A Review of Source Superposition and Conical Flow Methods in Supersonic Wing Theory

A T the end of World War I I the developments in £ * aircraft power plants reached the point where it became feasible to construct vehicles—first rockets and guided missiles and, later, manned aircraft—which were capable of flight a t supersonic speeds. At t h a t t ime it became essential for the aerodynamicist to apply in design practice the seemingly much more complex theory of a compressible fluid medium rather than the classical theory of an incompressible fluid which had been sufficiently accurate for almost all problems only a decade earlier. The engineering application of the techniques of analysis which had been developed in the transition decade suddenly disclosed the unexpected result t h a t many practical engineering problems tha t had been extremely difficult and impractical to solve in the incompressible fluid theory were relatively simple for supersonic speeds. Under these fortunate circumstances the aerodynamicist has been able to keep pace with the revolution of technology which has been required for the achievement of supersonic flight. I t is of interest to note t h a t the comprehensive review of the aerodynamics of a compressible fluid by G. I. Taylor and J. W. Maccoll, which was published in 1935, presents the oneand two-dimensional nonviscous theory of supersonic motion in a form which is essentially unchanged today. In particular they discussed Ackeret 's linearized two-dimensional theory, the twodimensional application by Prandt l , 3 Ackeret, and Busemann of the method of characteristics for the integration of a hyperbolic part ial differential equation by use of the hodograph transformation. They also treated the simplest flow with a degenerate hodograph, following Prandt l ' s 6 and Meyer 's solution of the expansion around a two-dimensional corner. The corresponding solution for the flow past a circular cone, which had been treated by Taylor and Maccoll, was the only three-dimensional example considered. Since supersonic aerodynamic configurations generally involve slender bodies and relatively low aspect ratio wings, three-dimensional effects are of pr imary importance in practical engineering problems. Several concepts which were introduced a t the t ime t ha t reference 1 was being written lead to the additional methods on which most of the current engineering theory is based. First, von Karman and Moore applied the linearized theory to axially symmetric bodies and solved the problem by the superposition of supersonic sources. Second, Busemann introduced the concept of a conical flow, which is a three-dimensional generalization of the Taylor and Maccoll cone solution since the degenerate hodograph is a function of two variables instead of one. Finally, von Karman, in a paper presented a t the Fifth Volta Congress which had the problems of high-speed flight as its general theme, developed a general theory of the drag of slender axially symmetric bodies and considered the solutions for a family of bodies related by a simple affine transformation in Cartesian coordinates to obtain a similarity theory for the linearized equations of motion. The combinations of these various concepts and their logical extensions form the basis of most of the current aerodynamic design methods. I t is the purpose of the present paper to review some of the phases of supersonic aerodynamic theory which have developed from these origins. Many of the problems discussed here have also been treated in his recent, much more extensive, review of linearized theory by G. N . Ward. 2 I t is hoped tha t this simple t reatment , writ ten from a different viewpoint, will be a useful supplement.