 37 5. CONCLUSIONS AND RECOMMENDATIONS There is no doubt that a kinetic study based on the conventional assumption of constant accommodation coefficients, is controversial and at best, is empirical in nature. Continuum analysis which involve the piecewise fitting together of free stream, shock wave, inviscid, and boundary layer regions is cumbersome and highly limited. The present analysis does not assume constant accommodation coefficients and is rela- tively simple. Due to the simplifying assumptions made throughout the course of analysis, the present theory may only provide a crude solution for the leading edge problem. The results give the expected trend of changes in the physical properties and agree fairly with the available experimental data in the kinetic flow regime. The agreement with the Monte Carlo solu- tion of Bird [1+5 ] is good (see Fig. 5). The results of this simple, crude analysis are so encouraging that one tends to believe that the half- range bimodal model has considerable promise a Future extension and refinement of the present analysis is, therefore, highly recommended. Several steps to improve and extend this analysis suggest themselves: (a) Reducing the number of simplifying assumptions and order of magnitude approximations and repeating the present analysis. (b) Iterating the solution of n~ by substituting the present results of n + and u + in an uncoupled equation and solving for n~. 38 (c) Using more than three unknowns in the assumed form of the distribution function. For example, one can assume T + = T + (x,y) to be unknown. For a more sophisticated and possibly more accurate extension, the molecular velocity components v + (x,y) and v _ (x,y) can be retained in the dis- tribution function. This will increase the num- ber of governing equations; hence, more moments of the Boltzmann equation can be satisfied. (d) Searching for a stable and convergent differencing scheme to solve the set of nonlinear partial dif- ferential equations numerically. (e) Extending the bimodal model to other simple geometries, e.g. the problem of the leading edge of cones or wedges. All these alternatives may make the mathematical analysis formidable and difficult. However, one should keep in mind that, the alternative con- tinuum approach of piecewise fitting together and matching various solu- tions, is an extremely difficult task and certainly limited in scope. 39 TABULATION OF SYMBOLS A (a) Pure number defined on page l6. a Proportionality constant in the inverse power intermolecular potential a/r . b Impact parameter (see Ref. k9, page 357)* c Molecular velocity. dcT Element of volume, dc dc dc , in the velocity space. x y z J * C Chapman-Rubesin coefficient u T /u T , (0.8 to l). oo W oo ' oo w C„ Friction coefficient. C Pressure coefficient 2(p /p - l)/yM . p \r oo " ' oo D Pure number Jjtm/2kT . f Full-range distribution function. f Maxwellian full-range distribution function. 00 d f Collision integral of the Boltzmann equation (rate of e o^r change in f due to molecular encounters). K Knudsen number \/x. n ' k Boltzmann constant. m Molecular mass. n Number of molecules per unit volume (number density) . p Pressure. R, and R Pure numbers defined on page 25 • R e Reynolds number (pux)/u. S Pure number defined on page l6. T Temperature. t Time . 40 u Horizontal mass velocity, v Vertical mass velocity. cA °°\j »' e oo,X v Rarefaction parameter M 00 x Streamwise coordinate. y Coordinate normal to x. 7 Ratio of specific heats (l.k for air). 5 Constant of integration or experimental parameter in the present theory. X Mean free path. \i Viscosity, p Mass density. X Deflection angle. 3 X Strong interaction parameter M C /R_ oo °° \j °° e oo,x o Horizontal momentum accommodation coefficient. Superscripts + Signifies molecule moving away from the -wall. Signifies molecule approaching the wall. Subscripts s Surface value. w Wall or surface value. oo Free stream value. \ s \ \ \ \ \ s N \ \ \ :< 8 a. O _J (f) o z o (/) (T LU Q. O I- o < a: LU I- £§ luo: UJ< 4 (f) ■ < CO LU s CD LU cc % < QC CO < CJ CO cc CD k2 if) (0 CD «■' I CVJ C cr CVJ o t o ? 8 cu CO en o > O □ < J_l O o CVJ a II* CO CM E CQ < 8 l> 9 CVJ £9 cc a CO CO LU cc Ou cc CO oi CO tz O O O O Q. 8 o CJ IT) h3 CVJ c ID CM < c I o > 0> Sr or x < D D CM II" to 8 ix 8 .8 Q_ CD o O CD CO m CO LU CJ cc CO CO CD A MOULIC-HOT WALL, Mqo = 6, T w /T =0.95 B VAS et al.; Moo = 24, T w /T *0.15 C VIDAL et al. D DESKINS; Moo* 20, T w /T f »0.10 E VAS 8 ALLEGRE (~ "»" " * > T '"«' °' 21 I — Moo* 26, T w /T = 0.21 A PRESENT THEORY CHOW(THEORY, REF 34) hk FIGURE 4 COMPARISON OF SURFACE PRESSURE DISTRIBUTION WITH VARIOUS EXPERIMENTAL DATA. h5 URVE SOURCE Moo T W /To NOTES A B C D E F G H CHENG, et al (REF. 22) PAN AND PROBSTEIN (REF. 17) OGUCHI (REF. 28) JAIN AND LI CHOW (REF. 34) CHARWAT (REF. 43) BIRD (REF. 45) HAYES AND PROBSTEIN (REF. 12) 10 10.15 »1.0 10.15 10 10.15 ~6.0 10.15 0.15 0.10 0.10 0.10 0.10 0.10 0.12 0.10 STRONG INTERACTION TRANSITION ANALYSIS SLIP SOLUTION OGUCHI MODEL WITHOUT SLIP INTEGRAL ANALYSIS FIRST COLLISION - SEMI-EMPIRICAL FIRST COLLISION FREE MOLECULAR LIMIT DIFFUSE REFLECTION a. o 10"* .06 .06 .05 .04 .03 z UJ C .02 u. UJ O O Ul or S io 2 x 53 VITA Rateb Jaber El-Ass ar was born in Palestine, on June 5 > 1937* He was awarded a United Nations (UNESCO) Scholarship to study at Cairo University, where he received the Bachelor of Mechanical Engineering degree in i960. He held the position of Assistant Lecturer at the University of Baghdad for a year and a half. In 1963 he was awarded a Fulbright Travel Grant and a Fellowship at Bucknell University where he received the Master of Science degree in Mechanical Engineering in 196k. He entered the University of Illinois in September 196^0 He held a Research Assistantship in the Mechanical Engineering Department from September 196^+ to August 1966. From June 1966 to June 1968, he was engaged in research activities with the ILLIAC IV Group, as a Research Assistant in the Computer Science Department. In June 1968 he was appointed Research Assistant Professor of Aerospace Engineering at Rutgers, The State University. RUG I 6 1S6B JUN 2 1969