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Nonlinear mode interactions in a counter-rotating split-cylinder flow

Published online by Cambridge University Press:  10 March 2017

Paloma Gutierrez-Castillo Open the ORCID record for Paloma Gutierrez-Castillo [Opens in a new window]  and
Juan M. Lopez Open the ORCID record for Juan M. Lopez [Opens in a new window]
Paloma Gutierrez-Castillo
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
Juan M. Lopez*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
*
†Email address for correspondence: juan.m.lopez@asu.edu
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Abstract

The flow in a split cylinder with each half in exact counter rotation is studied numerically. The exact counter rotation, quantified by a Reynolds number $\mathit{Re}$ based on the rotation rate and radius, imparts the system with an $O(2)$ symmetry (invariance to azimuthal rotations as well as to an involution consisting of a reflection about the mid-plane composed with a reflection about any meridional plane). The $O(2)$ symmetric basic state is dominated by a shear layer at the mid-plane separating the two counter-rotating bodies of fluid, created by the opposite-signed vortex lines emanating from the two endwalls being bent to meet at the split in the sidewall. With the exact counter rotation, the additional involution symmetry allows for steady non-axisymmetric states, that exist as a group orbit. Different members of the group simply correspond to different azimuthal orientations of the same flow structure. Steady states with azimuthal wavenumber $m$ (the value of $m$ depending on the cylinder aspect ratio $\unicode[STIX]{x1D6E4}$) are the primary modes of instability as $\mathit{Re}$ and $\unicode[STIX]{x1D6E4}$ are varied. Mode competition between different steady states ensues, and further bifurcations lead to a variety of different time-dependent states, including rotating waves, direction-reversing waves, as well as a number of slow–fast pulse waves with a variety of spatio-temporal symmetries. Further from the primary instabilities, the competition between the vortex lines from each half-cylinder settles on either a $m=2$ steady state or a limit cycle state with a half-period-flip spatio-temporal symmetry. By computing in symmetric subspaces as well as in the full space, we are able to unravel many details of the dynamics involved.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Instability: Nonlinear instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 816 , 10 April 2017 , pp. 719 - 745
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Copyright
© 2017 Cambridge University Press 

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Gutierrez-Castillo supplementary movie

Animation of RW state.

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Video 4.4 MB

Gutierrez-Castillo supplementary movie

Animation of DRW state.

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Video 446.2 KB

Gutierrez-Castillo supplementary movie

Animation of PWs state.

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Video 3.9 MB

Gutierrez-Castillo supplementary movie

Animation of PWa at Re=160

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Video 5.1 MB

Gutierrez-Castillo supplementary movie

Animation of PWa at Re=175.

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Video 2.1 MB

Gutierrez-Castillo supplementary movie

Animation of PWo state.

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Video 1.2 MB

Gutierrez-Castillo supplementary movie

Animation of LC state.

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Video 2.5 MB