Nonlinear Science and Fluid Dynamics

Summary

Introduction

This chapter contains supplementary material beyond the scope of the previous chapters. The choice of topics is, of course, determined largely by the author's own interests and knowledge (or lack thereof). No attempt is made, for example, to discuss supersonic and magnetohydrodynamic turbulence or turbulent convection. As for numerical simulation of turbulence, one of the most important current tools in turbulence research, it defies being summarized and requires a book of its own. Some topics are presented rather briefly, either because there is no need to elaborate them (Section 9.2 on further reading in turbulence and fluid mechanics) or because very good reviews are easily found in the literature (Section 9.3 on mathematics and Section 9.4 on dynamical systems). Other topics require more detailed presentation for lack of suitable review or just because the author's viewpoint is somewhat unusual. Section 9.5 is an introduction, occasionally rather critical, to closure, functional and diagrammatic methods. Section 9.6 is devoted to eddy viscosity, multiscale methods and renormalization; it includes some little known historical material on nineteenth century turbulence research. Finally, Section 9.7 deals mostly with recent developments in two-dimensional turbulence.

Books on turbulence and fluid mechanics

One of the earliest reviews of (mostly homogeneous) turbulence, was written by von Neumann (1949) in the form of a report to the Office of Naval Research, after he had attended a conference in Paris on Problems of Motion of Gaseous Masses of Cosmical Dimensions, organized jointly by the International Union of Theoretical and Applied Mechanics and the International Astronomical Union.

Summary

Introduction

In previous chapters we showed how it is possible to establish certain scaling laws for fully developed turbulence by starting from unproven but plausible hypotheses and then proceeding in a systematic fashion. By ‘phenomenology’ of fully developed turbulence one understands a kind of shorthand system whereby the same results can be recovered in a much simpler way, although, of course, at the price of less systematic arguments. Phenomenology of fully developed turbulence has some associated ‘mental images’, such as the ‘Richardson cascade’ (Section 7.3), which have played a very important role in the history of the subject. After recasting the K41 theory in phenomenological language and images, it also becomes possible to grasp intuitively some of the shortcomings which may be present. A considerable part of the existing work on turbulence rests on K41 phenomenology, particularly in applied areas such as the modeling of turbulent flow. Kolmogorov himself, with Ludwig Prandtl, was one of the pioneers of this important area of research, which is beyond the scope of this book (Kolmogorov 1942; see also Batchelor 1990, Spalding 1991 and Yaglom 1994). We shall give only some examples of what can be derived by phenomenology: counting degrees of freedom (Section 7.4), comparing macroscopic and microscopic length scales (Section 7.5), finding the probability distribution function of velocity gradients (Section 7.6) and finding the law of decay of the energy (Section 7.7).

Summary

There is presently no fully deductive theory which starts from the Navier–Stokes equation and leads to the two basic experimental laws reported in Chapter 5. Still, it is possible to formulate hypotheses, compatible with these laws and leading to additional predictions. This was the purpose of the celebrated Kolmogorov 1941 theory (in short K41). It will here be reformulated rather freely. In Section 6.1 we shall present a modern viewpoint with emphasis on postulated symmetries rather than on postulated universality, i.e. independence on the particular mechanism by the turbulence is generated. We thereby obtain a scaling theory with an undetermined scaling exponent. The latter is determined in Section 6.2 from the ‘four-fifths’ law, an exact relation derived by Kolmogorov, also in 1941. The main results of the K41 theory are presented in Section 6.3.

Kolmogorov 1941 and symmetries

In Section 2.2 we made a list of known symmetries for the Navier–Stokes equation (time- and space-translations, rotations, Galilean transformations, scaling transformations, etc). What are their implications for turbulence?

Let us begin with time-translations. At low Reynolds numbers, if the boundary conditions and any external driving force are time-independent, the flow is steady and thus does not break the time-invariance symmetry. When the Reynolds number is increased, an Andronov–Hopf bifurcation may occur. This makes the flow time-periodic and turns the continuous time-invariance symmetry into a discrete one. When the Reynolds number is increased further the flow will usually, at some point, become chaotic.

Summary

There is something predictable in a turbulent signal

In Chapter 1 we presented some pictures chosen to prompt the study of the symmetries of the Navier–Stokes equation. However important flow visualizations may be, experimental data on turbulence also include a considerable body of quantitative results. Velocimetry, the measurement of the flow velocity (or one component thereof) at a given point as a function of time, is by far the most common way of getting quantitative information. There are many different techniques of velocimetry which we shall not review here.

Let us turn directly to an example. Fig. 3.1(a) shows a one-second signal obtained from a hot-wire probe placed in the very large wind tunnel SI of ONERA. The signal is the ‘streamwise’ velocity (component parallel to the mean flow). It is sampled five thousand times per second (5 kHz). The mean flow has been subtracted so that the signal appears to fluctuate around zero.

What strikes us when looking at this signal?

Summary

Andrei Nikolaevich Kolmogorov's work in 1941 remains a major source of inspiration for turbulence research. Great classics, when revisited in the light of new developments, may reveal hidden pearls, as is the case with Kolmogorov's very brief third 1941 paper ‘Dissipation of energy in locally isotropic turbulence’ (Kolmogorov 1941c). It contains one of the very few exact and nontrivial results in the field, as well as very modern ideas on scaling, ideas which cannot be refuted by the argument Lev Landau used to criticize the universality assumptions of the first 1941 paper.

Revisiting Kolmogorov's fifty-year-old work on turbulence was one goal of the lectures on which this book is based. The lectures were intended for first-year graduate students in ‘Turbulence and Dynamical Systems’ at the University of Nice–Sophia–Antipolis. My presentation deliberately emphasizes concepts which are central in dynamical systems studies, such as symmetry-breaking and deterministic chaos. The students had some knowledge of fluid dynamics, but little or no training in modern probability theory. I have therefore included a significant amount of background material. The presentation uses a physicist's viewpoint with more emphasis on systematic arguments than on mathematical rigor. Also, I have a marked preference for working in coordinate space rather than in Fourier space, whenever possible.

Modern work on turbulence focuses to a large extent on trying to understand the reasons for the partial failure of the 1941 theory.

Summary

Introduction

This chapter is organized as follows. The basic concepts are introduced in Section 8.2. Experimental results about intermittency in the inertial range, based on velocity measurements, are presented in Section 8.3. Exact results, independent of any phenomenology, are presented in Section 8.4. Two broad classes of phenomenological models of intermittency are then discussed. In the first class (Section 8.5), intermittency is studied via velocity increments. It comprises the β-model (Section 8.5.1), the bifractal model (Section 8.5.2) and the multifractal model (Sections 8.5.3 and 8.5.4). Implications of the multifractal model for the dissipation range and for the skewness and flatness of velocity derivatives are presented in Sections 8.5.5 and 8.5.6, respectively. In the second class (Section 8.6), intermittency is studied via the fluctuation of the dissipation; inertial-range quantities are related to such fluctuations by a bridging ansatz, originally introduced by Obukhov and Kolmogorov (Section 8.6.2). Random cascade models are presented in Section 8.6.3; their multifractal behavior is shown to be a direct consequence of the probabilistic theory of ‘large deviations’, which is presented in an elementary fashion in Section 8.6.4. The lognormal model and its shortcomings are discussed in Section 8.6.5. Shell models, a class of deterministic nonlinear models which can display intermittency, are presented in Section 8.7.

The order chosen here for the presentation of the entire material on the theory of intermittency is pedagogical, not historical. Most of the latter aspects are discussed in Section 8.8. Recent trends in intermittency research are presented in Section 8.9.

Summary

Introduction to compressible flow

When the fluid velocity varies from place to place by amounts that are no longer small compared with the velocity of sound, the compressibility of the fluid cannot be ignored. Bernoulli's theorem does not have to be abandoned in these circumstances but it does need to be reformulated, and reformulation requires knowledge of the equation which relates pressure to density. The necessary equation is well known for ideal gases, and it is on gases – and on air in particular, which at normal temperatures and pressures conforms closely to the ideal model – that we focus attention in this chapter. As we shall see, the reformulated version of Bernoulli's theorem can be applied in an elementary way to a number of interesting phenomena which have to do with compressible flow of air. Among these are the shock fronts which develop following explosions and which accompany supersonic projectiles, and it is chiefly because these shock fronts are in many respects analogous to the tidal bores and hydraulic jumps discussed in §2.16 that this chapter stands where it does.

For the time being we shall continue to ignore viscosity. Neglect of viscosity is usually justified when the Reynolds Number is large compared with unity, for reasons which were outlined in §1.9, and most of the phenomena to be discussed below occur at flow rates where Re is 105 or more.

Summary

Introduction

Physics is a tree with many branches, and fluid dynamics is one of the older and sturdier ones. It began to form in the eighteenth century, when Euler and Daniel Bernoulli set out to apply the principles which Newton had enunciated for systems composed of discrete particles to liquids which are virtually continuous, and it has been in active growth ever since. Nowadays it is partially obscured from view by branches of more recent origin, such as relativity, atomic physics and quantum mechanics, and students of physics pay rather little attention to it. This is a pity, for several reasons. Firstly, because of the engineering applications of the subject, which are many and various: the design of aeroplanes and boats and automobiles, and indeed of any structure intended to move through fluid or propel fluid or simply to withstand the forces exerted by fluid, depends in a critical way upon the principles of the subject. Secondly, because fluid dynamics has important applications in other branches of physics and indeed in other realms of science, including astronomy, meteorology, oceanography, zoology and physiology: dripping taps, solitary waves on canals, vortices in liquid helium, seismic oscillations of the Sun, the Great Red Spot on Jupiter, small organisms that swim, the circulation of the blood – these are just a few of the very varied topics involving fluid dynamics which have been occupying research scientists and mathematicians of international reputation over the past few decades.

Summary

Shear stresses in Newtonian fluids

Throughout the last four chapters, the fact that real fluids possess viscosity has been almost completely ignored. We have supposed shear stress to be negligible and normal stress to be isotropic, and we have found that in so far as isotropic normal stress – the pressure p – depends upon fluid velocity u, it does so through formulae in which only the local magnitude of u and its rate of change with time appear. We cannot proceed much further on that simple, Eulerian, basis. The principal aims of the present chapter are firstly to establish the Newtonian formulae which relate the components of stress in viscous fluids to gradients of u, secondly to use these formulae to establish a more general equation of motion for fluids than Euler's equation, and thirdly to discuss a variety of relatively simple problems in which the effects of viscosity are dominant – so dominant in most cases that the fluid's inertia is negligible instead. The motion of fluids in such circumstances is sometimes referred to as creeping flow.

Newton himself may have considered only the simple situation illustrated by fig. 6.1, where planar laminae of fluid lying normal to the x2 axis are moving steadily in the x1 direction and sliding over one another, so that there exists a uniform velocity gradient ∂u1/∂x2.

Nonlinear Science and Fluid Dynamics

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Acknowledgments

Subject index

CHAPTER 5 - Two experimental laws of fully developed turbulence

CHAPTER 4 - Probabilistic tools: a survey

Author index

Frontmatter

References

CHAPTER 9 - Further reading: a guided tour

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CHAPTER 7 - Phenomenology of turbulence in the sense of Kolmogorov 1941

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CHAPTER 6 - The Kolmogorov 1941 theory

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Contents

CHAPTER 3 - Why a probabilistic description of turbulence?

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Preface

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CHAPTER 1 - Introduction

CHAPTER 2 - Symmetries and conservation laws

CHAPTER 8 - Intermittency

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3 - Gas dynamics

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1 - A bird's eye view

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Mathematical Conventions

6 - Viscosity

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Nonlinear Science and Fluid Dynamics

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