Fluid dynamics and solid mechanics

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.

Summary

Caminante, no hay camino.

Se hace camino al ander.

‘Traveller, there is no path

Paths are made by walking.’

A. Machado, popular song from Latin America

When bodies collide, they come together with some relative velocity at an initial point of contact. If it were not for the contact force that develops between them, the normal component of relative velocity would result in overlap or interference near the contact point and this interference would increase with time. This reaction force deforms the bodies into a compatible configuration in a common contact surface that envelopes the initial point of contact. Ordinarily it is quite difficult and laborious to calculate deformations that are geometrically compatible, that satisfy equations of motion and that give equal but opposite reaction forces on the colliding bodies. To avoid this detail, several different approximations have been developed for analyzing impact: rigid body impact theory, Hertz contact theory, elastic wave theory, etc. This book presents a spectrum of different theories for collision and describes where each is applicable. The question of applicability largely depends on the materials of which the bodies are composed (their hardness in the contact region and whether or not they are rate-dependent), the geometric configuration of the bodies and the incident relative velocity of the collision. These factors affect the relative magnitude of deformations in the contact region in comparison with global deformations.

A collision between hard bodies occurs in a very brief period of time.

Summary

Philosophy is written in this grand book – I mean the universe – which stands continuously open to our gaze, but cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics and its characters are triangles, circles and other geometric figures, without which it is humanly impossible to understand a single word of it; without these one is wandering about in a dark labyrinth.

Galileo Galilei, Two New Sciences, 1632

When a bat strikes a ball or a hammer hits a nail, the surfaces of two bodies come together with some relative velocity at an initial instant termed incidence. After incidence there would be interference or interpenetration of the bodies were it not for the interface pressure that arises in a small area of contact between the two bodies. At each instant during the contact period, the pressure in the contact area results in local deformation and consequent indentation; this indentation equals the interference that would exist if the bodies were not deformed.

At each instant during impact the interface or contact pressure has a resultant force of action or reaction that acts in opposite directions on the two colliding bodies and thereby resists interpenetration. Initially the force increases with increasing indentation and it reduces the speed at which the bodies are approaching each other.

Summary

This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge.

A. M. Turing, 1952

angle of incidence angle between the direction of the incident relative velocity of the contact points and the common normal direction. Direct or normal collisions have zero angle of incidence, whereas oblique collisions have a nonzero angle of incidence.

angle of rebound angle between the direction of the relative velocity of the contact points at separation and the common normal direction.

attractor steady state solution that is approached asymptotically with increasing time if the system has small dissipation.

coefficient of friction upper limit on ratio of tangential to normal force at contact.

coefficient of stick geometric parameter specifying ratio of tangential to normal force for stick.

collinear (or central) impact configuration colliding bodies oriented so that each center of mass is on common normal line passing through the point of initial contact.

common normal direction normal to common tangent plane that passes through contact point C.

common tangent plane If at least one of the bodies has a topologically smooth surface at the contact point, this is the plane that is tangent to the surface at the point of initial contact. Usually both bodies have smooth surfaces around their respective points of contact, so they have a common tangent plane.

Summary

Like a ski resort full of girls hunting for husbands, and husbands hunting for girls, the situation is not as symmetrical as it might seem.

Alan Lindsay Mackay, Lecture, Birkbeck College, University of London, 1984

Three-dimensional (3D, or nonplanar) changes in velocity occur in collisions between rough bodies if the configuration is not collinear and the initial direction of sliding is not in-plane with two of the three principal axes of inertia for each body. In collisions between rough bodies, dry friction can be represented by Coulomb's law. If there is a tangential component of relative velocity at the contact point (sliding contact) this law relates the normal and tangential components of contact force by a coefficient of limiting friction. The friction force acts in a direction opposed to sliding. For a collision with planar changes in velocity, sliding is in either one direction or the other on the common tangent plane. In general however, friction results in nonplanar changes in velocity. Nonplanar velocity changes give a direction of sliding that continuously changes, or swerves, during an initial phase of contact in an eccentric impact configuration. This chapter obtains changes in relative velocity during rigid body collisions as a function of the impulse P due to the normal component of the reaction force. The changes in velocity depend on two independent material parameters – the coefficient of friction and an energetic coefficient of restitution.

Fluid dynamics and solid mechanics

Refine search

Refine search

Actions for selected content:

7987 results in Fluid dynamics and solid mechanics

4 - Basic Theory and Illustrative Examples

6 - Spectral Analysis of Homogeneous Turbulence

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

Summary

2 - Statistical Tools

Summary

Index

1 - An Introduction to Turbulence

Summary

8 - Numerical Simulation of Turbulent Flows

Summary

3 - Space and Time Scales of Turbulence

Summary

Contents

General References

Frontmatter

Preface and Roadmap

Summary

Contents

References

Chapter 3 - Rigid Body Theory for Planar or 2D Collisions

Preface

Summary

Chapter 9 - Collision against Flexible Structures

Chapter 1 - Introduction to Analysis of Low Speed Impact

Summary

Appendix B - Glossary of Terms

Summary

Chapter 4 - 3D Impact of Rough Rigid Bodies

Summary

Fluid dynamics and solid mechanics

Refine search

Refine search

Actions for selected content:

Save Search

7987 results in Fluid dynamics and solid mechanics

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary