Estimating intermittency in three-dimensional Navier-Stokes turbulence

Abstract

The issue of why computational resolution in Navier-Stokes turbulence is so hard to achieve is addressed. It is shown that Navier-Stokes solutions can potentially behave differently in two distinct regions of space-time $\mathbb{R}^{\pm}$ where $\mathbb{R}^{-}$ is comprised of a union of disjoint space-time `anomalies'. Large values of $|\nabla\bom|$ dominate $\mathbb{R}^{-}$, which is consistent with the formation of vortex sheets or tightly-coiled filaments. The local number of degrees of freedom $\mathcal{N}^{\pm}$ needed to resolve the regions in $\mathbb{R}^{\pm}$ satisfies $$\mathcal{N}^{\pm}(\bx, t)\lessgtr c_{\pm}\mathcal{R}_{u}^{3}$$ where $\mathcal{R}_{u} = uL/ν$ is a Reynolds number dependent on the local velocity field $u(\bx, t)$.