More advanced properties of instabilities will be described in this chapter. The development of normal modes in space as well as time, the effect of weak nonlinearity and the energy budget will be explained.

*The Development of Perturbations in Space and Time

For partial differential systems, such as those describing fluid motions, it is valuable to analyse the nature of stability in more detail.

First, note that if a flow is bounded (and, of course, in practice all flows are bounded), then there is in general a countable infinity of normal modes, but that if the flow is unbounded then there is an uncountable infinity of normal modes; for the Poiseuille pipe flow of Example 2.11, which is unbounded in the x-direction, there is a continuum of modes with a continuous wavenumber k as well as discrete wavenumbers for θ- and r-variations, but for flow in a cube there would be three discrete wavenumbers to specify each normal mode. So for an unbounded flow the most unstable mode can be no more than first among equals, but for a bounded flow the growth rate of the most unstable mode will in general be substantially greater than that of the second most unstable mode. For bounded flows of large aspect ratio (or large Reynolds number), the most unstable modes are usually close together and so approximate a continuum.

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.

In this chapter the text begins with an informal introduction to the concept of stability and the nature of instability of a particular flow as a prototype – the flow along a pipe. The prototype illustrates the importance of instability as a prelude to transition to turbulence. Finally, the chief methods of studying instability of flows are briefly introduced.

Prelude

Hydrodynamic stability concerns the stability and instability of motions of fluids.

The concept of stability of a state of a physical or mathematical system was understood in the eighteenth century, and Clerk Maxwell (see Campbell & Garnett, 1882, p. 440) expressed the qualitative concept clearly in the nineteenth:

So hydrodynamic stability is an important part of fluid mechanics, because an unstable flow is not observable, an unstable flow being in practice broken down rapidly by some ‘small variation’ or another. Also unstable flows often evolve into an important state of motion called turbulence, with a chaotic three-dimensional vorticity field with a broad spectrum of small temporal and spatial scales called turbulence.

In this chapter, we shall draw together some general features of the onset of chaos and turbulence. The theory of dynamical systems, and in particular the theories of bifurcation and chaos, provide a mathematical framework with which we may interpret qualitatively the transition to turbulence without having to clutter our minds with a lot of detail. This framework can be used together with physical arguments of the mechanics of transition to understand the essence of instability of flows which may be so complicated geometrically as to defy solution except in numerical terms. However, the dynamics of fluids is very diverse, and the details of transition to turbulence depend on the details of the flow undergoing transition, and therefore can only be found by careful experiments and computational fluid dynamics of each case.

Evolution of Flows as the Reynolds Number Increases

The details of transition to turbulence not only are complicated but also vary greatly from flow to flow, so there is no possibility of a short summary of all transition. However, there are some unifying themes in the theory, and a few routes to turbulence essentially shared by many flows, even though the physical mechanisms of the same route may differ from one flow to another sharing the same route.

This text arose from notes on lectures delivered to M.Sc. students at the University of Bristol in the 1980s. The notes were revised and printed for a course of lectures delivered to postgraduates at the University of Tokyo in 1995. The latter course led to collaboration with Professor Tsutomu Kambe in writing in Japanese the book Ryutai Rikigaku – Anteisei To Ranyu (Fluid Dynamics – Stability and Turbulence), published by the University of Tokyo Press in 1998. The present book is an enlargement in English of the first part of the Japanese book. An advanced draft was prepared for a lecture course given to undergraduates and postgraduates at the University of Oxford in 2001. I am grateful to the many students, at Bristol, Tokyo and Oxford, for their stimulating me to clarify both my ideas and their expression, and their encouragement to learn more. I am especially grateful to Professor Kambe for what I learnt from him and put into the text.

The result is a textbook, not a research monograph. To be sure, many points of current research have been incorporated in the text, but there has been no attempt to lead the reader up to the frontier of current research. So the mathematical theory has been described as simply and briefly as was felt possible, and plenty of worked examples and accessible exercises for students have been included. I have cited many publications, perhaps because the habit of doing so is deeply ingrained, certainly not because I ever imagined that many students care about references, let alone follow them up. The overt intention of including the references is to encourage students' instructors to follow up various details and, most importantly, use still and moving pictures to supplement this book in their teaching.

It was pointed out in the Preface that methods of investigation of the uniqueness and solvability for the water-wave problem depend essentially on the type of obstacle in respect to its intersection with the free surface. Among various possibilities, the simplest one is the case in which the free surface coincides with the whole horizontal plane (and so rigid boundaries of the water domain are represented by totally submerged bodies and the bottom of variable topography); we restrict our attention to this case in the present chapter.

We begin with the method of integral equations (Section 2.1), which not only provides information about the unique solvability of the water-wave problem but also serves as one of the most frequently used tools for a numerical solution of the problem. In Section 2.2, various geometric criteria of uniqueness are obtained with the help of auxiliary integral identities. The uniqueness theorem established allows us to prove the solvability of the problem for various geometries of submerged obstacles in Section 2.3. The last section, Section 2.4, contains bibliographical notes.

Method of Integral Equations and Kochin's Theorem

When Green's function is constructed it is natural to solve the water-wave problem by applying integral equation techniques, which is a standard approach to boundary value problems. In doing so, a proof of the solvability theorem for an integral equation is usually based on Fredholm's alternative and the uniqueness of the solution to the boundary value problem.