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Surfing the edge: using feedback control to find nonlinear solutions

Published online by Cambridge University Press:  13 October 2017

A. P. Willis Open the ORCID record for A. P. Willis [Opens in a new window] ,
Y. Duguet ,
O. Omel’chenko  and
M. Wolfrum
A. P. Willis*
Affiliation:
School of Mathematics and Statistics, University of Sheffield S3 7RH, UK
Y. Duguet
Affiliation:
LIMSI-CNRS, UPR 3251, Université Paris-Saclay, F-91403, Orsay, France
O. Omel’chenko
Affiliation:
Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany
M. Wolfrum
Affiliation:
Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany
*
†Email address for correspondence: a.p.willis@sheffield.ac.uk
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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 compute edge states 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.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Flow Control: Instability control Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 831 , 25 November 2017 , pp. 579 - 591
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Copyright
© 2017 Cambridge University Press 

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