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34754 results in Cambridge Textbooks

Page 1679 of 1738

  • 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.

  • 7 - Stream and River Corridors

  • from Part III - Corridors
  • Richard T. T. Forman, Harvard University, Massachusetts
  • Foreword by Edward O. Wilson
  • Book:
    Land Mosaics
    Published online:
    12 October 2018
    Print publication:
    09 November 1995, pp 208-252

  • Summary

    Rivers … are the natural highways of all nations, not only leveling the ground and removing obstacles from the path of the traveller, quenching his thirst and bearing him on their bosoms, but conducting him through the most interesting scenery, the most populous portions of the globe, and where the animal and vegetable kingdoms attain their greatest perfection.

    Henry David Thoreau, A Week on the Concord and Merrimack Rivers, 1849

    A stream or river corridor is a strip of vegetation that encloses a channel with flowing water. The corridor may only include the channel and its adjacent banks, or may be wide enough to include a floodplain, hillslopes, and adjacent strips of upland. The focus here is not on the stream or river, nor on the network qualities of a river system, nor on the whole drainage basin, though general aspects of these are included. Rather, the emphasis is on the vegetation corridor, its components, functioning, and dynamics. Riparian corridor or vegetation generally refers to the floodplain portion of a stream or river corridor, although the concept sometimes is considered synonymous with stream or river corridor.130

    Landscape urologists, to use a medical analogy, can fill a bottle with stream or river water, analyze its contents, and tell much about the health of the landscape. Natural processes produce water that may be considered a control. Human activities, either directly in the corridor, or indirectly in the surrounding basin, alter the water in numerous telltale ways. A busy urologist, quietly filling bottles in many channels, can both locate and identify most human activities affecting the landscape.

    River corridors are so important to people that every component of society has its hand in the corridor. Its water is extracted for irrigation and drinking supplies. Water flow is altered for flood control, transportation, and hydroelectric dams that generate power. Wastes are carried away in this sewer of society. Fish and fishermen duel in the water. Recreation and aesthetics are enhanced with corridors. Sediments and mineral nutrients are absorbed, and move down the corridor. Beaver, livestock, and other large mammals alter its anatomy. Biodiversity and many rare wetland species are protected here. Agriculture, forestry, roads, and buildings are often rampant in river corridors. Like all corridors, width and connectivity are keys to each of these societal roles.

  • 6 - Windbreaks, Hedgerows, and Woodland Corridors

  • from Part III - Corridors
  • Richard T. T. Forman, Harvard University, Massachusetts
  • Foreword by Edward O. Wilson
  • Book:
    Land Mosaics
    Published online:
    12 October 2018
    Print publication:
    09 November 1995, pp 177-207

  • Summary

    These farms which I myself surveyed, these bounds which I have set up, appear dimly still as through a mist; but they have no chemistry to fix them … The world with which we are commonly acquainted leaves no trace, and it will have no anniversary.

    Henry David Thoreau, Walking, 1862

    Since the onset of agriculture, woody strips have spread to become the most conspicuous feature in many landscapes. Thus, the noted British ecologist, Charles Elton, observed (1958) that, ‘our own highly managed landscape is still interlaced with a wonderful network of hedgerows and roadside verges. These long winding strips of habitat by the road and land and field margins are the last really big remaining reserve we have’

    Hedgerows were originally created to keep livestock in a field, or to exclude livestock or wild herbivores from a field. Woody strips were used to clearly delineate boundaries. And windbreaks provided protection against wind, as well as defense against marauding humans.

    In contrast to trough corridors (chapter 5), woody strips have a canopy higher than the surroundings, and include windbreaks, hedgerows, and woodland corridors. Windbreaks or shelterbelts are planted to protect against wind. Hedgerows are narrow line corridors up to a few tree or shrub diameters in width that separate open areas. The term hedgerow is used in a generic sense to include examples produced by any mechanism, such as fencerows, planted hedges, and remnant lines of trees (see glossary in chapter 1 appendix). Woodland corridors are wider strips composed of natural vegetation. All wooded strips are of anthropogenic (human) origin, either directly planted or resulting from human activities on adjacent sides. Most are relatively straight, narrow, and costly to maintain. ‘Greenways’ that support recreation are included in this chapter, but ‘roadside natural strips’ are described in chapter 5, and stream and river corridors in chapter 7.

    Woods may produce wood products, but other economic and ecological values normally are more important. Gifford Pinchot, America's pioneering forester, pointed this out in 1905: ‘Not only does [the forest] sustain and regulate the streams, moderate the winds, and beautify the land, but it also supplies wood’ Thus, windbreaks are planted to reduce soil erosion, catch and hold snow, provide soil moisture in steppe areas, protect crops from desiccation, protect livestock, and reduce home energy costs.

  • 12 - Land transformation and fragmentation

  • from Part V - Changing Mosaics
  • Richard T. T. Forman, Harvard University, Massachusetts
  • Foreword by Edward O. Wilson
  • Book:
    Land Mosaics
    Published online:
    12 October 2018
    Print publication:
    09 November 1995, pp 405-434

  • Summary

    But when I consider that the nobler animals have been exterminated here - the cougar, panther, lynx, wolverine, wolf, bear, moose, deer, the beaver, the turkey, etc., etc. - I cannot but feel as if I lived in a tamed, and as it were, emasculated country … I listen to a concert in which so many parts are wanting … for instance, thinking that I have here the entire poem, and then, to my chagrin, I hear that it is but an imperfect copy that I possess and have read, that my ancestors have torn out many of the first leaves and grandest passages …

    Henry David Thoreau, Journal, 1856

    Robert Burns’ classic description of change as ‘Nature's mighty law; reminds us that whatever humans do, it is set within a changing land driven by natural processes. Furthermore, change, whether natural or human induced, is the norm. Short static or equilibrium phases are common within the broad dynamics. But persistent static phases are the exception, and they never persist indefinitely.

    The preceding chapters have explored the structural patterns and functional flows of mosaics during a static phase. This was useful for explanation, but in this and the following chapters we focus explicitly on the changing landscape or region. This chapter examines change driven by natural processes or by mainly unplanned human activities. The subsequent two chapters focus on change where human planning is also important.

    The subject of land transformation and fragmentation is significant to all human issues that involve land. Wise forestry, economics, biodiversity conservation, agriculture, landscape architecture, sociology, wildlife biology, soil science, and so forth explicitly recognize and deal with a dynamic land. These dynamics also characterize finer and broader scales, albeit at different rates, including succession within a patch1233,1029 and global climate change.1413,1158

    Many of the ideas and syntheses in this chapter were developed jointly with George F. Peterken; indeed, he is almost a coauthor of chapter 12. No overall review of the subject of this chapter exists. However, individual topics are reviewed as follows: fragmentation225,683,1793,1536,39; spatial processes683,501,1742,661; and geometric modeling of mosaic sequences.518

    Chapter 1 introduced the concepts of patch mosaic dynamics and landscape change.

  • 11 - Species movement in mosaics

  • from Part IV - Mosaics and Flows
  • Richard T. T. Forman, Harvard University, Massachusetts
  • Foreword by Edward O. Wilson
  • Book:
    Land Mosaics
    Published online:
    12 October 2018
    Print publication:
    09 November 1995, pp 364-402

  • Summary

    In our most trivial walks, we are constantly, though unconsciously, steering like pilots by certain well-known beacons and headlands, and if we go beyond our usual course we still carry in our minds the bearing of some neighboring cape; and not till we are completely lost … do we begin to find ourselves, and realize where we are and the infinite extent of our relations.

    Henry David Thoreau, Walden, 1854

    Dragons have played a key role in history and now are nearly extinct. This keystone species (or guild of species) apparently had population centers at least in Ethiopia, northern Europe, and China.738 The large ugly scaly reptile-relatives generally lived in deep caves, lakes, or rivers, and foraged singly across landscapes, both on the ground and in the air. They were attracted to virgins in distress, and repelled by heroes with swords. Chinese dragons were frightened by iron, centipedes, wax, leaves of the lien tree (Melia), and other objects. Populations fluctuated, with sighting records in Europe higher during the Middle Ages, in the seventeenth century when classification treatises were published, and in the nineteenth century when Darwin's evolutionary theory suggested relationships with known dinosaurs, such as pterodactyls. With increased human population and technologies including satellite images, remote habitats have decreased and dragon populations crashed. Most recent sighting reports have been of Chinese dragons, in areas where the species has traditionally received generosity and respect. But alas, too little information is available on how dinosaurs and these mythological dragons used pattern in moving across a landscape or region, the subject of this chapter. Therefore we must learn from movements of today's less fearsome animals and plants.

    The subject of species movement at this scale is critical in many arenas. Conserving biodiversity depends primarily on landscape and regional pattern. Forest regeneration in most regions depends on seed dispersal. Invasions of exotic species are inhibited or enhanced by landscape pattern. Plagues of insects or herds of herbivores attack crop fields one after another. And migratory fish move long distances through heterogeneous river systems.

    This chapter, along with the preceding one, is about movement across mosaics. It assumes familiarity with concepts in chapters 2, 3, 5, 8 and 9. Indeed, chapter 8 considers the resistance to species movement by a landscape, and chapter 9 examines species movement in mountains.

Page 1679 of 1738