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Relative periodic orbits form the backbone of turbulent pipe flow

Published online by Cambridge University Press:  06 November 2017

N. B. Budanur Open the ORCID record for N. B. Budanur [Opens in a new window] ,
K. Y. Short ,
M. Farazmand ,
A. P. Willis Open the ORCID record for A. P. Willis [Opens in a new window]  and
P. Cvitanović Open the ORCID record for P. Cvitanović [Opens in a new window]
N. B. Budanur
Affiliation:
Institute of Science and Technology (IST), Am Campus 1, 3400 Klosterneuburg, Austria Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA
K. Y. Short
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
M. Farazmand
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA
A. P. Willis*
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA
P. Cvitanović
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: ashleypwillis@gmail.com
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Abstract

The chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this is also the case for the infinite-dimensional dynamics of Navier–Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics – specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier–Stokes equations as they do in lower-dimensional systems – is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at $Re=2500$ , and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulent Flows

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Papers
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Journal of Fluid Mechanics , Volume 833 , 25 December 2017 , pp. 274 - 301
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© 2017 Cambridge University Press 

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