Development of Eulerian Theory of Turbulence within Kraichnan's Direct Interaction Approximation Framework

TL;DR

This paper introduces a new Eulerian turbulence theory within Kraichnan's DIA framework, providing closed equations compatible with Kolmogorov's spectrum and incorporating random Galilean invariance, offering a novel renormalized perturbation approach.

Contribution

It develops a variant of DIA (VDIA) that yields closed equations for velocity correlations and response tensors, compatible with Kolmogorov's spectrum and Galilean invariance.

Findings

The theory is consistent with Kolmogorov's energy spectrum.

The modified VDIA respects random Galilean transformation rules.

The approach offers a new perspective on renormalized perturbation theory.

Abstract

Within the framework of Kraichnan's Direct Interaction Approximation (DIA), we propose an Eulerian turbulence theory providing a closed set of equations for two-time and single-time velocity correlations, and second order correlations of infinitesimal response tensor $\hat{G}_{in}({\bf k};t,t')$. The proposed theory, namely variant of DIA (VDIA), is consistent with Kolmogorov's energy spectrum. The VDIA is further modified to make it compatible with random Galilean transformation rules. The closed set of equations does not contain equation for ensemble averaged response tensor $G_{in}({\bf k};t,t')=\langle\hat{G}_{in}({\bf k};t,t')\rangle$. The present theory can also be seen as a new remormalized perturbation theory having different method for renormalization.