Most of the physical processes illustrated in the previous chapter are conveniently described by a set of partial differential equations (PDEs) for a vector field Ψ(x, t) which represents the state of the system in phase space X. The coordinates of Ψ are the values of observables measured at position x and time t: the corresponding field theory involves an infinity of degrees of freedom and is, in general, nonlinear.

A fundamental distinction must be made between conservative and dissipative systems: in the former, volumes in phase space are left invariant by the flow; in the latter, they contract to lower dimensional sets, thus suggesting that fewer variables may be sufficient to describe the asymptotic dynamics. Although this is often the case, it is by no means true that a dissipative model can be reduced to a conservative one acting in a lower-dimensional space, since the asymptotic trajectories may wander in the whole phase space without filling it (see, e.g., the definition of a fractal measure in Chapter 5).

A system is conceptually simple if its evolution can be reduced to the superposition of independent oscillations. This is the integrable case, in which a suitable nonlinear coordinate change permits expression of the equations of motion as a system of oscillators each having its own frequency.

As we have described in Part I, attempts to build low-dimensional models of truly turbulent processes are likely to involve averaging or, more generally, modelling to account for neglected modes that are dynamically active in the sense that their states cannot be expressed as an algebraic function of the modes included in the model. Such models are in turn likely to involve probabilistic elements. Here, “neglected modes” may refer to (high wavenumber) modes in the inertial and dissipative ranges or to mid-range, active modes whose wavenumbers might be linearly unstable. They also may refer to spatial locations that are omitted, in selecting a subdomain of a large or infinite physical spatial extent. The boundary layer model of Chapter 9, for example, contains a forcing term representing a pressure field, unknown a priori, imposed on the outer edge of the wall region. While estimates of this term can be obtained from direct numerical simulations (e.g.), a natural simplification is to replace it with an external random perturbation of suitably small magnitude and appropriate power spectral content. More generally, many processes modelled by non-linear differential equations involve random effects, in either multiplicative form (coefficient variations) or additive form, and it is therefore worth making a brief foray into the field of stochastic dynamical systems to sample some of the tools available.

In this chapter we give a very selective and cursory description of how one can analyse the effect of additive white noise on a system linearised near an equilibrium point.

The proper orthogonal decomposition (POD) provides a basis for the modal decomposition of an ensemble of functions, such as data obtained in the course of experiments. Its properties suggest that it is the preferred basis to use in various applications. The most striking of these is optimality: it provides the most efficient way of capturing the dominant components of an infinite-dimensional process with only finitely many, and often surprisingly few, “modes.”

The POD was introduced in the context of turbulence by Lumley in. In other disciplines the same procedure goes by the names: Karhunen–Loéve decomposition, principal components analysis, singular systems analysis, and singular value decomposition. The basis functions it yields are variously called: empirical eigenfunctions, empirical basis functions, and empirical orthogonal functions. According to Yaglom (see), the POD was introduced independently by numerous people at different times, including Kosambi, Loéve, Karhunen, Pougachev, and Obukhov. Lorenz, whose name we have already met in another context, suggested its use in weather prediction. The procedure has been used in various disciplines other than fluid mechanics, including random variables, image processing, signal analysis, data compression, process identification and control in chemical engineering, and oceanography. Computational packages based on the POD are now becoming available.

In the bulk of these applications, the POD is used to analyse experimental data with a view to extracting dominant features and trends: coherent structures.

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 analogues 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 and Golubitsky, Stewart, and Schaeffer. In this chapter, as in the last, we shall attempt to sketch relevant parts of the theory using simple examples and without undue reliance on abstract mathematical ideas.

We have already met symmetric ODEs, in our discussion of the Lorenz equation (5.56) for example. This equation remains unchanged under the transformation (x1, x2, x3) → (−x1, −x2, x3), which corresponds to rotation of the phase space about the x3-axis through an angle π. This implies that, when the equilibrium 0 changes stability type from sink to saddle as ρ passes through 1, the resulting bifurcation is a pitchfork (see Equation (5.50) and Figure 5.8). Moreover, the global behavior, determined in part by the structure of the unstable manifold Wu(0) as we saw at the close of Chapter 5, is also influenced by symmetry in that the “left” and “right” branches of Wu(0) are mapped into each other by the symmetry.

This chapter and the following one provide a review of some aspects of the qualitative theory of dynamical systems that we will need in our analyses of low-dimensional models derived from the Navier–Stokes equations. Dynamical systems theory is a broad and rapidly growing field which, in its more megalomaniacal forms, might be claimed to encompass all of differential equations (ordinary, partial, and functional), iterations of mappings (real and complex), devices such as cellular automata and neural networks, as well as large parts of analysis and differential topology. Here our aim will be merely the modest one of introducing, with simple examples, some tools for the analysis of non-linear ordinary differential equations that may not be as familiar as, say, perturbation and asymptotic methods.

The viewpoint of dynamical systems theory is geometric, and invariant manifolds play a central rôle, but we shall not assume or require familiarity with differential topology. In the same way, symmetries are crucial in determining the behavior, and permitting the analysis, of the low-dimensional models of interest, but we shall avoid appeals to the subtleties of group theory in our introduction to symmetric bifurcations. Thus, it should be clear that these two chapters cannot substitute for a serious course (or, more likely, courses) in dynamical systems theory. The makings of such a course can be found in the books of Arnold, Guckenheimer and Holmes, Arrowsmith and Place, or Glendinning, and in other references cited below.

Turbulence

Turbulence is the last great unsolved problem of classical physics. Although temporarily abandoned by much of the community in favor of particle physics, the current popularity of chaos and dynamical systems theory (as well as funding problems in particle physics) is now drawing the physicists back. During the interim and up to the present, turbulence has been avidly pursued by engineers.

Turbulence has enormous intellectual fascination for physicists, engineers, and mathematicians alike. This scientific appeal stems in part from its inherent difficulty – most of the approaches that can be used on other problems in fluid mechanics are useless in turbulence. Turbulence is usually approached as a stochastic problem, yet the simplifications that can be used in statistical mechanics are not applicable – turbulence is characterised by strong dependency in space and in time, so that not much can be modelled usefully as a simple Markov process, for example. The non-linearity of turbulence is essential – linearisation destroys the problem. Many problems in fluid mechanics can be approached by supposing that the flow is irrotational – that is, that the vorticity is zero everywhere. In turbulence, the presence of vorticity is essential to the dynamics. In fact, the non-linearity, rotationality and the dimensionality interact dynamically to feed the turbulence – hence, to suppose that a realisation of the flow is two-dimensional also destroys the problem. There is more, but this is enough to make it clear that one faces the turbulence problem stripped of the usual arsenal of techniques, reduced to hand-to-hand combat.

The methods developed in this book may be applied rather generally to model the dynamics of coherent structures in spatially extended systems. They are gaining acceptance in many areas in addition to fluid mechanics, including mechanical vibrations, laser dynamics, non-linear optics, and chemical processes. They are even being applied to studies of neural activity in the human brain. Numerous studies of closed flow systems have been done using empirical eigenfunctions, some of which were discussed in Section 3.7. A considerable amount of work has also been done on model PDEs for weakly non-linear waves, such as the Ginzburg–Landau and Kuramoto–Sivashinsky equations, which falls largely outside the scope of this book. We do not have the abilities (or space) to provide a survey of these multifarious applications, but we do wish to draw the reader's attention to some of the other recent work on open fluid flows.

We restrict ourselves to studies in which empirical eigenfunctions are used to construct low-dimensional models and some attempt is made to analyse their dynamical behavior. There is an enormous amount of work in which the POD is applied and its results assessed in a “static,” averaged fashion. Some of this we have reviewed in Section 3.7. Yet even thus restricted, our survey cannot pretend to be complete: new applications to fluid flows are appearing at an increasing rate. We have selected five problems on which a reasonable amount of work has been done, one of which (the jet) is a “strongly” turbulent flow.

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 non-linearity, Burgers' equation, which was originally introduced as a model to illustrate some of the features of turbulence. 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.

In this chapter we shall describe the qualitative structure, in phase space, of some of the low-dimensional models derived in the preceding chapter. We will also discuss the physical implications of our findings. Drawing on the material introduced in Chapters 5–8, we shall solve for some of the simpler fixed points (steady, time-independent flows and travelling waves) and discuss their stability and bifurcations under variation of the loss parameters αj introduced in Section 9.1. We focus on the five mode model (N = 1, K1 = 0, K3 = 5) introduced in the original paper of Aubry et al., 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 10.1 and 10.2 we use the nesting properties of invariant subspaces, noted in Section 9.5, 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.

On physical grounds there is no doubt that the Navier–Stokes equations provide an excellent model for fluid flow as long as shock waves are relatively thick (in terms of mean free paths), and in such conditions of temperature and pressure that we can regard the fluid as a continuum. The incompressible version is restricted, of course, to lower speeds and more moderate temperatures and pressures. There are some mathematical difficulties – indeed, we still lack a satisfactory existence-uniqueness theory in three dimensions – but these do not appear to compromise the equations' validity. Why then is the “problem of turbulence” so difficult? We can, of course, solve these nonlinear partial differential equations numerically for given boundary and initial conditions, to generate apparently unique turbulent solutions, but this is the only useful sense in which they are soluble, save for certain non-turbulent flows having strong symmetries and other simplifications. Unfortunately, numerical solutions do not bring much understanding.

However, three fairly recent developments offer some hope for improved understanding. (1) The discovery, by experimental fluid mechanicians, of coherent structures in certain fully developed turbulent flows, (2) the suggestion that strange attractors and other ideas from finite-dimensional dynamical systems theory might play a rôle in the analysis of the governing equations, and (3) the introduction of the statistical technique of Karhunen–Loéve or proper orthogonal decomposition. This book introduces these developments and describes how the three threads can be drawn together to weave low-dimensional models that address the rôle of coherent structures in turbulence generation.

As we near the end of our story, the reader will now appreciate that there are many steps in the process of reducing the Navier–Stokes equations to a low-dimensional model for the dynamics of coherent structures. Some of these involve purely mathematical issues, but most require an interplay among physical considerations, judgement, and mathematical tractibility. While our development of a general strategy for constructing low-dimensional models has been based on theoretical developments such as the POD and dynamical systems methods, the general theory is still sketchy and, in specific applications, many details remain unresolved.

The mathematical techniques we have drawn on lie primarily in probability and dynamical systems theory. In this closing chapter we review some aspects of the reduction process and attempt to put them into context. Some prospects for rigor in the reduction process are also mentioned. This is by no means a comprehensive review or discussion of future work; instead, we have chosen to highlight a few recent applications of dynamical and probabilistic ideas to illustrate lines along which a general theory might be further developed.

We start by discussing some desirable properties for low-dimensional models, and criteria by which they might be judged. We then outline in Section 12.2 an a priori short-term tracking estimate which describes, in a probabilistic context, how rapidly typical solutions of the model equations are expected to diverge from those of the full Navier–Stokes equations restricted to the model domain.