Accuracy of differential approximation for radiative ionization in cold precursor of axisymmetric detached shock

Abstract

Radiative ionization in the cold precursor of the shock preceding a blunt body immersed in a hypersonic monatomic gas flow is studied using the Milne-Eddington type differential approximation. In order to assess the accuracy of the differential approximation for a multidimensional, axisymmetric geometry, the results are compared with earlier work that presents an exact formulation. The precursor gas is non-emitting with a prescribed frequency dependent absorption coefficient, and the governing differential equation describing the radiation in the precursor is the (elliptic) Helmholtz equation; specification of the boundary conditions pose an interesting dilemma. An integral formulation is proposed and the resulting integral equation is solved to yield the spectral radiant energy density. Calculations of the volumetric photon absorption and the degree of ionization for argon in the near precursor are presented for a downstream degree of ionization on the stagnation streamline of 0.8, an upstream temperature of 300°K, and an upstream pressure of 10 -3 atm. The results are found to be sufficiently accurate for many purposes when the influenced point is only a few photon mean free paths from the surface and is close to the shock axis.