Self-regularization in turbulence from the Kolmogorov 4/5-law and alignment

Abstract

A defining feature of three-dimensional hydrodynamic turbulence is that the rate of energy dissipation is bounded away from zero as viscosity is decreased (Reynolds number increased). This phenomenon—anomalous dissipation—is sometimes called the ‘zeroth law of turbulence’ as it underpins many celebrated theoretical predictions. Another robust feature observed in turbulence is that velocity structure functions Sp(ℓ):=⟨|δℓu|p⟩ exhibit persistent power-law scaling in the inertial range, namely Sp(ℓ)∼|ℓ|ζp for exponents ζp>0 over an ever increasing (with Reynolds) range of scales. This behaviour indicates that the velocity field retains some fractional differentiability uniformly in the Reynolds number. The Kolmogorov 1941 theory of turbulence predicts that ζp=p/3 for all p and Onsager’s 1949 theory establishes the requirement that ζp≤p/3 for p≥ 3 for consistency with the zeroth law. Empirically, ζ2⪆2/3 and ζ3⪅1, suggesting that turbulent Navier–Stokes solutions approximate dissipative weak solutions of the Euler equations possessing (nearly) the minimal degree of singularity required to sustain anomalous dissipation. In this note, we adopt an experimentally supported hypothesis on the anti-alignment of velocity increments with their separation vectors and demonstrate that the inertial dissipation provides a regularization mechanism via the Kolmogorov 4/5-law. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.