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.

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.

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.

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.

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.

In this chapter lumped parameter models for compliance of the deforming region are used to examine the influence of factors which previously in this book were assumed to be negligibly small – namely the effects of (a) a viscoelastic or rate-dependent normal compliance relation and (b) tangential compliance. Because these factors depend on the interaction force and not simply the impulse, the analysis of their effects necessarily uses time rather than normal impulse as an independent variable.

Before publication of Newton's Principia (1687), dynamics was an empirical science; i.e., it consisted of propositions that described observed behavior without any explanation for the forces that caused motion. For example, Kepler's laws are kinematic relations that describe orbital motion. Kepler (1571–1630) discovered these relations by laboriously fitting various possibilities to the voluminous measurements of planetary motion that had been recorded by Tycho Brahe. Likewise Galileo stated propositions describing the motion of freely falling bodies. The propositions are based on relating transit times for different drop heights to clear ideas of distance, time and translational velocity. These savants, however, possessed only the vaguest notion of force.

While Galileo recognized that there must be some extended cause for acceleration or retardation, he did not realize that a uniform acceleration was a consequence of a steady force; he recognized that there was a cause for acceleration but was not able to relate cause and effect. Indeed, it is difficult to imagine how there could be progress in this direction before the creation of calculus.

At the time of Galileo, the topics at the forefront of dynamics were percussion, projectile ballistics and celestial mechanics. Each of these topics had technological importance for warfare, industrial development or navigation. Percussion in particular was concerned with the terminal ballistics of musket balls as well as the effect of a forging hammer on a workpiece.

Two bodies, labeled B and B′, collide when they come together with an initial difference in velocity. Ordinarily they first touch at a point that will be termed the contact point C. During a very brief period of contact, the point C on the surface of body B is coincident with point C′ on the surface of body B′. If at least one of the bodies, B or B′, has a surface that is topologically smooth at the contact point (i.e., the surface has continuous curvature), there is a plane tangent to this surface at C; the coincident contact points C and C′ lie in this tangent plane. If both bodies are convex and the surfaces have continuous curvature near the contact point, then this tangent plane is tangential to both surfaces that touch at C; i.e., the surfaces of the colliding bodies have a common tangent plane. The direction of the normal to the tangent plane is specified by a unit vector n; this direction is termed the common normal direction. The contact force and changes in relative velocity at the contact point C will be resolved into components normal and tangential to the common tangent plane.

In practice the bodies that are colliding are composed of elastic, elastic–plastic or viscoplastic materials, so that the large contact forces acting during a collision induce both local deformations near the contact point and global deformations (vibrations) of the entire body. This chapter focuses on the local deformations in a contact region that can be represented as an elastic–perfectly plastic solid; the additional effect of global deformations will be introduced in Chapter 7.

For collisions between hard bodies, the analysis of changes in velocity during collision is simplified by assuming that the initial point of contact is surrounded by an infinitesimally small deforming region. For other purposes, however, it is necessary to consider deformations in the small region surrounding a finite area of contact.

Axial impact on a deformable body results in a disturbance which initially propagates away from the impact site at a specific speed. This disturbance is a pulse or wave of particle displacement (and consequent stress). Wave propagation relates to propagation of a coherent pulse of stress and particle displacement through a medium at a finite speed. Familiar manifestations of this phenomenon are the transmission of sound through air, water waves across the surface of the sea and seismic tremors through the earth; thus, waves exist in gases, liquids and solids. Sources of excitation may be either concentrated or distributed spatially, and brief or extended functions of time. The unifying characteristic of waves is propagation of a disturbance through a medium. Properties of the medium that result in waves and determine the speeds of propagation are the density ρ and moduli of deformability (Young's modulus E, shear modulus G, bulk modulus K, etc.).

Longitudinal Wave in Uniform Elastic Bar

Consider a uniform slender elastic bar of cross-sectional are A, elastic modulus E and density ρ; the bar contains a region with axial stress σ(x, t) that is propagating in the positive x direction as shown in Fig. 7.1.

Impact against a mechanism composed of nearly rigid bodies is a feature of many practical machines. These systems may include mechanisms where the relative velocities at joints are initially zero and finally must vanish or they can be another type of system such as a gear train or an agglomerate of unconnected bodies where at each contact the normal component of terminal relative velocity must be separating. These two classes of multibody impact problems – mechanisms and separate bodies that are touching – are distinguished from analyses of two-body impacts by the addition of constraint equations that describe limitations on relative motion at each point of contact between bodies. These constraint equations express the linkage between separate elements. Books on dynamics typically analyze the impulsive response of systems composed of rigid bodies linked by frictionless or nondissipative pinned joints.

A ball that falls in a gravitational field before colliding against a flat surface will rebound from the surface with a loss of energy that depends on the coefficient of restitution. If the ball is free, it will continue bouncing on the surface in a series of collisions; these arise because in each collision the ball is partly elastic and during the period between collisions the ball is attracted towards the surface by gravity. In Chapter 2 it was shown that an inelastic ball (0 < e* < 1) which is bouncing on a level surface in a gravitational field has both a period of time between collisions and a bounce height that asymptotically approach zero as the number of collisions increases. In other words, with increasing time this dissipative system asymptotically approaches a stable attractor – the equilibrium configuration where the ball is resting on the level surface.

Some other systems can experience energy input during each cycle of impact and flight; consequently these systems exhibit more complex behavior. For example, a pencil has a regular hexagonal cross-section with six vertices. If the pencil rolls down a plane, the mean translational speed of the axis asymptotically approaches a steady mean speed of rolling where the kinetic energy dissipated by the collision of a vertex against the plane equals the loss in gravitational potential energy as the pencil rolls from one flat side to the next.