A Rigorous Entropy Law for the Turbulent Cascade

Abstract

There is a lack of high precision results for turbulence. Here we present a non-equilibrium thermodynamical approach to the turbulent cascade and show that the entropy generation $\varDelta S_{tot}$ of the turbulent cascade fulfills in high precision the rigorous integral fluctuation theorem $\langle e^{-\varDelta S_{tot}} \rangle _{u(\cdot )} = 1$. To achieve this result the turbulent cascade has to be taken as a stochastic process in scale, for which Markov property is given and for which an underlying Fokker-Planck equation in scale can be set up. For one exemplary data set we show that the integral fluctuation theorem is fulfilled with an accuracy better than $10^{-3}$. Furthermore, we show that other basic turbulent features are well taking into account like the third order structure function or the skewness of the velocity increments.