Optimum form of lifting bodies at hypersonic speeds

Abstract

The variational problem of the form of bodies with minimum drag for given lift force, volume, and other constraints in general leads to a second-order partial differential equation even for the simplest methods of drag calculation (Newton law and averaged friction coefficient). The solution of this equation is not justified; in its place an approximate solution is suggested which consists of: a) selection of a “scheme” characterized by certain parameters which are determined from the solution of the extremal problem, b) determination of the optimal surface form for the selected “scheme” with the aid of the system of ordinary Euler equations. This paper presents a comparison of the body “schemes” with minimum drag and maximum L/D and presents the solution of several variational problems. At the present time we have quite complete information on the form of minimum-drag bodies for zero lift (nonlifting bodies), and both approximate and quite rigorous methods are known for solving the corresponding variational problems. This cannot be said at all of the form of lifting bodies, for which the requirements are numerous, differing essentially for vehicles of different application, and are generally not limited to a single flight regime. Account for all the mandatory requirements in solving the variational problems is not possible; therefore in the majority of cases these solutions do not yield answers which are directly suitable in practice; rather they yield limiting estimates. The natural tendency to utilize for lifting bodies the axisymmetric form which is customary for nonlifting bodies leads to the study of axisymmetric bodies at angle of attack, axisymmetric bodies with skewed base, sections of axisymmetric bodies cut by planes, etc. In order to obtain a broader view of the optimum forms of lifting bodies we must, obviously, drop the limitation to axisymmetric bodies and bodies with similar cross sections. However, in the case of an arbitrary extremal surface the Euler equation is a second-order partial differential equation, and its simple solution is difficult. In practice it seems wise to solve those variational problems whose Euler equation may be reduced to a system of ordinary differential equations. Therefore, we propose the following method for selecting the optimum forms: a) we select a scheme, a form, which is formed by a set of planes and cylindrical, conical, spherical surfaces and which is defined by parameters that are found from the solution of the extremal problem; b) for the selected “scheme” the generators of the “scheme” surface are found from the solution of the variational problem. For the calculation of the air pressure on the body surface we use the empirical Newton law, which yields in the majority of cases results which are very close to the results of the more rigorous methods. It is assumed that the pressure may vanish only at the trailing edge of the body. The frictional drag coefficient, averaged over the body surface, is assumed to be independent of the body shape. In the case of a body of simple form the hottest portion is the frontal portion and account for the thermal protection requirements reduces to the selection of suitable dimensions of this portion of the body. In the general case the problem is stated as follows: find the form of the minimum-drag body for a given lift force, volume, length, and other conditions. To the particular case of the body with maximum L/D corresponds the value of the Lagrange multiplier λ=−1/k. All the results of calculations presented in the paper are intended only to illustrate the method. After the present paper was submitted for publication, another study [3] appeared which also proposes a method for determining the optimal parameters.