Stability of nonparallel flows: 'Minimal composite' theories

Abstract

The theory of linear stability of shear flows has been studied extensively over much of the last century. Most studies have been based on the Orr-Sommerfeld equation for parallel flows, but in recent decades there have been several attempts at more general theories, including the use of parabolized stability equations. As shear flows tend in general to be nonparallel, the question has remained about the formulation of a proper theory accounting for flow nonparallelism. Introducing the concept of minimal composite equations, with the use of similarity coordinates, it has been possible during the last ten years to develop a hierarchy of stability equations ranging from an ordinary differential equation like the Orr-Sommerfeld (but not identical to it) to partial differential equations like the PSE. The approach through minimal composite equations has now been extended to include effects of wing sweep and compressibility, and we present a review of these developments and their implications.