Introduction To Renormalization Group Modeling Of Turbulence

The renormalization group (RNG) and related e-expansion methods are a powerful technique that allow the systematic derivation of coarse-grained equations of motion for turbulent flows and, in particular, the derivation of sophisticated turbulence models based on the fundamental underlying physics. The RNG method provides a convenient calculus for the analysis of complex physical effects in complex flows. The details of the RNG method applied to fluid mechanics differ in some crucial respects from how renormalization group techniques are applied to field theories in other branches of physics. At the present time, the RNG methods for fluid dynamics are by no means rigorously justified, so their utility must be based on the quality and quantity of results to which they lead. In this paper we discuss the basis for the RNG method and then illustrate its application to a variety of turbulent flow problems, emphasizing those points where further analysis is needed. The application of a field-theoretic method like the RNG technique to turbulence is based on the fundamental assumption of universality of small scales in turbulent flows. Such universal behavior was first suggested over 50 years ago in the seminal work of A. N. Kolmogorov who argued that the small-scale spectrum of incompressible turbulence is universal and characterized by two numbers, the rate of energy dissipation ε per unit mass and the kinematic viscosity v.