Self-Regularization in turbulence from the Kolmogorov 4/5-Law and Alignment

TL;DR

This paper explores how the Kolmogorov 4/5-law and velocity increment alignment contribute to turbulence regularization, linking empirical observations with theoretical predictions of energy dissipation and structure function scaling.

Contribution

It introduces a novel perspective on turbulence regularization by connecting the Kolmogorov 4/5-law with velocity increment alignment and inertial dissipation mechanisms.

Findings

Inertial dissipation acts as a regularization mechanism in turbulence.

Velocity increment anti-alignment influences energy cascade and dissipation.

The Kolmogorov 4/5-law underpins the proposed regularization process.

Abstract

A defining feature of 3D 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 $S_p(\ell) :=\langle |\delta_\ell u|^p\rangle$ exhibit persistent power-law scaling in the inertial range, namely $S_p(\ell) \sim |\ell|^{\zeta_p}$ for exponents $\zeta_p>0$ over an ever-increasing (with Reynolds) range of scales. This behavior indicates that the velocity field retains some fractional differentiability uniformly in the Reynolds number. The Kolmogorov 1941 theory of turbulence predicts that $\zeta_p=p/3$ for all $p$ and Onsager's 1949 theory establishes the requirement that $\zeta_p\leq p/3$ for $p\geq 3$ for consistency with the zeroth law. Empirically, $\zeta_2 \gtrapprox 2/3$ and $\zeta_3 \lessapprox 1$, 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.