High-speed Aerodynamics Research Articles

Development of some hypersonic benchmark flows using CFD and experiment

Abstract. This paper presents a set of test cases in high speed aerodynamics that describe our perceived relationship between experiment and computation. Computational fluid dynamics, with sensible interpretation, can guide experimental design, so that wind tunnel studies can focus better on fundamental ‘benchmark’ studies. Likewise experimental data may be used as feed back to evaluate codes and to improve their physical modelling. Here we present several test cases, developed in our laboratory, that we regard as basic ‘building blocks’ for high speed aerodynamics. These include: design for boundary-layer/pressure-gradient interaction; cavity flows; shock-wave/boundary-layer interactions; techniques for a graduated and controlled study of three-dimensional separated flows.

Analysis of Readers' Questions: A Case Study

Mentions the need for the analysis of the subjects of questions put by the specialist readers to document finding systems. Gives data on the facet structures of the subjects of 323 questions in the fields of High-Speed Aerodynamics and Aeronautical Engineering. On the whole, such questions appear to be fairly well confined to the core subjects. The subjects could be patterned into 34 facet structures. About 73 per cent of the subjects were based on three similar facet structures with 3, 4, and 5 facets respectively. The maximum number of Levels of [P] in anyone facet structure was 3; while that of the Levels of [IMP] in anyone facet structure was 5. The maximum number of facets in anyone subject was 12 and the minimum 2, with an average of 5·4. The maximum number of kernel ideas in anyone subject was IS and the minimum 3, with an average of S. This appears to conform to the findings of psychology of the Span of Immediate Memory and the Span of Absolute Judgment. The average frequency of incidence of (P), (M), and (E) isolates was in the ratio of 4·2: 5·6: 1. The frequency of incidence of isolates from the second and later Rounds of [P] and [M] was quite small. The number of different (lPl) isolates incident in the subjects of the questions was 48; 20 such isolates accounted for a 90-per cent incidence of (1P1) isolates. The number of different (1MP) isolates incident in the subjects was 152. 20 such isolates accounted for a 62-per cent and 50 isolates for an 80-per cent incidence of (1MP) isolates. The number of different (IE) isolates incident in the subjects was 24. They were all common isolates. A principle is mentioned as an aid in differentiating whether an idea is to be deemed a manifestation of Matter (Property) or of Energy, when the idea is associa!ed with an action. A total of 178 qualifiers occuring 266 times qualified 80 ideas in [1P1].A total of 274 qualifers occurring 577 times qualified 98 ideas in [1MP]. 22·5 per cent of the isolates in [1P1] and 88 per cent of the isolates in [1MP] did not have qualifiers. There were 76 Complex Subjects in the subjects of the questions. The influence Phase Relation had the highest incidence (68 %) among the Complex Subjects. Concludes mentioning a few subjects for further investigation.

Shock phenomena in high speed aerodynamics: still a source of major concern

AbstractShockwaves are present in a flow as soon as the Mach number becomes supersonic. Being viscous phenomena, Shockwaves are a source of drag which can be predominant when the Mach number is significantly higher than one. In supersonic air intakes, the production of entropy by shocks is felt as a loss in efficiency. At high Mach numbers, Shockwaves produce a considerable temperature rise leading to severe heating problems, complicated by real gas effects. The intersection - or interference - of two shocks gives rise to complex wave patterns containing slip-lines and associated shear layers whose impingement on a nearby surface can cause detrimental pressure and heat transfer loads. The impact of a Shockwave on a boundary layer is the origin of strong viscous interactions which remain a limiting factor in the design of transonic wings, supersonic air intakes, propulsive nozzles and compressor cascades. More effort is needed to improve prediction of these interactions and to devise new techniques to control such phenomena.

Interactions of Space Bodies With Atmospheres of Planets

In this review article, a modern theory of meteor body motion in the atmosphere is described. The theory is based on the results of high-speed aerodynamics. The author understands that such an application is difficult and unusual. The difference between space vehicle aerodynamics and bolide physics is, first of all, in absence of knowledge about shapes of meteoroids and of their physical-mechanical properties. On the other hand, the main aerodynamical laws are the same for vehicles and meteoroids. This permits one to achieve some new developments in meteor physics. In the first section, a simple analytical model for meteor trajectories is developed. The second and fourth sections of the article contain high-speed aerodynamics results: inviscid gas flow past bodies, and convective and radiative heat transfer. The third section is devoted to one of the difficult problems in meteor physics-the theory of meteoroid destruction due to aerodynamical loading. Finally, the fifth section contains some common questions on bolide phenomema. For the solution of meteor physics problems, both analytical and numerical methods are used. As such problems contain a number of determining parameters which are not accurately known, the analytical methods are preferable. The numerical methods are necessary for calculation of complex gasdynamical structures forming during the motion of the meteor bodies in the atmosphere. There are 50 references included at the end of this article.

Knowledge-based hybrid expert system for automated interferometric data reduction

In order to accomplish the ultimate goal of automated interferogram evaluation, the utilization of artificial neural networks or expert systems in artificial intelligence has been expected. In this paper, a new PC-based fringe evaluation technique, which is based on hybridization of an expert system and a conventional fringe tracking method, is presented together with some test results of noise-ridden interferograms in high-speed aerodynamics. This technique incorporates algorithmic image processing with expert-system-driven knowledge manipulation in a single structure to provide an optimal solution for noisy interferogram evaluation. A new approach for regional phase unwrapping based on the technique is also proposed. This concept is applicable to interferometry such as phase shifting and Fourier transform that provides phase-wrapped information. It is based on conservative decision making in defining phase-wrapped regions by utilizing global/regional knowledge in the expert system. The approach is thus believed to reduce computation time and memory space in processing complex or noisy interferograms that require frequent operator interactions.

Automated interferogram analysis based on an integrated expert system

Interferometric data, either from single-frame fringe-tracking and Fourier-transform techniques or from multiframe phase-shifting techniques, pose a problem of 2π ambiguity, that is, wrapped-phase information. As the degree of noise level increases, especially in high-speed aerodynamics, these techniques encounter difficulties in phase extraction to provide continuous unwrapped-phase information. Here, a new hybrid approach, called the integrated expert system, which is developed primarily for aerodynamic interferogram evaluation, is presented. The integrated expert system utilizes interferometric-specific knowledge rules to compensate for the limitations associated with conventional techniques. It integrates in a single structure an expert system and algorithmic programming to provide, as much as possible, a unified approach for all the interferogram evaluation techniques. This initial attempt may provide a useful groundwork for future development in intelligent interferogram processing.

Laplace interaction law for the computation of viscous airfoil flow in low- and high-speed aerodynamics

A new interaction law that accelerates convergence of interactive boundary-layer methods is presented. The law is derived from the Laplace equation and allows the automatic implementation of the Kutta condition for lifting airfoil flows. The accurate influence coefficients of the Laplace interaction law (LIL) facilitate efficient prediction of high-lift flows about multi-element airfoil by direct numerical simulation.Taking into account suitable turbulent closure relations for boundary-layer equations, a fast and reliable computer code has been developed for industrial applications at Deutsche Aerospace Airbus.

Accuracy and Efficiency Assessments for a Weak Statement CFD Algorithm for High-Speed Aerodynamics

A bilinear finite element, implicit Runge-Kutta space-time discretization has been established for an aerodynamics weak statement CFD algorithm. The algorithm admits real-gas effect simulation, for reliable hypersonic flow characterization, via an equilibrium reacting air model. The terminal algebraic system is solved using an efficient block-tridiagonal quasi-Newton linear algebra procedure that employs tensor matrix product factorizations within a lexicographic mesh-sweeping protocol. A block solution-adaptive remeshing, for totally arbitrary convex elements, is also utilized to facilitate accurate shock and/or boundary layer flow resolution. Numerical validations are presented for representative benchmark supersonic-hypersonic aerodynamics problem statements.

High-speed aerodynamics of several blunt-cone configurations

The experimental techniques and results from NASA Ames research into the aerodynamics of blunt shapes are described. Two facilities are used: the Hypersonic Free Flight Aerodynamic Facility (HFFAF) and the Pressurized Ballistic Range (PBR). The HFFAF features a 23 m test section through which projectiles can be fired by light-gas guns to achieve velocities up to 9 km/sec in pressures from 1 atm down to 20 microns Mg. Aerodynamic flight data are recorded with shadowgraphs. The 62 m long PBR, with 24 orthogonal spark shadowgraph stations, allows testing in up to 6 atm pressures but with only half the velocity of the HFFAF. Tests have been performed on the Galileo probe to be dropped into the Jovian atmosphere. The Galileo data for a 45 deg cone probe have been similar to Pioneer Venus probe data. Ballistic data are presently being obtained for the probe at Re below 1000, when the drag coefficients are expected to increase dramatically as the probe enters the Jovian atmosphere and begins braking from 4700 m/s speed. Sample test data from an OTV which can change orbits through aerodynamic maneuvers and for several asymmetric configurations ae also provided.

Early Development of an "Acoustic Analogy" Approach to Aeroacoustic Theory

I is a special pleasure to contribute with this brief historical paper to AIAA's celebrations of its 50th anniversary. I am, also, particularly glad that the paper gives me an opportunity to commemorate the energy and vision shown in 1949 by the late H. B. Irving, in his then position as Assistant Director of Scientific Research (Air) within Britain's government machine. Part of his job was to think about the future of aeronautics, with the object of identifying serious problems likely to arise as part of that future; and to draw attention to new research fields that needed to be opened up without delay if timely solutions to those problems were to be found. Remarkable vision was shown by Irving in recognizing so clearly, as early as 1949, that jet engines were bound to cause severe noise problems when (as must ultimately happen) they penetrated the civil transport market. This may have made him the first person in the world to identify research on jet noise and means for its reduction as a major new field of research to which effort urgently needed to be devoted. Great energy, too, was shown by Irving when he followed up that insight by a vigorous campaign waged with the object of setting up substantial research groups in this field within British university institutions. These groups (supported, where necessary, by contributions from the modest funds at Irving's disposal) were invited to pursue basic experimental and theoretical studies of the noise of jets; studies that might be usefully complementary to a large-scale program of empirical studies of jet-engine noise and schemes for its reduction initiated at Rolls Royce Ltd. Irving ensured, too, that the university groups created in this way (especially, the group including G. M. Lilley at the College of Aeronautics in Cranfield, the University of Southampton group that included E. J. Richards and A. Powell, and the University of Manchester group including myself on the theoretical side and J. H. Gerrard on the experimental side) would all take part together in frequent meetings with the outstanding Rolls Royce team under the leadership of F. B. Greatrex. These regular meetings very powerfully influenced the progress of the research work. I remember this in my own case: as the participant mainly concerned with developing theoretical approaches, I was helped by constant contact with several growing bodies of experimental data to adopt the necessary critical and selective approach to my own growing collection of theoretical ideas. The only Satisfactory theories are those that simultaneously accord with empirical data and with scientific logic!—a formidable pair of requirements. I recall Irving coming up to my attic room in the deeply blackened old Owens College building (since revealed by modern stone-cleaning technology as a charming blend of light brown and blue shades!) at Manchester University. In 1949, I was a 25-year-old Senior Lecturer in Applied Mathematics, already copiously involved in compressibleflow theory (especially, high-speed aerodynamics and shockwave dynamics). Somehow, Irving convinced me that jet noise was an exceptionally exciting theoretical challenge. I recognized that a jet was one of the classical turbulent flows, characterized to be sure by a complex pattern of vorticity; but not (according to views then current) by any dilatation. Compressibility had not been regarded hitherto as a significant factor in vortex motions or turbulence; yet it must be playing a part if any radiated sound were to result. Next morning I had to go to London, which in 1949 necessitated a four-hour train journey. Once in the train J could not stop thinking about the j£t-noise conundrum. It is, furthermore, the literal truth that the only piece of paper I had on me was the proverbial back of an envelope'M How fortunate that was. Sitting in the impersonal confines of the railway compartment, after having awakened that morning still more clearly conscious of what an exceptionally exciting theoretical problem in mechanics of fluids had been put to me, I had four hours to think about it while prevented from succumbing to the theoretician's besetting weakness; namely, rushing into filling sheets of paper with endless equations.... The essential idea of the acoustic analogy approach came before my journey's end. Iremember that the first part of the journey was entirely devoted to considering what would be the appropriate dependent variable in a system of equations to describe jet noise. I had found in many problems of compressible flow that the key to producing tractable equations was the correct choice of dependent variable. Often the pressure was an optimum choice (for example, in the aerodynamics of disturbances to a nonuniform oncoming stream). However, I soon decided against use of the pressure,

Fluid Mechanics Research in 1986

In 1965, a book was published with the title Research Frontiers in Fluid Dynamics, edited by Raymond Seeger and G. Temple. It was intended to give a panoramic view of some exciting vistas in fluid dynamics. It covered the following areas: (1) High-speed aerodynamics; (2) magnetophydrodynamics (MHD); (3) physics of fluids (low or high density, low or high temperature, etc); (4) constitutive properties of fluids (viscosity, viscoelasticity, etc.); (5) oceanography and meterology; (6) astrophysical and planetary fluid mechanics; and (7) mathematical aspects and numerical aspects. A few of these areas, such as MHD, have blossomed and faded away within a short decade. Some others, such as high-speed aerodynamics, have reached maturity and hope to keep their momentum. In the intervening years, we have witnessed that a number of fields in fluid mechanics have revived from their old times into a new life; still, some have emerged with brand new growth. For instance, the subject of long waves has had a colorful development, with the result of improving our understanding of at least seven different physical phenomena, though originally the solitary water wave was its home base. Low Reynolds number flows have again received new stimuli from many needed applications such as aerosol physics, two-phase flows, rheology, geophysics of the earth interior, as well as micro and molecular biology. Oil exploration has motivated various aspects of marine-related research and development, giving ever-increasing activities in ocean engineering. The energy program, a new glamorous field by its own importance, has brought forth investigations of fluid mechanical problems pertaining to nuclear, geothermal, solar, wind, ocean wave, and other forms of energy sources. Riding on the waves of these broad movements that have carried us thus far, we now hope to forecast the future of fluid mechanics research in 1986. We may like to put the focus at a slightly different depth and ask: What will be the most significant areas of fluid mechanics that by 1986 will enjoy the best prospects of vigorous development, most rewarding not only to the fluid dynamicist but also to mankind, and by then, still offer the expectation of longevity into the 1990's? The task is almost as hard as to make a prophecy on what the political world will be in 1986.

Linearized thin-wing theory of gas-centrifuge scoops

We study a steady hypersonic rotating flow of a perfect gas past a system of thin stationary scoops in a gas centrifuge of annulus type. The gas is assumed inviscid; its ratio of specific heats is assumed to be approximately 1.The scoops are set at zero angle of attack and are periodic with respect to the azimuthal variable. The flow is assumed to be a three-dimensional small perturbation on a basic state of rigid-body rotation. New scaling laws are proposed as appropriate to realistic operating conditions of gas centrifuges. Basic equations, boundary conditions and shock conditions are linearized for a weakly hypersonic flow by an analytical procedure similar to that used in the thin-wing approximation in high speed aerodynamics. The solution of the basic equations is obtained by the eigenfunction expansion method. The solution provides us a simple addition theorem for the scoop drag which makes the resultant drag of a system of several scoops equal to the product of the number of scoops and the drag of a standard system with a single scoop. The solution makes it clear that despite the above addition theorem, the scoops interact in their effects on the flow.

Errata: Remarks on Thin Airfoil Theory

Fig. 3 Angles of attack for vanishing leading-edge singularity and for zero lift. References E., High-Speed Aerodynamics, Pergamon Press, London/New York/Paris, 1956, pp. 619-624. 2 Holla, V. S. and Krishnaswamy, T. N., Conically Cambered Triangular Wings, Journal of the Aeronautical Society of India, Vol. 22, May 1970, pp. 94-107. Bera, R. K., Slender Delta Wings with Conical Camber, Journal of Aircraft, Vol. 11, April 1974, pp. 245-247. Evvard, J. C., Effects of Yawing Thin Pointed Wings at Supersonic Speeds, NACA TN 1429, 1947. Hancock, G. J., Note on the Extension of Evvard's Method to Wings with Subsonic Leading Edges Moving at Supersonic Speeds, Aeronautical Quarterly, Vol. VIII, 1957, pp. 87-102. Wagner, B., Untersuchungen uber konische Uberschalltragflachen mit Unterschallvorderkanten, Institut fur Flugtechnik, T. H. Darmstadt, Rept. No. 8/75, 1975.

Reply by Author to M. J. Walsh

Henderson, C. B., Coefficients of Spheres in Continuum and Rarefied Flows, AIAA Journal, Vol. 14, June 1966, pp. 707708. Bailey, A. B. and Hiatt, J., Free-Flight Measurements of Sphere Drag at Subsonic, Transonic, Supersonic, and Hypersonic Speeds for Continuum, Transition, and Near-Free-Molecular Flow Conditions, Arnold Engineering Development Center, Tullahoma, Tenn., AEDCTR-70-291, March 1971. Walsh, M. J., Coefficient Equations for Small Particles in High Speed Flows, AIAA Journal, Vol. 13, Nov. 1975, pp. 15261528. Walsh, M. J., Influence of Particle Drag Coefficient on Particle Motion in High-Speed Flow with Typical Laser Velocimetry Applications, NASATN D-8120, Feb. 1976. Zarin, N. A., Measurement of Non-Continuum and Turbulence Effects on Subsonic Sphere Drag, NASA CR-1585, June 1970. Emmons, H. W., Ed., Fundamentals of Gas Dynamics, Vol. HI. High Speed Aerodynamics and Jet Propulsion, Princeton University Press, Princeton, N.J., 1958, p. 689.

Orthogonality of generally normalized eigenvectors and eigenrows

References Pansier, K. S., Analysis of Boundary Layer on Moving Wall and Its Application to a Spinning Cylinder in Crossflow, Ph.D. thesis, May 1974, Dept. of Applied Sciences, University of Delaware, Newark, Del. Fansler, K. S. and Danberg, J. E., Development on Moving Walls Using an Integral Theory, AIAA Journal, Vol. 14, Aug. 1976, pp. 1137-1139. Danberg, J. E. and Fansler, K. S., Solutions of the Boundary-Layer Equations for Flow Over a Moving Wall, Rept. 1714, April 1974, Ballistic Laboratories, Aberdeen Proving Ground, Md. Moore, F. K., On the Separation of the Unsteady Boundary-Layer, Proceedings of Symposium on Boundary-Layer Research, Aug. 1957; edited by H. Gortler, Springer-Verlag Press, Berlin, 1958, pp. 296-311. Rott, N., Theory of Flows: High-Speed Aerodynamics and Jet Propulsion, Vol. 4, edited by F. K. Moore, Princeton University Press, Princeton, N. J., 1964, p. 432. Danberg, J. E. and Fansler, K. S., Separation-Like Similarity Solutions on Two-Dimensional Moving Walls, AIAA Journal, Vol. 13, Jan. 1975, pp. 110-112. Vidal, R. J., Research on Rotating Stall in Axial Flow Compressors Part I I I — Experiments on Separation from a Moving Wall, WADC TR-59-75, Pt. 3, Jan. 1959, Wright Air Development Center, U. S. Air Force, Wright-Patterson Air Force Base, Ohio. Brady, W. G. and Ludwig, G. R., Research in Unsteady Stall of Axial-Flow Compressors, Rept. CAL AM-1762-S-4, Nov. 1963, Cornell Aeronautical Lab., Buffalo, N. Y. Swanson, W. M., An Experimental Investigation of the TwoDimensional Magnus Effect, Final Rept., Office of Ordnance Contract No. DA-33-019-ORD-1434, Dec. 1956, Dept. of Mechanical Engineering, Case Institute of Technology. Telionis, D. P. and Werle, J. J., Separation from Moving Boundaries, Journal of Applied Mechanics, Transactions of ASME, Ser. E, June 1973, pp. 369-373. Goldstein, S., On Boundary-Layer Flow Near a Position of Separation, Quarterly Journal of Mechanics, Vol. 1, 1948, pp. 43-69. Tsahalis, D. Th., Laminar Boundary Layer Separation from an Upstream Moving Wall, AIAA Paper 76-377, July 1976, San Diego, Calif. Brady, W. G. and Ludwig, G. R., Studies of Rotating Stall and an Investigation of Flow-Instability Sensing Devices, Part I. Basic Studies of Rotating Stall Flow Mechanisms, AFAPL-TR-65115, Oct. 1965, Air Force Aero Propulsion Lab., Wright-Patterson AFB, Ohio. Hartunian, R. A. and Moore, F. K., Research on Rotating Stall in Axial Flow Compressors Part II-On the Separation of the Unsteady Boundary Layer, WADC TR-59-75, Part II, Jan. 1959, Cornell Aeronautical Laboratory, Buffalo, N. Y.

A critical review of development of experimental methods in high-speed aerodynamics

A critical review of development of experimental methods in high-speed aerodynamics

Three-Dimensional Flow Separations on Aircraft and Missiles

William J. Rainbird; received his B.E.(Mech) at Canterbury University, Christchurch, New Zealand in 1947, and was awarded a Government scholarship to the College of Aeronautices, Cranfield, England, where he left in 1950 with a Post Graduate Diploma, with distinction, specializing in aerodynamics. After a year in Stockholm, Sweden working on transonic and supersonic wind-tunnel design at F.F.A., he returned to New Zealand and lectured at Canterbury University until September 1954. Returning to the College of Aeronautics, Cranfield that year, he was Lecturer in General and Experimental Aerodynamics for 3 years. Mr. Rainbird worked at the National Aeronautical Establishment, a Division of the National Research Council, Ottawa, from January 1958 to July 1970 and was a Senior Research Officer and Head of the High Speed Aerodynamics Section. In July 1970, Mr. Rainbird was appointed Professor of Engineering and Chairman of the Division of Aerothermodynamics, at Carleton University, Ottawa. In 1967, Mr. Rainbird spent a year’s study and refresher leave at the University of California, San Diego, Department of Aerospace and Mechanical Engineering Sciences. He is a Fellow of the C ASI, Associate Fellow of the Royal Aeronautical Society, Member of the AIAA, and was a Canadian Member of the AGARD Fluid Dynamics Panel from 1964-1969. His main field of specialization and interest in aerodynamics is in threedimensional boundary layers and separation. David J. Peake is a graduate of University of Bristol, England, Department of Aeronautical Engineering [B.Sc., 1960 (First Class Honors); M.Sc., 1962], Concurrent with University training, Mr. Peake completed an Engineering Student Apprenticeship at National Gas Turbine Establishment, Pyestock, Farnborough, England, during 1956- 1960. He remained at NGTE until May 1962 as Scientific Officer, working with Dr. Brian Stratford on the control of supersonic turbulent boundary layers with tangential air injection, and supersonic inlet design. Mr. Peake moved to National Research Council, Ottawa, Canada in May 1962, joining the Council's National Aeronautical Establishment in January 1964, in the High Speed Aerodynamics Laboratory. He worked with W. J. Rainbird in the investigation of three-dimensional separations in both subsonic and supersonic flow; transonic jet-flapped aerofoils, and axisymmetric turbulent boundary layers, all at high Reynolds numbers. Mr. Peake is studying for Ph.D. (parttime) at Carleton University, Ottawa and has membership in AIAA, RAeS and CASI. He has recently been awarded a Ph.D. from the University of Bristol for submitted published work. Edward G. Atraghji is a graduate of University of London (B.Sc., 1962). He conducted research on slender delta wing configurations at Queen Mary College, London, between 1962 and 1966. He joined the National Research Council, NAE Division, High Speed Aerodynamics Laboratory, 1966. Topics of research have included the flows about various aircraft and missile configurations.

Problems of High-Speed Hydrodynamics

A hard look at the problems provoked by the quest for higher speeds of underwater craft shows that they can indeed be solved in a practical manner using the knowledge and principles derived in the evolution of the science of high-speed aerodynamics. Hydrodynamic heating, turbulent heat transfer and friction, boiling and cavitation, friction reduction by additives, normal and oblique shock waves, and rocket propulsion are topics brought under discussion. Propulsion remains the prime area of future endeavor in order to achieve the speeds demanded by modern times.

Strongly radiating hypersonic flow over a sphere - An asymptotic sketch.

14 Pridmore Brown, B. N. and Franks, W. J., A method of characteristics solution in three independent variables/' Northrop Norair Rept NOR 65-42 (1965); also Aerospace Research Laboratories Rept. ARL 65-124 (1965). 15 Sauerwein, H. and Sussman, M., Numerical stability of three-dimensional method of characteristics, AIAA J. 2, 387389 (1964). 16 Heie, H. and Leigh, D. C., Numerical stability of hyperbolic equations in three independent variables, AIAA J. 3, 10991103 (1965). 17 Heaslet, N. A. and Lomax, H., Supersonic and transonic small perturbation theory, General Theory of High Speed Aerodynamics, edited by W. R. Sears (Princeton University Press, Princeton, N. J., 1954), p. 330. 18 Der, J., Jr., Niemann, A. F., Jr., Powers, S. A., and Pridmore Brown, B. N., The determination of combined viscid-inviscid flow fields over hypersonic lifting reentry configurations, Wright Patterson Air Force Base RTD-TDR-64 (1964). 19 Chu, C. W., Numerical integration by modified spline fit, Northrop Norair Rept. NOR 65-316 (1965).

Theory of Laminar Flow (vol. IV, High speed Aerodynamics and Jet Propulsion). Edited by F. K. MOORE. Princeton University Press, 1964 869 pp. £10.

<b>Theory of Laminar Flow</b> (vol. IV, <i>High speed Aerodynamics and Jet Propulsion</i>). Edited by F. K. M<scp>OORE</scp>. Princeton University Press, 1964 869 pp. £10.