Turbulence and Random Processes in Fluid Mechanics

Most flows encountered in nature and in engineering practice are turbulent. It is therefore important to understand the fundamental mechanisms at work in such flows. Turbulent flows are unsteady and contain fluctuations that are random in space and time. An important characteristic is the richness of scales of eddy motion present in such flows: In a fully developed turbulent flow all scales appear to be fully occupied or saturated in a sense, from the largest ones that can fit within the size of the flow region down to the smallest scale allowed by dissipative processes. Turbulent flows are also highly vortical, a consequence of vortex stretching and tilting by larger random vorticity fields.

The reason turbulence is so prevalent in fluids of low viscosity is that steady laminar flows tend to become unstable at high Reynolds or Rayleigh numbers and therefore cannot be maintained indefinitely as steady laminar flows. Instability to small disturbances is an initial step in the process whereby a laminar flow goes through transition to turbulence. In investigations of instability of flows that are homogeneous in one or more spatial dimensions, one usually formulates a linear problem of the evolution of an infinite train of small-amplitude waves so as to find whether such waves will grow or decay with time. More general disturbances may be analyzed by Fourier superposition.

A complete description of the transition process requires one to consider the development of disturbances of finite amplitudes. This is generally a difficult theoretical task since it leads to nonlinear problems. A few simplified model problems, giving some insight into the nature of the transition process, are tractable, however, such as the evolution of a finiteamplitude wave train in a parallel shear flow, finite-amplitude density interface waves, weak nonlinear interaction of several wave trains, and the influence of distortion by large-scale motion on smaller-scale wave trains. The treatment of non-wave-like amplitude disturbances lacking spatial and temporal periodicity is more difficult.

Because of the random nature of fully developed turbulent flow fields, statistical methods are usually employed for their description. However, in the statistical averages much of the information that may be relevant to the understanding of the turbulent mechanisms may be lost, especially phase relationships. This may not seem too serious for flows in which the motion appears to be completely disorganized, such as in nearly isotropic or homogeneous turbulent flows.