The role of a length constraint in the design of minimum-drag bodies

Abstract

The key role of a constraint on the length of a designed body is demonstrated by taking the example of symmetric profiles which achieve minimum wave drag in a supersonic flow. As a result of this an optimal body can contain a rear-base which emerges as a section of a boundary extremum. By assumption, there is no gas flow around the rear base, and the “base” pressure p + which acts on them is specified and is independent of the form of the required contour and the ordinate y. When designing profiles, in addition to their length, it is customary to specify the area of the longitudinal section F and other isoperimetric conditions. Even when p + = 0, which, naturally, does not occur, it is now necessary to introduce a rear base for extremely small F. When p + > 0, a base appears earlier still. The replacement of the optimal contours with a rear base by a “pseudo-optimal” contours with a sharp edge leads to an increase in drag of tens and hundreds of percent. Special attention has been paid to cases in which p +, due to heat supply in the base domain, for example, exceeds the free-stream pressure. Here, there is always a rear base and, when F < F 0, where F 0 depends on p +/ p ∞, the form of the optimal contours is the same as in the problem without a specified F. In this case, the optimal configuration is a hollow or partially hollow “checkmark”.