Fourth-order statistical moments of the velocity gradient tensor in homogeneous, isotropic turbulence

TL;DR

This paper derives a unified, compact expression for fourth-order statistical moments of the velocity gradient tensor in homogeneous, isotropic turbulence, aligning with some prior results and offering a method extendable to higher moments.

Contribution

It presents a new, unified formula for fourth-order moments of the velocity gradient tensor, reconciling previous expressions and providing an extendable algorithm for higher moments.

Findings

The derived expression agrees with Siggia's results.

Discrepancies are noted with Phan-Thien and Antonia's expression.

The algorithm can be extended to higher order moments.

Abstract

A compact expression of fourth-order statistical moments of the velocity gradient tensor in homogeneous, isotropic, incompressible turbulence is obtained as a function of its invariants and of generic components of the velocity gradient. This single, compact expression is in full agreement with the four different expressions previously obtained by Siggia as functions of the same invariants and of generic components of the vorticity vector and the strain tensor; however, some discrepancies arise with respect to a similar, single expression obtained by Phan-Thien and Antonia. The used algorithm may be easily extended to handle higher order statistical moments of the velocity gradient.