Viscous vortex layers subject to more general strain and comparison to isotropic turbulence

TL;DR

This paper analyzes viscous vortex layers under general strain conditions, compares steady solutions, and relates findings to isotropic turbulence, revealing conditions for vorticity decay, steady states, and stability.

Contribution

It introduces a comprehensive analysis of vortex layers under various strain configurations and compares these to isotropic turbulence, highlighting stability conditions and steady-state behaviors.

Findings

Vorticity decays over time for strain parameter a<1.

Steady states exist only with boundary vorticity supply for a<1.

Super-Townsend case (a>1) exhibits opposite vorticity sheaths.

Abstract

Viscous vortex layers subject to a more general uniform strain are considered. They include Townsend's steady solution for plane strain (corresponding to a parameter $a = 1$) in which all the strain in the plane of the layer goes toward vorticity stretching, as well as Migdal's recent steady asymmetric solution for axisymmetric strain ($a = 1/2$) in which half of the strain goes into vorticity stretching. In addition to considering asymmetric, symmetric and antisymmetric steady solutions $\forall a \ge 0$, it is shown that for $a < 1$, i.e., anything less than the Townsend case, the vorticity inherently decays in time: only boundary conditions that maintain a supply of vorticity at one or both ends lead to a non-zero steady state. For the super-Townsend case $a > 1$, steady states have a sheath of opposite sign vorticity. Comparison is made with homogeneous-isotropic turbulence in which case the average vorticity in the strain eigenframe is layer-like, has wings of opposite vorticity, and the strain configuration is found to be super-Townsend. Only zero-integral perturbations of the $a > 1$ steady solutions are stable; otherwise, the solution grows. Finally, the appendix shows that the average flow in the strain eigenframe is (apart from an extra term) the Reynolds-averaged Navier-Stokes equation.