On higher-order derivative ratios in turbulent flows

TL;DR

This study investigates higher-order derivative ratios in turbulent flows, revealing their potential as indicators of energy dissipation peaks through a rigorous mathematical framework applied to the 3D Taylor-Green vortex.

Contribution

It introduces a novel power law relation for derivative ratios near enstrophy peaks, linking turbulence dissipation mechanisms with spatial analyticity bounds.

Findings

Power law relation enables bounds on spatial analyticity radius.

Higher-order derivative ratios may identify energy dissipation peaks.

Turbulent dissipation engages via harmonic measure maximum principle.

Abstract

A computational study of higher-order derivative ratios on a time interval leading to the enstrophy peak is presented in the case of the 3D Taylor-Green vortex, a benchmark problem in the simulation of turbulent flows. The main finding is that the power law relating the ratios at time $t$ to $T^*-t$ where $T^*$ is the peak enstrophy time is of a form that allows the machinery of dynamic interpolation-sparseness to produce a lower bound on the radius of spatial analyticity sufficient to overcome an upper bound on the scale of sparseness of the super-level sets in view. As a consequence, the mechanism of turbulent dissipation engages via the harmonic measure maximum principle, furnishing a rigorous explanation for the subsequent slump of the enstrophy. This indicates that the higher-order derivative ratios -- which could be viewed as higher-order analogs of the classical Taylor and Kraichnan scales in turbulence phenomenology -- may be reasonable identifiers of the peak of the energy dissipation rate.