Hypersonic shapes of maximum drag for a given heat-transfer rate

Abstract

The problem of determining the shape of a blunt, axisymmetric body which maximizes the drag for a given heat-transfer rate and diameter is considered. For blunt bodies, pressure drag predominates and is estimated from modified Newtonian flow considerations. With regard to heat transfer, it is assumed that the body is operating in the range of hypersonic speeds where the radiative heating rate can be neglected with respect to the convective heating rate. The latter is estimated from the boundary-layer analysis due to Lees. Optimum power-law shapes as well as variational shapes are determined and are shown to yield almost identical results. For low-to-moderate values of the convective heat-transfer parameter, the optimum shape is very flat and is approximately a one-half power-law shape; in this range, spherical segments are approximately one-half power-law shapes and, hence, are nearly maximum drag shapes. There exists a maximum value of the convective heat-transfer parameter for which maximum drag shapes exist, and the corresponding optimum shape is a cone, or a power-law shape of exponent unity. This limiting shape is shown to be that which maximizes the convective heat-transfer rate for a given diameter.