1 Introduction

Most flows encountered in engineering practice and in nature are turbulent. The boundary layer on an aircraft wing is likely to be turbulent, the atmospheric boundary layer over the earth's surface is turbulent, and the major oceanic currents are turbulent. In this chapter we shall discuss certain elementary ideas about the dynamics of turbulent flows. We shall see that such flows do not allow a strict analytical study, and one depends heavily on physical intuition and dimensional arguments. In spite of our everyday experience with it, turbulence is not easy to define precisely. In fact, there is a tendency to confuse turbulent flows with "random flows". With some humor, Lesieur (1987) said that "turbulence is a dangerous topic which is at the origin of serious fights in scientific meetings since it represents extremely different points of view, all of which have in common their complexity, as well as an inability to solve the problem. It is even difficult to agree on what exactly is the problem to be solved". Some characteristics of turbulent flows are the following:

1. Randomness: Turbulent flows seem irregular, chaotic, and unpredictable.

2. Nonlinearity: Turbulent flows are highly nonlinear. The nonlinearity serves two purposes. First, it causes the relevant nonlinearity parameter, say the Reynolds number Re, the Rayleigh number Ra,or the inverse Richardson num-ber Ri ^-1, to exceed a critical value. In unstable flows small perturbations grow spontaneously and frequently equilibrate as finite amplitude disturbances. On further exceeding the stability criteria, the new...