Flow of an ideal fluid past a wedge

3.1.1 The problem statement and a direct application of dimensional analysis

In the heat conduction and gas dynamic problems discussed in the preceding chapter, we were able to establish that the intermediate-asymptotic solutions were self-similar and to determine the self-similar variables using dimensional analysis alone. However, this is very frequently not the case. It often turns out that a problem has a self-similar intermediate-asymptotic solution, but that dimensional analysis alone is insufficient to obtain it. We shall presently see this for a simple example – the flow of an ideal fluid (one having no internal friction) past a wedge. Even this simple example will help us to find out why dimensional analysis on its own is sufficient for constructing self-similar solutions in some cases but not in others. Further examples will demonstrate other, more complicated situations. Thus, in this section, we shall consider the steady symmetric flow of an ideal fluid of constant density past a wedge-shaped body with uniform velocity U at infinity perpendicular to the leading edge of the wedge (Figure 3.1). We select a system of rectangular coordinates as shown in Figure 3.1 (a), with the x-axis in the plane of symmetry of the wedge, parallel to the flow velocity, the z-axis along the leading edge of the wedge, and the y-axis perpendicular to these two axes. It may be assumed that the motion is identical in all planes perpendicular to the leading edge of the wedge. Thus, the z-component of the velocity, uz, can be set equal to zero, while the longitudinal and transverse components of the velocity,ux and uy, depend only on the coordinates x and y. The region in which the fluid moves is very large; we shall assume that it is infinite. It is also an important assumption; however, for the present case it is not critical.

Dimensional analysis and physical similarity

The starting point of this book is dimensional analysis and it is used throughout. Like unhappy families, every unfortunate scientific idea is unfortunate in its own way. Many of those who have taught dimensional analysis (or have merely thought about how it should be taught) have realized that it has suffered an unfortunate fate.

In fact, the idea on which dimensional analysis is based is very simple, and can be understood by everybody: physical laws do not depend on arbitrarly chosen basic units of measurement. An important conclusion can be drawn from this simple idea, using a simple argument: the functions that express physical laws must possess a certain fundamental property, which in mathematics is called generalized homogeneity or symmetry. This property allows the number of arguments in these functions to be reduced, thereby making it simpler to obtain them (by calculating them or determining them experimentally). This is, in fact, the entire content of dimensional analysis – there is nothing more to it.

Nevertheless, using dimensional analysis, researchers have been able to obtain remarkably deep results that have sometimes changed entire branches of science. The mathematical techniques required to derive these results turn out to be simple and accessible to all. The list of great names involved runs from Newton and Fourier to Maxwell, Rayleigh and Kolmogorov. Among recent developments, it is sufficient to recall the triumph of the Kolmogorov–Obukhov theory in turbulence.

Everyone would like to score a classical triumph. Many people therefore attacked what one would think were almost identical problems using the same simple dimensional analysis approach. Alas, they almost always failed. Dimensional analysis was cursed and reproached for being untrustworthy and unfounded, even mystical. Paradoxically, the reason for this lack of success was that only a few people understood the content and real abilities of dimensional analysis.

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.