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.

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.

An acoustic pulse propagating with a current travels faster than one propagating against the current. Ocean currents are typically of order 10 cm/s rms or less, except in strong western boundary currents such as the Gulf Stream, whereas ocean sound-speed perturbations are typically of order 5 m/s rms. Travel-time perturbations due to ocean currents are correspondingly one to two orders of magnitude smaller than travel-time signals due to sound-speed perturbations. It is nonetheless possible to measure ocean currents using acoustic techniques, by differencing the travel times of signals traveling in opposite directions. As was briefly summarized in chapter 1, travel-time signals due to sound-speed perturbations cancel in the difference travel time, leaving only the effect of currents.

Section 3.1 describes ray theory as applied to moving media. The presence of a current introduces anisotropy. Perturbation expressions for the sum and difference of reciprocal travel times are then presented in section 3.2. When the flow is in geostrophic balance, the current and sound-speed fields are related. In section 3.3, quantitative estimates of their relative sizes are made, confirming the rough orders of magnitude cited earlier.

Using a horizontal-slice approximation, section 3.4 shows that the averaging properties of acoustic travel times make acoustic techniques uniquely suited for measuring the fluid circulation by integrating around a closed contour. By Stokes's theorem, the circulation is equivalent to the areal-average relative vorticity. This result is then generalized to show that differential travel times are sensitive to the solenoidal component of the flow, from which relative vorticity can be mapped, but are not functions of the irrotational component of the flow between the transceivers, which is needed to map the horizontal flow divergence.

The idea for acoustic tomography arose abruptly. Because we can mark a clear conceptual start, the following brief chronicle of the development of ocean acoustic tomography, given from our perspective as participants, may be of interest.

Some hopes for “Monitoring the oceans acoustically” were voiced at the thirtieth anniversary of the founding of the Office of Naval Research (ONR) (Munk and Worcester, 1976). A “Preliminary report on ocean acoustic monitoring” was prepared during the JASON Summer Study (JSN-77-8) by Garwin, Munk, and Wunsch. That work was expanded into “Ocean acoustic tomography: a scheme for large scale monitoring” (Munk and Wunsch, 1979), which examined the acoustic and inverse theoretical requirements for mapping the oceans with mesoscale resolution. It concluded that “an acoustic tomographic system appears to be both practical and useful.” The name “ocean acoustic tomography” was deliberately chosen to arouse the reader's curiosity as to what it is all about. Response by the oceanographic community was varied; those with a background in inverse theory regarded the inverse problem as trivial, but found the acoustic applications to be of interest, whereas the marine acousticians were interested in the inverse problem.

Overture

Advances in several fields were prerequisites for the development of ocean acoustic tomography: an understanding of underwater sound propagation, a statistical description of oceanic processes (especially internal waves and other fine structure), and the availability of inverse methods for inference from measurements. In this section we focus on the history of the acoustic developments crucial to tomography.