On the probability distribution of power fluctuations in turbulence

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.