On universal features of the turbulent cascade in terms of non-equilibrium thermodynamics
Features of the turbulent cascade are investigated for various datasets from three different turbulent flows. The analysis is focused on the question as to whether developed turbulent flows show universal small scale features. To answer this question, 2-point statistics and joint multi-scale statistics of longitudinal velocity increments are analysed. Evidence of the Markov property for the turbulent cascade is shown, which corresponds to a 3-point closure that reduces the joint multi-scale statistics to simple conditional probability density functions (cPDF). The cPDF are described by the Fokker-Planck equation in scale and its Kramers-Moyal coefficients (KMCs). KMCs are obtained by a self-consistent optimisation procedure from the measured data and result in a Fokker-Planck equation for each dataset. The knowledge of these stochastic cascade equations enables to make use of the concepts of non-equilibrium thermodynamics and thus to determine the entropy production along individual cascade trajectories. In addition to this new concept, it is shown that the local entropy production is nearly perfectly balanced for all datasets by the integral fluctuation theorem (IFT). Thus the validity of the IFT can be taken as a new law of the turbulent cascade and at the same time independently confirms that the physics of the turbulent cascade is a memoryless Markov process in scale. IFT is taken as a new tool to prove the optimal functional form of the Fokker-Planck equations and subsequently to investigate the question of universality of small scale turbulence. The results of our analysis show that the turbulent cascade contains universal and non-universal features. We identify small scale intermittency as a universality breaking feature. We conclude that specific turbulent flows have their own particular multi-scale cascade, with other words their own stochastic fingerprint.