Singularities encountered in three-dimensional boundary layers under an adverse or favourable pressure gradient

Abstract

Singularities in solutions of the classical boundary-layer equations are considered, numerically and analytically, in an example of steady hypersonic flow along a flat plate with three-dimensional surface roughness. First, a wide parametric study of the breakdown of symmetry-plane flow is performed for two particular cases of the surface geometry. Emphasis is put on the structural stability of the singularities’ development to local/global variation of the pressure distribution. It is found that, as usual, the solution behaviour under an adverse pressure gradient involves the Goldstein- or marginal-type singularity at a point of zero streamwise skin friction. As the main alternative, typical of configurations with favourable or zero pressure forcing, an inviscid breakdown in the middle of the flow is identified. Similarly to unsteady flows, the main features of the novel singularity include infinitely growing boundary-layer thickness and finite limiting values of the skin-friction components. Subsequent analytical extensions of the singular symmetry-plane solution then suggest two different scenarios for the global boundary-layer behaviour: one implies inviscid breakdown of the flow at some singular line, the other describes the development of a boundary-layer collision at a downstream portion of the symmetry plane. In contrast with previous studies of the collision phenomenon in steady flows, the present theory suggests logarithmic growth of boundary-layer thickness on both sides of the discontinuity. Finally, an example of numerical solution of the full three dimensional boundary layer equations is given. The flow régime chosen corresponds to inviscid breakdown of a centreplane flow under a favourable pressure gradient and development of the discontinuity/collision downstream. The numerical results near the origin of the discontinuity are found to be supportive, producing quantitative agreement with the local analytical description.