Introduction

Turbulent motion in fluids is a familiar phenomenon from our everyday experience, but it is nevertheless an extremely difficult thing to define quantitatively. For the most part, the best that can be done to define turbulence is to list some of its characteristics: It is unsteady chaotic flow, apparently random, with fluid motions distributed over a relatively wide range of length and time scales. The complicated spatio-temporal structure of turbulent velocity fields renders their analytical description impossible, and the large number of degrees of freedom and the wide range of scales in turbulent flows result in difficult problems for numerical analysis, taxing both the speed and memory capacities of present day computers.

Statistical turbulence theory and the closure problem

Because of the effectively random behavior of turbulent flows, it is natural to attempt a statistical formulation. This is the classical approach to turbulence theory. The idea is to decompose a turbulent velocity vector field into a mean and a fluctuating part in an attempt to extract the relevant mean physical quantities. The “mean” in this approach may be a time average – appropriate for a steady configuration which, although fluctuating at all times, has well behaved time averaged characteristics – or an ensemble average where the average is over initial conditions in some class. To illustrate this decomposition we will consider only a steady state turbulent flow, assuming that well defined time averages of all quantities exist.