On the strength of the nonlinearity in isotropic turbulence

TL;DR

This paper investigates how nonlinearity diminishes in fully developed isotropic turbulence at high Reynolds numbers, using a modified approximation to analyze the spectral behavior and non-Gaussian effects.

Contribution

It provides a closed-form expression for the mean-square nonlinearity spectrum in turbulence and examines its scaling and depletion mechanisms at high Reynolds numbers.

Findings

Nonlinearity depletion is constant across inertial range scales.

Sweeping effects are weaker than random sweeping predictions.

Mean-square nonlinearity exceeds Gaussian estimates due to non-Gaussian Reynolds-stress fluctuations.

Abstract

Turbulence governed by the Navier-Stokes equations shows a tendency to evolve towards a state in which the nonlinearity is diminished. In fully developed turbulence this tendency can be measured by comparing the variance of the nonlinear term to the variance of the same quantity measured in a Gaussian field with the same energy distribution. In order to study this phenomenon at high Reynolds numbers, a version of the Direct Interaction Approximation is used to obtain a closed expression for the statistical average of the mean-square nonlinearity. The wavenumber spectrum of the mean-square nonlinear term is evaluated and its scaling in the inertial range is investigated as a function of the Reynolds number. Its scaling is dominated by the sweeping by the energetic scales, but this sweeping is weaker than predicted by a random sweeping estimate. At inertial range scales, the depletion of nonlinearity as a function of the wavenumber is observed to be constant. At large it is observed that the mean-square nonlinearity is larger than its Gaussian estimate, which is shown to be related to the non-Gaussianity of the Reynolds-stress fluctuations at these scales.