Small scale structures of turbulence in terms of entropy and fluctuation theorems

Abstract

We present experimental evidence that, together with the integral fluctuation theorem, which is fulfilled with high accuracy, a detailed-like fluctuation theorem holds for large entropy values in cascade processes in turbulent flows. Based on experimental data, we estimate the stochastic equations describing the scale-dependent cascade process in a turbulent flow by means of Fokker-Planck equations, and from the corresponding individual cascade trajectories an entropy term can be determined. Since the statistical fluctuation theorems set the occurrence of positive and negative entropy events in strict relation, we are able to verify how cascade trajectories, defined by entropy consumption or entropy production, are linked to turbulent structures: Trajectories with entropy production start from large velocity increments at large scale and converge to zero velocity increments at small scales; trajectories with entropy consumption end at small scale velocity increments with finite size and show a lower bound for small scale increments. A linear increase with the magnitude of the negative entropy value is found. This indicates a tendency to local discontinuities in the velocity field. Our findings show no lower bound of negative entropy values and thus for the corresponding piling up velocity differences of the small scale structures.