Leonid M. Brekhovskikh

Elements of the Theory of Turbulence

All the flows considered above were regular or “laminar” ones. The origin of the term is due to the fact that if one marks fluid with dye, then one can see laminar currents in such flows. Flows are always laminar, according to experiment, if their velocities are sufficiently small. On the other hand, a laminar flow always turns into a turbulent one if the velocity increases. Turbulent flow is essentially irregular. Its velocity, pressure and other parameters vary in a random way at any point, even when the boundary and all other external conditions are regular. This chapter contains the basic theory of turbulence.

Nonlinear Effects in Wave Propagation

Linear theory for different kinds of waves in fluids has so far been developed. The superposition principle holds in this theory, i.e., waves of different kinds as well as different modes of waves of the same kind propagate without interaction with each other. One must bear in mind, however, that linear equations are only approximate ones and the original hydrodynamic equations are substantially nonlinear. It is important to clarify the conditions under which a linear approximation is adequate and to consider new effects caused by nonlinear terms in the equations. To this very question is this chapter devoted.

Elastic Waves in Solids

We will now continue to investigate elastic waves in solids. Using the general equation of motion obtained in the previous chapter, we will show that two kinds of waves can exist in an elastic medium—longitudinal (compressional) and transverse (shear) waves. The simplest case—plane harmonic waves—will be considered in detail as well as their reflection at a plane boundary. The surface Rayleigh and Love waves will be discussed, too.

Basic Laws of Ideal Fluid Dynamics

We begin with ideal fluids, i.e. with those without internal friction, and hence without the transformation of mechanical energy into heat. Moreover, we assume that heat transfer between volumes of the fluid can also be neglected. Hence, the entropy of a material volume of the fluid is assumed to be constant and stresses in the fluid can be specified in terms of only one scalar quantity, namely the pressure p.

Surface and Internal Waves in Fluids

This chapter begins the study of waves in fluids, a very important kind of motion with numerous applications. The main characteristic feature of waves is the possibility of energy transport over considerable distances without mass transport. The diversity of forces acting on the particle in a fluid causes a variety of different types of waves. In the following four chapters we will consider these waves using the linear approximation so that interaction between waves will be disregarded.

Waves in Rods, Vibrations of Rods

Static forces and static deformations have been considered in the previous chapter. If, however, an elastic body is under the action of a force varying in time, different types of waves can arise in this body. Each type of wave can transfer disturbances from one part of the body to the other at a finite speed.

General Theory of Stress and Strain

Let us now proceed to the general theory of the behaviour of an elastic body under the action of external forces. Our discussion in this chapter is not restricted to isotropic bodies, i.e., bodies whose characteristics are the same in all directions, but is also applicable to crystals of arbitrary symmetry.

Flows of Viscous Fluids

It was shown above that the flow of an ideal fluid in the field of a potential force produces no vortexes,—i.e., a potential flow remains potential at any time. In real flows, however, we regularly observe a production and destruction of vortexes. This is because the real fluid is a viscous one. The assumption that there is a slip at the boundary between a flowing fluid and a solid (as in the case of an ideal fluid) also turns out to be incorrect for a real fluid. All components of the velocity vanish on the surface of a body at rest, according to experiments. That is why the dust accumulates on the surfaces of bodies even when flow exists past these bodies (the dust on blades of a fan, for example).

The Main Types of Strain in Elastic Solids

In this chapter we will consider basic laws of the theory of elasticity such as Hooke’s law and Poisson’s relation. The main types of deformations of elastic bodies (tension of a rod, shearing, torsion, bending of a beam) will be treated by applying of these laws.

Waves in Rotating Fluids

Gravity waves were discussed in the previous chapter. These waves are of great importance for the dynamics of the ocean and the atmosphere. No less important are waves related to the rotation of the Earth. The theory of such waves, as well as the development of the theory of gravity waves which takes the Earth’s rotation into account, is the main purpose of this chapter.