Introduction

The essential problems of hydrodynamic stability were recognized and formulated in the nineteenth century, notably by Helmholtz, Kelvin, Rayleigh and Reynolds. It is difficult to introduce these problems more clearly than in Osborne Reynolds's (1883) own description of his classic series of experiments on the instability of flow in a pipe.

For nearly a century now, hydrodynamic stability has been recognized as one of the central problems of fluid mechanics. It is concerned with when and how laminar flows break down, their subsequent development, and their eventual transition to turbulence. It has many applications in engineering, in meteorology and oceanography, and in astrophysics and geophysics. Some of these applications are mentioned, but the book is written from the point of view intrinsic to fluid mechanics and applied mathematics. Thus, although we have emphasized the analytical aspects of the theory, we have also tried, wherever possible, to relate the theory to experimental and numerical results.

Our aim in writing this book has been twofold. Firstly, in Chapters 1–4, to describe the fundamental ideas, methods, and results in three major areas of the subject: thermal convection, rotating and curved flows, and parallel shear flows. Secondly, to provide an introduction to some aspects of the subject which are of current research interest. These include some of the more recent developments in the asymptotic theory of the Orr–Sommerfeld equation in Chapter 5, some applications of the linear stability theory in Chapter 6 and finally, in Chapter 7, a discussion of some of the fundamental ideas involved in current work on the nonlinear theory of hydrodynamic stability.

Each chapter ends with a number of problems which often extend or supplement the main text as well as provide exercises to help the reader understand the topics.

Introduction

The approximations to the solutions of the Orr–Sommerfeld equation which were derived in § 27 suffer from two major defects. First, the approximations (except for the regular inviscid solution) are not uniformly valid in a full neighbourhood of the critical point and, second, the theory does not lead to a systematic method of obtaining higher approximations. In this chapter, therefore, we shall describe some of the attempts which have been made to overcome these deficiencies of the heuristic theory. These improved theories have generally been based on either the comparison-equation method or the method of matched asymptotic expansions. Although neither method is entirely satisfactory by itself, both have played important roles in the development of the subject.

The comparison-equation method has been extensively studied by Wasow (1953b), Langer (1957, 1959), Lin (1957a,b, 1958), Lin & Rabenstein (1960, 1969) and others. In all of this work the major aims have been to obtain asymptotic approximations to the solutions of the Orr–Sommerfeld equation which are uniformly valid in a bounded domain containing one simple turning point and to develop an algorithm by which higher approximations can be obtained systematically.

Of the problems in the linear theory of hydrodynamic stability which have been solved, only a few have been presented in the previous chapters. These few were selected both to include the major results and to illustrate the fundamental ideas of the theory. A few more problems are treated briefly in this chapter, some to indicate the variety of the applications of the theory and others to cover more advanced topics.

Instability of parallel flow of a stratified fluid

Introduction

In Chapter 4 we developed the theory of stability of parallel flow of an inviscid fluid because it has direct applications and because it is fundamental to the theory of a viscous fluid. However, it is fundamental also to the theory of stability of parallel flow of an inviscid fluid under the action of external forces. There are many force fields important in diverse applications, but lack of space obliges us to confine our attention to a single external force, and we have chosen the force of buoyancy because it is as important as any for both theoretical developments and practical applications, notably to meteorology and oceanography.

Introduction

Thermal instability often arises when a fluid is heated from below. The classic example of this, described in this chapter, is a horizontal layer of fluid with its lower side hotter than its upper. The basic state is then one of rest with light fluid below heavy fluid. When the temperature difference across the layer is great enough, the stabilizing effects of viscosity and thermal conductivity are overcome by the destabilizing buoyancy, and an overturning instability ensues as thermal convection. This convective instability may be distinguished from free convection, such as that due to a hot vertical plate, for which hydrostatic equilibrium is impossible. Again, a basic flow of free convection may itself be unstable. Our concern here with convection is only with thermal instability. Convective instability seems to have been first described by James Thomson (1882), the elder brother of Lord Kelvin, but the first quantitative experiments were made by Bénard (1900).

Rayleigh (1916a) wrote that

Introduction

Instability also occurs in a homogeneous fluid owing to the dynamical effects of rotation or of streamline curvature. Three important examples of flows which exhibit this type of centrifugal instability are shown in Fig. 3.1. They are Couette flow, in which the fluid is contained between two rotating coaxial cylinders; flow in a curved channel due to a pressure gradient acting around the channel; and the flow in a boundary layer on a concave wall. In the absence of curvature, the last two of these examples may exhibit the type of instability associated with parallel shear flows, which will be discussed in Chapter 4.

The instability of rotating flows was first considered by Rayleigh (1880, 1916b). He considered a basic swirling flow of an inviscid fluid which moves with angular velocity Ω(r), an arbitrary function of the distance r from the axis of rotation. By a simple physical argument Rayleigh then derived his celebrated criterion for stability.

The study of hydrodynamic stability goes back to the theoretical work of Helmholtz (1868), Kelvin (1871) and Rayleigh (1879, 1880) on inviscid flows and, above all, the experimental investigations of Reynolds (1883), which initiated the systematic study of viscous shear flows. Reynolds's work stimulated the theoretical investigations of Orr (1907) and Sommerfeld (1908), who independently considered small, traveling-wave disturbances of an otherwise steady, parallel flow and derived (what is now known as) the Orr–Sommerfeld equation.

Early attempts to solve the Orr–Sommerfeld equation for the flow associated with the uniform, relative motion of two parallel plates (plane Couette flow) led to the prediction of stability for all Reynolds numbers, in apparent disagreement with experiment (although the prediction is correct for infinitesimal disturbances). G. I. Taylor (1923), referring to this work, remarked that:

Introduction

In many internal flows there are only limited regions in which the velocity can be considered irrotational; i.e. in which the motion is such that particles travel without local rotation. In an irrotational, or potential, flow the velocity can be expressed as the gradient of a scalar function. This condition allows great simplification and, where it can be employed, is of enormous utility. Although we have given examples of its use, potential flow theory has a narrower scope in internal flow than in external flow and the description and analysis of non-potential, or rotational, motions plays a larger role in the former than in the latter. One reason for this difference is the greater presence of bounding solid surfaces and the accompanying greater opportunity for viscous shear forces to act. Even in those internal flow configurations in which the flow can be considered inviscid, however, different streamtubes can receive different amounts of energy (from fluid machinery, for example), resulting in velocity distributions which do not generally correspond to potential flows. Because of this, we now examine two key fluid dynamic concepts associated with rotational flows: vorticity, which has to do with the local rate of rotation of a fluid particle, and circulation, a related, but more global, quantity.

Before formally introducing these concepts, it is appropriate to give some discussion concerning the motivation for working with them, rather than velocity and pressure fields only. The equations of motion for a fluid contain expressions of forces and acceleration, derived from Newton's laws.

Introduction

In this chapter we address three-dimensional flows in which streamwise vorticity is a prominent feature. Three main topics are discussed. The first, and principal, subject falls under the general label of secondary flows, cross-flow plane (secondary) circulations which occur in flows that were parallel at some upstream station. The second is the enhancement of mixing by embedded streamwise vorticity and the accompanying motions normal to the bulk flow direction (see for example Bushnell (1992)). The third is the connection between vorticity generation and fluid impulse.

The different topics are linked in at least three ways. First, the class of fluid motions described are truly three-dimensional. Second, focus on the vortex structure in these flows is a way to increase physical insight. The perspective of the chapter is that the flows of interest are rotational and three-dimensional, and the appropriate tools for capturing their quantitative behavior are three-dimensional numerical simulations (e.g. Launder (1995)). Results from such computations, as well as from experiments, are used to illustrate the overall features. To complement detailed simulations and experiments, however, it is often helpful to have a simplified description of the motion which can guide the interrogation and scope of the computations, enable understanding of why different effects are seen, and suggest scaling for different mechanisms. The ideas about vorticity evolution and vortex structure, introduced in Chapter 3, provide a skeleton for this type of description.

Introduction

In this chapter the discussion of fluid component and system response to disturbances, begun in Chapter 6, is extended to a broader class of flow non-uniformities. Whereas Chapter 6 considered primarily one-dimensional disturbances, that restriction is now dropped and we address more general (two- and three-dimensional) non-uniformities with variations transverse to the bulk flow direction. Examples of interest are turbomachines subjected to circumferentially varying inlet conditions and the behavior of components with geometry generated non-uniformity, such as is caused by a contraction or a bend in close proximity.

Three important issues relating to these situations can be identified. One is the effect of the fluid component on the flow non-uniformity, or distortion: how are the non-uniformities altered by passage through the component? A second is the effect of the non-uniformity on the component: how does the distortion modify the component performance? The approaches needed to address these two questions are fundamentally different. For the former, qualitative aspects, and even many quantitative features, can be resolved within the framework of a linearized description. For the latter, however, the problem is inherently nonlinear and a different level of analysis is needed. Beyond component performance there is a third issue. Because fluid components typically occur as part of an overall system, what changes in interactions with the rest of the system arise due to the non-uniformity?

Several integrating themes thread through the different applications discussed. The first is that fluid components do not passively accept non-uniform flow but play a major role in modifying the velocity distribution.