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

Academic Article


  • 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.
  • Authors

  • Gibson, John
  • Schneider, TM
  • Status

    Publication Date

  • May 2016
  • Has Subject Area

    Published In


  • nonlinear dynamical systems
  • pattern formation
  • transition to turbulence
  • Digital Object Identifier (doi)

    Start Page

  • 530
  • End Page

  • 551
  • Volume

  • 794