Radiative interactions in boundary layers

Abstract

The flow of a compressible, viscous, heat-conducting, radiating gray gas parallel to a gray semi-infinite flat plate is considered. The radiation field is described by using the differential approximation. N, the ratio of photon mean free path to molecular mean free path and Bo, the Boltzmann number, the ratio of a representative convective energy flux to radiative energy flux, are identified as the two new non-dimensional parameters that enter the problem. Calculations for high-temperature air indicate the useful asymptotic limits in terms of these two parameters. For these as well as for other asymptotic limits we make an asymptotic analysis of the problem employing a formalism originally suggested by Kruskal and subsequently employed by Solan and Cohen in their study of the radiating Rayleigh problem. The Kruskal formalism leads to what are called optimal scales for the flow, which, while maximising the information about the flow, achieve considerable simplifications for the original set of equations. A feature of the present work is the recognition of the existence of non-optimal scales of the flow which lead to useful engineering approximations. For N > 1, Bo > 1 and asymptotically finite Mach number we have the most interesting case of dominant interaction between radiation and other modes of energy transport. The total flow field can be viewed in terms of simpler asymptotic descriptions. The viscous boundary layer close to the plate is optically thin and outside this is an inviscid region of arbitrary optical thickness which provides a sink for the radiation leaving the plate and the boundary layer. As an important result we recognise the existence of a conduction layer close to the plate, for asymptotically large distance along the flow. This region represents a balance between radiation and conduction with no flow to the leading order. Outside the conduction layer is an optically thick inviscid region. The radiation field is adequately described everywhere by a one-dimensional formulation. As a case of practical importance, a weakly interacting radiating boundary layer flow receives a detailed treatment. Relevant higher-order effects are taken into account and solution is obtained by the method of matched asymptotic expansions. Comparison with earlier work in the literature demonstrates the need for a realistic description of long-range radiative interaction. Solutions for the strong interaction case are provided by using the simplified descriptions resulting from the asymptotic analysis. The simpler case of linear radiation is considered both by exact and approximate methods with a view to extend the analysis to the case of nonlinear radiation. Exact numerical methods are also used in the case of nonlinear radiation. The boundary layer equations are solved by the assumption of local similarity. The conduction layer equation is easily reduced to an initial value problem requiring a numerical solution. The results presented speak for the usefulness of the approximate methods developed in the present study and also demonstrate the merits of using the differential approximation for radiative transfer in boundary layer flows. When Mach number is asymptotically active the flow shows many new details. An asymptotic analysis shows close agreement with the accepted descriptions of hypersonic interaction theory. When radiation is present the value of Boltzmann number determines the nature of interaction. Solution is provided for a simple case corresponding to equal wall and free-stream temperatures and dominant radiation interaction in the weak shock limit.