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Summary

In this chapter we briefly review the Navier–Stokes equations which govern the flow of constant-property Newtonian fluids. More comprehensive accounts can be found in the texts of Batchelor (1967), Panton (1984), and Tritton (1988). Two topics that are important in the study of turbulent flows, that are not extensively discussed in these texts, are the Poisson equation for pressure (Section 2.5), and the transformation properties of the Navier–Stokes equations (Section 2.9). The equations of fluid motion are expressed either in vector notation or in Cartesian tensor notation, which is reviewed in Appendix A.

Continuum fluid properties

The idea of treating fluids as continuous media is both natural and familiar. It is, however, worthwhile to review the continuum hypothesis – that reconciles the discrete molecular nature of fluids with the continuum view – so as to avoid confusion when quantities such as ‘fluid particles’ and ‘infinitesimal material elements’ are introduced.

The length and time scales of molecular motion are extremely small compared with human scales. Taking air under atmospheric conditions as an example, the average spacing between molecules is 3 × 10−9 m, the mean free path, λ, is 6 × 10−8 m, and the mean time between successive collision of a molecule is 10−10 s. In comparison, the smallest geometric length scale in a flow, ℓ, is seldom less than 0.1 mm = 10−4 m, which, for flow velocities up to 100 m s−1, yields a flow timescale larger than 10−6 s.

Summary

Introduction

In large-eddy simulation (LES), the larger three-dimensional unsteady turbulent motions are directly represented, whereas the effects of the smallerscale motions are modelled. In computational expense, LES lies between Reynolds-stress models and DNS, and it is motivated by the limitations of each of these approaches. Because the large-scale unsteady motions are represented explicitly, LES can be expected to be more accurate and reliable than Reynolds-stress models for flows in which large-scale unsteadiness is significant – such as the flow over bluff bodies, which involves unsteady separation and vortex shedding.

As discussed in Chapter 9, the computational cost of DNS is high, and it increases as the cube of the Reynolds number, so that DNS is inapplicable to high-Reynolds-number flows. Nearly all of the computational effort in DNS is expended on the smallest, dissipative motions (see Fig 9.4 on page 351), whereas the energy and anisotropy are contained predominantly in the larger scales of motion. In LES, the dynamics of the larger-scale motions (which are affected by the flow geometry and are not universal) are computed explicitly, the influence of the smaller scales (which have, to some extent, a universal character) being represented by simple models. Thus, compared with DNS, the vast computational cost of explicitly representing the small-scale motions is avoided.

Summary

This book is primarily intended as a graduate text on turbulent flows for engineering students, but it may also be valuable to students in atmospheric sciences, applied mathematics, and physics, as well as to researchers and practicing engineers.

The principal questions addressed are the following.

(i) How do turbulent flows behave?
(ii) How can they be described quantitatively?
(iii) What are the fundamental physical processes involved?
(iv) How can equations be constructed to simulate or model the behavior of turbulent flows?

In 1972 Tennekes and Lumley produced a textbook that admirably addresses the first three of these questions. In the intervening years, due in part to advances in computing, great strides have been made toward providing answers to the fourth question. Approaches such as Reynolds-stress modelling, probability-density-function (PDF) methods, and large-eddy simulation (LES) have been developed that, to an extent, provide quantitative models for turbulent flows. Accordingly, here (in Part II) an emphasis is placed on understanding how model equations can be constructed to describe turbulent flows; and this objective provides focus to the first three questions mentioned above (which are addressed in Part I). However, in contrast to the book by Wilcox (1993), this text is not intended to be a practical guide to turbulence modelling. Rather, it explains the concepts and develops the mathematical tools that underlie a broad range of approaches.

There is a vast literature on turbulence and turbulent flows, with many worthwhile questions addressed by many different approaches.

Summary

Three important classes of statistically steady turbulent flows will be examined in some detail in this chapter, namely jets, wakes, and boundary layers. These flows are of considerable practical interest and are illustrated in Figure 5.1, together with a shear layer, which is not specifically considered here, but whose treatment is not essentially different from that of jets and wakes. Much is known about these flows from numerous experimental and theoretical studies, stretching back over many decades, and the resulting theoretical models are semiempirical, being based on the experimentally observed properties of the flows, such as the small angles of divergence of jets and wakes and the existence of self-similar behavior far downstream of the nozzle, in the case of jets, or body, in the case of wakes. We attempt to give a unified presentation of these models, emphasizing their common origins in the boundary-layer approximation and various types of mean-flow self-similarity.

The basis of the theories developed in this chapter is the mean-flow equations, (4.9), to which simplifying assumptions, similar to those of laminar boundary-layer theory, are applied, leading to neglect of some of the terms in the equations. This turbulent boundary-layer approximation, which applies to jets and wakes as well, is justified by slowness of mean-flow development with streamwise distance, as witnessed by the small angles of divergence of turbulent jets and wakes. Since the approximations used are similar to Prandtl's theory of laminar boundary layers, with which we assume the reader is familiar, and since, despite their quite different mechanisms of growth, the behavior of turbulent boundary layers in many ways resembles that of laminar ones, we devote Section 5.1 to reminding the reader of the principal characteristics and theory of laminar boundary layers, before developing the turbulent analog of Prandtl's theory in Section 5.2.

Having applied the boundary-layer approximation to simplify the mean-flow equations, one is still faced with their lack of closure, apparent from the Reynolds stress terms. Here, empirical information is introduced, whose details depend on the class of flows considered. In the case of jets and wakes, we use the observed fact that the mean flow becomes self-similar sufficiently far downstream, together with an eddy-viscosity approximation and an assumed eddy viscosity that does not vary across the flow, but only with streamwise distance.

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.

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.

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:

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.

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.

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.

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2 - The equations of fluid motion

Summary

Appendix B - Properties of second-order tensors

10 - Turbulent-viscosity models

13 - Large-eddy simulation

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List of tables

Appendix E - Spectral representation of stationary random processes

Preface

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5 - Classical Models of Jets, Wakes, and Boundary Layers

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4 - Basic Theory and Illustrative Examples

6 - Spectral Analysis of Homogeneous Turbulence

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

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2 - Statistical Tools

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Index

1 - An Introduction to Turbulence

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8 - Numerical Simulation of Turbulent Flows

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3 - Space and Time Scales of Turbulence

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Contents

General References

Frontmatter

Preface and Roadmap

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