Interaction Between a Shock Wave and a Turbulent Boundary Layer in Transonic Flow

Abstract

Interaction between a shock wave and an unseparated turbulent boundary layer is considered. The method of matched asymptotic expansions is used, with solutions valid in the double limit as Reynolds number tends to infinity and Mach number tends to unity. The shock is weak enough that interaction effects can be considered as perturbations to the undisturbed flow; the case considered is that where the sonic line is near the outer edge of the boundary layer. The interaction region consists of four layers; the external flow region, and three boundary layer regions consisting of the two usual boundary layer regions (velocity defect and wall layers) and an intermediate Reynolds stress sublayer. It is shown that, with order estimates for Reynolds stress perturbations based on Ribner’s [17] calculations, the induced wall pressure distribution can be calculated using only the two outer regions, independent of a specific closure condition, and that this solution is in fact a turbulent free interaction solution. A detailed analysis of the two inner regions, for which an eddy viscosity model for the Reynolds shear stress is used, provides a description of the variations in velocity, temperature and density near and at the wall. The analysis suggests that the condition for incipient separation is $M_e - 1 = O( 1 )$, independent of Reynolds number, where $M_e $ is the external flow Mach number. Numerical calculations are presented for the case of an oblique incoming shock where the flow is everywhere supersonic; for this case it is shown that all of the shock induced flow variation occurs upstream of the shock impingement point.