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Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow

Published online by Cambridge University Press:  28 March 2013

Gary J. Chandler  and
Rich R. Kerswell
Gary J. Chandler
Affiliation:
School of Mathematics, University of Bristol, University walk, Bristol, UK
Rich R. Kerswell*
Affiliation:
School of Mathematics, University of Bristol, University walk, Bristol, UK
*
†Email address for correspondence: [email protected]
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Abstract

We consider long-time simulations of two-dimensional turbulence body forced by $\sin 4y\hat {\boldsymbol{x}} $ on the torus $(x, y)\in \mathop{[0, 2\mathrm{\pi} ] }\nolimits ^{2} $ with the purpose of extracting simple invariant sets or ‘exact recurrent flows’ embedded in this turbulence. Each recurrent flow represents a sustained closed cycle of dynamical processes which underpins the turbulence. These are used to reconstruct the turbulence statistics using periodic orbit theory. The approach is found to be reasonably successful at a low value of the forcing where the flow is close to but not fully in its asymptotic (strongly) turbulent regime. Here, a total of 50 recurrent flows are found with the majority buried in the part of phase space most populated by the turbulence giving rise to a good reproduction of the energy and dissipation p.d.f. However, at higher forcing amplitudes now in the asymptotic turbulent regime, the generated turbulence data set proves insufficiently long to yield enough recurrent flows to make viable predictions. Despite this, the general approach seems promising providing enough simulation data is available since it is open to extensive automation and naturally generates dynamically important exact solutions for the flow.

JFM classification

Nonlinear Dynamical Systems: Chaos Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 722 , 10 May 2013 , pp. 554 - 595
Copyright
©2013 Cambridge University Press

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Supplementary material: Image

Chandler and Kerswell supplementary movie

This is a video of the unstable periodic orbit P4 (as listed in Table 3 of Chandler & Kerswell, 2013) found from DNS carried out at Re=100.

Download Chandler and Kerswell supplementary movie(Image)
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Supplementary material: Image

Chandler and Kerswell supplementary movie

This is a video of the DNS segment at Re=100 which suggested the presence of the periodic orbit P4 and from which the initial velocity field guess was taken to converge P4 as an exact recurrent flow

Download Chandler and Kerswell supplementary movie(Image)
Image 2.7 MB