A Method for Analyzing the Heat Insulating Properties of the Laminar Compressible Boundary Layer

Abstract

In some cooling problems associated with high energy flows it may be convenient to localize strongly the cooling, as for example by injecting a coolant through an upstream porous strip, and to depend on the insulating properties of the boundary layer to reduce, or to eliminate completely the need for further cooling on the surface downstream of the highly cooled section. This upstream cooling technique may be of interest in connection with optical windows in hypersonic wind tunnels, and on radomes, wings, and bodies of high-speed aircraft and missiles. In this paper a method for investigating the insulating properties of a laminar compressible boundary layer on a two-dimensional surface with zero heat transfer is presented. The physical situation considered thus corresponds to the case in which the heat transfer downstream of the strongly cooled section is completely eliminated. Of practical concern is how the temperature of the uncooled surface varies in the downstream direction from its low initial value and thus how the low energy layer established by the upstream cooling insulates the downstream surface. The Karman integral method extended to both the momentum and energy partial differential equations of the boundary layer has been used. The station, at which cooling and/or injection ceases, corresponds to a discontinuity in boundary conditions and thus in solutions. At this point the flux of mass, momentum, and energy within the boundary layer has been made continuous by the introduction of three additional parameters in the velocity and stagnation enthalpy profiles. Thus the velocity and stagnation enthalpy profiles have both been taken as sixth degree polynomials. The resulting two integral-differential equations are then solved for two unknown functions of the distance along the wall. These two functions are related to the boundary-layer thickness and to the wall temperature. Initial conditions corresponding to a given initial wall temperature and an initial boundary-layer thickness are prescribed. Exact closed-form solutions for the case of zero axial pressure gradient are obtained. For flows with significant pressure gradients, numerical solutions are required in general. Several numerical examples of practical interest are presented. H M P R S T T t u U V W