Velocity-Gradient Dynamics in Turbulence: Effect of Viscosity and Forcing

The restricted Euler equation is a promising but incomplete model for velocity-gradient dynamics in turbulent flows. While it captures many of the geometric features of the vorticity vector and the strain rate tensor, viscous and anisotropic pressure Hessian effects are not accounted for satisfactorily. Inadequate viscous-effect modeling causes velocity gradients to diverge in finite time, rendering the restricted Euler model unsuitable for practical applications. We perform a Lagrangian frame analysis to comprehend fully the physics of the viscous relaxation time scale and propose a variable time-scale model that can adequately account for deformation history. Most importantly, the finite-time singularity (divergence of velocity gradients) problem is fully resolved with the present model. We also model the effects of forcing that is used in numerical simulations to sustain stationary isotropic turbulence. Detailed comparison of the new model with DNS data reveals good agreement.