The axisymmetric nose shape of minimum wave drag for given size and volume

Abstract

Within the limits of Newton's formula, a solution of the problem of designing axisymmetric nose shapes that minimize wave drag for given volume and maximum admissible size is presented. The solution is obtained both in the slender-body approximation and in the complete formulation (without assuming a small angle between the axis of the body and the tangent to its contour). In both formulations, the optimum contours, besides a two-sided extremum segment, may contain a cylindrical generator — a boundary extremum segment with respect to the maximum admissible radial coordinate. In addition, in the complete formulation the optimum contours may include a leading or trailing flat end — a boundary extremum segment in the sense of both the longitudinal coordinate and the limit of applicability of Newton's formula. Determination of the drag coefficient of the nose shapes designed using numerical integration of the equations of the axisymmetric flow of an ideal gas confirmed the advantages of convex configurations. In the same approximation, the drag of optimum nose shapes (within the limits of Newton's formula) with concave segments may exceed the drag of equivalent cones. The formulation of the variational problem modified for such cases (in particular, with the fixed specification of the volume replaced by a lower bound) yields Newton's solution with a free volume.