Large Scale Structures a Gradient Lines: the case of the Trkal Flow

TL;DR

This paper develops an asymptotic expansion for large Reynolds number flows, revealing how large-scale structures in Trkal solutions of Navier-Stokes equations form as gradient lines of energy, influenced by initial conditions.

Contribution

It introduces a multi-scale asymptotic analysis for non-stationary anisotropic helical flows, linking large-scale structures to initial conditions through gradient lines of energy.

Findings

Large scale structures are gradient lines of energy.

Asymptotic expansion valid for Reynolds number scaled by square root.

Initial conditions determine vortex-velocity tube boundaries.

Abstract

A specific asymptotic expansion at large Reynolds numbers (R)for the long wavelength perturbation of a non stationary anisotropic helical solution of the force less Navier-Stokes equations (Trkal solutions) is effectively constructed of the Beltrami type terms through multi scaling analysis. The asymptotic procedure is proved to be valid for one specific value of the scaling parameter,namely for the square root of the Reynolds number (R).As a result large scale structures arise as gradient lines of the energy determined by the initial conditions for two anisotropic Beltrami flows of the same helicity.The same intitial conditions determine the boundaries of the vortex-velocity tubes, containing both streamlines and vortex lines