Minimal composite equations and the stability of non-parallel flows

Abstract

The stability of a laminar boundary layer has classically been analysed in terms of the solutions of the Orr-Sommerfeld equation, which assumes that the flow is parallel. The purpose of this paper is to summarize the principles underlying the work done by the authors on non-parallel flows. This work adopts an asymptotic approach that involves the formulation of what we shall call 'minimal composite equations' in the limit of large Reynolds numbers. These equations include every term that is important somewhere, and none that is important nowhere, 'importance' being defined in terms of errors to some prescribed order in the local Reynolds number. This approach leads to a hierarchy of stability equations of successively increasing accuracy, including, in the lowest order, an ordinary differential equation for similarity flows, a low-order parabolic partial differential equation in the next order, and finally a 'full nonparallel' equation which is equivalent to the parabolized stability (partial differential) equations of Bertolotti et al.¹. The o.d.e., written here in similarity variables, is similar to but not identical with the Orr-Sommerfeld. Typical results from the present approach are given to illustrate the nature of the stability 'surface' derived from the present theory, and the accuracy of the computed amplitude distributions.