Supersonic flow past finite-length bodies in dispersive hydrodynamics A. V. Gurevich Theoretical Section of the P. N. Lebedev Physics Institute, Russan Academy of Sciences, 117924 Moscow, Russia A. L. Krylov, and V. V. Khodorovski , O. Yu. Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, 123810 Moscow, Russia. G. A. E ´ l’ Institute of Terrestrial Magnetism, The Ionosphere, and the Propagation of Radio Waves, Russian Academy of Sciences, 142092 Troitsk, Moscow Province, Russia Submitted 27 August 1995 Zh. E ´ ksp. Teor. Fiz. 109, 1316–1334 April 1996 Using the Gurevich–Pitaevski approach, an analytic study is made of two-dimensional supersonic flow past slender bodies of finite length in nondissipative dispersive hydrodynamics. The problem of recovering the shape of the body from data on the wake at infinity is solved. It is shown that under conditions when the KdV approximation is valid the nonlinearity and dispersion do not affect the macroscopic properties of the flow—the drag and lift. © 1996 American Institute of Physics. S1063-77619601904-X 1. INTRODUCTION The present paper is a continuation of Ref. 1 and is devoted to the study of supersonic flow of a nondissipative weakly dispersive fluid past pointed bodies. If the bodies are slender, then the general equations of steady two- dimensional dispersive hydrodynamics reduce to the Korteweg–de Vries KdV equation, in which the role of the time is played by the spatial coordinate at right angles to the direction of flow. The initial data for the KdV equation are determined by an arbitrary function y ( x ), which specifies the profile of the body. 1,2 The most important feature of supersonic dispersive flow past bodies is the formation of a nondissipative shock wave, a wedge shaped region of space occupied by small-scale nonlinear oscillations described by modulated elliptic func- tions. The oscillations have the shape of solitons on the front facing the oncoming flow and the shape of harmonic oscil- lations of infinitesimally small amplitude on the opposite front toward the body. In Ref. 1, the Gurevich–Pitaevski approach 3 was used to make an analytic study of the struc- ture of the nondissipative shock waves that arise in the case of flow past infinite wedge-shaped bodies with y  ( x ) 0. Such profiles correspond to monatomic initial data in the Gurevich–Pitaevski problem the case of a nondissipative shock wave with intensity that does not decrease with time. However, it is important that by virtue of the supersonic nature of the motion the expressions obtained in Ref. 1 have finite ‘‘domains of influence’’ and can be used in the de- scription of the flow past finite sections of bodies of a more complicated shape. In this paper, we consider flow past thin pointed bodies possessing in profile a section of finite or infinite extent with y  ( x ) 0, in particular we consider flow past finite- length bodies. Such a change in the geometry of the body, which would appear to be a minor one compared with Ref. 1, leads to a significant modification in the solution to the problem. Difficulties arise manily because of the nonmono- tonicity of the initial data in the corresponding Gurevich– Pitaevski evolution problem. In addition, it is often the case that the typical shapes of the bodies around which the flow takes place see, for example, Fig. 1c correspond to initial data that are not at all characteristic of the Gurevich– Pitaevski problem Fig. 1d. Finally, flow past bounded bod- ies is accompanied by the formation of two nondissipative shock waves that possess different asymptotic properties. At the same time, it is clear that precisely these cases are the ones of greatest interest from the point of view of applica- tions. We note also that, since the supersonic nature of the flow makes it possible to study the flow in the upper half-plane independently, our problem can be interpreted as the prob- lem of the flow past a convex or concave inhomogeneity on the bottom of a flat channel. As in Ref. 1, to describe the rapidly oscillating region of the nondissipative shock wave we use Whitham’s method of averaging. 4 Whitham’s system for the KdV equation with the Gurevich–Pitaevski matching conditions is integrated by means of a generalized hodograph transformation 5,6 and the ‘‘scalar potential’’ technique, 7–9 but by virtue of the non- monotonicity of the initial data the corresponding transforms have a ‘‘two-sheeted’’ nature. 10–12 The relatively simple as- ymptotic behavior of the solution for localized initial data makes it possible to efficiently solve the problem of recov- ering the shape of the body in the flow from data on the wake at infinity recall that we are studying the purely nondissipa- tive situation. We show that in the case of flow past a slen- der body under conditions for which the KdV approximation is valid the nonlinearity and the dispersion do not affect the drag and lift, which are the most important macroscopic characteristics of the flow Ref. 13, § 125. 709 709 JETP 82 (4), April 1996 1063-7761/96/040709-10$10.00 © 1996 American Institute of Physics