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Metric for attractor overlap

Published online by Cambridge University Press:  12 July 2019

Rishabh Ishar Open the ORCID record for Rishabh Ishar [Opens in a new window] ,
Eurika Kaiser Open the ORCID record for Eurika Kaiser [Opens in a new window] ,
Marek Morzyński Open the ORCID record for Marek Morzyński [Opens in a new window] ,
Daniel Fernex Open the ORCID record for Daniel Fernex [Opens in a new window] ,
Richard Semaan Open the ORCID record for Richard Semaan [Opens in a new window] ,
Marian Albers Open the ORCID record for Marian Albers [Opens in a new window] ,
Pascal S. Meysonnat Open the ORCID record for Pascal S. Meysonnat [Opens in a new window] ,
Wolfgang Schröder Open the ORCID record for Wolfgang Schröder [Opens in a new window]  and
Bernd R. Noack Open the ORCID record for Bernd R. Noack [Opens in a new window]
Rishabh Ishar
Affiliation:
Department of Mechanical Engineering, Punjab Engineering College, Chandigarh 160012, India
Eurika Kaiser
Affiliation:
University of Washington, Department of Mechanical Engineering, Stevens Way, Box 352600, Seattle, WA 98195, USA
Marek Morzyński
Affiliation:
Poznań University of Technology, Chair of Virtual Engineering, Jana Pawla II 24, PL 60-965 Poznań, Poland
Daniel Fernex
Affiliation:
Institut für Strömungsmechanik, Technische Universität Braunschweig, Hermann-Blenk-Str. 37, 38108 Braunschweig, Germany
Richard Semaan
Affiliation:
Institut für Strömungsmechanik, Technische Universität Braunschweig, Hermann-Blenk-Str. 37, 38108 Braunschweig, Germany
Marian Albers
Affiliation:
Institute of Aerodynamics, RWTH Aachen University, Wüllnerstr. 5a, 52062 Aachen, Germany
Pascal S. Meysonnat
Affiliation:
Institute of Aerodynamics, RWTH Aachen University, Wüllnerstr. 5a, 52062 Aachen, Germany
Wolfgang Schröder
Affiliation:
Institute of Aerodynamics, RWTH Aachen University, Wüllnerstr. 5a, 52062 Aachen, Germany Forschungszentrum Jülich, JARA-High-Performance Computing, 52425 Jülich, Germany
Bernd R. Noack
Affiliation:
Institut für Strömungsmechanik, Technische Universität Braunschweig, Hermann-Blenk-Str. 37, 38108 Braunschweig, Germany LIMSI, CNRS, Université Paris-Saclay, Bât 507, rue du Belvédère, Campus Universitaire, F-91403 Orsay, France Institut für Strömungsmechanik und Technische Akustik (ISTA), Technische Universität Berlin, Müller-Breslau-Straße 8, 10623 Berlin, Germany Institute for Turbulence-Noise-Vibration Interaction and Control, Harbin Institute of Technology, Shenzhen Campus, 518055 Shenzhen, China
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Abstract

We present the first general metric for attractor overlap (MAO) facilitating an unsupervised comparison of flow data sets. The starting point is two or more attractors, i.e. ensembles of states representing different operating conditions. The proposed metric generalizes the standard Hilbert-space distance between two snapshot-to-snapshot ensembles of two attractors. A reduced-order analysis for big data and many attractors is enabled by coarse graining the snapshots into representative clusters with corresponding centroids and population probabilities. For a large number of attractors, MAO is augmented by proximity maps for the snapshots, the centroids and the attractors, giving scientifically interpretable visual access to the closeness of the states. The coherent structures belonging to the overlap and disjoint states between these attractors are distilled by a few representative centroids. We employ MAO for two quite different actuated flow configurations: a two-dimensional wake with vortices in a narrow frequency range and three-dimensional wall turbulence with a broadband spectrum. In the first application, seven control laws are applied to the fluidic pinball, i.e. the two-dimensional flow around three circular cylinders whose centres form an equilateral triangle pointing in the upstream direction. These seven operating conditions comprise unforced shedding, boat tailing, base bleed, high- and low-frequency forcing as well as two opposing Magnus effects. In the second example, MAO is applied to three-dimensional simulation data from an open-loop drag reduction study of a turbulent boundary layer. The actuation mechanisms of 38 spanwise travelling transversal surface waves are investigated. MAO compares and classifies these actuated flows in agreement with physical intuition. For instance, the first feature coordinate of the attractor proximity map correlates with drag for the fluidic pinball and for the turbulent boundary layer. MAO has a large spectrum of potential applications ranging from a quantitative comparison between numerical simulations and experimental particle-image velocimetry data to the analysis of simulations representing a myriad of different operating conditions.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models Wakes/Jets: Wakes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 874 , 10 September 2019 , pp. 720 - 755
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Copyright
© 2019 Cambridge University Press 

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Footnotes

†

Present address: Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. Email address for correspondence: [email protected]

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