A thermodynamical approach to dissipation range turbulence

A model to explain the statistics of the velocity gradients in the dissipation range of a turbulent flow is presented. The experimentally observed non-gaussian statistics is theoretically predicted by means of a thermodynamical analogy using the maximum entropy principle of ordinary statistical mechanics. PACS numbers: 47.27.Gs The dynamics of fluids at high Reynolds number is one of the most interesting subject of statistical physics. In the limit of infinite Reynolds number, the celebrated theory of Kolmogorov (K41) [1] suggests that the small scales statistics is characterized only by the mean rate of energy dissipation per unit mass ǫ and the scale l. The K41 theory is based upon the concept of selfsimilarity of the inertial range [ld, L] and implies that the velocity gradients δvl(x) = |v(x+ l)− v(x)| scale as < δv l >∼ (ǫl). (1) The experimental evidence of the breakdown of eq.(1), see [2, 3, 4], due to the presence of intermittency, induced Kolmogorov and Obukhov [5, 6] to modify the K41 theory introducing the fluctuation of the mean flux of energy ǫ. Indeed, as pointed out by Siggia [7], the turbulent flow can be described as the superposition of two fluid states, the coherent structures and the random fluctuations. Hence, in the inertial range, the flow preserves organized structures as, for instance, vortex sheets and filaments. The coherent structures are “rare” events that contribute to the non-gaussian high-amplitude fluctuations. Many models for such superposition have been proposed, either based