Interaction Between a Normal Shock Wave and a Turbulent Boundary Layer at High Transonic Speeds. Part II: Wall Shear Stress

When a shock wave impinges upon a wall, it penetrates the boundary layer along the surface and both the shock wave and the boundary layer are changed from their undisturbed states. If the boundary layer remains unseparated, these mutually induced changes take place in a small interaction region. For a turbulent boundary layer, it has been established [1-8] that an asymptotic description of the interaction region requires a three layer structure. In the outermost layer, comprising most of the boundary layer, pressure forces are much larger than forces resulting from Reynolds or viscous stresses so the governing equations are those for an inviscid flow. For the limit process to be considered, the solutions for this inviscid flow region are those given in Part I of this paper [9], hereafter referred to as (I). Immediately adjacent to the wall is the wall layer, in which viscous and Reynolds stresses dominate to lowest order. Between these two layers is the Reynolds stress sublayer [1] (referred to as the blending layer in reference [2]) in which momentum transfer toward the wall is carried out by turbulent means (Reynolds stresses); the dominant terms in the equation of motion are the Reynolds stress, pressure gradient, and inertia terms. This paper is concerned with the analysis of the flow in the two inner layers, the Reynolds stress sublayer and the wall layer, the goal being the calculation of the shear stress at the wall in the interaction region. As indicated above, the limit processes considered are those used in (I). Thus, if E is equal to the nondimensional difference between the velocity and the critical sound speed in the flow external to the boundary layer, and u~ is the nondimensional friction velocity, we consider limit processes such that u~ << ~ << 1. In previous analyses for E << u~ (Reference [1]) and ~ = O(u 0 (Reference [2]) it was found that it was not possible to formulate an asymptotic criterion for shock induced separation. Here, it will be shown that even for E >> u~ there is no apparent asymptotic separation criterion. However, example calculations