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Phase-response analysis of synchronization for periodic flows

Published online by Cambridge University Press:  03 May 2018

Kunihiko Taira Open the ORCID record for Kunihiko Taira [Opens in a new window]  and
Hiroya Nakao
Kunihiko Taira*
Affiliation:
Department of Mechanical Engineering, Florida State University, Tallahassee, FL 32310, USA
Hiroya Nakao
Affiliation:
Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
*
†Email address for correspondence: ktaira@fsu.edu
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Abstract

We apply phase-reduction analysis to examine synchronization properties of periodic fluid flows. The dynamics of unsteady flows is described in terms of the phase dynamics, reducing the high-dimensional fluid flow to its single scalar phase variable. We characterize the phase response to impulse perturbations, which can in turn quantify the influence of periodic perturbations on the unsteady flow. These insights from phase-based analysis uncover the condition for synchronization. In the present work, we study as an example the influence of periodic external forcing on an unsteady cylinder wake. The condition for synchronization is identified and agrees closely with results from direct numerical simulations. Moreover, the analysis reveals the optimal forcing direction for synchronization. Phase-response analysis holds potential to uncover lock-on characteristics for a range of periodic flows.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Wakes/Jets: Wakes
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 846 , 10 July 2018 , R2
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Copyright
© 2018 Cambridge University Press 

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References

Arnold, V. I. 1997 Mathematical Methods of Classical Mechanics. Springer.Google Scholar
Buck, J. 1988 Synchronous rhythmic flashing of fireflies. II. Q. Rev. Biol. 63 (3), 265–289.Google Scholar
Canuto, D. & Taira, K. 2015 Two-dimensional compressible viscous flow around a circular cylinder. J. Fluid Mech. 785, 349–371.CrossRefGoogle Scholar
Colonius, T. & Taira, K. 2008 A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions. Comput. Meth. Appl. Mech. Engng 197, 2131–2146.Google Scholar
Ermentrout, G. B. & Terman, D. H. 2010 Mathematical Foundations of Neuroscience. Springer.Google Scholar
Guckenheimer, J. 1975 Isochrons and phaseless sets. J. Math. Biol. 1 (3), 259–273.Google Scholar
Kawamura, Y. & Nakao, H. 2013 Collective phase description of oscillatory convection. Chaos 23, 043129.Google Scholar
Kawamura, Y. & Nakao, H. 2015 Phase description of oscillatory convection with a spatially translational mode. Physica D 295–296, 11–29.Google Scholar
Kuramoto, Y. 1984 Chemical Oscillations, Waves, and Turbulence. Springer.Google Scholar
Linnick, M. N. & Fasel, H. F. 2005 A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains. J. Comput. Phys. 204, 157–192.Google Scholar
Liu, C., Zheng, X. & Sung, C. H. 1998 Preconditioned multigrid methods for unsteady incompressible flows. J. Comput. Phys. 139, 35–57.Google Scholar
Mauroy, A. & Mezić, I. 2012 On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics. Chaos 22, 033112.Google Scholar
Munday, P. M. & Taira, K. 2013 On the lock-on of vortex shedding to oscillatory actuation around a circular cylinder. Phys. Fluids 25, 013601.CrossRefGoogle Scholar
Nakao, H. 2016 Phase reduction approach to synchronisation of nonlinear oscillators. Contemp. Phys. 57 (2), 188–214.Google Scholar
Pikovsky, A., Rosenblum, M. & Kurths, J. 2001 Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press.Google Scholar
Roma, A. M., Peskin, C. S. & Berger, M. J. 1999 An adaptive version of the immersed boundary method. J. Comput. Phys. 153, 509–534.Google Scholar
Strogatz, S. H. 2000 From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 1–20.Google Scholar
Taira, K. & Colonius, T. 2007 The immersed boundary method: a projection approach. J. Comput. Phys. 225, 2118–2137.Google Scholar
Tokumaru, P. T. & Dimotakis, P. E. 1991 Rotary oscillation control of a cylinder wake. J. Fluid Mech. 224, 77–90.Google Scholar
Winfree, A. T. 1967 Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16 (1), 15–42.Google Scholar
Yamaguchi, Y., Suzuki, T., Mizoro, Y., Kori, H., Okada, K., Chen, Y., Fustin, J.-M., Yamazaki, F., Mizuguchi, N., Zhang, J. et al. 2013 Mice genetically deficient in vasopressin V1a and V1b receptors are resistant to jet lag. Science 342, 85–90.Google Scholar