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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: [email protected]
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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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References

Aamo, O. M. & Krstić, M. 2002 Flow Control by Feedback: Stabilization and Mixing. Springer.Google Scholar
Balajewicz, M., Dowell, E. & Noack, B. R. 2013 Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation. J. Fluid Mech. 729, 285–308.CrossRefGoogle Scholar
Boberg, L. & Brosa, U. 1988 Onset of turbulence in a pipe. Z. Naturforsch. A 43, 697–726.CrossRefGoogle Scholar
Cazemier, W., Verstappen, R. W. C. P. & Veldman, A. E. P. 1998 Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Phys. Fluids 7, 1685–1699.CrossRefGoogle Scholar
Chernyshenko, S. I., Goulart, P., Huang, D. & Papachristodoulou, A. 2014 Polynomial sum of squares in fluid dynamics: a review with a look ahead. Phil. Trans. R. Soc. Lond. A 372, 20130350.Google ScholarPubMed
Cordier, L., Noack, B. R., Daviller, G., Lehnasch, G., Tissot, G., Balajewicz, M. & Niven, R. 2013 Control-oriented model identification strategy. Exp. Fluids 54, 1580.CrossRefGoogle Scholar
Deane, A. E., Kevrekidis, I. G., Karniadakis, G. E. & Orszag, S. A. 1991 Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Phys. Fluids A 3, 2337–2354.CrossRefGoogle Scholar
Doering, C. R. & Constantin, P. 1994 Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E 49, 4087–4099.CrossRefGoogle ScholarPubMed
Drazin, P. G. & Reid, H. W. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Fletcher, C. A. J. 1984 Computational Galerkin Methods. Springer.CrossRefGoogle Scholar
Galdi, G. P. & Padula, M. 1990 A new approach to the energy theory in the stability of fluid motion. Arch. Rat. Mech. Anal. 22, 163–184.Google Scholar
Galletti, G., Bruneau, C. H., Zannetti, L. & Iollo, A. 2004 Low-order modelling of laminar flow regimes past a confined square cylinder. J. Fluid Mech. 503, 161–170.CrossRefGoogle Scholar
Gerhard, J., Pastoor, M., King, R., Noack, B. R., Dillmann, A., Morzyński, M. & Tadmor, G.2003 Model-based control of vortex shedding using low-dimensional Galerkin models. AIAA Paper 2003-4262.CrossRefGoogle Scholar
Gershenfeld, N. 2006 The Nature of Mathematical Modeling. Cambridge University Press.Google Scholar
Goulart, P. J. & Chernyshenko, S. 2012 Global stability analysis of fluid flows using sum-of-squares. Physica D 241, 692–704.CrossRefGoogle Scholar
Guckenheimer, J. & Holmes, P. 1986 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.Google Scholar
Holmes, P., Lumley, J. L., Berkooz, G. & Rowley, C. W. 2012 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Horn, R. A. & Johnson, C. R. 2013 Matrix Analysis, 2nd edn. Cambridge University Press.Google Scholar
Joseph, D. D. 1976 Stability in Fluid Motions I, II, Springer Tracts in Natural Philosophy, vol. 27, 28. Springer.Google Scholar
Khalil, H. K. 2002 Nonlinear Dynamics. Dover.Google Scholar
Kim, J. & Bewley, T. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383–417.CrossRefGoogle Scholar
Kolda, T. G. & Bader, B. W. 2009 Tensor decompositions and applications. SIAM Rev. 51 (3), 455–500.CrossRefGoogle Scholar
Kryloff, N. & Bogoliouboff, N. 1935 La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire. Ann. Maths 38 (1), 65–113.CrossRefGoogle Scholar
Ladyz̆henskaya, O. A. 1963 The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach.Google Scholar
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.2.0.CO;2>CrossRefGoogle Scholar
Lyapunov, A. M. 1892 Stability of Motion. Academic Press.Google Scholar
Manneville, P. 2004 Instabilites, Chaos and Turbulence. Imperial College Press.CrossRefGoogle Scholar
McComb, W. D. 1991 The Physics of Fluid Turbulence. Clarendon.Google Scholar
Meiss, J. D. 2007 Differential Dynamical Systems, Monographs on Mathematical Modelling and Computation, vol. 14. SIAM.CrossRefGoogle Scholar
Moehlis, J., Faisst, H. & Eckardt, B. 2004 A low-dimensional model for turbulent shear flows. New J. Phys. 6, 56.CrossRefGoogle Scholar
Noack, B. R., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.CrossRefGoogle Scholar
Noack, B. R. & Fasel, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.CrossRefGoogle Scholar
Noack, B. R., Morzyński, M. & Tadmor, G. 2011 Reduced-Order Modelling for Flow Control, CISM International Centre for Mechanical Sciences, vol. 528. Springer.CrossRefGoogle Scholar
Noack, B. R., Papas, P. & Monkewitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339–365.CrossRefGoogle Scholar
Parrilo, P. A. 2003 Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96, 293–320.CrossRefGoogle Scholar
Rempfer, D. & Fasel, H. F. 1994 Dynamics of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech. 275, 257–283.CrossRefGoogle Scholar
Rummler, B.2000 Zur Lösung der instationären inkompressiblen Navier–Stokesschen Gleichungen in speziellen Gebieten. Habilitation thesis, Fakultät für Mathematik, Otto-von-Guericke-Universität, Magdeburg.Google Scholar
Rummler, B. & Noske, A. 1998 Direct Galerkin approximation of plane-parallel-Couette and channel flows by Stokes eigenfunctions. Notes Numer. Fluid Mech. 64, 3–19.Google Scholar
Saltzman, B. 1962 Finite amplitude free convection as an initial value problem. J. Atmos. Sci. 19, 329–341.2.0.CO;2>CrossRefGoogle Scholar
Samimy, M., Debiasi, M., Caraballo, E., Serrani, A., Yuan, X., Little, J. & Myatt, J. 2007 Feedback control of subsonic cavity flows using reduced-order models. J. Fluid Mech. 579, 315–346.CrossRefGoogle Scholar
Schlegel, M., Noack, B. R., Comte, P., Kolomenskiy, D., Schneider, K., Farge, M., Scouten, J., Luchtenburg, D. M. & Tadmor, G. 2009 Reduced-order modelling of turbulent jets for noise control. In Numerical Simulation of Turbulent Flows and Noise Generation (ed. Brun, C., Juvé, D., Manhart, M. & Munz, C.-D.), Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), vol. 104, pp. 3–27. Springer.CrossRefGoogle Scholar
Schlegel, M., Noack, B. R., Jordan, P., Dillmann, A., Gröschel, E., Schröder, W., Wei, M., Freund, J. B., Lehmann, O. & Tadmor, G. 2012 On least-order flow representations for aerodynamics and aeroacoustics. J. Fluid Mech. 697, 367–398.CrossRefGoogle Scholar
Schlichting, H. 1968 Boundary-Layer Theory, 3rd English edn. McGraw-Hill.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, Applied Mathematical Sciences, vol. 142. Springer.CrossRefGoogle Scholar
Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open flows: a linearized approach. Appl. Mech. Rev. 63, 030801.CrossRefGoogle Scholar
Sirisup, S. & Karniadakis, G. E. 2004 A spectral viscosity method for correcting the long-term behavior of POD models. J. Comput. Phys. 194, 92–116.CrossRefGoogle Scholar
Straughan, B. 2004 The Energy Method, Stability and Nonlinear Convection, 2nd edn. Applied Mathematical Sciences, vol. 94. Springer.CrossRefGoogle Scholar
Swinnerton-Dyer, P. 2000 A note on Liapunov’s method. Dyn. Stab. Syst. 15, 3–10.CrossRefGoogle Scholar
Swinnerton-Dyer, P. 2001 Bounds for trajectories of the Lorenz equations: an illustration of how to choose Liapunov functions. Phys. Lett. A 281, 161–167.CrossRefGoogle Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra. Princeton University Press.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.CrossRefGoogle ScholarPubMed
Waleffe, F. 1995 Transition in shear flows. Nonlinear normality versus nonnormal linearity. Phys. Fluids 7, 3060–3066.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.CrossRefGoogle Scholar
Wei, M. & Rowley, C. W. 2009 Low-dimensional models of a temporally evolving free shear layer. J. Fluid Mech. 618, 113–134.CrossRefGoogle Scholar
Willcox, K. & Megretski, A. 2005 Fourier series for accurate, stable, reduced-order models in large-scale applications. SIAM J. Sci. Comput. 26 (3), 944–962.CrossRefGoogle Scholar