Andrei Nikolaevich Kolmogorov's work in 1941 remains a major source of inspiration for turbulence research. Great classics, when revisited in the light of new developments, may reveal hidden pearls, as is the case with Kolmogorov's very brief third 1941 paper ‘Dissipation of energy in locally isotropic turbulence’ (Kolmogorov 1941c). It contains one of the very few exact and nontrivial results in the field, as well as very modern ideas on scaling, ideas which cannot be refuted by the argument Lev Landau used to criticize the universality assumptions of the first 1941 paper.

Revisiting Kolmogorov's fifty-year-old work on turbulence was one goal of the lectures on which this book is based. The lectures were intended for first-year graduate students in ‘Turbulence and Dynamical Systems’ at the University of Nice–Sophia–Antipolis. My presentation deliberately emphasizes concepts which are central in dynamical systems studies, such as symmetry-breaking and deterministic chaos. The students had some knowledge of fluid dynamics, but little or no training in modern probability theory. I have therefore included a significant amount of background material. The presentation uses a physicist's viewpoint with more emphasis on systematic arguments than on mathematical rigor. Also, I have a marked preference for working in coordinate space rather than in Fourier space, whenever possible.

Modern work on turbulence focuses to a large extent on trying to understand the reasons for the partial failure of the 1941 theory.

Introduction

This chapter is organized as follows. The basic concepts are introduced in Section 8.2. Experimental results about intermittency in the inertial range, based on velocity measurements, are presented in Section 8.3. Exact results, independent of any phenomenology, are presented in Section 8.4. Two broad classes of phenomenological models of intermittency are then discussed. In the first class (Section 8.5), intermittency is studied via velocity increments. It comprises the β-model (Section 8.5.1), the bifractal model (Section 8.5.2) and the multifractal model (Sections 8.5.3 and 8.5.4). Implications of the multifractal model for the dissipation range and for the skewness and flatness of velocity derivatives are presented in Sections 8.5.5 and 8.5.6, respectively. In the second class (Section 8.6), intermittency is studied via the fluctuation of the dissipation; inertial-range quantities are related to such fluctuations by a bridging ansatz, originally introduced by Obukhov and Kolmogorov (Section 8.6.2). Random cascade models are presented in Section 8.6.3; their multifractal behavior is shown to be a direct consequence of the probabilistic theory of ‘large deviations’, which is presented in an elementary fashion in Section 8.6.4. The lognormal model and its shortcomings are discussed in Section 8.6.5. Shell models, a class of deterministic nonlinear models which can display intermittency, are presented in Section 8.7.

The order chosen here for the presentation of the entire material on the theory of intermittency is pedagogical, not historical. Most of the latter aspects are discussed in Section 8.8. Recent trends in intermittency research are presented in Section 8.9.

Introduction to compressible flow

When the fluid velocity varies from place to place by amounts that are no longer small compared with the velocity of sound, the compressibility of the fluid cannot be ignored. Bernoulli's theorem does not have to be abandoned in these circumstances but it does need to be reformulated, and reformulation requires knowledge of the equation which relates pressure to density. The necessary equation is well known for ideal gases, and it is on gases – and on air in particular, which at normal temperatures and pressures conforms closely to the ideal model – that we focus attention in this chapter. As we shall see, the reformulated version of Bernoulli's theorem can be applied in an elementary way to a number of interesting phenomena which have to do with compressible flow of air. Among these are the shock fronts which develop following explosions and which accompany supersonic projectiles, and it is chiefly because these shock fronts are in many respects analogous to the tidal bores and hydraulic jumps discussed in §2.16 that this chapter stands where it does.

For the time being we shall continue to ignore viscosity. Neglect of viscosity is usually justified when the Reynolds Number is large compared with unity, for reasons which were outlined in §1.9, and most of the phenomena to be discussed below occur at flow rates where Re is 105 or more.

Introduction

Physics is a tree with many branches, and fluid dynamics is one of the older and sturdier ones. It began to form in the eighteenth century, when Euler and Daniel Bernoulli set out to apply the principles which Newton had enunciated for systems composed of discrete particles to liquids which are virtually continuous, and it has been in active growth ever since. Nowadays it is partially obscured from view by branches of more recent origin, such as relativity, atomic physics and quantum mechanics, and students of physics pay rather little attention to it. This is a pity, for several reasons. Firstly, because of the engineering applications of the subject, which are many and various: the design of aeroplanes and boats and automobiles, and indeed of any structure intended to move through fluid or propel fluid or simply to withstand the forces exerted by fluid, depends in a critical way upon the principles of the subject. Secondly, because fluid dynamics has important applications in other branches of physics and indeed in other realms of science, including astronomy, meteorology, oceanography, zoology and physiology: dripping taps, solitary waves on canals, vortices in liquid helium, seismic oscillations of the Sun, the Great Red Spot on Jupiter, small organisms that swim, the circulation of the blood – these are just a few of the very varied topics involving fluid dynamics which have been occupying research scientists and mathematicians of international reputation over the past few decades.

Shear stresses in Newtonian fluids

Throughout the last four chapters, the fact that real fluids possess viscosity has been almost completely ignored. We have supposed shear stress to be negligible and normal stress to be isotropic, and we have found that in so far as isotropic normal stress – the pressure p – depends upon fluid velocity u, it does so through formulae in which only the local magnitude of u and its rate of change with time appear. We cannot proceed much further on that simple, Eulerian, basis. The principal aims of the present chapter are firstly to establish the Newtonian formulae which relate the components of stress in viscous fluids to gradients of u, secondly to use these formulae to establish a more general equation of motion for fluids than Euler's equation, and thirdly to discuss a variety of relatively simple problems in which the effects of viscosity are dominant – so dominant in most cases that the fluid's inertia is negligible instead. The motion of fluids in such circumstances is sometimes referred to as creeping flow.

Newton himself may have considered only the simple situation illustrated by fig. 6.1, where planar laminae of fluid lying normal to the x2 axis are moving steadily in the x1 direction and sliding over one another, so that there exists a uniform velocity gradient ∂u1/∂x2.

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.