Origin and growth of structures in boundary layer flows

Abstract

A central goal of high Reynolds number fluid dynamics is to gain a comprehensive understanding of the origin and growth of perturbations in shear flows in which the rate of strain of the background velocity provides the source of energy of the perturbations. In all cases the availability of energy for perturbation growth can be determined by linearizing the equations of motion about the appropriate background flow and searching for growing perturbations. If all possible perturbations are examined and only decaying ones are found then it is certain that the background flow will persist when subjected to a sufficiently small disturbance. However, determining the potential for growth of all possible perturbations has not been the historical course of inquiry in stability theory. Rather, traditional stability theory follows the program of Rayleigh (1880) according to which instability is traced to the existence of exponentially growing modes of the linearized dynamic equations. The classical application of the normal mode paradigm envisions unstable modes growing exponentially from infinitesimal beginnings over a large number of e-foldings so that the exponential mode of greatest growth eventually emerges as a finite amplitude wave. This assumption of undisturbed growth is necessary to ensure the asymptotic dominance of the most rapidly growing normal mode which in turn permits the theory to make predictions concerning the structures of finite amplitude. Acceptance of the theory of small oscillations was encouraged by its success in application to problems such as the Rayleigh-Benard and Rayleigh-Taylor problems.