A low-order theory for stability of non-parallel boundary layer flows

As a sequel to the earlier analysis of Govindarajan & Narasimha, we formulate here the lowest-order rational asymptotic theory capable of handling the linear stability of spatially developing two-dimensional boundary layers. It is shown that a new ordinary differential equation, using similarity-transformed variables in Falkner-Skan flows, provides such a theory correct upto (but not including) O(R-2/3), where R is the local boundary layer thickness Reynolds number. The equation so derived differs from the Orr-Sommerfeld in two respects: the terms representing streamwise diffusion of vorticity are absent; but a new term for the advection of disturbance vorticity at the critical layer by the mean wall-normal velocity was found necessary. Results from the present lowest-order theory show reasonable agreement with the full O(R-1) theory. Stability loops at different wall-normal distances, in either theory, show certain peculiar characteristics that have not been reported so far but are demonstrated here to be necessary consequences of flow non-parallelism.