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

Rama Govindarajan, R. Narasimha

Abstract

As a sequel to the earlier analysis of Govindarajan and 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.