Is the singularity at separation removable?

Abstract

Many numerical integrations support Goldstein's theory of the structure of the solution of a laminar boundary layer near the point of separation O when the mainstream is prescribed, and in particular confirm that the solution is singular there. The existence of the singularity, however, implies that the hypotheses of the boundary layer break down in the neighbourhood of O, and it has been suggested that the disturbance to the mainstream near O is sufficient to smooth out the singularity and enable the solution to pass over into another conventional boundary layer downstream of O containing a region of reversed flow. The aim of this paper is to explore this possibility in detail using the methods of the triple-deck, developed by the author and others, which have proved successful in somewhat related problems.Granted the hypothesis that the interaction between the boundary layer and the mainstream is significant near separation and manifests itself through a triple deck, it is found that its streamwise extent is O(ε2l) where ε−8 is a characteristic Reynolds number, ε [Lt ] 1, and l a characteristic length of the problem. The upper deck is of width O(ε2l), lies entirely outside the boundary layer, and in it the flow is inviscid. The main deck is of width O(ε4l) and constitutes the majority of the boundary layer near O, and the perturbations in the velocity are largely inviscid. Finally, the lower deck is of lateral extent and is controlled by a linear equation of boundary-layer type. The whole structure is found to be consistent provided a certain integro-differential equation can be solved, which takes different forms according as the mainstream is supersonic or subsonic. When the mainstream is subsonic it is found that there is no solution to this equation that is sufficiently smooth on the downstream side of the triple deck. When the mainstream is supersonic it is found that the triple deck can at best postpone the breakdown of the assumed structure which still must occur within a distance O(ε2l) of O.It is concluded that the singularity is not removable by the methods proposed and it is inferred that the singularity is a real phenomenon terminating the flow which, at high Reynolds number, exists upstream of O.