Transonic Region of a Supersonic Boundary Layer Turning a Sharp Corner

Abstract

T paper presents solutions for the flow variables in the transonic region of a steady, laminar, supersonic boundary layer turning a corner. The analysis holds in those cases where the corner is sharp and the pressure drop across the corner is large enough that separation occurs, viscous effects in the corner region are important only in a negligibly thin sublayer, and the base pressure to which the flow expands is less than the sonic pressure, P*, corresponding to the flow conditions at the wall upstream of the corner. It has been shown' that under these conditions, in viscid flow equations may be employed in a region extending a few boundary-layer thicknesses upstream of the corner; the sonic line intersects the corner to the scale of the inviscid region. Since this analysis is concerned only with the transonic flow in the inviscid region, it is valid in a region with dimensions small compared to those associated with the inviscid region, yet large compared to those associated with the viscous sublayer. It supplements Refs. 1 and 2 by supplying detailed transonic solutions not given in these references. Hence, the solutions given here fulfill the same role in the problem of the turning of a supersonic boundary layer around a corner, that Vaglio-Laurin's transonic corner flow solutions do in the problem of an inviscid flow turning a corner. Physical applications are the corner flows occurring at the lip of a rocket nozzle, at a rearward facing step, or at the base corner of a re-entry vehicle, for example. A systematic method of treating rotational transonic inviscid flow around a convex corner was presented in a previous paper by the present authors. In that paper, the method was applied to the transonic region in the case where the boundary layer approaching the corner had no pressure gradient or heat transfer. In this paper, the method is extended to cover the case of a laminar supersonic boundary layer with pressure gradient and heat transfer; it is demonstrated that the shape of the sonic line and the pressure gradient are fundamentally changed from the simple case. Y Y-4