The vortex dynamics in incompressible viscous turbulent flows

TL;DR

This paper explores the geometric aspects of turbulence in incompressible viscous flows, deriving quantitative statistical insights, including entropy decay and vorticity dynamics, under the assumption of universal small time scales.

Contribution

It introduces a geometric framework for turbulence analysis and establishes new inequalities and characterizations of vorticity behavior in viscous flows.

Findings

Entropy decay inequality involving turbulent kinetic energy and enstrophy

Identification of small time scales for vorticity breakdown and creation

Quantitative statements about turbulence statistics

Abstract

In this paper, we consider turbulence from a geometric perspective based on the vorticity equations for incompressible viscous fluid flows. We derive several quantitative statements about the statistics of turbulent flows. In particular we establish an entropy decay inequality which involves the turbulent kinetic energy and $L^{1}$-enstrophy, and we identify the small time scale of the vorticity broken down and the vorticity creation under the universality assumption of small time scales of turbulence flows.