A Rigorous Entropy Law for the Turbulent Cascade

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 {\$}{\$}{\backslash}varDelta S{\_}{\{}tot{\}}{\$}{\$}of the turbulentFuchs, A.Reinke, N.Nickelsen, D.Peinke, J. cascade fulfills in high precision the rigorous integral fluctuation theorem {\$}{\$}{\backslash}langle e^{\{}-{\backslash}varDelta S{\_}{\{}tot{\}}{\}} {\backslash}rangle {\_}{\{}u({\backslash}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