A Solution of Shock-Induced Boundary-Layer Interaction Problems by an Integral Method

Abstract

An integral technique is developed to solve a general class of shock-induced boundary-layer interaction problems. Included in this class is the boundary layer which grows downstream of the leading edge of a semi-infinite flat plate with a shock wave propagating over it, and the boundary-layer region in a shock tube that is dependent upon both the shock wave and the expansion wave. The integral equations used to solve the Howarth transformed (incompressible) momentum equation are formulated in terms of a general two-parameter family of velocity profiles. These equations are solved for a velocity profile which is a linear combination of the two exact solutions valid at either end of the interaction region. The relative proportion of these two solutions is controlled by a shape factor similar to the Karman-Pohlhausen one in that it is controlled by the degree of unsteadiness in the boundary layer rather than by the pressure gradient. The solutions generated are in excellent agreement with published exact solutions, and all discontinuities in the slope of the shear stress present in earlier similar integral solutions are eliminated. The momentum and displacement thicknesses and the wall shear stress are predicted to within one percent of the exact values.