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Recursive dynamic mode decomposition of transient and post-transient wake flows

Published online by Cambridge University Press:  21 November 2016

Bernd R. Noack Open the ORCID record for Bernd R. Noack [Opens in a new window] ,
Witold Stankiewicz ,
Marek Morzyński  and
Peter J. Schmid
Bernd R. Noack*
Affiliation:
Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur, LIMSI-CNRS, Rue John von Neumann, Campus Universitaire d’Orsay, Bât 508, F-91403 Orsay, France Institut für Strömungsmechanik, Technische Universität Berlin, Hermann-Blenk-Straße 37, D-38108 Braunschweig, Germany
Witold Stankiewicz
Affiliation:
Chair of Virtual Engineering, Poznań University of Technology, ul. Jana Pawła II 24, 60-965 Poznań, Poland
Marek Morzyński
Affiliation:
Chair of Virtual Engineering, Poznań University of Technology, ul. Jana Pawła II 24, 60-965 Poznań, Poland
Peter J. Schmid
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
*
†Email address for correspondence: bernd.noack@limsi.fr
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Abstract

A novel data-driven modal decomposition of fluid flow is proposed, comprising key features of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). The first mode is the normalized real or imaginary part of the DMD mode that minimizes the time-averaged residual. The $N$ th mode is defined recursively in an analogous manner based on the residual of an expansion using the first $N-1$ modes. The resulting recursive DMD (RDMD) modes are orthogonal by construction, retain pure frequency content and aim at low residual. Recursive DMD is applied to transient cylinder wake data and is benchmarked against POD and optimized DMD (Chen et al., J. Nonlinear Sci., vol. 22, 2012, pp. 887–915) for the same snapshot sequence. Unlike POD modes, RDMD structures are shown to have purer frequency content while retaining a residual of comparable order to POD. In contrast to DMD, with exponentially growing or decaying oscillatory amplitudes, RDMD clearly identifies initial, maximum and final fluctuation levels. Intriguingly, RDMD outperforms both POD and DMD in the limit-cycle resolution from the same snapshots. Robustness of these observations is demonstrated for other parameters of the cylinder wake and for a more complex wake behind three rotating cylinders. Recursive DMD is proposed as an attractive alternative to POD and DMD for empirical Galerkin models, in particular for nonlinear transient dynamics.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Wakes/Jets: Wakes

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Papers
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Journal of Fluid Mechanics , Volume 809 , 25 December 2016 , pp. 843 - 872
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© 2016 Cambridge University Press 

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