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34754 results in Cambridge Textbooks

Page 1639 of 1738

  • 7 - Kolmogorov's and Other Theories Based on Spectral Analysis

  • Jean Mathieu, Ecole Centrale de Lyon, Julian Scott, Ecole Centrale de Lyon
  • Book:
    An Introduction to Turbulent Flow
    Published online:
    28 May 2018
    Print publication:
    26 June 2000, pp 283-326

  • Summary

    This chapter covers a variety of different subjects, of which the most important is the Kolmogorov theory of inertial and dissipative range scales in its original (Kolmogorov 1941a, b, c) version. The Kolmogorov theory is discussed in Section 7.3 and is one of the foundation stones of the theory of turbulence, leading to the famous E α k -5/3 inertial-range energy spectrum, which is now a benchmark for measurements and theoretical models alike. However, the original theory is not without its problems, as we will see in Section 7.5, where it is reformulated to avoid Landau's objection to the original version, and the possibility of intermittency corrections is introduced. To illustrate intermittency effects, we use the y6-model here, not because it gives a fully satisfactory account of intermittency, but because it is relatively simple to describe, understand, and analyze.

    Aside from the theory of the small scales (high wavenumbers), we also discuss spectral behavior at small wavenumbers (in the course of Section 7.1), the final phase of viscous decay (Section 7.2), in which the turbulent Reynolds number is small, and the consequences of large-scale self-similarity for high-Reynolds-number gridlike turbulence (Section 7.4). Throughout the chapter, the case of high-Reynolds-number turbulence will form our main preoccupation, and we will suppose developed turbulence, that is, that the cascade has been allowed the time to reach equilibrium over the full range of scales from the large, energy-containing scales, down to the small, dissipative ones. We further assume homogeneous, isotropic turbulence without mean flow1 in the analyses of this chapter, although these assumptions are not really essential to all the conclusions reached, in particular Kolmogorov's theory of the small scales is observed to be quite robust, as we will discuss later.

    Properties of the Energy Spectrum and Velocity Correlations

    We first want to recall some relevant results from the spectral analysis of homogeneous, isotropic turbulence, which were derived in the previous chapter.

  • 2 - Statistical Tools

  • Jean Mathieu, Ecole Centrale de Lyon, Julian Scott, Ecole Centrale de Lyon
  • Book:
    An Introduction to Turbulent Flow
    Published online:
    28 May 2018
    Print publication:
    26 June 2000, pp 31-56

  • Summary

    The apparently random character of turbulent flows strongly suggests that statistical methods will be fruitful. In this chapter, we discuss some statistical techniques, in preparation for later chapters. The reader is presumed to have a modest background in probability theory and this chapter summarizes the main elements, while placing statistical ideas in the context of turbulent flows (those whose knowledge of the basic theory is rusty or nonexistent should perhaps keep one of the many textbooks on the mathematics of probability and statistics to hand; Lumley (1970) and Monin and Yaglom (G 1971) describe the theory as applied to turbulence).

    As discussed in Chapter 1, despite being generally considered as deterministic, turbulent flows are highly nonunique in practice. Thus, if an experiment is repeatedly carried out, a different velocity field is obtained in each realization, even if the experimental conditions are nominally the same (i.e., the experimenter endeavors to reproduce the same flow). This is because the detailed behavior of the flow in any one realization is extremely sensitive to small changes in the initial or boundary conditions, which the experimenter cannot control to infinite precision. This type of problem is ideally suited to statistical methods. Indeed, it was the nonrepeatability of an experiment such as tossing a coin that led to the idea of probabilities in the first place. By looking at the statistical properties of an ensemble of different flow realizations, all obtained using the same nominal conditions, one hopes to extract useful quantities, namely probabilities and averages, which depend only on parameters that the experimenter controls. To take an example, the detailed behavior of the turbulent wake of a sphere placed in a uniform flow may vary with tiny perturbations in the incoming stream and small vibrations of the sphere, to name just two possible extraneous factors, but one hopes and expects that, for instance, the average velocity is well defined. Of course, the flow statistics depend on the gross experimental conditions used, for example, a turbulent boundary layer can have quite different thickness and drag with suction than without, but we expect them not to change significantly with small variations in those conditions.

  • 1 - An Introduction to Turbulence

  • Jean Mathieu, Ecole Centrale de Lyon, Julian Scott, Ecole Centrale de Lyon
  • Book:
    An Introduction to Turbulent Flow
    Published online:
    28 May 2018
    Print publication:
    26 June 2000, pp 1-30

  • Summary

    This book, intended as an introductory course in turbulence, presumes the reader to be well acquainted with basic fluid mechanics and to have a reasonable level of mathematical sophistication (the General References following the Preface include some textbooks on fluid dynamics to which the reader may want to refer if necessary). In this chapter we take a critical look at the concepts underlying a description of turbulence and lay the foundations for the chapters to follow.

    Most flows occurring in nature and in industrial applications are said to be turbulent. Everyday life gives us an intuitive understanding of turbulence in fluids and thus the concept is often accepted without further discussion. Among other properties, typical turbulent flows have apparently random velocity fluctuations with a wide range of different length and time scales. Thus, for instance, a graph of turbulent velocity as a function of time at a fixed point in space shows random variations. The graph has a “furry” appearance and successively higher magnifications of the “fur” show smaller fluctuations of shorter time scales, until finally the shortest is reached and the velocity is revealed as a smooth function. The same is true if the velocity is plotted as a function of one of the spatial coordinates at fixed time. However, these characteristics of turbulence do not provide a rigorous definition allowing one to objectively distinguish turbulence from a complicated laminar flow. In fact, it is difficult to give a watertight definition of turbulence, although it is easy to give specific examples in which turbulence is present in at least part of the flow:

  • • the rapid flow of fluid around a bluff body or airfoil,

  • • the majority of terrestrial atmospheric and oceanic currents,

  • • the motion of the atmospheres of the sun and planets,

  • • the flow inside most industrial plant.

    • It would certainly help in the systematic study of turbulence if there were a complete and succinct definition of a turbulent flow, but no fully satisfactory one has been found to date and so, rather than attempt a strict definition, we will describe specific examples and some general properties of turbulence in the course of this chapter. We begin by considering how turbulence arises in the first place, via the instability of laminar flows.

  • 8 - Numerical Simulation of Turbulent Flows

  • Jean Mathieu, Ecole Centrale de Lyon, Julian Scott, Ecole Centrale de Lyon
  • Book:
    An Introduction to Turbulent Flow
    Published online:
    28 May 2018
    Print publication:
    26 June 2000, pp 327-368

  • Summary

    The ability of modern computers to store and rapidly manipulate large quantities of data has allowed the development of flow simulations that complement, and sometimes even replace, traditional laboratory experiments. In industry, the considerable expense of constructing and exploiting physical models motivates the search for reliable simulation techniques to reduce the cost of the design and development phases. Full, detailed coverage of numerical methods for turbulent flows goes well beyond the scope of this book and would require a volume to itself. This chapter is therefore limited to a relatively brief overview of the three main types of numerical techniques, represented by the following three sections, in which the reader desiring further details of numerical methods will find references to more specialized works.

    Direct Numerical Simulation (DNS)

    The first, and most obvious, approach to the numerical simulation of turbulent flows treats them like any others, solving the unsteady, three-dimensional Navier-Stokes equations numerically without further ado. Such methods are referred to as direct simulations, in contrast with those based on the statistically averaged equations considered in Section 8.3, which involve closure approximations, but are much less computationally costly than DNS at high Reynolds numbers. Direct simulation allows numerical experiments to be carried out that are often difficult or impossible to realize in the laboratory, and yields detailed information concerning the flow field in individual realizations and its statistical properties that again often surpasses what can be measured in real flows. Furthermore, unlike other simulation techniques for turbulent flows, DNS does not involve approximations, other than those due to discretization, which are inherent in any numerical solution of differential equations, and so provides benchmarks against which less costly simulation methods may be evaluated and parameterized, just as they can by using the results of physical experiments. However, the continuum of different spatial (and temporal) scales intrinsic to turbulence, a continuum whose width increases with the turbulent Reynolds number, ReL, leads to computer time and storage requirements that increase rapidly with ReL. In consequence, DNS is limited to relatively low Reynolds numbers in practice and it is currently very expensive to conduct a DNS calculation at even moderately high Reynolds number.

  • 3 - Space and Time Scales of Turbulence

  • Jean Mathieu, Ecole Centrale de Lyon, Julian Scott, Ecole Centrale de Lyon
  • Book:
    An Introduction to Turbulent Flow
    Published online:
    28 May 2018
    Print publication:
    26 June 2000, pp 57-72

  • Summary

    Besides its apparent randomness, one of the important characteristics of turbulent flows at high Reynolds number (which we presume throughout this chapter) is the wide range of different space and time scales which they contain. As illustrated in Figure 3.1a, the velocity measured at a given point within a turbulent flow fluctuates randomly and does not yield a smooth function of time, but rather one with finescale “fur.” If part of such a graph is examined at successively higher magnifications, instead of approaching a straight line, as would a smooth function, the fur is revealed as being itself “furry,” and so on, through a succession of smaller and smaller time scales, until eventually the smallest of all is reached. At this scale, the velocity finally appears as a smooth function of time, with no further structure revealed by additional magnification. On the other hand, the largest time scales of turbulence are characterized by the time interval required for statistical decorrelation of the turbulent velocity fluctuations. At high Reynolds number, the separation of scales between the largest and smallest is wide, and gets wider as the Reynolds number increases.

    The continuum of time scales described above is also reflected in the spatial structure of the flow. The order of size of the largest spatial scales of turbulence is measured by the velocity correlation length L, which characterizes the spatial separation required for significant decorrelation of the velocity fluctuations. Successively higher magnification of the spatial structure of the velocity field shows smaller and smaller scales down to the smallest, known as the Kolmogorov scale, at which it appears smooth. Thus, turbulence contains a continuum of scales, ranging from the large ones, whose size is of order the correlation length, to those of order the Kolmogorov scale. Both the Kolmogorov scale, r\, and correlation scale, L, are best considered as orders of magnitude: one talks of scales comparable with, much smaller than, or much larger than these quantities. The spatial and temporal “furriness” of the velocity field are related; indeed, one can think of the small time scales as resulting from convection of fine spatial structures past the measurement probe, as we will see later.

    If one attempts to take the derivative of the velocity field, either with respect to time or one of the spatial coordinates, its furry nature becomes markedly more apparent.

An Introduction to Turbulent Flow
  • Jean Mathieu, Julian Scott
  • Published online:
    28 May 2018
    Print publication:
    26 June 2000
  • Most natural and industrial flows are turbulent. The atmosphere and oceans, automobile and aircraft engines, all provide examples of this ubiquitous phenomenon. In recent years, turbulence has become a very lively area of scientific research and application, attracting many newcomers who need a basic introduction to the subject. An Introduction to Turbulent Flow, first published in 2000, offers a solid grounding in the subject of turbulence, developing both physical insight and the mathematical framework needed to express the theory. It begins with a review of the physical nature of turbulence, statistical tools, and space and time scales of turbulence. Basic theory is presented next, illustrated by examples of simple turbulent flows and developed through classical models of jets, wakes, and boundary layers. A deeper understanding of turbulence dynamics is provided by spectral analysis and its applications. The final chapter introduces the numerical simulation of turbulent flows. This well-balanced text will interest graduate students in engineering, applied mathematics, and the physical sciences.

  • Preface and Roadmap

  • Jean Mathieu, Ecole Centrale de Lyon, Julian Scott, Ecole Centrale de Lyon
  • Book:
    An Introduction to Turbulent Flow
    Published online:
    28 May 2018
    Print publication:
    26 June 2000, pp vii-ix

  • Summary

    The theory of turbulence has seen considerable progress in the last fifty years, although these advances are, at first sight, less marked than might have been hoped at the outset. Prior to that period, Reynolds, Taylor, Prandtl, Von Karman, and Kolmogorov, to name just some of the eminent workers, had laid the foundations of the subject and a number of excellent books and surveys appeared in the 1950s. Batchelor's book, The Theory of Homogeneous Turbulence, was first published in 1953 and mainly concerned the development of spectral methods, quickly gaining a following among existing and aspiring specialists in turbulence. Shortly afterwards (1956), another classic appeared in the shape of Townsend's The Structure of Turbulent Shear Flow, whose approach was based more on physical reasoning and placed more emphasis on the more realistic, but less theoretically tractable case of inhomogeneous turbulent flows.

    Many other excellent works have appeared since then, among them the books given in the General References list following this preface. In each of them, the authors view turbulence from their own perspective, different from that of other authors, for, owing to the nature of the subject, an author's personal viewpoint colors the presentation perhaps more than in most areas of science. The book by Tennekes and Lumley is somewhat different from most of the others because it gives considerable weight to physical ideas, order-of-magnitude discussions, and other means for gaining intuition about turbulent flows, while the authors remind us that “several dozen introductory texts in general fluid dynamics exist, but the gap between monographs and advanced texts in turbulence is wide.” The book gives the reader excellent food for thought before tackling more specialized ones, such as Monin and Yaglom's two-volume work, which remains the undisputed (if slightly dated) bible for the subject.

    The present book does not seek to replace these existing works, but to complement them, giving the reader a solid grounding in the theory of turbulence and developing both physical insight and the mathematical framework needed to express the theory. Turbulence is treated using one-point statistics in the early chapters of the book, while multipoint statistics, analyzed by spectral methods, are introduced later, the two approaches being complementary and mutually illuminating.

  • Introduction

  • Thomas Paine
  • Edited by Bruce Kuklick, University of Pennsylvania
  • Book:
    Paine: Political Writings
    Published online:
    14 September 2019
    Print publication:
    13 April 2000, pp vii-xxiii

  • Summary

    Thomas Paine was a unique political thinker. Writing at the end of the eighteenth Century while a revolutionary tide swept over North America and Europe, he had an unrivalled reputation not just as a democratic polemicist but also as an activist politician. He lived in England, the new United States, and France, and was hailed by all those eager for an anti-monarchical political order in the Atlantic world. Hated and feared by conservatives in England and America, Paine was nonetheless suspect to the most radical French revolutionaries. Extraordinary because he came not from the cultured classes but from the common folk, he expressed their political feelings and ushered in a new era of political rhetoric and public commitment.

    Little is known of the details of Paine's life, especially of the first thirty-seven years he spent in England. He was born at Thetford in Norfolk, in 1737, and was early on subject to the unconventional religious influence of his Quaker father. For a time he followed his father's occupation as a maker of corsets, but also went to sea as a common sailor for two years, ran a small tobacco shop, and had a minor job with the English government collecting customs duties, an “excise officer.”

    Despite the absence of information, Paine's biographers have rightly speculated on what shaped his life in England in the hope of understanding his later career. For after what appeared to be an entirely ordinary and nondescript life in England, filled with the failures and woes of the lower ranks of that time, he left for the American colonies in 1774, where he began to write in favor of their Separation from the Mother Country. Within a short time he became known in both the Old and New Worlds as a political theorist of great significance. What had gone on in England to prepare him for his novel role?

    Paine had obtained enough education to master reading, writing, and arithmetic in Norfolk, a tribute to his family's zeal for education and a sign of the growing aspirations of the class of craftspeople in England to which his family belonged. As an adult, in addition to his other employments, Paine had even briefly been a school teacher. In his early manhood he supplemented his rudimentary knowledge by attending populär lectures on the physics of Isaac Newton that had transformed eighteeenth-century intellectual life.

  • The Age of Reason, Part I (1794)

  • Thomas Paine
  • Edited by Bruce Kuklick, University of Pennsylvania
  • Book:
    Paine: Political Writings
    Published online:
    14 September 2019
    Print publication:
    13 April 2000, pp 265-318

  • Summary

    It has been my intention for several years past to publish my thoughts upon Religion. I am well aware of the difficulties that attend the subject; and from that consideration had reserved it to a more advanced period of life. I intended it to be the last offering I should make to my fellow-citizens of all nations, and that at a time when the purity of the motive that induced me to it could not admit of a question, even by those who might disapprove the work.

    The circumstance that has now taken place in France, of the total abolition of the whole national order of priesthood and of everything appertaining to compulsive Systems of religion and compulsive articles of faith, has not only precipitated my intention, but rendered a work of this kind exceedingly necessary; lest, in the general wreck of superstition, of false Systems of government, and false theology, we lose sight of morality, of humanity, and of the theology that is true. As several of my colleagues, and others of my fellow-citizens of France, have given me the example of making their voluntary and individual profession of faith, I also will make mine; and I do this with all that sincerity and frankness with which the mind of man communicates with itself.

    I believe in one God, and no more; and I hope for happiness beyond this life.

    I believe in the equality of man, and I believe that religious duties consist in doing justice, loving mercy, and endeavoring to make our fellow-creatures happy.

    But lest it should be supposed that I believe many other things in addition to diese, I shall, in the progress of this work, declare the things I do not believe and my reasons for not believing them.

    I do not believe in the creed professed by the Jewish church, by the Roman church, by the Greek church, by the Turkish church, by the Protestant church, nor by any church that I know of. My own mind is my own church.

    All national institutions of churches - whether Jewish, Christian, or Turkish - appear to me no other than human inventions set up to terrify and enslave mankind and monopolize power and profit.

  • Rights of Man, Part I (1791)

  • Thomas Paine
  • Edited by Bruce Kuklick, University of Pennsylvania
  • Book:
    Paine: Political Writings
    Published online:
    14 September 2019
    Print publication:
    13 April 2000, pp 57-154

  • Summary

    From the part Mr. Burke took in the American Revolution it was natural that I should consider him a friend to mankind; and as our acquaintance commenced on that ground, it would have been more agreeable to me to have had cause to continue in that opinion than to change it.

    At the time Mr. Burke made his violent speech last winter in the English Parliament against the French Revolution and the National Assembly, I was in Paris, and had written to him but a short time before to inform him how prosperously matters were going on. Soon after this I saw his advertisement of the pamphlet he intended to publish: As the attack was to be made in a language but little studied and less understood in France, and as everything suffers by translation, I promised some of the friends of the Revolution in that country that whenever Mr. Burke's pamphlet came forth, I would answer it. This appeared to me the more necessary to be done when I saw the flagrant misrepresentations which Mr. Burke's pamphlet contains; and that while it is an outrageous abuse on the French Revolution and the principles of Liberty, it is an imposition on the rest of the world.

    I am the more astonished and disappointed at this conduct in Mr. Burke, as (from the circumstances I am going to mention) I had formed other expectations.

    I had seen enough of the miseries of war to wish it might never more have existence in the world, and that some other mode might be found out to settle the differences that should occasionally arise in the neighborhood of nations. This certainly might be done if courts were disposed to set honestly about it, or if countries were enlightened enough not to be made the dupes of courts. The people of America had been bred up in the same prejudices against France which at that time characterized the people of England; but experience and an acquaintance with the French nation have most effectually shown to the Americans the falsehood of those prejudices; and I do not believe that a more cordial and confidential intercourse exists between any two countries than between America and France.

Page 1639 of 1738