Construction of the Three-Dimensional Minimum-Wave-Drag Forebody of Given Length and with a Circular Base (Review)

Abstract

We consider the problems occurring in constructing the nose parts (NP) of bodies of revolution realizing the wave drag minimum at a given body length and when the number of the planes of symmetry n ≥ 2. The axisymmetric solution of this problem had been given by I. Newton in his Philosophiae Naturalis Principia Mathematica within the framework of his own Newton formula (NF) for the pressure on the windward side of a body in flow suggested in the same study. The solution given by Newton without any explanations was not at once understood by aerodynamicists who turned to the solution of this problem and its certain generalizations at the mid-twentieth century. Newton’s Principia were translated into Russian by A.N. Krylov, who gave also detailed commentaries to this treatise, including the problem under discussion. However, even these comments could not help to understand Newton’s solution and it was only the Soviet aerodynamicists who could cope with them. Nevertheless, before long the three-dimensional NPs could be constructed within the framework of the NF; the drag of those bodies, which had firstly star-shaped and then circular bases, was smaller than that of the axisymmetric Newtonian NPs of the same length at n ≥ 2. The mathematicians, who started to deal with this problem at the end of the 20th century and the beginning of the 21st century, and had known nothing about the studies of aerodynamicists, turned to the same NF, prohibiting concave regions of the surfaces of the required NFs. The main result of their investigations within the framework of the NF is the NP that consists of n ≥ 2 identical inclined planes, adjoining lineate surfaces, and a leading flat face—a regular n-gon (a rectilinear segment at n = 2). However, for both the theory and applications it is important to know how they behave in at least an inviscid flow. The results of calculations of these flows according to Euler equations presented below are intended to reply to this question.