On the origin of small-scale intermittency in turbulence

Turbulent flows are notoriously difficult to describe, understand, and model based on first principles. One reason is that turbulence contains highly intermittent bursts of vorticity and strain-rate with highly non-Gaussian statistics. Quantitatively, intermittency is manifested in highly elongated tails in the probability density functions of the velocity increments between pairs of points. A long-standing open issue has been to predict the origin of intermittency and non-Gaussian statistics directly from the Navier-Stokes equations. Here we use a simplified version of the exact dynamics, namely the restricted Euler equations, to describe the generation of intermittency. By adopting a Lagrangian viewpoint, we derive a simple nonlinear dynamical system for the time evolution of longitudinal and transverse velocity increments. From this simple system (the ``advected delta-vee'' equations), we are able to show that the ubiquitous non-Gaussian tails in turbulence have their origin in the inherent self-amplification of longitudinal velocity increments, and cross amplification of the transverse velocity increments. Using direct numerical simulation filtered at various length-scales, we quantify and comment upon the dynamical effects of terms that are neglected in the advected delta-vee system, namely pressure Hessian, subgrid-scale stress tensor, and viscous forces.