High Reynolds number steady separated flow past a wedge of negative angle

Abstract

According to classical boundary-layer theory, uniform flow past a semi-infinite wedge, inclined at a negative angle ½πβ to the direction of the free stream, does not separate unless β ≤−0·1988. It has been assumed, therefore, that, in the range −0·1988 < β < 0, the flow within the boundary layer is represented by the Falkner-Skan equation, which, as was shown by Stewartson (1954), has two admissible solutions. All such solutions for p < 0 appear to be somewhat unsatisfactory, however, because they require an adverse pressure gradient, which, by becoming infinite as the corner of the wedge is approached, could lead to separation even if, β > −0·1988. In addition, the structure of the high Reynolds number flow for β < −0·1988 has remained, to date, unresolved.

We present here a fundamentally different solution to this classical problem which eliminates the singularity in the potential region by allowing the flow to separate at the leading edge of the inclined surface. The associated flow field is then characterized by an essentially uniform free stream flowing over an inviscid and, to a high approximation, irrotational region of reverse flow in which the velocity is of O(R^−½) in magnitude, R being the Reynolds number. Mixing of these two streams is confined to a free shear boundary layer, of O(R^−½) in thickness, extending downstream from the leading edge and parallel to the direction of the undisturbed main flow. Finally, an additional boundary layer, of O(R^−½) in thickness, is shown to exist between the separated region and the surface of the wedge. Owing to the absence of a characteristic length in the problem, similar solutions to the appropriate equations describing the flow in each region are obtained and are valid for all β < 0, provided that the Reynolds number is sufficiently large. The analysis is then extended to higher order in R to increase its range of validity and to demonstrate that the proposed structure of the flow field remains self-consistent. Although the solution is developed only for a semi-infinite wedge with β < 0, it is believed that certain of its features may be of value in the analysis of other problems involving high Reynolds number separated flows.