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Triadic resonances in precessing rapidly rotating cylinder flows

Published online by Cambridge University Press:  30 July 2015

T. Albrecht ,
H. M. Blackburn ,
J. M. Lopez ,
R. Manasseh  and
P. Meunier
T. Albrecht*
Affiliation:
Department of Mechanical and Aerospace Engineering, Monash University, VIC 3800, Australia
H. M. Blackburn
Affiliation:
Department of Mechanical and Aerospace Engineering, Monash University, VIC 3800, Australia
J. M. Lopez
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
R. Manasseh
Affiliation:
Department of Mechanical and Product Design Engineering, Swinburne University of Technology, VIC 3122, Australia
P. Meunier
Affiliation:
IRPHE, CNRS, Aix–Marseille Université, 49 Rue Joliot-Curie, 13013 Marseille, France
*
†Email address for correspondence: [email protected]
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Abstract

Direct numerical simulations of flows in cylinders subjected to both rapid rotation and axial precession are presented and analysed in the context of a stability theory based on the triadic resonance of Kelvin modes. For a case that was chosen to provide a finely tuned resonant instability with a small nutation angle, the simulations are in good agreement with the theory and previous experiments in terms of mode shapes and dynamics, including long-time-scale regularization of the flow and recurrent collapses. Cases not tuned to the most unstable triad, but with the nutation angle still small, are also in quite good agreement with theoretical predictions, showing that the presence of viscosity makes the physics of the triadic-resonance model robust to detuning. Finally, for a case with $45^{\circ }$ nutation angle for which it has been suggested that resonance does not occur, the simulations show that a slowly growing triadic resonance predicted by theory is in fact observed if sufficient evolution time is allowed.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Nonlinear instability Geophysical and Geological Flows: Waves in rotating fluids
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 778 , 10 September 2015 , R1
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Copyright
© 2015 Cambridge University Press 

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References

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Albrecht et al. supplementary movie

For the flow in figure 2 (b), Γ=1.62, α=1°, Po=-0.153, Re=7670, this movie shows positive (red) and negative (yellow) isosurfaces of helicity, visualised in the gimbal frame of reference, but with helicity computed after the background rotation was removed. As indicated by the vertical lines in figure 2 (b), the movie shows one cycle of the terminal low-frequency oscillation, over ≈40 cylinder rotations.

Download Albrecht et al. supplementary movie(Video)
Video 2 MB

Albrecht et al. supplementary movie

For the flow in figure 2 (b), Γ=1.62, α=1°, Po=-0.153, Re=7670, this movie shows positive (red) and negative (yellow) isosurfaces of helicity, visualised in the gimbal frame of reference, but with helicity computed after the background rotation was removed. As indicated by the vertical lines in figure 2 (b), the movie shows one cycle of the terminal low-frequency oscillation, over ≈40 cylinder rotations.

Download Albrecht et al. supplementary movie(Video)
Video 1.9 MB

Albrecht et al. supplementary movie

This shows axial vorticity for the flow represented in figure 2 (a): Γ=1.62, α=0.8°, Po=-0.153, Re=5310 after the final saturated state is reached. The flow was computed and visualised in the cylinder frame of reference. We filtered the velocity field to azimuthal wave number m=1 (a), m=5 (b), and m=6 (c). Then we extracted contours of axial vorticity at their maximum planes, i.e., one-quarter of the cylinder height for m=1 and m=5, and one-half form=6. As predicted for a triadic resonance, the forced mode rotates at -Ω1, while the parasitic modes m=5 and m=6 counter-rotate.

Download Albrecht et al. supplementary movie(Video)
Video 941.8 KB

Albrecht et al. supplementary movie

This shows axial vorticity for the flow represented in figure 2 (a): Γ=1.62, α=0.8°, Po=-0.153, Re=5310 after the final saturated state is reached. The flow was computed and visualised in the cylinder frame of reference. We filtered the velocity field to azimuthal wave number m=1 (a), m=5 (b), and m=6 (c). Then we extracted contours of axial vorticity at their maximum planes, i.e., one-quarter of the cylinder height for m=1 and m=5, and one-half form=6. As predicted for a triadic resonance, the forced mode rotates at -Ω1, while the parasitic modes m=5 and m=6 counter-rotate.

Download Albrecht et al. supplementary movie(Video)
Video 1.6 MB

Albrecht et al. supplementary movie

The same flow and equivalent data processing as for movie 2, but computed and visualised in the gimbal frame of reference, shows a steady forced mode and co-rotating parasitic modes.

Download Albrecht et al. supplementary movie(Video)
Video 1.3 MB

Albrecht et al. supplementary movie

The same flow and equivalent data processing as for movie 2, but computed and visualised in the gimbal frame of reference, shows a steady forced mode and co-rotating parasitic modes.

Download Albrecht et al. supplementary movie(Video)
Video 2.2 MB