Dynamics of velocity gradient invariants in turbulence: Restricted Euler and linear diffusion models

Abstract

A complete system of dynamical equations for the invariants of the velocity gradient, the strain rate, and the rate-of-rotation tensors is deduced for an incompressible flow. The equations for the velocity gradient invariants R and Q were first deduced by Cantwell [Phys. Fluids A 4, 782 (1992)] in terms of Hij, the tensor containing the anisotropic part of the pressure Hessian and the viscous diffusion term in the velocity gradient equation. These equations are extended here for the strain rate tensor invariants, RS and QS, and for the rate-of-rotation tensor invariant, QW, using HijS and HijW, the symmetric and the skew-symmetric parts of Hij, respectively. In order to obtain a complete system, an equation for the square of the vortex stretching vector, Vi≡Sijωj, is required. The resulting dynamical system of invariants is closed using a simple model for the velocity gradient evolution: an isotropic approximation for the pressure term and a linear model for the viscous diffusion term. The local topology and the resulting statistics implied by this model reproduce a number of trends similar to known results from numerical experiments for the small scales of turbulence.