Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-20T22:24:46.515Z Has data issue: false hasContentIssue false
  • English
  • Français

Rapidly rotating precessing cylinder flows: forced triadic resonances

Published online by Cambridge University Press:  25 January 2018

Juan M. Lopez Open the ORCID record for Juan M. Lopez [Opens in a new window]  and
Francisco Marques
Juan M. Lopez*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
Francisco Marques
Affiliation:
Departament de Fisica Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
*
†Email address for correspondence: juan.m.lopez@asu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Rapidly rotating cylinder flows subjected to low-amplitude precessional forcing are studied numerically over a range of cylinder and precessional rotation rates. For sufficiently small rotation rates, viscous effects lead to a forced overturning flow that is steady in the precession (table) frame of reference. Increasing the rotation rates, this forced flow loses stability in a Hopf bifurcation, which can be either supercritical or subcritical, and may preserve or break the symmetry of the system, depending on the parameter regime studied. Regardless of these details of the Hopf bifurcation, it is found that the Hopf instability is associated with a slightly detuned triadic resonance between the forced overturning flow and two free Kelvin modes (inviscid eigenmodes of the rotating cylinder). Further increases in rotation rates lead to a sequence of secondary instabilities which also follow a generic pattern irrespective of the parameter regime investigated. The relationship between this sequence of instabilities and the resultant nonlinear dynamics with the experimentally observed phenomenon of resonant collapse is discussed.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Instability: Nonlinear instability Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 839 , 25 March 2018 , pp. 239 - 270
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abshagen, J., Lopez, J. M., Marques, F. & Pfister, G. 2005a Mode competition of rotating waves in reflection-symmetric Taylor–Couette flow. J. Fluid Mech. 540, 269–299.CrossRefGoogle Scholar
Abshagen, J., Lopez, J. M., Marques, F. & Pfister, G. 2005b Symmetry breaking via global bifurcations of modulated rotating waves in hydrodynamics. Phys. Rev. Lett. 94, 074101.CrossRefGoogle ScholarPubMed
Abshagen, J., Lopez, J. M., Marques, F. & Pfister, G. 2008 Bursting dynamics due to a homoclinic cascade in Taylor–Couette flow. J. Fluid Mech. 613, 357–384.Google Scholar
Äkervik, E., Brandt, L., Henningson, D. S., Hœpffner, J., Marxen, O. & Schlatter, P. 2006 Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18, 068102.Google Scholar
Albrecht, T., Blackburn, H. M., Lopez, J. M., Manasseh, R. & Meunier, P. 2015 Triadic resonances in precessing rapidly rotating cylinder flows. J. Fluid Mech. 778, R1.CrossRefGoogle Scholar
Albrecht, T., Blackburn, H. M., Meunier, P., Manasseh, R. & Lopez, J. M. 2016 Experimental and numerical investigation of a strongly-forced precessing cylinder flow. Intl J. Heat Fluid Flow 61, 68–74.CrossRefGoogle Scholar
Altmeyer, S., Do, Y., Marques, F. & Lopez, J. M. 2012 Symmetry-breaking Hopf bifurcations to 1-, 2-, and 3-tori in small-aspect-ratio counter-rotating Taylor–Couette flow. Phys. Rev. E 86, 046316.Google Scholar
Blackburn, H. M., Marques, F. & Lopez, J. M. 2005 Symmetry breaking of two-dimensional time-periodic wakes. J. Fluid Mech. 522, 395–411.Google Scholar
Dauxois, T., Joubaud, S., Odier, P. & Venaille, A. 2018 Instabilities of internal gravity wave beams. Annu. Rev. Fluid Mech. 50, 1–28.CrossRefGoogle Scholar
Eloy, C., Le Gal, P. & Le Dizès, S. 2003 Elliptic and triangular instabilities in rotating cylinders. J. Fluid Mech. 476, 357–388.Google Scholar
Giesecke, A., Albrecht, T., Gundrum, T., Herault, J. & Stefani, F. 2015 Triadic resonances in nonlinear simulations of a fluid flow in a precessing cylinder. New J. Phys. 17, 113044.CrossRefGoogle Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Guckenheimer, J. & Holmes, P. 1997 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.Google Scholar
Guckenheimer, J., Krauskopf, B., Osinga, H. M. & Sandstede, B. 2015 Invariant manifolds and global bifurcations. Chaos 25, 097604.CrossRefGoogle ScholarPubMed
Herault, J., Gundrum, T., Giesecke, A. & Stefani, F. 2015 Subcritical transition to turbulence of a precessing flow in a cylindrical vessel. Phys. Fluids 27, 124102.CrossRefGoogle Scholar
Lord Kelvin 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Kerswell, R. R. 1999 Secondary instabilities in rapidly rotating fluids: inertial wave breakdown. J. Fluid Mech. 382, 283–306.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.Google Scholar
Kobine, J. J. 1995 Inertial wave dynamics in a rotating and precessing cylinder. J. Fluid Mech. 303, 233–252.Google Scholar
Kobine, J. J. 1996 Azimuthal flow associated with inertial wave resonance in a precessing cylinder. J. Fluid Mech. 319, 387–406.Google Scholar
Kuznetsov, Y. A. 2004 Elements of Applied Bifurcation Theory, 3rd edn. Springer.Google Scholar
Lagrange, R., Eloy, C., Nadal, F. & Meunier, P. 2008 Instability of a fluid inside a precessing cylinder. Phys. Fluids 20, 081701.CrossRefGoogle Scholar
Lagrange, R., Meunier, P. & Eloy, C. 2016 Triadic instability of a non-resonant precessing fluid cylinder. C. R. Méc. 344, 418–433.Google Scholar
Lagrange, R., Meunier, P., Nadal, F. & Eloy, C. 2011 Precessional instability of a fluid cylinder. J. Fluid Mech. 666, 104–145.CrossRefGoogle Scholar
Le Bars, M., Cebron, D. & Le Gal, P. 2015 Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47, 163–193.CrossRefGoogle Scholar
Leung, J. J. F., Hirsa, A. H., Blackburn, H. M., Marques, F. & Lopez, J. M. 2005 Three-dimensional modes in a periodically driven elongated cavity. Phys. Rev. E 71, 026305.Google Scholar
Lin, Y., Noir, J. & Jackson, A. 2014 Experimental study of fluid flows in a precessing cylindrical annulus. Phys. Fluids 26, 046604.Google Scholar
Lopez, J. M. 2006 Rotating and modulated rotating waves in transitions of an enclosed swirling flow. J. Fluid Mech. 553, 323–346.CrossRefGoogle Scholar
Lopez, J. M. & Gutierrez-Castillo, P. 2016 Three-dimensional instabilities and inertial waves in a rapidly rotating split-cylinder flow. J. Fluid Mech. 800, 666–687.Google Scholar
Lopez, J. M. & Marques, F. 2004 Mode competition between rotating waves in a swirling flow with reflection symmetry. J. Fluid Mech. 507, 265–288.Google Scholar
Lopez, J. M. & Marques, F. 2005 Finite aspect ratio Taylor–Couette flow: Shil’nikov dynamics of 2-tori. Physica D 211, 168–191.Google Scholar
Lopez, J. M. & Marques, F. 2011 Instabilities and inertial waves generated in a librating cylinder. J. Fluid Mech. 687, 171–193.Google Scholar
Lopez, J. M. & Marques, F. 2014 Rapidly rotating cylinder flow with an oscillating sidewall. Phys. Rev. E 89, 013019.Google ScholarPubMed
Lopez, J. M. & Marques, F. 2016a Inertial waves in rapidly rotating flows: a dynamical systems perspective. Phys. Scr. 91, 124001.CrossRefGoogle Scholar
Lopez, J. M. & Marques, F. 2016b Nonlinear and detuning effects of the nutation angle in precessionally-forced rotating cylinder flow. Phys. Rev. Fluids 1, 023602.Google Scholar
Lopez, J. M., Welfert, B. D., Wu, K. & Yalim, J. 2017 Transition to complex dynamics in the cubic lid-driven cavity. Phys. Rev. Fluids 2, 074401.Google Scholar
Manasseh, R. 1992 Breakdown regimes of inertia waves in a precessing cylinder. J. Fluid Mech. 243, 261–296.Google Scholar
Manasseh, R. 1994 Distortions of inertia waves in a rotating fluid cylinder forced near its fundamental mode resonance. J. Fluid Mech. 265, 345–370.Google Scholar
Manasseh, R. 1996 Nonlinear behaviour of contained inertia waves. J. Fluid Mech. 315, 151–173.Google Scholar
Marques, F. & Lopez, J. M. 1997 Taylor–Couette flow with axial oscillations of the inner cylinder: Floquet analysis of the basic flow. J. Fluid Mech. 348, 153–175.Google Scholar
Marques, F. & Lopez, J. M. 2001 Precessing vortex breakdown mode in an enclosed cylinder flow. Phys. Fluids 13, 1679–1682.Google Scholar
Marques, F. & Lopez, J. M. 2006 Onset of three-dimensional unsteady states in small-aspect ratio Taylor–Couette flow. J. Fluid Mech. 561, 255–277.CrossRefGoogle Scholar
Marques, F. & Lopez, J. M. 2015 Precession of a rapidly rotating cylinder flow: traverse through resonance. J. Fluid Mech. 782, 63–98.CrossRefGoogle Scholar
Marques, F., Lopez, J. M. & Blackburn, H. M. 2004 Bifurcations in systems with Z 2 spatio-temporal and O (2) spatial symmetry. Physica D 189, 247–276.Google Scholar
Mason, D. M. & Kerswell, R. R. 2002 Chaotic dynamics in a strained rotating flow: a precessing plane fluid layer. J. Fluid Mech. 471, 71–106.Google Scholar
McEwan, A. D. 1970 Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40, 603–640.Google Scholar
Mercader, I., Batiste, O. & Alonso, A. 2010 An efficient spectral code for incompressible flows in cylindrical geometries. Comput. Fluids 39, 215–224.Google Scholar
Meunier, P., Eloy, C., Lagrange, R. & Nadal, F. 2008 A rotating fluid cylinder subject to weak precession. J. Fluid Mech. 599, 405–440.Google Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103–144.Google Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167–192.CrossRefGoogle Scholar
Wood, W. W. 1965 Properties of inviscid, recirculating flows. J. Fluid Mech. 22, 337–346.Google Scholar
Zhang, K. & Liao, X. 2017 Theory and Modeling of Rotating Fluids: Convection, Inertial Waves and Precession. Cambridge University Press.Google Scholar

Lopez et al. supplementary movie 1

Helcity isosurfaces of LCa (Po=-0.3745).

Download Lopez et al. supplementary movie 1(Video)
Video 3.1 MB

Lopez et al. supplementary movie 2

Helicity isosurfaces of unstable LCs, computed in symmetry subspace (Po=-0.3745)

Download Lopez et al. supplementary movie 2(Video)
Video 3.1 MB

Lopez et al. supplementary movie 3

Helicity isosurfaces of LCs (Po=-0.4).

Download Lopez et al. supplementary movie 3(Video)
Video 3.6 MB

Lopez et al. supplementary movie 4

Helicity isosurface of LCa (Po=-0.4).

Download Lopez et al. supplementary movie 4(Video)
Video 3.8 MB

Lopez et al. supplementary movie 5

Helicity isosurface of SC (Po=-0.3745).

Download Lopez et al. supplementary movie 5(Video)
Video 15.7 MB