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On long-term boundedness of Galerkin models

Published online by Cambridge University Press:  21 January 2015

Michael Schlegel  and
Bernd R. Noack
Michael Schlegel*
Affiliation:
Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Département Fluides, Thermique, Combustion, CEAT, 43 Rue de l’Aérodrome, 86036 Poitiers CEDEX, France Institut für Strömungsmechanik und Technische Akustik, Technische Universität Berlin MB1, Straße des 17 Juni 135, 10623 Berlin, Germany Department II, Mathematics – Physics – Chemistry, Beuth University of Applied Sciences Berlin, Luxemburger Straße 10, 13353 Berlin, Germany Fachbereich 1, Ingenieurwissenschaften – Energie und Information, Hochschule für Technik und Wirtschaft Berlin, Treskowallee 8, 10318 Berlin, Germany
Bernd R. Noack
Affiliation:
Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Département Fluides, Thermique, Combustion, CEAT, 43 Rue de l’Aérodrome, 86036 Poitiers CEDEX, France Institute für Strömungsmechanik, Technische Universität Braunschweig, Hermann–Blenck–Straße 37, 38108 Braunschweig, Germany
*
†Email address for correspondence: michael.schlegel@cfd.tu-berlin.de
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Abstract

We investigate linear–quadratic dynamical systems with energy-preserving quadratic terms. These systems arise for instance as Galerkin systems of incompressible flows. A criterion is presented to ensure long-term boundedness of the system dynamics. If the criterion is violated, a globally stable attractor cannot exist for an effective nonlinearity. Thus, the criterion can be considered a minimum requirement for control-oriented Galerkin models of viscous fluid flows. The criterion is exemplified, for example, for Galerkin systems of two-dimensional cylinder wake flow models in the transient and the post-transient regime, for the Lorenz system and for wall-bounded shear flows. There are numerous potential applications of the criterion, for instance, system reduction and control of strongly nonlinear dynamical systems.

JFM classification

Flow Control: Flow Control Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 765 , 25 February 2015 , pp. 325 - 352
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Copyright
© 2015 Cambridge University Press 

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