TY - JOUR

T1 - Koopman analysis of the long-term evolution in a turbulent convection cell

AU - Giannakis, Dimitrios

AU - Kolchinskaya, Anastasiya

AU - Krasnov, Dmitry

AU - Schumacher, Jörg

N1 - Funding Information:
D.G. received support from DARPA grant HR0011-16-C-0116, NSF grant DMS-1521775, ONR grant N00014-14-1-0150 and ONR YIP grant N00014-16-1-2649. The work of A.K. is supported by grant nos SCHU 1410/18 and GRK 1567 of the Deutsche Forschungsgemeinschaft, and the work of J.S. by the Priority Programme on Turbulent Superstructures which is funded by the Deutsche Forschungsgemeinschaft by grant no. SPP 1881. D.K. is supported by the Helmholtz Research Alliance ‘Liquid Metal Technologies’, which is funded by the Helmholtz Association and by grant no. SCHU 1410/29 of the Deutsche Forschungsgemeinschaft. We acknowledge support with computer time by the large-scale project HIL12 of the John von Neumann Institute for Computing (NIC). We would like to thank B. Eckhardt, N. Foroozani and K. R. Sreenivasan for helpful discussions.
Publisher Copyright:
© 2018 Cambridge University Press.

PY - 2018/7/25

Y1 - 2018/7/25

N2 - 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.

AB - 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.

KW - Bénard convection

KW - low-dimensional models

KW - turbulent convection

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U2 - 10.1017/jfm.2018.297

DO - 10.1017/jfm.2018.297

M3 - Review article

AN - SCOPUS:85047879177

SN - 0022-1120

VL - 847

SP - 735

EP - 767

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -