Lagrangian Dynamics and Models of the Velocity Gradient Tensor in Turbulent Flows

Many fundamental and intrinsic properties of small-scale motions in turbulence can be described using the velocity gradient tensor. This tensor encodes interesting geometric and statistical information such as the alignment of vorticity with respect to the strain-rate eigenvectors, rate of deformation and shapes of fluid material volumes, non-Gaussian statistics, and intermittency. In the inertial range of turbulence, similar properties can be described using the coarse-grained or filtered velocity gradient tensor. In this article we review various models that aim at understanding these phenomena using a small number of ordinary differential equations, written either as a low-dimensional dynamical system or as a set of stochastic differential equations. Typically these describe the Lagrangian evolution of the velocity gradient tensor elements following fluid particles and require models for the pressure Hessian and viscous effects. Sample results from various models are shown, and open challenges are highlighted.