In numerical simulations of turbulence, one can only integrate a finite set of differential equations or, equivalently, seek solutions on a finite spatial grid. One method that converts an infinite-dimensional evolution equation or partial differential equation into a finite set of ordinary differential equations is that of Galerkin projection. In this procedure the functions defining the original equation are projected onto a finite-dimensional subspace of the full phase space. In deriving low-dimensional models we shall ultimately wish to use subspaces spanned by (small) sets of empirical eigenfunctions, as described in the previous chapter. However, Galerkin projection can be used in conjunction with any suitable set of basis functions, and so we discuss it first in a general context.

After a brief description of the method in Section 4.1, we apply it in Section 4.2 to a simple problem: the linear, constant-coefficient heat equation in both one- and two-space dimensions. We recover the classical solutions, which are often obtained by separation of variables and Fourier series methods in introductory applied mathematics courses. We then consider an equation with a quadratic nonlinearity, Burgers’ equation, which was originally introduced as a model to illustrate some of the features of turbulence [65]. The remainder of the chapter is devoted to the Navier–Stokes equations. In Section 4.3 we describe Fourier mode projections for fluid flows in simple domains with periodic boundary conditions, paying particular attention to the way in which the incompressibility condition is addressed. The final Section 4.4 focuses on the use of empirical eigenfunctions and introduces some issues that arise in making the drastic truncations necessary to obtain low-dimensional models.

Physical systems often exhibit symmetry: we have already remarked on the symmetries of spanwise translation and reflection in boundary layers and shear layers and of rotations in circular jets. One could cite many more such cases. Of course, symmetric systems do not always, or even typically, exhibit symmetric behavior, and the study of spontaneous symmetry breaking is an important field in physics. These physical phenomena have their analogs in dynamical systems and in particular in ODEs, as we describe in this chapter.

The theory of symmetric dynamical systems and their bifurcations relies heavily on group theory and especially the notions of invariant functions and equivariant vector fields. The major references are the two volumes by Golubitsky and Schaeffer [134] and Golubitsky et al. [136]. In this chapter, as in the last, we attempt to sketch relevant parts of the theory using simple examples and without undue reliance on abstract mathematical ideas.

In this chapter we describe the qualitative structure, in phase space, of some of the low-dimensional models derived in the preceding chapter. We also discuss the physical implications of our findings. Drawing on the material introduced in Chapters 6–9, we solve for some of the simpler fixed points (steady, time-independent flows and traveling waves) and discuss their stability and bifurcations under variation of the loss parameters αj introduced in Section 10.1. We focus on the five mode model (N = 1, K1 = 0, K3 = 5) introduced in the original paper of Aubry et al. [22], and referred to there as the “six mode model,” the k3 = 0 mode being implicitly included in the model of the slowly varying mean flow. The full range of dynamical behavior of even such a draconian truncation as this is bewilderingly complex and still incompletely understood, but we are able to give a fairly complete account of a particular family of solutions – attracting heteroclinic cycles – which appear especially relevant to understanding the burst/sweep cycle which was described in Section 2.5.

In Sections 11.1 and 11.2 we use the nesting properties of invariant subspaces, noted in Section 10.5, to solve a reduced system, containing only two (even) complex modes, for fixed points. We exhibit the bifurcation diagram and discuss the stability of a particular branch of fixed points corresponding to streamwise vortices of the appropriate spanwise wavenumber. Due to the spanwise translation invariance (Section 10.3), circles of such equilibria occur in phase space.

In the Preface to the first edition, we commented on the benefits and drawbacks of interdisciplinary research; the contributions of specialists to advance our understanding and the difficulty for the non-specialist in understanding these advances. We were thinking particularly about the mechanics of the circulation and the contributions that had been made by engineers, physicists and mathematicians working in collaboration with physiologists and medical doctors. Our goal in writing the book was to alleviate the problem of understanding these advances by providing an introductory text on the mechanics of the circulation that was accessible to physiologists and medical practitioners.

The three decades since the book was published have seen an explosive growth in research on the cardiovascular system. In 1978, bioengineering did not exist as a separate academic discipline and the field of cardiovascular mechanics was relatively small, although it had a long and distinguished history extending over more than three centuries. Today, bioengineering is widely recognized as an academic discipline and interdisciplinary research is generally accepted as essential to progress.

Our understanding of the circulation is immeasurably greater today than it was in 1978, but many problems remain unsolved and cardiovascular disease is still the largest single cause of death world-wide. Again, however, these advances have brought increased difficulty in understanding. We believe that the need for an introductory text on the mechanics of the circulation that is accessible to the non-specialist is even greater now than it was when the book was first published.

We saw in the last chapter that in the large arteries blood may be treated as a homogeneous fluid and its particulate structure ignored. Furthermore, fluid inertia is a dominant feature of the flow in the larger vessels since the Reynolds numbers are large. The fluid mechanical reasons for treating the circulation in two separate parts, with a division at vessels of 100μm diameter, were also given in that chapter. In the microcirculation, which comprises the smallest arteries and veins and the capillaries, conditions are very different from those in large arteries and it is appropriate to consider the flow properties within them separately.

First, it is no longer possible to think of the blood as a homogeneous fluid; it is essential to treat it as a suspension of red cells and other formed elements in plasma. As will be seen later in the chapter, this comes about because even the largest vessels of the microcirculation are only approximately 15 red cells in diameter. Second, in all vessels, viscous rather than inertial effects dominate and the Reynolds numbers are very low; typical Reynolds numbers in 100μm arteries are about 0.5 and in a 10μm capillary they fall to less than 0.005 (see Table I).

In larger arteries, the Womersley parameter α (p. 60) is always considerably greater than unity. In the microcirculation, however, α is very small; in the dog (assuming a heart rate of 2Hz) it is approximately 0.08 in 100μm vessels and falls to approximately 0.005 in capillaries. This means that everywhere in these small vessels the flow is in phase with the local pressure gradient and conditions are quasi-steady.

This chapter provides an insight into the physical nature of turbulence and the mathematical framework that is used in numerical simulations of turbulent flows. The aim is to explain why turbulence must be modelled and how turbulence can be modelled, and also to explain what is modelled with different turbulence models. In addition, the limitations of the turbulence models are discussed. The intention is to give you such an understanding of turbulence modelling that you can actually suggest appropriate turbulence models for different kinds of turbulent flows depending upon their complexity and the required level of description.

The physics of fluid turbulence

Turbulence is encountered in most flows in nature and in industrial applications. Natural turbulent flows can be found in oceans, in rivers and in the atmosphere, whereas industrial turbulent flows can be found in heat exchangers, chemical reactors etc. Most flows encountered in industrial applications are turbulent, since turbulence significantly enhances heat- and mass-transfer rates. In industry a variety of turbulent multiphase flows can be encountered. Turbulence plays an important role in these types of flows since it affects processes such as break-up and coalescence of bubbles and drops, thereby controlling the interfacial area between the phases. Thus, turbulence modelling becomes one of the key elements in CFD.

Simple harmonic motion

When blood is ejected from the heart during systole, the pressure in the aorta and other large arteries rises, and then during diastole it falls again. The pressure rise is associated with outward motions of the walls, and they subsequently return because they are elastic. This process occurs during every cardiac cycle, and it can be seen that elements of the vessel walls oscillate cyclically, with a frequency of oscillation equal to that of the heartbeat. The blood, too, flows in a pulsatile manner, in response to the pulsatile pressure. In fact, as we shall see in Chapter 12, a pressure wave is propagated down the arterial tree. It is therefore appropriate in this chapter to consider the mechanics of pulsatile phenomena in general, and the propagation of waves in particular.

Let us examine first the oscillatory motion of a single particle. Suppose that the particle can be in equilibrium at a certain point, say P, but when it is disturbed from this position, it experiences a restoring force, tending to return it to P. There are many examples of this situation, as when a particle is hanging from a string and is displaced sideways (a simple pendulum) or when the string is elastic and the particle is pulled down below its equilibrium position. In cases like these, the restoring force increases as the distance by which the particle is displaced from P increases. In fact, for sufficiently small displacements, the restoring force is approximately proportional to the distance from P (see p. 124). If the particle is displaced and then released, it will return towards P, but will overshoot because of its inertia.

It soon becomes clear to any student of physiology that there are many systems of units and forms of terminology. For example, respiratory physiologists measure pressures in centimetres of water and cardiovascular physiologists use millimetres of mercury. As the study of any single branch of physiology becomes increasingly sophisticated, more and more use is made of other disciplines in science. As a result, the range of units has increased to such an extent that conversion between systems takes time and can easily cause confusion and mistakes.

We see also frequent misuse of terminology which can only confuse; for example, the partial pressure of oxygen in blood is often referred to as the ‘oxygen tension’, when in reality tension means a tensile force and is hardly the appropriate word to use.

In order to combat a situation which is deteriorating, considerable effort is being made to reorganize and unify the systems of nomenclature and units as employed in physiology. For any agreed procedure to be of value, it must be self-consistent and widely applicable. Therefore, it has to be based upon a proper understanding of mathematical principles and the laws of physics.

The system of units which has been adopted throughout the world and is now in use in most branches of science is known as the Système International or SI (see p. 28).

The study of the mechanics of blood flow in veins has been far less extensive than that of blood flow in arteries. However, virtually all the blood ejected by the left ventricle must return to the right atrium through the veins; they normally contain almost 80% of the total volume of blood in the systemic vascular system and have an important controlling influence on cardiac output. It is therefore important to understand their mechanics.

The venous system resembles the arterial system, in that it consists of a tree-like network of branching vessels; the main trunks are the venae cavae, which come together and lead into the heart. However, it is fundamentally different from the arterial system in several respects:

1. (1) As can be seen from Fig. 12.11, p. 257, the pressure in a vein is normally much lower than that in an artery at the same level, and may be less than atmospheric (for example in veins above the level of the heart).

2. (2) The vessels have thinner walls and their distensibility varies over a much wider range than that of arteries at physiological pressures.

3. (3) The blood flows from the periphery towards the heart, and the flow rate into a vein is determined by the arterio-venous pressure difference and the resistance of the intervening microcirculation.

4. (4) Many veins contain valves which prevent backflow.

The mammalian heart consists of two pumps, connected to each other in series, so that the output from each is eventually applied as the input to the other. Since they are developed, embryologically, by differentiation of a single structure, it is not surprising that the pumps are intimately connected anatomically, and that they share a number of features. These include a single excitation mechanism, so that they act almost synchronously; a unique type of muscle, cardiac muscle, which has an anatomical structure similar to skeletal muscle, but some important functional differences; and a similar arrangement of chambers and one-way valves. Not surprisingly, the assumption has often been made that the function of the two pumps will also be similar. Thus it has become common practice to examine the properties of one pump, usually the left, and to assume that the results apply to the other also. This may often be unjustified, particularly in studies of cardiac mechanics, with the result that our knowledge of the mechanics of the right heart and the pulmonary circulation remains very incomplete. It must also be remembered that the scope for experiments on the human heart is very limited, and we must rely heavily on experimental information from animal studies. Thus the descriptions which follow apply primarily to the dog heart.

Many factors which affect the performance of the heart are not our concern in this chapter, among the most important being the wide range of reflexes which act on the heart. For example, nerve endings in the aortic wall and carotid sinus are sensitive to stretch, and thus to changes in arterial pressure.