Modified snaking in plane Couette flow with wall-normal suction
Abstract
A specific family of spanwise-localised invariant solutions of plane Couette flow exhibits homoclinic snaking, a process by which spatially localised invariant solutions of a nonlinear partial differential equation smoothly grow additional structure at their fronts while undergoing a sequence of saddle-node bifurcations. Homoclinic snaking is well understood in the context of simpler pattern forming systems such as the one-dimensional Swift-Hohenberg equation with cubic-quintic nonlinearity, whose solutions remarkably well resemble the snaking solutions of plane Couette flow. We study the structural stability of the characteristic snakes-and-ladders structure associated with homoclinic snaking for flow modifications that break symmetries of plane Couette flow. We demonstrate that wall-normal suction modifies the bifurcation structure of three-dimensional plane Couette solutions in the same way, a symmetry-breaking quadratic term modifies solutions of the one-dimensional Swift-Hohenberg equation. These modifications are related to the breaking of the discrete rotational symmetry. At large amplitudes of the symmetry-breaking wall-normal suction the connected snakes-and-ladders structure is destroyed. Previously unknown solution branches are created and can be parametrically continued to vanishing suction. This yields new localised solutions of plane Couette flow that exist in a wide range of Reynolds number.
Cite
@article{arxiv.2004.00127,
title = {Modified snaking in plane Couette flow with wall-normal suction},
author = {Sajjad Azimi and Tobias M. Schneider},
journal= {arXiv preprint arXiv:2004.00127},
year = {2021}
}