A point explosion in a cold exponential atmosphere. Part 2. Radiating flow

Abstract

The problem considered is that of a strong shock propagating from a point energy source into a cold exponential atmosphere with radiative heat transfer in the flow behind the shock. The radiation mean free path is taken to be small compared to the shock radius, so that the shock may be treated as discontinuous and the radiative heat flux represented by the Rosseland diffusion approximation. The solution obtained is an approximate one based upon the ‘local radiality’ assumption and integral method previously utilized by Laumbach & Probstein for the case of adiabatic flow. The ‘thin shock’ concepts, which underlie the integral method, are extended to the present case of a radiating flow enabling an approximate integral of the differential energy equation to be obtained. A radiation parameter is developed, which provides an index as to when the effects of radiation may be neglected and the flow taken to be adiabatic. The physical interpretation of this parameter is that of the ratio of a characteristic radiation energy flux to a characteristic kinetic energy flux. When the value of this parameter is less than about 0·1, radiation effects may be neglected. It is shown that, when the radiation mean free path varies as a power of the temperature (Tⁿ), where $n = -\frac{17}{6}$, the infinity of solutions for various polar angles can be transformed into two distinct solutions thereby essentially eliminating the parametric dependence on the polar angle and the atmospheric scale height. For fixed values of the radiation parameter the dependence on the explosion energy and the atmospheric density at the point of explosion is also eliminated. The results presented are for the mean free path–temperature variation indicated, but the technique of solution does not have this restriction, though for other temperature dependences some of the scaling advantages are sacrificed. The solution demonstrates the existence of an independence principle in which the flow becomes isothermal and independent of the detailed radiation mechanism when the radiation parameter becomes large. The limiting results of the present analysis for a uniform density atmosphere correspond quite, well with the exact solutions of Elliott & Korobeinikov. The radiating far field behaviour of the descending shock is shown to approach that for adiabatic flow, and, consequently, the asymptotic adiabatic solution obtained by Raizer is an appropriate limit to the present solution. However, the asymptotic results for the ascending shock show that the radiating flow does not approach an adiabatic one but rather an isothermal one.