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

  • Gary J. Chandler (a1) and Rich R. Kerswell (a1)

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.

Copyright

Corresponding author

Email address for correspondence: R.R.Kerswell@bristol.ac.uk

References

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JFM classification

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UNKNOWN
Movies

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.

 Unknown (176 KB)
176 KB
UNKNOWN
Movies

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

 Unknown (2.7 MB)
2.7 MB

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