Estimating intermittency in three-dimensional Navier–Stokes turbulence

• Published in: Journal of fluid mechanics 2009-04, Vol.625, p.125-133
• Main Author: GIBBON, J. D.
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fluid dynamics; Fluid mechanics; Fundamental areas of phenomenology (including applications); Physics; Turbulence; Turbulence simulation and modeling; Turbulent flows, convection, and heat transfer
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/S0022112009006089

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 $\mathcal{N}^{\pm}(\bx,\,t)\lessgtr 3\sqrt{2}\,\mathcal{R}_{u}^{3},$, where u = uL/ν is a Reynolds number dependent on the local velocity field u(x, t).