Turbulence and Random Processes in Fluid Mechanics

Flows of fluids of low viscosity may become unstable when large gradients of kinetic and/or potential energy are present. The flow field set up by the instability generally tends to smooth out the velocity and temperature differences causing it. The available kinetic or potential energy released by the instability may be so large that transition to a fully developed turbulent flow occurs.

Transition is influenced by many parameters. An important one is the level of preexisting disturbances in the fluid; a high level would generally cause early transition. Another cause for early transition in the case of wall-bounded shear flows is surface roughness. The manner in which transition occurs may also be very sensitive to the detailed flow properties.

For shear flows the basic nondimensional flow parameter measuring the tendency toward instability and transition is the Reynolds number; for high Re values, kinetic energy differences can be released faster into fluctuating motion than viscous diffusion will have time to smooth them out. For a heated fluid subject to gravity the Rayleigh number is the main stability parameter.

Of crucial importance for the tendency of a flow to become unstable and go through transition is the detailed distribution of mean velocity and/or temperature in the field. The analysis that follows is intended to illustrate this.

Although the flow processes involved in instability and transition might at a first glance appear to have only a slight resemblance to those observed in fully developed turbulence, they are nevertheless related to it in important ways. In a gross sense turbulence may be regarded as a manifestation of flow instability occurring randomly in space and time. The linear instability problem is the simplest flow model incorporating the interaction between unsteady fluctuations and a background shear or density distribution. With the aid of nonlinear instability theory one may also possibly be able to clarify some of the mechanisms whereby turbulence is maintained.

Instability to small disturbances

Because of the mathematical difficulties in the analysis of flow instability, only idealized cases for which the basic fluid flow properties vary with one spatial coordinate can be analyzed in a reasonably simple manner.