Spatially localized solutions of shear flows

TL;DR

This paper introduces new spatially localized equilibrium and traveling-wave solutions in shear flows, revealing concentrated vortex structures and their properties, advancing understanding of coherent structures in fluid dynamics.

Contribution

The paper develops novel methods to construct localized solutions in shear flows and analyzes their symmetry, decay, and scaling properties at high Reynolds numbers.

Findings

Localized vortex structures are confined to near-wall regions.

Solutions exhibit exponential tail decay and scale separation.

Critical layers develop at large Reynolds numbers.

Abstract

We present several new spatially localized equilibrium and traveling-wave solutions of plane Couette and channel flows. The solutions exhibit strikingly concentrated regions of vorticity that are flanked on either side by high-speed streaks. For several traveling-wave solutions of channel flow, the concentrated vortex structures are confined to the near-wall region and form particularly isolated and elemental coherent structures in the near-wall region of shear flows. The solutions are constructed by a variety of methods: application of windowing functions to previously known spatially periodic solutions, continuation from plane Couette to channel flow conditions, and from initial guesses obtained from turbulent simulation data. We show how the symmetries of localized solutions derive from the symmetries of their periodic counterparts, analyze the exponential decay of their tails, examine the scale separation and scaling of their streamwise Fourier modes, and show that they develop critical layers for large Reynolds numbers.