Modified snaking in plane Couette flow with wall-normal suction

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. The Swift-Hohenberg solutions closely resemble the snaking solutions of plane Couette flow, yet this remarkable resemblance and the mechanisms supporting homoclinic snaking within the three-dimensional Navier-Stokes equations remain to be fully understood. Studies of Swift-Hohenberg revealed the central importance of discrete symmetries for homoclinic snaking to be supported by an equation. We therefore study the structural stability of the characteristic snakes-and-ladders structure associated with homoclinic snaking in three-dimensional plane Couette flow for flow modifications that break symmetries of the 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 numbers.

Automated summary

The study investigates the impact of symmetry breaking on the three-dimensional plane Couette flow homoclinic snaking, revealing that wall-normal suction can modify the bifurcation structure of solutions. This modification disrupts the connected snakes-and-ladders structure and leads to the creation of previously unknown solution branches.