An Approximate Solution of the Direct Supersonic Blunt-Body Problem for Arbitrary Axisymmetric Shapes

Abstract

The integral method of Belotserkovskii has been carried out to the first approximation for arbitrary blunt axisymmetric bodies in supersonic or hypersonic flight. This method is direct, in that it gives the surface-pressure distribution and shock shape for a prescribed body. Results obtained by numerical integration for several body shapes at several Mach numbers are compared to experimental results with good agreement. I t is also shown, that the method can be successfully applied to pointed bodies with attached shock. In the stagnation region, simple relationships are found from the equations of the first approximation which connect the surface-velocity gradient, shock curvature, shock-detachment distance, and body curvature. These relations are also correlated with experiment for a variety of shapes as a function of Mach number. The conelations permit a rapid estimate of the stagnation-point velocity gradient, important for heat-transfer calculations, for any blunt body from the shock stand-off distance. A method for a higher approximation is described, for which, in contrast to the higher approximations of Belotserkovskii, a large number of simultaneous total differential equations with unknown parameters does not occur. One form of this method has been studied numerically. Results are given which, though only partially successful, indicate the amount of improvement to be expected from a higher approximation.