An Introduction and Overview of Various Theoretical Approaches to Turbulence

Abstract

Theoretical methods to treat turbulence may be organized along two principal lines: (1) statistical, in which the distribution function of the flow (or equivalently, various moments of the velocity or vorticity field) serve as the basic ingredient; and (2) the dynamical, in which the mechanisms of instability and turbulent coherent structures serve as the primary focus. We include in the former second-order modeling, the two-point moment closures, and methods based on the distribution function. Central to this approach is the assumption that complete knowledge of the flow is not necessary for an approximate knowledge of low-order moments, or other simple features of the flow’s distribution function. The dynamical approach focuses upon some key aspect of turbulence such as coherent structure (vortices) and follows their evolution in detail by employing suitable but perhaps heavily approximate dynamical equations. Practitioners argue that averaging (ensemble or time) may be a final needed step, but may legitimately be done only after developing some understanding of how to represent the relevant structures.