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.

In the preceding eight chapters we have developed our basic tools and techniques. In this chapter and the next we shall 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 modelled, unlike a large eddy simulation (LES), in which only the neglected high modes are modelled. 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 modelled. 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 10 we shall describe the use of the dynamical systems ideas, presented in Chapters 5 through 8, in the analysis of these models, and interpret their solutions in terms of the dynamical behavior of the fluid flow.

Introduction

Fluids, that is gases and liquids, are self–evidently prerequisites for normal life. They also play a major role in the production of many artefacts and in the operation of much of the equipment upon which modern life depends. Occasionally, a fluid is the ultimate result of a technological process, such as a liquid or gaseous fuel, so that its existence impinges directly on the public consciousness. More often, fluids are intermediates in processes yielding solid materials or objects, and are then contained within solid objects so that their public image is very much less and their significance not fully appreciated. Nevertheless, every single component of modern life relies upon a fluid at some point and therefore upon our understanding of the fluid state.

The gross behavioral features of a fluid are well understood in the sense that it is easy to grasp that a gas has the property to completely fill any container and that a liquid can be made to flow by the imposition of a very small force. However, beyond these qualitative features lie a wide range of thermophysical and thermochemical properties of fluids that determine their response to external stimuli. This analysis concentrates exclusively on thermophysical properties and will not consider any process that involves a change to the molecular entities that comprise the fluid.

Introduction

The power and versatility of corresponding–states (CS) methods as a prediction tool has been pointed out by Mason and Uribe in Chapter 11 of this volume. Here, however, the strong point of corresponding–states principles is stressed: that methods based on the principle are theoretically based and predictive, rather than empirical and correlative. Thus, while CS cannot always reproduce a set of data within its experimental accuracy, as can an empirical correlation, it should be able to represent data to a reasonable degree but, more important, do what a correlation cannot do – estimate the properties beyond the range of existing data. In this chapter a particular corresponding–states method is reviewed that can predict the viscosity and thermal conductivity of pure fluids and their mixtures over the entire phase range from the dilute gas to the dense liquid with a minimum number of parameters. The method was proposed several years ago by Hanley (1976) and also by Mo & Gubbins (1974, 1976). It led to a computer program known as TRAPP (‘TRAnsport Properties Prediction’ (Ely & Hanley 1981a)). The method is also the basis for two NIST Standard Reference Databases – NIST Standard Reference Database 4 (SUPERTRAPP; Ely & Huber 1990) and NIST Standard Reference Database 14 (DDMIX; Friend 1992). Here, the original TRAPP procedure will be discussed as well as some more recent modifications to it. The performance of the model for viscosity and thermal conductivity prediction will also be examined for selected pure fluids and mixtures.