Method of curved bodies in problems of unsteady hypersonic flow past slender bodies

The method of curved bodies involves replacing the unsteady flow past a body by steady flow past a different body obtained from the original body by suitable curvature of its form. The idea of the method was proposed by Vetchinkin in 1918 and was first carried out in 1]. Here the authors started from the assumption that the pressure on the body surface is determined only by its local angle of attack.We know that this method is justified only for circular motion of a slender body with constant velocity within the framework of subsonic or supersonic linearized theory.It will be shown below that the method of curved bodies is rigorously justified for hypersonic unsteady flow past slender pointed bodies within the framework of the law of plane sections, which is often used to study unsteady flows, for example 2, 3]. Here the idea of the method involves the selection of a body of form such that for uniform translational motion its wake in a stationary, normally intersected plane coincides in time with the wake of the original body.The general theory is presented for arbitrary bodies, in particular for bodies of the type of slender oscillating wings, but attention is devoted primarily to the motion of a rigid body of rotation. In this case, in the hypersonic approximation (of the type of 4, 5]) the method also extends to slender blunted bodies.In the general case this method reduces the four-dimensional unsteady problem to a three-dimensional steady problem, which presents no particular difficulty in view of the existence of suitable methods and programs (for example 6]). Here, in contrast with the classical version of the method 1], in the general case the original body is replaced at very moment of time by a one-parameter (with parameter t₀) family of curved bodies.In the case which is most often encountered in practice of slow oscillation of the body surface, when the unsteady component of the solution is small in comparison with the steady compoent, the small-parameter method is used, which allows us to represent the solution in a simple form with an explicit linear dependence on the parameter t₀.The basic notation L body length - ₀ body characteristic relative thickness or angle of attack - ₀ characteristic Strouhal number - r₀ maximal radius of the blunt nose - ,a undisturbed medium density and speed of sound - V and M velocity and Mach number of the center of rotation or of the point x₀ - T₀ characteristic time of the unsteady motion (for example, the period of the oscillation) - T=L/V time for the body to pass a fixed plane - V²p pressure The author wishes to thank A. V. Antonets and Yu. M. Lipnitskii for carrying out the calculations and analyzing their results.