Synthesis

Introduction

The plan of this text has to been to describe the important general concepts and methods of hydrodynamic stability in the opening chapters, and then to apply them to selected flows in the later chapters. The flows have been selected partly for their mathematical simplicity, partly for their historical importance (and these two reasons are connected), and partly for their physical value. Many of the resultant problems are very idealized; yet all of the problems are much more widely applicable than their precise form might at first sight suggest. The theory of Rayleigh–Bénard convection, for example, may be used to interpret not just instability of an infinite thin horizontal layer of fluid heated below, but many convective instabilities of flows which locally resemble a thin layer of fluid heated from below. The theory of Taylor vortices may be used to interpret instabilities of flows with curved streamlines such that there is a local centrifugal force. The theory of Görtler vortices can be applied to interpret the local instabilities of flows whose streamlines are convex, so that this mechanism is complementary to the mechanism of Taylor vortices, to be applied when the streamlines or the wall ‘bend the other way’. The theory of instability of parallel flows, with Rayleigh's inflection-point theorem and the Orr–Sommerfeld problem, may be used to interpret instabilities of flows that are nearly parallel, at least locally; indeed, it has already been used to interpret instabilities of boundary layers, jets and free shear layers. The use of these idealized problems to interpret instabilities of more complicated flows is valuable, but is not easy until one has a lot of experience of hydrodynamic instability.