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 tractability. 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 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 13.2 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. Here and in the following section we view low-dimensional models as perturbations of the full evolution equations. Section 13.3 also addresses reproduction of statistics by low-dimensional models.

In the preceding nine chapters we have developed our basic tools and techniques. In this chapter and the next we illustrate their use in the derivation and analysis of low-dimensional models of the wall region of a turbulent boundary layer. First, the Navier– Stokes equations are rewritten in a form that highlights the dynamics of the coherent structures (CS) and their interaction with the mean flow. To do this, both the neglected (high) wavenumber modes and the mean flow must be modeled, unlike a large eddy simulation (LES), in which only the neglected high modes are modeled. Second, using physical considerations, we select a family of empirical subspaces upon which to project the equations. Galerkin projection is then carried out. In doing this, we restrict ourselves to a small physical flow domain, and so the response of the (quasi)local mean flow to the coherent structures must also be modeled. This chapter describes each step of the process in some detail, drawing on material presented in Chapters 2, 3, and 4. After deriving the family of low-dimensional models, in the last three sections we discuss in more depth the validity of assumptions used in their derivation. In Chapter 11 we describe use of the dynamical systems ideas presented in Chapters 6 through 9 in the analysis of these models, and interpret their solutions in terms of the dynamical behavior of the fluid flow.

Our presentation is based on a series of papers, beginning with [22] and including [24, 43, 44, 158, 161]. We have selected the boundary layer as our main illustrative example largely because we are most familiar with it.

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 [220]. In other disciplines the same procedure goes by the names: Karhunen–Loève 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 [221]), the POD was introduced independently by numerous people at different times, including Kosambi [197], Loève [215], Karhunen [183], Pougachev [285], and Obukhov [272]. Lorenz [216], 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 [275], image processing [313], signal analysis [5], data compression [7], process identification and control in chemical engineering [118,119], and oceanography [286]. Computational packages based on the POD are now readily available (an early example appeared in [11]).

In the bulk of these applications, the POD is used to analyze experimental data with a view to extracting dominant features and trends – in particular coherent structures. In the context of turbulence and other complex spatio-temporal fields, these will typically be patterns in space and time. However, our goal is somewhat different.

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– Loève 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 shall see in Parts III and IV, the techniques of proper orthogonal decomposition and Galerkin projection can be powerful tools for obtaining low-order models that capture the qualitative behavior of complex, high-dimensional systems. However, for certain systems, the resulting models can perform poorly: even if a large fraction of energy (over 99%) is captured by the modes used for projection, the resulting low-order models may still have completely different qualitative behavior. The transients may be poorly captured, and the stability types of equilibria can even be different.

In this chapter, we present a method which can dramatically outperform projection onto traditional energy-based empirical eigenfunctions described in Chapter 3.We focus primarily (though not exclusively) on linear systems, for several reasons. Many of the pitfalls of traditional proper orthogonal decomposition can be demonstrated for linear systems, without the additional complexity of nonlinearities. Furthermore, for linear systems, one can use operator norms to quantify the difference between a detailed model and its reduced order approximation. Most importantly, for linear systems, there are established tools for performing model reduction, for instance using balanced truncation, which is described in Section 5.1. In contrast, while some modest extensions to nonlinear systems have been attempted, model reduction of nonlinear systems is still an active area of research.

The techniques described in this chapter also differ from those in Chapters 3 and 4 in that they are formulated for input–output systems. The inputs represent the external influences on the system, for instance from external disturbances, or from actuators in a flow-control setting.

Much work has been done on low-dimensional models of turbulence and fluid systems in the 16 years since the first edition of this book appeared. In preparing the second edition, we have not attempted a comprehensive review: indeed, we doubt that this is possible, or even desirable. Rather, we have added one chapter and several sections and subsections on some new developments that are most closely related to material in our first edition. We have also made minor corrections and clarifications throughout, and added comments in several places, as well as correcting a number of errors that readers have pointed out. Here, to orient the reader, we outline the major changes.

Clancy Rowley (the new member of our team) has contributed a chapter on balanced truncation, a technique from linear control theory that chooses bases that optimally align inputs and outputs. Over the past ten years this has led to the method of balanced proper orthogonal decomposition (BPOD), which is especially useful for systems equipped with sensors and actuators. Since low-dimensional models provide a computational means for studying control of turbulence, we feel that BPOD has considerable potential. This new chapter (5) now closes the first part of the book (readers familiar with the first edition must therefore remember to add 1 to correctly identify the following eight chapters). The only other entirely new sections are 7.5, a discussion of traveling modes in translation-invariant systems, 12.6, a review of work on coherent structures in internal combustion engines, and 12.7, which gathers a miscellany of recent results.

This chapter and the following one provide a review of some aspects of the qualitative theory of dynamical systems that we 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 is merely the modest one of introducing, with simple examples, some tools for analysis of nonlinear 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 do 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 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 [15,17], Guckenheimer and Holmes [144], Arrowsmith and Place [18], or Glendinning [129], and in other references cited below. In particular we omit entirely any discussion of partial differential equations, which may seem scandalous, since this book ostensibly treats turbulence as described by the Navier–Stokes equations.

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 characterized by strong dependency in space and in time, so that not much can be modeled usefully as a simple Markov process, for example. The nonlinearity of turbulence is essential – linearization 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 nonlinearity, rotationality, and the dimensionality interact dynamically to feed the turbulence – hence, to suppose that a realization 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. One is forced to find unexpected chinks in its armor almost by necromancy, and to fabricate new approaches from whole cloth. This is its fascination.

As we have described in Part One, attempts to build low-dimensional models of truly turbulent processes are likely to involve averaging or, more generally, modeling 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 10, 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. [244]), 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 modeled by nonlinear 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 analyze the effect of additive white noise on a system linearized near an equilibrium point.

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.