A three-dimensional boundary-layer separation

Abstract

A nonlinear three-dimensional boundary-layer problem governing the flow upstream of a particular disturbance (e.g. a shallow obstacle) at the wall is considered. The upstream response, a free interaction, takes place under zero displacement of the boundary layer, and the solution is found numerically using Fourier series truncation and varying the number of terms kept in the series. In one part of the flow field regular separation is encountered, beyond which the motion becomes strongly attached to the wall elsewhere in the flow field. Analytically, local structural investigations then suggest that the attached part of the upstream response terminates at a line singularity, while the separated part can continue indefinitely far downstream. The former structure leads to a new set of similarity solutions of the three-dimensional boundary-layer equations, while the latter develops a vortex sheet formation. The three-dimensional flow problem has most relevance to pipe flows, but some connexion also with external flows, and the implications for these are discussed.