Estimating intermittency in three-dimensional Navier–Stokes turbulence

Abstract

The issue of why computational resolution in Navier–Stokes turbulence is hard to achieve is addressed. Under the assumption that the three-dimensional Navier–Stokes equations have a global attractor it is nevertheless shown that solutions can potentially behave differently in two distinct regions of space–time$\mathbb{S}$±where$\mathbb{S}$−is comprised of a union of disjoint space–time ‘anomalies’. If$\mathbb{S}$−is non-empty it is dominated by large values of |∇ω|, which is consistent with the formation of vortex sheets or tightly coiled filaments. The local number of degrees of freedom±needed to resolve the regions in$\mathbb{S}$±satisfies$\mathcal{N}^{\pm}(\bx,\,t)\lessgtr 3\sqrt{2}\,\mathcal{R}_{u}^{3},$, whereu=uL/ν is a Reynolds number dependent on the local velocity fieldu(x,t).