Two-dimensional high speed flow of a radiating gas

Abstract

In high speed gas dynamics the analysis of two-dimensional flows of hot gases, where the effect of radiation as a mode of energy transfer cannot be neglected, is extremely difficult due to the integro-differential nature of the governing equations. To date, most of the existing literature on radiation gas dynamics is confined to the study of the two asymptotic cases of optically thin or thick gas. For arbitrary optical properties, the integro-differential equations can be reduced to partial differential equations by applying the differential or the so-called Milne-Eddington approximation which is well-known in the astrophysical literature. In the case of supersonic flows, even this approximation does not simplify the analysis because the governing equations are elliptic in character. This elliptic nature of the equations does not allow the use of the methods already developed for non-radiating gaseous flows where the equations are hyperbolic in nature. The present paper deals with certain flow situations where two characteristic lengths are available in the system. An order of magnitude analysis, based on two lengths and the mean free path of radiation, has been carried out and the basic equations of the flow have been simplified to equations that are parabolic in character. This (after applying the Milne-Eddington approximation) is then applied to the supersonic flow past a wavy wall. Within the parametric range of the applicability, a comparison of the results of the present investigation with the exact known results of Cheng shows good agreement. The error introduced by the use of the Milne-Eddington approximation is also evaluated. Applications to supersonic and hypersonic flow past thin aerofoils are also discussed in detail.