Scale dependence of the coarse-grained velocity derivative tensor structure in turbulence

Abstract

Velocity fluctuations in hydrodynamic turbulence have a nontrivial structure, characterized by correlations of the velocity gradient tensor. In this paper, we consider a phenomenological model, incorporating the main features of hydrodynamic fluid turbulence, aimed at predicting the structure of the velocity gradient tensor coarse grained at a spatial scale r. This model [M. Chertkov, A. Pumir, and B.I. Shraiman, Phys. Fluids 11, 2394 (1999)] is formulated as a set of stochastic ordinary differential equations, with three dimensionless parameters, characterizing the reduction of the nonlinearity induced by the pressure term, the reisotropization effect of the small scale velocity field, and the influence of the small scales on the coarse-grained velocity derivative tensor. Semiclassical solutions of this model are obtained and compared with direct numerical simulations (DNS) data. The DNS data show that the joint probability distribution function of the second and third invariants of M becomes increasingly skewed as the scale r decreases in the inertial range. The model results correctly reproduce this behavior provided the parameter that controls nonlinearity reduction is finely tuned; the influence of the other parameters in the model is much weaker.