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Optimal mode decomposition for unsteady flows

Published online by Cambridge University Press:  24 September 2013

A. Wynn ,
D. S. Pearson ,
B. Ganapathisubramani  and
P. J. Goulart
A. Wynn*
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
D. S. Pearson
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
B. Ganapathisubramani
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK Aerodynamics and Flight Mechanics Group, University of Southampton, Southampton SO17 1BJ, UK
P. J. Goulart
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK Automatic Control Laboratory, ETH Zürich, 8092 Zurich, Switzerland
*
†Email address for correspondence: [email protected]
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Abstract

A new method, herein referred to as optimal mode decomposition (OMD), of finding a linear model to describe the evolution of a fluid flow is presented. The method estimates the linear dynamics of a high-dimensional system which is first projected onto a subspace of a user-defined fixed rank. An iterative procedure is used to find the optimal combination of linear model and subspace that minimizes the system residual error. The OMD method is shown to be a generalization of dynamic mode decomposition (DMD), in which the subspace is not optimized but rather fixed to be the proper orthogonal decomposition (POD) modes. Furthermore, OMD is shown to provide an approximation to the Koopman modes and eigenvalues of the underlying system. A comparison between OMD and DMD is made using both a synthetic waveform and an experimental data set. The OMD technique is shown to have lower residual errors than DMD and is shown on a synthetic waveform to provide more accurate estimates of the system eigenvalues. This new method can be used with experimental and numerical data to calculate the ‘optimal’ low-order model with a user-defined rank that best captures the system dynamics of unsteady and turbulent flows.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 733 , 25 October 2013 , pp. 473 - 503
Copyright
©2013 Cambridge University Press 

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