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

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 ∀a≥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.