Development of high vorticity structures in incompressible 3D Euler equations

TL;DR

This paper investigates the formation and evolution of high vorticity pancake structures in 3D Euler equations, revealing their self-similar behavior and role in energy cascade leading to Kolmogorov spectrum.

Contribution

It provides a systematic numerical analysis of pancake structures, their self-similar evolution, and their connection to the Kolmogorov energy spectrum in inviscid flows.

Findings

Pancake structures exhibit self-similar evolution with universal ratios.

Energy transfer to small scales occurs through pancake structures.

Vorticity scales as _{ ext{max}}  \u2212 2/3 with pancake scale .

Abstract

We perform the systematic numerical study of high vorticity structures that develop in the 3D incompressible Euler equations from generic large-scale initial conditions. We observe that a multitude of high vorticity structures appear in the form of thin vorticity sheets (pancakes). Our analysis reveals the self-similarity of the pancakes evolution, which is governed by two different exponents $e^{-t/T_{\ell}}$ and $e^{t/T_{\omega}}$ describing compression in the transverse direction and the vorticity growth respectively, with the universal ratio $T_{\ell}/T_{\omega} \approx 2/3$. We relate development of these structures to the gradual formation of the Kolmogorov energy spectrum $E_{k}\propto\, k^{-5/3}$, which we observe in a fully inviscid system. With the spectral analysis we demonstrate that the energy transfer to small scales is performed through the pancake structures, which accumulate in the Kolmogorov interval of scales and evolve according to the scaling law $\omega_{\max} \propto \ell^{-2/3}$ for the local vorticity maximums $\omega_{\max}$ and the transverse pancake scales $\ell$.