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Phase-response analysis of synchronization for periodic flows

  • Kunihiko Taira (a1) and Hiroya Nakao (a2)
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.

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Email address for correspondence: ktaira@fsu.edu
References
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Arnold, V. I. 1997 Mathematical Methods of Classical Mechanics. Springer.
Buck, J. 1988 Synchronous rhythmic flashing of fireflies. II. Q. Rev. Biol. 63 (3), 265289.
Canuto, D. & Taira, K. 2015 Two-dimensional compressible viscous flow around a circular cylinder. J. Fluid Mech. 785, 349371.
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, 21312146.
Ermentrout, G. B. & Terman, D. H. 2010 Mathematical Foundations of Neuroscience. Springer.
Guckenheimer, J. 1975 Isochrons and phaseless sets. J. Math. Biol. 1 (3), 259273.
Kawamura, Y. & Nakao, H. 2013 Collective phase description of oscillatory convection. Chaos 23, 043129.
Kawamura, Y. & Nakao, H. 2015 Phase description of oscillatory convection with a spatially translational mode. Physica D 295–296, 1129.
Kuramoto, Y. 1984 Chemical Oscillations, Waves, and Turbulence. Springer.
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, 157192.
Liu, C., Zheng, X. & Sung, C. H. 1998 Preconditioned multigrid methods for unsteady incompressible flows. J. Comput. Phys. 139, 3557.
Mauroy, A. & Mezić, I. 2012 On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics. Chaos 22, 033112.
Munday, P. M. & Taira, K. 2013 On the lock-on of vortex shedding to oscillatory actuation around a circular cylinder. Phys. Fluids 25, 013601.
Nakao, H. 2016 Phase reduction approach to synchronisation of nonlinear oscillators. Contemp. Phys. 57 (2), 188214.
Pikovsky, A., Rosenblum, M. & Kurths, J. 2001 Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press.
Roma, A. M., Peskin, C. S. & Berger, M. J. 1999 An adaptive version of the immersed boundary method. J. Comput. Phys. 153, 509534.
Strogatz, S. H. 2000 From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 120.
Taira, K. & Colonius, T. 2007 The immersed boundary method: a projection approach. J. Comput. Phys. 225, 21182137.
Tokumaru, P. T. & Dimotakis, P. E. 1991 Rotary oscillation control of a cylinder wake. J. Fluid Mech. 224, 7790.
Winfree, A. T. 1967 Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16 (1), 1542.
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, 8590.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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