Hypersonic flow over spherically blunted cone capsules for atmospheric entry. Part 1. The sharp cone and the sphere

Abstract

Depending on the cone half-angle and the inverse normal-shock density ratio $\unicode[STIX]{x1D700}$, hypersonic flow over a spherically blunted cone exhibits two regimes separated by an almost discontinuous jump of the body end of the sonic line from a point on the spherical nose to the shoulder of the cone, here called sphere behaviour and cone behaviour. The inflection point of the shock wave in sphere behaviour is explained. In Part 1 we explore the two elements of the capsule shape, the sphere and the sharp cone with detached shock, theoretically and computationally, in order to put the treatment of the full capsule shape on a sound basis. Starting from the analytical expression for the shock detachment angle of a cone given by Hayes & Probstein (Hypersonic Flow Theory, 1959, Academic Press) we make a hypothesis for the sharp cone, about the functional form of the dependence of dimensionless quantities on $\unicode[STIX]{x1D700}$ and a cone angle parameter, $\unicode[STIX]{x1D702}$. In the critical part of atmospheric entry the shock shape and drag of the capsule are insensitive to viscous effects, so that much can be learned from inviscid studies. Accordingly, the hypothesis is tested by making a large number of Euler computations to cover the parameter space: Mach number, specific heat ratio and cone angle. The results confirm the hypothesis in the case of the dimensionless shock stand-off distance as well as for the drag coefficient, yielding accurate analytical functions for both. This reduces the number of independent parameters of the problem from three to two. A functional form of the shock stand-off distance is found for the transition from the $90^{\circ }$ cone to the sphere. Although the analysis assumes a calorically perfect gas, the results may be carried over to the high-enthalpy real-gas situation if the normal-shock density ratio is replaced by the density ratio based on the average density along the stagnation streamline (see e.g. Stulov, Izv. AN SSSR Mech. Zhidk. Gaza, vol. 4, 1969, pp. 142–146; Hornung, J. Fluid Mech., vol. 53, 1972, pp. 149–176; Wen & Hornung, J. Fluid Mech., vol. 299, 1995, pp. 389–405).