A Rigorous Entropy Law for the Turbulent Cascade

One important question of turbulence theory is to get a profound understanding of turbulence as a cascade process, that can be understood as the evolution of turbulent structures on different spatial or temporal scales. There have been many works to achieve a better understanding but rigorous results, like the Káarmán-Howarth equations or Kolmogorov's 4/5th law, are still rare. Although these laws are derived from the Navier Stokes equation, the experimental verification is not of high precision.

In this work, we present a non-equilibrium thermodynamical approach to the turbulent cascade process. This new statistical approach, which take the turbulent cascade as Markov processes in scale, enables to apply concepts of stochastic thermodynamics to turbulent flows and to define the thermodynamical quantity of entropy ΔS for a cascade path. Most interestingly the entropy fluctuations are large like known from micro system. As a new feature we find that the turbulent entropy fluctuations fulfill in high precision the rigorous integral fluctuation theorem <exp(-ΔS)>=1, which is a fundamental entropy law of non-equilibrium thermodynamics. We investigate these features for a macroscopic system by analyzing measurements of different turbulent flows like free jets or wake flows of grids.