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1680 results in Nonlinear Science and Fluid Dynamics

Page 71 of 84

  • CHAPTER SIX - Experimental characterization, prediction, and modification of chaotic states

  • Gregory L. Baker, Academy of the New Church College, Jerry P. Gollub, Haverford College, Pennsylvania
  • Book:
    Chaotic Dynamics
    Published online:
    05 June 2012
    Print publication:
    26 January 1996, pp 133-165

  • Summary

    Chaotic states occur widely in natural phenomena, but closed form mathematical models are rarely available. This situation leads to a number of related problems that bear on the use of experimental data. First, how is it possible to tell whether a set of apparently noisy data in fact arises from chaotic dynamics? Second, how can chaotic data be used to make short-term predictions or forecasts? Finally, how can experimental data be used to influence and control nonlinear systems? We address these questions in the present chapter.

    Characterization of chaotic states

    In this section, we consider the use of experimental data to test for the existence of chaos, to reconstruct the chaotic attractor if it exists, and to characterize its structure quantitatively.

    One rarely has complete information about all of the degrees of freedom in a complex dynamical system. For example, in a chaotic fluid system, this information would include the velocity of the fluid at many different positions as a function of time. Even for the pendulum, a complete specification would seem to require measurement of three distinct time-dependent quantities (the angle, the angular velocity, and the phase of the forcing function).

    Although this information can easily be obtained for a pendulum, it is more informative to use the pendulum to learn how to handle situations where a fuller description is unavailable. It is frequently possible to learn a considerable amount from a single time series, a list of successive values of one dynamical quantity.

  • CHAPTER SEVEN - Chaos broadly applied

  • Gregory L. Baker, Academy of the New Church College, Jerry P. Gollub, Haverford College, Pennsylvania
  • Book:
    Chaotic Dynamics
    Published online:
    05 June 2012
    Print publication:
    26 January 1996, pp 166-189

  • Summary

    The primary physical example discussed up to this point is the driven pendulum, which provides an elementary pedagogical example of a chaotic system. Its behavior is extraordinarily complex. Varying the parameters leads to an intricate pattern of periodic and chaotic states with several types of transitions between them. In the chaotic regions, nearby orbits diverge exponentially from each other, with consequent long-term unpredictability. We have also discussed how a single experimental time series can be used to characterize chaotic states, to control them, or to achieve short-term predictability.

    Chaos occurs widely in nature. In this chapter we briefly describe examples of chaotic behavior in lasers, chemical reactions, fluid dynamics, interfacial growth, and earthquake models. We emphasize the fact that natural systems are often spatially extended and therefore have intrinsically many degrees of freedom. Thus, their chaotic behavior may be more complicated than that of the pendulum and other systems with only a few degrees of freedom. Still, the main concepts of nonlinear dynamics continue to play a significant role in organizing our knowledge of spatially extended nonlinear systems. At the end of this final chapter, we consider the impact of our understanding of chaos on two major fields of theoretical physics: quantum mechanics and statistical mechanics.

    Chaos in lasers

    Since the early days of laser technology, instabilities in laser action have been apparent. That is, the light output need not be time-independent (Harrison and Biswas, 1986).

  • CHAPTER 9 - Further reading: a guided tour

  • Uriel Frisch, Observatoire de la Cote d'Azur
  • Book:
    Turbulence
    Published online:
    28 May 2018
    Print publication:
    30 November 1995, pp 195-254

  • 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.

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

  • Uriel Frisch, Observatoire de la Cote d'Azur
  • Book:
    Turbulence
    Published online:
    28 May 2018
    Print publication:
    30 November 1995, pp 100-119

  • 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).

  • CHAPTER 6 - The Kolmogorov 1941 theory

  • Uriel Frisch, Observatoire de la Cote d'Azur
  • Book:
    Turbulence
    Published online:
    28 May 2018
    Print publication:
    30 November 1995, pp 72-99

  • 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.

  • CHAPTER 3 - Why a probabilistic description of turbulence?

  • Uriel Frisch, Observatoire de la Cote d'Azur
  • Book:
    Turbulence
    Published online:
    28 May 2018
    Print publication:
    30 November 1995, pp 27-39

  • 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?

  • (i) The signal appears highly disorganized and presents structures on all scales.

  • (ii) The signal appears unpredictable in its detailed behavior.

  • (iii) Some properties of the signal are quite reproducible.

    • Regarding item (i), we observe that in contrast to the signal shown in Fig. 2.2 which had only two scales present, the signal shown here displays structures on all scales: the eye directly perceives structures with time-scales of the order of one second, of one-tenth of a second, of one-hundredth of a second, and possibly smaller.

  • Preface

  • Uriel Frisch, Observatoire de la Cote d'Azur
  • Book:
    Turbulence
    Published online:
    28 May 2018
    Print publication:
    30 November 1995, pp xi-xiv

  • 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.

  • CHAPTER 8 - Intermittency

  • Uriel Frisch, Observatoire de la Cote d'Azur
  • Book:
    Turbulence
    Published online:
    28 May 2018
    Print publication:
    30 November 1995, pp 120-194

  • 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.

  • 3 - Gas dynamics

  • T. E. Faber, University of Cambridge
  • Book:
    Fluid Dynamics for Physicists
    Published online:
    05 June 2012
    Print publication:
    17 August 1995, pp 78-118

  • 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.

Page 71 of 84