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Streamwise and doubly-localised periodic orbits in plane Poiseuille flow

Published online by Cambridge University Press:  25 November 2014

Stefan Zammert  and
Bruno Eckhardt
Stefan Zammert*
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, Renthof 6, D-35032 Marburg, Germany
Bruno Eckhardt
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, Renthof 6, D-35032 Marburg, Germany J. M. Burgerscentrum, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands
*
†Email address for correspondence: [email protected]
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Abstract

We study localised exact coherent structures in plane Poiseuille flow that are relative periodic orbits. They are obtained from extended states in smaller periodically continued domains, by increasing the length to obtain streamwise localisation and then by increasing the width to achieve spanwise localisation. The states maintain the travelling wave structure of the extended states, which is then modulated by a localised envelope on larger scales. In the streamwise direction, the envelope shows exponential localisation, with different exponents on the upstream and downstream sides. The upstream exponent increases linearly with Reynolds number $\mathit{Re}$, but the downstream exponent is essentially independent of $\mathit{Re}$. In the spanwise direction the decay is compatible with a power-law localisation. As the width increases the localised state undergoes further bifurcations which add additional unstable directions, so that the edge state, the relative attractor on the boundary between the laminar and turbulent motions, in the system becomes chaotic.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Waves/Free-surface Flows: Channel flow Turbulent Flows: Turbulent transition
Type
Papers
Information
Journal of Fluid Mechanics , Volume 761 , 25 December 2014 , pp. 348 - 359
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Copyright
© 2014 Cambridge University Press 

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