Analytical theories of turbulence

TLDR

The paper surveys the current state of analytical attempts at a theory of turbulence. It discusses formulating turbulence in terms of moments, introduces eddy relaxation times and the direct‑interaction approximation, and examines their properties. It highlights the closure problem difficulty via the quasinormal approximation and compares numerical solutions of turbulence equations with direct numerical simulation of the Navier–Stokes equations.

Abstract

This paper surveys the current state of analytical attempts at a theory of turbulence. The formulation of the problem in terms of moments is discussed. The difficulty posed by the closure problem is examined in detail using the quasinormal approximation as an example. The notion of dynamical relaxation by non-linear scrambling leads to the introduction of eddy relaxation times and the direct-interaction approximation. The properties of the direct-interaction approximation are indicated. Finally, a comparison is made between numerical solution of the equations of turbulence their and direct numerical simulation of the Navier–Stokes equations.