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Quasi-periodic intermittency in oscillating cylinder flow

Published online by Cambridge University Press:  12 September 2017

Bryan Glaz*
Affiliation:
Vehicle Technology Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA
Igor Mezić
Affiliation:
Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, CA 93106, USA
Maria Fonoberova
Affiliation:
Aimdyn, Inc., Santa Barbara, CA 93106, USA
Sophie Loire
Affiliation:
Aimdyn, Inc., Santa Barbara, CA 93106, USA
*
Email address for correspondence: [email protected]

Abstract

Fluid dynamics induced by periodically forced flow around a cylinder is analysed computationally for the case when the forcing frequency is much lower than the von Kármán vortex shedding frequency corresponding to the constant flow velocity condition. By using the Koopman mode decomposition approach, we find a new normal form equation that extends the classical Hopf bifurcation normal form by a time-dependent term for Reynolds numbers close to the Hopf bifurcation value. The normal form describes the dynamics of an observable and features a forcing (control) term that multiplies the state, and is thus a parametric – i.e. not an additive – forcing effect. We find that the dynamics of the flow in this regime is characterized by alternating instances of quiescent and strong oscillatory behaviour and that this pattern persists indefinitely. Furthermore, the spectrum of the associated Koopman operator is shown to possess quasi-periodic features. We establish the theoretical underpinnings of this phenomenon – that we name quasi-periodic intermittency – using the new normal form model and show that the dynamics is caused by the tendency of the flow to oscillate between the unstable fixed point and the stable limit cycle of the unforced flow. The quasi-periodic intermittency phenomenon is also characterized by positive finite-time Lyapunov exponents that, over a long period of time, asymptotically approach zero.

Type
Papers
Copyright
© Cambridge University Press 2017. This is a work of the U.S. Government and is not subject to copyright protection in the United States. 

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References

Bagheri, S. 2013 Koopman-mode decomposition of the cylinder wake. J. Fluid Mech. 726, 596623.Google Scholar
Bagheri, S. 2014 Effects of weak noise on oscillating flows: linking quality factor, floquet modes, and Koopman spectrum. Phys. Fluids 26, 094104.Google Scholar
Bajaj, A. K. 1986 Resonant parametric perturbations of the Hopf bifurcation. J. Math. Anal. Appl. 115, 214224.CrossRefGoogle Scholar
Brunton, S. L. & Noack, B. R. 2015 Clsoed-loop turbulence control: progress and challenges. Annu. Mech. Rev. 67, 050801.Google Scholar
Budisic, M. & Mezić, I. 2012 Applied Koopmanism. Chaos 22, 047510.Google Scholar
Cetiner, O. & Rockwell, D. 2001 Streamwise oscillations of a cylinder in steady current. Part 1. Locked-on states of vortex formation and loading. J. Fluid Mech. 427, 128.CrossRefGoogle Scholar
Chen, K. K., Tu, J. H. & Rowley, C. W. 2011 Variants of dynamic mode decomposition: boundary conditions, Koopman, and Fourier analyses. J. Nonlinear Sci. 22 (6), 887915.Google Scholar
Gabale, A. P. & Sinha, S. C. 2009 A direct analysis of nonlinear systems with external periodic excitations via normal forms. Nonlinear Dyn. 55 (1), 7993.Google Scholar
Gal, P. L., Nadim, A. & Thompson, M. 2001 Hysteresis in the forced Stuart–Landau equation: application to vortex shedding from an oscillating cylinder. J. Fluids Struct. 15, 445457.CrossRefGoogle Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.CrossRefGoogle Scholar
Hilborn, R. C. 1994 Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22 (1), 473537.Google Scholar
Konstantinidis, E. & Balabani, S. 2007 Symmetric vortex shedding in the near wake of a circular cylinder due to streamwise perturbations. J. Fluids Struct. 23, 10471063.CrossRefGoogle Scholar
Koopman, B. O. 1931 Hamiltonian systems and transformation in Hilbert space. Proc. Natl Acad. Sci. USA 17 (5), 315318.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, vol. 2. Pergamon.Google Scholar
Leontini, J. S., Jacono, D. L. & Thompson, M. C. 2011 A numerical study of an inline oscillating cylinder in a free stream. J. Fluid Mech. 688, 551568.Google Scholar
Leontini, J. S., Jacono, D. L. & Thompson, M. C. 2013 Wake states and frequency selection of a streamwise oscillating cylinder. J. Fluid Mech. 730, 162192.Google Scholar
Lin, K. K. & Young, L. S. 2008 Shear-induced chaos. Nonlinearity 21, 899922.Google Scholar
Luchtenburg, D. M., Gunther, B., Noack, B. R., King, R. & Tadmor, G. 2009 A generalized mean-field model of the natural and high frequency actuated flow around a high-lift configuration. J. Fluid Mech. 623, 283316.CrossRefGoogle Scholar
McCroskey, W. J. 1982 Unsteady airfoils. Annu. Rev. Fluid Mech. 14 (1), 285311.Google Scholar
Mezić, I. 2005 Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41 (1), 309325.Google Scholar
Mezić, I. 2013 Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357378.Google Scholar
Nayfeh, A. H. 2011 The Method of Normal Forms. Wiley.CrossRefGoogle Scholar
Perdikaris, P. G., Kaiktsis, L. & Triantafyllou, G. S. 2009 Chaos in a cylinder wake due to forcing at the Strouhal frequency. Phys. Fluids 21, 101705.Google Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Benard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 122.Google Scholar
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115127.Google Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167192.CrossRefGoogle Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 528.Google Scholar
Schmid, P. & Sesterhenn, J. 2008 Dynamic mode decomposition of numerical and experimental data. In 61st Annual Meeting of the APS Division of Fluid Dynamics.Google Scholar
Singh, R. K. & Manhas, J. S. 1993 Composition Operators on Function Spaces. North-Holland.Google Scholar
Sipp, D. 2012 Open-loop control of cavity oscillations with harmonic forcings. J. Fluid Mech. 708, 439468.Google Scholar
Sreenivasan, K. R., Strykowski, P. J. & Olinger, D. J. 1987 Hopf bifurcation, Landau equation and vortex shedding behind circular cylinders. In Proc. Forum Unsteady Separation (ed. Ghia, K. N.), vol. 52. ASME.Google Scholar
Susuki, Y. & Mezić, I. 2012 Nonlinear Koopman modes and a precursor to power system swing instabilities. Power Systems, IEEE Trans. 27 (3), 11821191.CrossRefGoogle Scholar
Tsarouhas, G. E. & Ross, J. 1987 Explicit solutions of normal form of driven oscillatory systems. J. Chem. Phys. 87 (11), 65386543.Google Scholar
Tsarouhas, G. E. & Ross, J. 1988 Explicit solutions of normal form of driven oscillatory systems in entrainment bands. J. Chem. Phys. 88 (9), 57155720.CrossRefGoogle Scholar
Vance, W., Tsarouhas, G. & Ross, J. 1989 Universal bifurcation structures of forced oscillators. Prog. Theor. Phys. Suppl. 99, 331338.CrossRefGoogle Scholar
Wang, Q. & Young, L. S. 2003 Strange attractors in periodically-kicked limit cycles and Hopf bifurcations. Commun. Math. Phys. 240, 509529.Google Scholar
Wiggins, S. 1990 Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer.Google Scholar
Wynn, A., Pearson, D. S., Ganapathisubramani, B. & Goulart, P. J. 2013 Optimal mode decomposition for unsteady flows. J. Fluid Mech. 733, 473503.CrossRefGoogle Scholar