Surfing the edge: using feedback control to find nonlinear solutions

Abstract

Many transitional wall-bounded shear flows are characterised by the coexistence in state-space of laminar and turbulent regimes. Probing the edge boundary between the two attractors has led in the last decade to the numerical discovery of new (unstable) solutions to the incompressible Navier-Stokes equations. However, the iterative bisection method used to achieve this can become prohibitively costly for large systems. Here we suggest a simple feedback control strategy to stabilise edge states, hence accelerating their numerical identification by several orders of magnitude. The method is illustrated for several configurations of cylindrical pipe flow. Travelling waves solutions are identified as edge states, and can be isolated rapidly in only one short numerical run. A new branch of solutions is also identified. When the edge state is a periodic orbit or chaotic state, the feedback control does not converge precisely to solutions of the uncontrolled system, but nevertheless brings the dynamics very close to the original edge manifold in a single run. We discuss the opportunities offered by the speed and simplicity of this new method to probe the structure of both state space and parameter space.

12 pages, 5 figures

References in corpus (12)

• Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
• Turbulence transition and the edge of chaos in pipe flow
• Laminar-turbulent boundary in plane Couette flow
• Recurrence of Travelling Waves in Transitional Pipe Flow
• The Openpipeflow Navier--Stokes Solver
• Relative periodic orbits in transitional pipe flow
• Highly-symmetric travelling waves in pipe flow
• An adjoint-based approach for finding invariant solutions of Navier-Stokes equations
• Edge states as mediators of bypass transition in boundary-layer flows
• Symmetry reduction in high dimensions, illustrated in a turbulent pipe
• Emergence of spatio-temporal dynamics from exact coherent solutions in pipe flow
• Optimising energy growth as a tool for finding exact coherent structures

Cited by in corpus (7)

• Stabilisation of exact coherent structures in two-dimensional turbulence using time-delayed feedback
• Linear feedback control of invariant solutions in channel flow
• New equilibrium solution branches of plane Couette flow discovered using a project-then-search method
• A model for the collapse of the edge when two transitions routes compete
• Stabilising nonlinear travelling waves in pipe flow using time-delayed feedback
• Dynamic feedback control through wall suction in shear flows
• Stabilizing two-dimensional turbulent Kolmogorov flow via selective modification of inviscid invariants