Confined Vortex Surface and Irreversibility. 3. Nested Tubes and Energy Cascade

TL;DR

This paper presents exact solutions of the Confined Vortex Surface equations featuring nested vortex tubes, modeling the energy cascade in turbulence and revealing how vortex hierarchy scales with Reynolds number.

Contribution

It introduces a new family of exact vortex solutions with nested tubes, providing a physical model for the energy cascade in turbulent flows.

Findings

Nested vortex solutions with scale-invariant cross-sections.

The size spectrum of vortex shells minimizes a conserved surface dissipation Hamiltonian.

Tube thickness decreases as a power law with Reynolds number.

Abstract

We find a new family of exact solutions of the Confined Vortex Surface equations (The Euler equations with extra boundary conditions coming from the stability of the Navier-Stokes equations in the local tangent plane). This family of solutions has an infinite number of nested tubes of varying diameters. The shape of the boundary cross-section is the same up to a scale. This Russian doll implements in physical space the scenario of the energy cascade from an eddy to a smaller eddy. This hierarchy of vortex shells is not wishful thinking but rather an exact solution of the Euler (CVS) equations. The spectrum of the size of the shells is determined from the minimization of the effective Hamiltonian of our turbulent statistics. This effective Hamiltonian is given by a surface dissipation integral, conserved in the \NS{} dynamics in virtue of the \CVS{} conditions. The thickness of each tube goes to zero as a power of Reynolds number $\R^{-\frac{3}{4}}$, compared to the average distance between tubes in the turbulent flow. Thus, at finite viscosity, there will be a logarithmic number of inner tubes nested inside the external one.