Compressibility effects in thin channels with injection

Abstract

A theoretical analysis of the inviscid flow between a porous plate and a parallel impermeable plate is performed for small values of the ratio of the plate separation distance to the lateral extent of the plates, for both planar and axisymmetric geometries. The problem of computing the flowfield is reduced to the solution of a single integral equation, which is accomplished numerically. The ratio of specific heats 7 is a parameter of the solution, and parametric results are presented from 7 = 1.0 to 1.67. The flow exhibits choking at a critical value of the lateral extent of the plate, in the vicinity of which the Mach number approaches unity. The results are needed in providing external boundary-layer conditions for studying the flame structure in the viscous region between two counterflowing streams when compressibility is important. UPERSONIC combustion in turbulent mixing layers lo- cally can involve counterflowing streams in which com- pressibility is important, with the viscous and chemical effects restricted to a narrow layer at the stagnation plane. The outer portions of such counterflows are inviscid and do not experi- ence exothermicity. As a first step toward analyzing the exothermic viscous layer, descriptions are needed of the invis- cid counter flow. The present paper provides a solution for a model of the inviscid outer region. The model addresses the flow between a porous plate and a parallel impermeable plate for small values of the ratio of the separation distance between the plates to the lateral extent of the plates. Two such flows can describe the inviscid counter flow, with the viscous layer located at the impermeable plate between them. In addition, laboratory experiments can be designed, involving opposed flow between two porous plates, for studying diffusion-flame combustion under conditions in which compressibility is im- portant. The present analysis provides the external-flow con- ditions required for the viscous-layer study. Theoretical analyses of flows between porous plates at high Reynolds numbers began with Proudman's study1 of an in- compressible, nonreacting fluid in a two-dimensional channel with porous walls through which the fluid is injected uni- formly. Extensions of this work addressed both heat transfer2 and combustion3 in the same type of configuration. The last of these studies was directed toward facilitating quantitative in- terpretation of results of experimental measurements on coun- ter flow diffusion flames.46 Similar experimental configura- tions have been employed in much earlier work on linear pyrolysis of vaporizable or decomposable materials, in which steady regression rates were measured when the material was pressed against a heated, impermeable flat plate.7 The poten- tial importance of compressibili ty in such experiments has been pointed out by Cantrell,8 who had earlier9 completed an analysis of such flows including compressibility effects but not considering the simplifications that arise in the limit of large Reynolds numbers, addressed herein.