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Koopman analysis of the long-term evolution in a turbulent convection cell

Published online by Cambridge University Press:  29 May 2018

Dimitrios Giannakis ,
Anastasiya Kolchinskaya ,
Dmitry Krasnov  and
Jörg Schumacher Open the ORCID record for Jörg Schumacher [Opens in a new window]
Dimitrios Giannakis
Affiliation:
Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Anastasiya Kolchinskaya
Affiliation:
Institut für Thermo- und Fluiddynamik, Postfach 100565, Technische Universität Ilmenau, D-98684 Ilmenau, Germany
Dmitry Krasnov
Affiliation:
Institut für Thermo- und Fluiddynamik, Postfach 100565, Technische Universität Ilmenau, D-98684 Ilmenau, Germany
Jörg Schumacher*
Affiliation:
Institut für Thermo- und Fluiddynamik, Postfach 100565, Technische Universität Ilmenau, D-98684 Ilmenau, Germany
*
†Email address for correspondence: joerg.schumacher@tu-ilmenau.de
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Abstract

We analyse the long-time evolution of the three-dimensional flow in a closed cubic turbulent Rayleigh–Bénard convection cell via a Koopman eigenfunction analysis. A data-driven basis derived from diffusion kernels known in machine learning is employed here to represent a regularized generator of the unitary Koopman group in the sense of a Galerkin approximation. The resulting Koopman eigenfunctions can be grouped into subsets in accordance with the discrete symmetries in a cubic box. In particular, a projection of the velocity field onto the first group of eigenfunctions reveals the four stable large-scale circulation (LSC) states in the convection cell. We recapture the preferential circulation rolls in diagonal corners and the short-term switching through roll states parallel to the side faces which have also been seen in other simulations and experiments. The diagonal macroscopic flow states can last as long as 1000 convective free-fall time units. In addition, we find that specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced oscillatory fluctuations for particular stable diagonal states of the LSC. The corresponding velocity-field structures, such as corner vortices and swirls in the midplane, are also discussed via spatiotemporal reconstructions.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulent convection
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 847 , 25 July 2018 , pp. 735 - 767
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Copyright
© 2018 Cambridge University Press 

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