Lagrangian finite time- and conditionally-averaged fluctuation relation in isotropic turbulence

The entropy generation rate in isotropic turbulence can be defined using local properties from the Kolmogorov-Hill equation, featuring the energy cascade rate as well as the `temperature of turbulence' at a prescribed inertial-range length-scale (Ref. 1). The fluctuation relation (FR) from non-equilibrium thermodynamics that predicts exponential behavior of positive to negative entropy production rate PDFs has been tested using instantaneous flow fields from isotropic turbulence data at Reynolds numbers $Re_\lambda = 1250$ and $Re_\lambda = 433$. We now test the finite-time averaged FR over intervals extending from one to several eddy turnover times, finding that the FR holds and the exponential trend of probability ratios remains valid. Results suggest a minor redefinition of the `temperature of turbulence' that includes a 1/3 factor corresponding to the kinetic energy for each velocity component separately. A key result is that finite-time averaging must be performed within the Lagrangian framework, i.e. integrating along fluid trajectories using the filtered convective fluid velocity. In contrast, the FR fails when using the Eulerian framework, i.e. time-averaging at fixed positions. Finally, we test the single-time FR adherance to the Kolmogorov refined similarity hypothesis, confirming that FR holds (approximately) even when conditioning on different values of locally averaged molecular dissipation rates (Ref. 2).