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

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 of plane Couette flow, 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 establish the parameter regions over which snaking occurs and show that the finite-size effects of the travelling wave solution are due to a coupling between its fronts and interior that results from its shift-reflect symmetry. A new winding solution of plane Couette flow is derived from a strongly skewed localized equilibrium.