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Low-order model for successive bifurcations of the fluidic pinball

Published online by Cambridge University Press:  17 December 2019

Nan Deng Open the ORCID record for Nan Deng [Opens in a new window] ,
Bernd R. Noack Open the ORCID record for Bernd R. Noack [Opens in a new window] ,
Marek Morzyński  and
Luc R. Pastur
Nan Deng*
Affiliation:
Institute of Mechanical Sciences and Industrial Applications, ENSTA-Paris,Institut Polytechnique de Paris, 828 Bd des Maréchaux, F-91120Palaiseau, France LIMSI, CNRS, Université Paris-Saclay, Bât 507, rue du Belvédère, Campus Universitaire, F-91405Orsay, France
Bernd R. Noack
Affiliation:
LIMSI, CNRS, Université Paris-Saclay, Bât 507, rue du Belvédère, Campus Universitaire, F-91405Orsay, France Institute for Turbulence-Noise-Vibration Interaction and Control, Harbin Institute of Technology, Shenzhen Graduate School, University Town, Xili, Shenzhen518058, PR China Institut für Strömungsmechanik und Technische Akustik (ISTA), Technische Universität Berlin, Müller-Breslau-Straße 8, D-10623Berlin, Germany
Marek Morzyński
Affiliation:
Chair of Virtual Engineering, Poznań University of Technology, Jana Pawla II 24, PL 60-965Poznań, Poland
Luc R. Pastur
Affiliation:
Institute of Mechanical Sciences and Industrial Applications, ENSTA-Paris,Institut Polytechnique de Paris, 828 Bd des Maréchaux, F-91120Palaiseau, France
*
†Email address for correspondence: nan.deng@ensta-paris.fr
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Abstract

We propose the first least-order Galerkin model of an incompressible flow undergoing two successive supercritical bifurcations of Hopf and pitchfork type. A key enabler is a mean-field consideration exploiting the symmetry of the mean flow and the asymmetry of the fluctuation. These symmetries generalize mean-field theory, e.g. no assumption of slow growth rate is needed. The resulting five-dimensional Galerkin model successfully describes the phenomenogram of the fluidic pinball, a two-dimensional wake flow around a cluster of three equidistantly spaced cylinders. The corresponding transition scenario is shown to undergo two successive supercritical bifurcations, namely a Hopf and a pitchfork bifurcation on the way to chaos. The generalized mean-field Galerkin methodology may be employed to describe other transition scenarios.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Low-dimensional models Wakes/Jets: Wakes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 884 , 10 February 2020 , A37
Copyright
© 2019 Cambridge University Press

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Deng et al. supplementary movie 1

Quasi-periodic dynamics at ReD=105 after the transition. The base-bleeding jet oscillates around the deflected position with a lower frequency. The current time is expressed in convective time units.

Download Deng et al. supplementary movie 1(Video)
Video 2.6 MB

Deng et al. supplementary movie 2

Chaotic dynamics at ReD=130 after the transition. The base-bleeding jet switches randomly in time between the two symmetric deflected positions. The current time is expressed in convective time units.

Download Deng et al. supplementary movie 2(Video)
Video 3.9 MB