On the probability distribution of power fluctuations in turbulence

TL;DR

This paper models local power fluctuations in turbulence using a Gaussian-based distribution, deriving an exact large deviation function that satisfies a fluctuation relation, and compares predictions with experimental data.

Contribution

It provides an exact analytical form for the power fluctuation distribution and demonstrates its fluctuation relation in turbulence simulations.

Findings

Distribution modeled by product of two Gaussian variables

Exact large deviation function derived and shown to satisfy a fluctuation relation

Deviations from the model linked to tail behavior of velocity distribution

Abstract

We study local power fluctuations in numerical simulations of stationary, homogeneous, isotropic turbulence in two and three dimensions with Gaussian forcing. Due to the near-Gaussianity of the one-point velocity distribution, the probability distribution function (pdf) of the local power is well modelled by the pdf of the product of two joint normally distributed variables. In appropriate units, this distribution is parameterised only by the mean dissipation rate, $\epsilon$. The large deviation function for this distribution is calculated exactly and shown to satisfy a Fluctuation Relation (FR) with a coefficient which depends on $\epsilon$. This FR is entirely statistical in origin. The deviation from the model pdf are most pronounced for positive fluctuations of the power and can be traced to a slightly faster than Gaussian decay of the tails of the one-point velocity pdf. The resulting deviations from the FR are consistent with several recent experimental studies.