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Revealing the state space of turbulent pipe flow by symmetry reduction

Published online by Cambridge University Press:  19 March 2013

A. P. Willis ,
P. Cvitanović  and
M. Avila
A. P. Willis*
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
P. Cvitanović
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
M. Avila
Affiliation:
Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany Institute of Fluid Mechanics, Fridriech-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 4, 91058 Erlangen, Germany
*
†Email address for correspondence: [email protected]
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Abstract

Symmetry reduction by the method of slices is applied to pipe flow in order to obtain a quotient of the streamwise translation and azimuthal rotation symmetries of turbulent flow states. Within the symmetry-reduced state space, all travelling wave solutions reduce to equilibria, and all relative periodic orbits reduce to periodic orbits. Projections of these solutions and their unstable manifolds from their infinite-dimensional symmetry-reduced state space onto suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows. Visualizations of the flow within the slice and its linearization at equilibria enable us to trace out the unstable manifolds, determine close recurrences, identify connections between different travelling wave solutions and find, for the first time for pipe flows, relative periodic orbits that are embedded within the chaotic saddle, which capture turbulent dynamics at transitional Reynolds numbers.

JFM classification

Instability: Instability Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 721 , 25 April 2013 , pp. 514 - 540
Copyright
©2013 Cambridge University Press

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Willis et al. supplementary movie

Cross-section movie for the orbit extracted from turbulent flow, RPO$_{36.72}$. Within the slice, the orbit is seen to close after one period. It exhibits quiescent and bursting phases.

Download Willis et al. supplementary movie(Video)
Video 4.8 MB