Entropy and fluctuation relations in isotropic turbulence

TL;DR

This paper explores the concept of entropy in isotropic turbulence using a generalized Kolmogorov-Hill equation, confirming the fluctuation relation from non-equilibrium thermodynamics through numerical simulations.

Contribution

It introduces a novel definition of turbulence entropy and demonstrates the validity of fluctuation relations in the inertial range of turbulent flows.

Findings

Confirmation of fluctuation relation validity in turbulence

Probability density ratios follow exponential behavior

Entropy generation rate linked to turbulence temperature

Abstract

Based on a generalized local Kolmogorov-Hill equation expressing the evolution of kinetic energy integrated over spheres of size $\ell$ in the inertial range of fluid turbulence, we examine a possible definition of entropy and entropy generation for turbulence. Its measurement from direct numerical simulations in isotropic turbulence leads to confirmation of the validity of the fluctuation relation (FR) from non-equilibrium thermodynamics in the inertial range of turbulent flows. Specifically, the ratio of probability densities of forward and inverse cascade at scale $\ell$ is shown to follow exponential behavior with the entropy generation rate if the latter is defined by including an appropriately defined notion of ``temperature of turbulence'' proportional to the kinetic energy at scale $\ell$.