A uniformly valid solution for the three-dimensional inviscid supersonic flow past blunt bodies with strong compression in the shock wave

Abstract

An analytical uniformly valid approximate solution is developed for the steady threedimensional supersonic flow past blunt bodies. The inverse problem is investigated, i.e. the shock shape is prescribed. Viscosity and heat conduction are neglected. The approximation is based on two main assumptions: i) the density ratio across the shock is very small, ii) the pressure does not change its order of magnitude along a normal to the shock surface. The pattern of the streamlines projected on to the shock surface is calculated from an ordinary differential equation of second order by taking into consideration the pressure gradients. By evaluating two integrals the flow quantities and the streamlines are determined in the shock layer together with the body shape. The solution is also valid for sharp nosed bodies. The method is applied to paraboloidal or hyperboloidal shock shapes of various cross sections. Results are presented for the streamline projections in the entire flow field. The flow quantities and the streamline pattern in the shock layer are calculated in the symmetry plane of the flow field. The streamlines differ from the geodesics, i.e. the solution according to the Newtonian model. The flow quantities on the body and the body shape show good agreement with numerical results of the direct problem by Rusanov.