Boundary-layer flow along a ridge: alternatives to the Falkner-Skan solutions

We consider the laminar boundary-layer flow past a semi-infinite plate with a streamwise ridge. We seek similarity solutions to the problem, when the freestream velocity takes the form x*n, where x* denotes the distance from the leading edge of the plate; such solutions may exist if the transverse and lateral scales of the ridge develop in the streamwise direction at the same rate as the boundary-layer thickness grows. In deriving the necessary far-field boundary conditions for these calculations, we are led to a consideration of a class of flows of the Falkner-Skan type, but which may possess a cross-flow component of velocity (which grows linearly in the cross-flow direction). This new class of how is a three-dimensional alternative to the Falkner-Skan family. Wall transpiration effects are also addressed and portions of the solution curves correspond to separated flows. Solutions for the flow along a ridge for both the aforementioned classes of far-field behaviour are presented. A study of the effects of relaxing the similarity constraint on both the classical solution and new families of solution is also made. It is found that the problem is (frequently) complicated by the existence of spatially developing eigensolutions (originating from the leading edge), which have the effect of rendering standard parabolic marching procedures ill posed.