Flow in symmetric bifurcations

Most branchings in the cardiovascular system are asymmetric, the only major exception in man being the bifurcation where the aorta divides to form the iliac arteries. This is in contrast to the bifurcating airways of the lung, for which the assumption of symmetry is more appropriate, and which have been the subject of extensive research (Pedley, 1977). Furthermore, the precise definition of an asymmetric bifurcation requires the specification of several more parameters than that of a symmetric one (e.g. the ratios of the flow-rates in, and the diameters of, the two daughter tubes, as well as the different angles of branching). There is, therefore, considerably more fluid mechanical information available on the subject of symmetric bifurcations, and this chapter begins with a survey of it (taken largely from the review by Pedley, 1977). It should be said at the start, however, that the problem is still very complicated, and most of the data have been obtained experimentally not theoretically, with steady rather than unsteady flow. Clearly much work remains to be done.

In all the investigations described in this section and the next, the geometry of the bifurcations is taken to be fully three-dimensional. There has been relatively extensive theoretical and experimental work on two-dimensional bifurcations, but since that geometry rules out all secondary motions it is unlikely to have much relevance to the cardiovascular system.

We now turn to the second main feature of the thoracic aorta: its curvature. The aim is to describe flow near the entrance of a curved tube in the same way that the previous section described flow near the entrance of a straight tube. However, we immediately come up against the major difficulty that the fully developed flow to which the entry flow tends, and which in a straight tube is Poiseuille flow (the mean flow) plus an easily calculated oscillatory component, is very complicated, and even the steady component is not yet completely understood. In the next three sections, therefore, we concentrate on fully developed flow in curved tubes, leaving a discussion of entry flow to §§ 4.4 and 4.5.

The reason why the flow in a curved tube is difficult to calculate lies in the fact that the motion cannot be everywhere parallel to the curved axis of the tube, but transverse (or secondary) components of velocity must be present. This follows because in order for a fluid particle to travel in a curved path of radius R with speed w it must be acted on by a lateral force (provided by the pressure gradients in the fluid) to give it a lateral acceleration w2/R. Now the pressure gradient acting on all particles will be approximately uniform, but the velocity of those particles near the wall will be much lower than that of particles in the core, as a result of the no-slip condition.

In this chapter we will consider the processes of formation of mixed layers by externally driven turbulence, and entrainment of fluid across the interfaces bounding such layers. Mixing across a density interface has so far been treated only in the special context of double-diffusive convection (§8.3), but more general stirring mechanisms must now be discussed. The problems of interest here may be identified with cases (d) or (e) of fig. 4.19, in which stirring at one level is used to produce mixing across an interface located some distance below or above the source of turbulent energy.

Various laboratory experiments which have shed light on the mechanism of entrainment at a density interface will be described first. Mixing can be generated by mechanical stirring, or by the production of a mean turbulent flow in the surface layer, and both methods have been used. The observed structure of the surface layers of the ocean and atmosphere discussed in §10.1 suggest that mixing in those regions is dominated by the boundary processes. The laboratory results can immediately be applied to these geophysical examples, and, with the addition of results from the earlier chapters on convection, they can be extended to take account of convective as well as mechanical mixing. It is assumed throughout this chapter that the mixing processes under consideration can be treated as one-dimensional in depth, implying that mixing is uniform (in the mean) over a large horizontal area, with no significant contribution from large scale lateral convection.

The mechanisms responsible for mixing in the interior of a stratified fluid are even less well understood than those described in chapter 9, since the sources of energy are not so obvious, and several different processes must be taken into account simultaneously. The important ideas have already been introduced in earlier sections, but it seems appropriate in this final chapter to take a broader and less detailed view, and to consider together the whole array of mixing phenomena which can be relevant in large natural bodies of stratified fluid. Geophysical examples have figured prominently in this book, and indeed the range of subject matter has been chosen with this final synthesis in mind. The basic facts requiring explanation are somewhat scattered, however, and in order to collect them together and to define the problems to be treated here, a brief summary will now be given of the observed structure of the ocean and atmosphere.

This field is developing rapidly, and the interpretation given here must necessarily be a somewhat tentative and personal one. Nevertheless, it seems useful to sketch how our present knowledge of the separate components can be fitted into a self-consistent picture, at the same time extending some of the earlier arguments so that they can be applied in this wider context.

The observational data

Routine density profiles made using reversing bottles and thermometers show that the ocean is everywhere stably stratified, except for limited regions where bottom water forms intermittently as the water column becomes convectively unstable.

A comparatively recent development in the field of convection has been the study of fluids in which there are gradients of two (or more) properties with different molecular diffusivities. When the concentration gradients have opposing effects on the vertical density distribution, a number of surprising things can happen, and these are the subject of the present chapter. The phenomena were first studied with an application to the ocean in mind (see §8.2.4), and because heat and salt (or some other dissolved substance) are then the relevant properties, the process has been called ‘thermohaline’ or ‘thermosolutal’ convection. Related effects have now been observed in the laboratory using a pair of solutes, and in solidifying metal alloys, and the name ‘double-diffusive convection’ has been chosen to encompass this wider range of phenomena.

The stability problem will first be reviewed, somewhat more fully than was done in previous chapters because of the comparative novelty of the double-diffusive phenomena. It will then be shown that when two components contribute to the vertical density gradient, a series of steps tends to form, with well-mixed layers separated by sharper density interfaces. The detailed structure of these interfaces and measurements of the coupled fluxes across them will also be described.

The stability problem

The mechanism, of instability

In such a system with opposing gradients, the existence of a net density distribution which decreases upwards is not a guarantee of stability.

Buoyancy forces arise as a result of variations of density in a fluid subject to gravity, and produce a wide range of phenomena of importance in many branches of fluid mechanics. Progress in this field has been made largely through the desire to solve very practical problems, arising for instance in meteorology or in hydraulic engineering. This emphasis on particular applications has meant that parallel developments have often been made in different disciplines without much cross reference to related work, and some results, well understood in one context, are less familiar in another where they might be used to advantage. In this book I have attempted to write a coherent account of the various fluid motions which can be driven or influenced by the presence of small density differences. It is intended as a general introduction to the subject and its literature, in which the physical understanding of the phenomena is emphasized, rather than the applications on the one hand or detailed mathematical theory on the other.

The selection of subject matter must always be a personal one, however, and my own research interests have certainly influenced the topics chosen and the amount of space given to each of them. I have worked with laboratory models of small scale processes in the ocean and atmosphere, and so laboratory and geophysical examples come most readily to mind, but comparisons have also been made with results from various fields of engineering where possible.

In this chapter we consider various mechanisms whereby laminar flows of a stably stratified fluid can break down and become turbulent. The first task is to summarize some results of hydrodynamic stability theory as it applies in this context, that is, the investigation of the conditions under which small disturbances to the motion can grow. The logical development of the preceding chapters will be followed by restricting the discussion to dynamic instabilities due to shearing motions of a statically stable initial stratification. ‘Convective’ instability associated with an increase of density with height will be left until chapter 7, and two-component systems, in which different diffusivities play a vital role, will be treated separately in chapter 8.

The problem of instability of layers across which both density and velocity are rapidly varying functions of height is given special attention here, since such shear layers are very common in the atmosphere and ocean, and the vertical transports of properties such as heat and salt depend strongly on what happens near them. The topics covered now will, however, go beyond what is usually meant by the term ‘instability’ in the strict sense. It seems useful at the same time to outline other ways in which energy can be fed into limited regions of a more extensive flow, at a rate sufficient to cause a local breakdown. Some of the phenomena have already been mentioned in the context of steady flows, and the wider geophysical implications of the results will be discussed in chapter 10.

The topics to be discussed

It seems useful to begin by outlining the range of subjects covered in this book, to give a broad picture of the way in which the several parts of the field have developed, and at the same time some explanation of the theme which has been used to connect them. The phenomena studied all depend on gravity acting on small density differences in a non-rotating fluid. Often the undisturbed fluid has a density distribution which varies in the vertical but is constant in horizontal planes; this will be called a stratified system whether the density changes smoothly or discontinuously. Special attention will be given to the problems of buoyant convection (arising from an unstable density distribution) and to various mechanisms of mixing when the stratification is stable.

Chapters 2 and 3 summarize relevant results on internal waves, and these were also historically the first phenomena to be studied. The original applications of the methods of perfect fluid theory to motion under gravity were to the problems of small amplitude surface waves and tides (subjects which will not be discussed here). These were soon extended to the case of two layers of uniform density with a density discontinuity between them. Some of the basic results had already been obtained by 1850 (notably by Stokes 1847), and they were applied to phenomena such as the drag experienced by a ship when it creates a wave on an interface close to the surface (Ekman 1904), and to internal seiches in lakes.

Natural bodies of fluid such as the atmosphere, the oceans and lakes are characteristically stably stratified: that is, their mean (potential) density decreases as one goes upwards, in most regions and for most of the time. When they are disturbed in any way, internal waves are generated. These ubiquitous motions take many forms, and they must be invoked to explain phenomena ranging from the temperature fluctuations in the deep ocean to the formation of clouds in the lee of a mountain. In this chapter we summarize the results which can be obtained using linear theory (i.e. when the amplitudes are assumed to be small), and in §3.1 extend some of them to describe waves of large amplitude.

Many of the elementary properties of infinitesimal wave motions in stratified fluids can be introduced conveniently by considering waves at an interface between two superposed layers, and so this case is treated first in §2.1. These waves are analogous to waves on a free water surface, and therefore seem very familiar. It should be emphasized at the outset, however, that they are not the most general wave motions which can occur in a continuously stratified fluid. Energy can propagate through such a fluid at an angle to the horizontal, not just along surfaces of constant density, and our intuition based on surface waves is of little help here. The more general theory, and a comparison between the two descriptions, is given in §2.2.

The previous chapter was based on equations of motion made linear by assuming that the amplitude of wave-like disturbances of the fluid remained infinitesimal. We now consider various large amplitude phenomena which require the inclusion of the non-linear terms for their explanation. First, some of the inviscid wave problems already treated will be extended to finite amplitude, and the essentially non-linear phenomenon of internal solitary waves will be discussed. Then various quasi-steady flows which arise in nature and in civil engineering applications will be treated, using a generalization of free surface hydraulic theory (and thus relating the properties of such flows to the waves which can form on them). Internal hydraulic jumps, the flow of a thin layer down a slope, and the nose at the front of a gravity current come under this heading. Finally we introduce the effects that viscosity and diffusion can have on slow steady motions in a stratified fluid, describing upstream wakes and boundary layers and the process of selective withdrawal.

Internal waves of finite amplitude

Interfacial waves

We refer again to the statement made in §2.1.2, that (2.1.8) is valid for finite amplitude long waves. (Cf. Lamb 1932, p. 278.) This implies that the highest point of any disturbance will move fastest, and so the forward slope of a wave of finite amplitude will tend to steepen, an effect called ‘amplitude dispersion’. This is in contrast with the result of the frequency or wavenumber dispersion previously described by (2.1.7) which is valid for general depths.

The buoyancy effects discussed so far have for the most part been stabilizing, or have been assumed to produce a small modification of an existing turbulent flow. Now we turn to convective flows, in which buoyancy forces play the major role because they are the source of energy for the mean motion itself. The usual order of presentation will be reversed: it is convenient to set aside for the present the discussion of the mean properties of a convecting region of large horizontal extent, and flows near solid bodies, and in this chapter to treat various models of the individual convective elements which carry the buoyancy flux. (See Turner (1969a) for a review of this work and a more extensive bibliography.)

Such models can be broadly divided into two groups, those which assume the motion to be in the form of ‘plumes’ or of ‘thermals’. (See fig. 6.1.) In both of them motions are produced under gravity by a density contrast between the source fluid and its environment; the velocity and density variations are interdependent, and occupy a limited region above or below the source. Plumes, sometimes called buoyant jets, arise when buoyancy is supplied steadily and the buoyant region is continuous between the source and the level of interest. The term thermal is used in the sense which has become common in the meteorological literature to denote suddenly released buoyant elements.