Cowl Shapes of Minimum Drag in Supersonic Flow

Guderley, Armitage and Valentine have computed the inlet and closed body contours which form the forepart of an axially symmetric body, of given length and fineness ratio, having minimum pressure drag. The solution is not based on a simplified pressure law, such as the Newtonian impact law, because by a suitable choice of control surface for mass flow and momentum they are able to employ the general flow equations. It is clear, however, from an analysis of their tabulated results that their cowl shapes fall on a single curve for a given value of Δ=didt, the ratio of the initial to the final diameter of the cowl, when plotted in terms of a dimensionless length ξ=x/l and thickness η=y/dt, as in Fig. 1. Furthermore Fig. 1 shows that, except for small values of Δ, the Guderley shapes are indistinguishable from the optimum shapes calculated from Newtonian impact theory. The shape and characteristics of the Newtonian duct of given length and thickness, offering minimum drag to the external stream, are derived using the slender-body approximation. Ducts for which Δ > 0.04 are shown to be sufficiently slender. The slopes of those with 0 ≤ Δ < 0.04 are too large only in a small critical region near the nose. Thus slender body theory will give a close approximation to the exact Newtonian solution even in these cases. For the larger values of Δ likely to be used in practice slender body theory is valid everywhere.