Homoclinic snaking in plane Couette flow: bending, skewing, and finite-size effects

TL;DR

This paper investigates homoclinic snaking in plane Couette flow, revealing new solution features like bending, skewing, and finite-size effects, and connects pattern formation theory with shear flow dynamics.

Contribution

It generalizes the study of homoclinic snaking in plane Couette flow beyond fixed wavelengths, discovering new solution features and deriving a novel winding solution.

Findings

Finite-size effects are linked to shift-reflect symmetry.

New solution features include bending and skewing.

Parameter regions for snaking are identified.

Abstract

Invariant solutions of shear flows have recently been extended from spatially periodic solutions in minimal flow units to spatially localized solutions on extended domains. One set of spanwise-localized solutions of plane Couette flow exhibits homoclinic snaking, a process by which steady-state solutions grow additional structure smoothly at their fronts when continued parametrically. Homoclinic snaking is well understood mathematically in the context of the one-dimensional Swift-Hohenberg equation. Consequently, the snaking solutions of plane Couette flow form a promising connection between the largely phenomenological study of laminar-turbulent patterns in viscous shear flows and the mathematically well-developed field of pattern-formation theory. In this paper we present a numerical study of the snaking solutions, generalizing beyond the fixed streamwise wavelength of previous studies. We find a number of new solution features, including bending, skewing, and finite-size effects. We show that the finite-size effects result from the shift-reflect symmetry of the traveling wave and establish the parameter regions over which snaking occurs. A new winding solution of plane Couette flow is derived from a strongly skewed localized equilibrium.