Introduction

It is not infrequently claimed that the subject of turbulence contains the last great unsolved problems that classical physics has to offer. What are these problems, and how important are they? These are not easy questions to answer in a short space, and the answers sketched under four headings below are partial in two senses of the word. They are partial in that they are incomplete, and they are also partial in that they reflect the prejudices of someone whose understanding of the subject derives at second hand from what others have written about it.

The development of turbulence

Turbulence is often triggered by one of the instabilities discussed in chapter 8, and these have been exhaustively studied and seem well enough understood. Relatively little is known, however, about the processes which link trigger and explosion, i.e. which lead from an infinitesimal perturbation in one part of a fluid system to genuine turbulence downstream. Most fluid dynamicists probably believed until the 1970's that there were few general principles to be discovered in this area, apart from the essentially qualitative idea that once a state of laminar flow has been corrupted by one perturbation it tends to provide a breeding ground within which perturbations on a smaller scale may grow.

Introduction

Fluids were defined in §1.2 as materials which cannot withstand a shear stress, however small, without deforming, and it was suggested there that glaziers' putty should be classified as a plastic solid rather than a fluid because it appears to hold its shape indefinitely unless subjected to appreciable force. But does it really do so? If we were to watch it for a very long time (and to find some way of preventing it from drying out during the process) might we not see putty flow under its own weight? After all, lead pipes flow visibly under their own weight given a century or two in which to do so, as anyone in Cambridge may verify by inspection of some of the older buildings there. The process by which lead flows, known as creep, involves vacancy diffusion, and provided that a specimen of lead is poly crystalline on a fine scale its creep rate should be proportional to the shear stress acting upon it; if so, then according to the definition given in §1.2 it is a fluid – a liquid rather than a gas, of course – though its viscosity is certainly enormous, greater than the viscosity of water by many orders of magnitude. If apparently solid materials such as polycrystalline lead are really liquid, is putty really liquid too? And if putty is not, what about chewing gum, or toothpaste, or yoghurt, or mayonnaise, or a host of similar substances which do not appear to flow under their own weight but which flow readily enough when squeezed?

The model

The title of this chapter refers to the idealised model discussed in chapter 1, on which Euler and Bernoulli based their contributions to fluid dynamics. An Euler fluid by definition has zero viscosity and zero compressibility. A fluid without viscosity cannot sustain shear stress, and the pressure p within it is therefore isotropic at all points. A fluid without compressibility has a density ρ which is unaffected by variations of p from place to place. The model need not exclude small variations of density due to thermal expansion if the temperature is nonuniform, but such variations are normally irrelevant except in so far as they may drive thermal convection currents in the fluid. Consideration of the topic of convection is deferred to chapter 8. For the time being we may regard temperature as something which has no influence on the flow behaviour of our model fluid and which may therefore be ignored.

Some of the conditions which need to be satisfied if the model is to match the behaviour of real fluids have been discussed in chapter 1. The reader may wish to refer back to that, and to the summary in § 1.16 in particular.

The continuity condition

It is usually safe to assume that fluids remain continuous, and in that case the mass of fluid which occupies any volume V whose boundaries are fixed in space is just the integral over this volume of ρdx, where dx is a volume element.

Lines of vorticity

Most of this chapter concerns incompressible flow past solid obstacles, and the drag and lift forces which they experience, at values of the Reynolds Number which are too large compared with unity for the approximations employed in chapter 6, e.g. in the derivation of Stokes's law for the drag force on a solid sphere, to be valid. The effects to be discussed depend critically on the behaviour of boundary layers, and boundary layers, as we have seen, are layers within which the fluid is contaminated by vorticity. To understand these effects properly we need to understand how vorticity behaves, and that is why the chapter has ‘Vorticity’ as its heading.

The properties of free vortex lines, set in otherwise vorticity-free fluid, have already been described in §§4.13 and 4.14, but we can explore the subject of vorticity dynamics in a more general fashion now that we have the Navier–Stokes equation to use as a starting point. The first point to note is that because Ω is defined as the curl of another vector its divergence is necessarily zero everywhere; vorticity, like the electromagnetic fields E and B in free space and like the velocity u of an incompressible fluid, is what is called a solenoidal vector. This means that its spatial variation can be described by continuous field lines whose direction coincides everywhere with the local direction of Ω and whose density is proportional to the magnitude of Ω.

Every physicist should know some fluid dynamics, and every university physics department should include the subject in its core curriculum. Those propositions can readily be justified by pointing out the usefulness of the subject – its relevance to diverse areas of contemporary research and to a vast range of problems of practical importance. What counts as much for me, however, is that most of the students I have known at Cambridge have enjoyed their limited exposure to it. The notion that the only way to arouse the enthusiasm of physicists is to teach them about quarks and black holes is in my view a myth.

I hope that this book will slightly increase the chance that future generations of physicists will be taught the subject systematically, in a way that I and my contemporaries were not. However, since newer branches of physics may continue to displace it, I have tried to write something that may be read for pleasure as well as for instruction by physicists of any age and at almost any level of sophistication who want to learn fluid dynamics for themselves. They do, of course, have many books to choose from already, but most of them were written for mathematicians or engineers. Students of all three disciplines – mathematics, physics and engineering – speak the same language and have many objectives in common, but they differ in their approach to new problems because their intuition has been honed in different ways, and they also tend to differ in what they find interesting.

What is nonlinearity?

Introductory science textbooks – and much of our educational system, for that matter – are built on the idea that a natural system subjected to well-defined external conditions will follow a unique course and that a slight change in these conditions will likewise induce a slight change in the system's response. Owing undoubtedly to its cultural attractiveness, this idea, along with its corollaries of reproducibility and unlimited predictability and hence of ultimate simplicity, has long dominated our thinking and has gradually led to the image of a linear world: a world in which the observed effects are linked to the underlying causes by a set of laws reducing for all practical purposes to a simple proportionality.

Appealing and reassuring as it may sound, this perennial idea is now being challenged and shown to provide, at best, only a partial view of the natural world. In many instances – and as a matter of fact in most of those interfering with our everyday experience – we witness radical, qualitative deviations from the regime of proportionality. This book has to do with nonlinearity, that is to say, the phenomena that can take place under these conditions.

A striking difference between linear and nonlinear laws is whether the property of superposition holds or breaks down.

The Poincaré map

As we have seen throughout this monograph, in a nonlinear dynamical system the first bifurcation from a fixed point leads to fixed point or to limit cycle behavior. Chaotic behavior. which according to the experimental data surveyed in Chapter 1 is abundant in nature, can therefore arise smoothly from simple fixed point behavior only through a sequence of bifurcations involving high order (tertiary etc.) transitions. As a rule, at some stage of this sequence of transitions a periodic solution loses its stability, a fact that is also reflected in the experimental data where chaotic behavior seems to be much more intimately intertwined with periodic rather than steady-state behavior.

The above comments suggest that to gain an insight into the onset of chaos it is necessary to analyze the loss of stability and the subsequent bifurcation behavior of periodic solutions. Unfortunately, this task is unattainable. First, the analytic form of these solutions in the interesting parameter region is not known except in a number of exceptional situations. Second, even if the analytic form were known one would be led to study dynamical systems of the form of eq. (3.26) in which both the linearized operator and the nonlinear part h contain an explicit periodic dependence in time.