Estimating intermittency in three-dimensional Navier-Stokes turbulence

TL;DR

This paper investigates the complexity of resolving three-dimensional Navier-Stokes turbulence, revealing that solutions exhibit distinct behaviors in different regions, with implications for understanding intermittency and computational challenges.

Contribution

It introduces a framework distinguishing regions with different turbulence behaviors and quantifies the local degrees of freedom needed for resolution based on Reynolds number.

Findings

Regions with high gradients correspond to vortex sheets or filaments.

The local degrees of freedom scale with the cube of the Reynolds number.

Different regions require different computational resolutions.

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 $\mathcal{R}_{u} = uL/\nu$ is a Reynolds number dependent on the local velocity field $u(\bx, t)$.