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Multi-scale proper orthogonal decomposition of complex fluid flows

Published online by Cambridge University Press:  15 May 2019

M. A. Mendez Open the ORCID record for M. A. Mendez [Opens in a new window] ,
M. Balabane  and
J.-M. Buchlin
M. A. Mendez*
Affiliation:
von Karman Institute for Fluid Dynamics, Environmental and Applied Fluid Dynamics Department, Rhode-St-Genèse, 1640, Belgium
M. Balabane
Affiliation:
Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Villetaneuse, 93430, France
J.-M. Buchlin
Affiliation:
von Karman Institute for Fluid Dynamics, Environmental and Applied Fluid Dynamics Department, Rhode-St-Genèse, 1640, Belgium
*
†Email address for correspondence: mendez@vki.ac.be
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Abstract

Data-driven decompositions are becoming essential tools in fluid dynamics, allowing for tracking the evolution of coherent patterns in large datasets, and for constructing low-order models of complex phenomena. In this work, we analyse the main limits of two popular decompositions, namely the proper orthogonal decomposition (POD) and the dynamic mode decomposition (DMD), and we propose a novel decomposition which allows for enhanced feature detection capabilities. This novel decomposition is referred to as multi-scale proper orthogonal decomposition (mPOD) and combines multi-resolution analysis (MRA) with a standard POD. Using MRA, the mPOD splits the correlation matrix into the contribution of different scales, retaining non-overlapping portions of the correlation spectra; using the standard POD, the mPOD extracts the optimal basis from each scale. After introducing a matrix factorization framework for data-driven decompositions, the MRA is formulated via one- and two-dimensional filter banks for the dataset and the correlation matrix respectively. The validation of the mPOD, and a comparison with the discrete Fourier transform (DFT), DMD and POD are provided in three test cases. These include a synthetic test case, a numerical simulation of a nonlinear advection–diffusion problem and an experimental dataset obtained by the time-resolved particle image velocimetry (TR-PIV) of an impinging gas jet. For each of these examples, the decompositions are compared in terms of convergence, feature detection capabilities and time–frequency localization.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 870 , 10 July 2019 , pp. 988 - 1036
Copyright
© 2019 Cambridge University Press 

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Mendez et al. supplementary movie 1

Animation of the synthetic test case 1 described in section 5. This simple dataset consists of three modes having equal energy but a different spatial location and temporal evolution. Despite its basic form, this test case is sufficient to challenge standard decompositions such as DMD, POD, and DFT. The mPOD naturally distinguish the introduced modes.

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Mendez et al. supplementary movie 2

Animation of the second dataset considered, discussed in Section 6. This is a numerical simulation of a nonlinear advection-diffusion problem featuring coherent and random sources, together with random noise. This poses serious problems to the DMD, and yield severe spectral mixing to the POD. The proposed mPOD correctly identifies all the sources introduced.

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Video 7.6 MB

Mendez et al. supplementary movie 3

Animation of the third dataset considered, discussed in Section 7. This is an experimental characterization via TR-PIV of an impinging jet in transitional conditions. Although the dataset is stationary, the presence of periodic phenomena at largely different frequencies in different portions of the flow domain yields poor convergence of the DMD and spectral-mixing for the POD. The proposed mPOD avoids both limitations.

Download Mendez et al. supplementary movie 3(Video)
Video 11.9 MB