Bifurcations analysis of turbulent energy cascade

TL;DR

This paper analyzes the turbulent energy cascade mechanism using bifurcation theory applied to Navier-Stokes equations, providing insights into turbulence characteristics, length scales, and critical Reynolds numbers.

Contribution

It introduces a statistical bifurcation property of Navier-Stokes equations and offers a spatial bifurcation representation related to turbulence onset.

Findings

Local deformation can be faster than fluid variables

Energy cascade explained via bifurcation property

Estimates of critical Reynolds number and bifurcation count

Abstract

This note studies the mechanism of turbulent energy cascade through an opportune bifurcations analysis of the Navier--Stokes equations, and furnishes explanations on the more significant characteristics of the turbulence. A statistical bifurcations property of the Navier--Stokes equations in fully developed turbulence is proposed, and a spatial representation of the bifurcations is presented, which is based on a proper definition of the fixed points of the velocity field. The analysis first shows that the local deformation can be much more rapid than the fluid state variables, then explains the mechanism of energy cascade through the aforementioned property of the bifurcations, and gives reasonable argumentation of the fact that the bifurcations cascade can be expressed in terms of length scales. Furthermore, the study analyzes the characteristic length scales at the transition through global properties of the bifurcations, and estimates the order of magnitude of the critical Taylor--scale Reynolds number and the number of bifurcations at the onset of turbulence.