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

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 $S_p(\ell) :=\langle |δ_\ell u|^p\rangle$ exhibit persistent power-law scaling in the inertial range, namely $S_p(\ell) \sim |\ell|^{ζ_p}$ for exponents $ζ_p>0$ 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 $ζ_p=p/3$ for all $p$ and Onsager's 1949 theory establishes the requirement that $ζ_p\leq p/3$ for $p\geq 3$ for consistency with the zeroth law. Empirically, $ζ_2 \gtrapprox 2/3$ and $ζ_3 \lessapprox 1$, 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.