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On triadic resonance as an instability mechanism in precessing cylinder flow

Published online by Cambridge University Press:  27 February 2018

Thomas Albrecht Open the ORCID record for Thomas Albrecht [Opens in a new window] ,
Hugh M. Blackburn ,
Juan M. Lopez Open the ORCID record for Juan M. Lopez [Opens in a new window] ,
Richard Manasseh  and
Patrice Meunier Open the ORCID record for Patrice Meunier [Opens in a new window]
Thomas Albrecht*
Affiliation:
Department of Mechanical and Aerospace Engineering, Monash University, VIC 3800, Australia
Hugh M. Blackburn
Affiliation:
Department of Mechanical and Aerospace Engineering, Monash University, VIC 3800, Australia
Juan M. Lopez
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
Richard Manasseh
Affiliation:
Department of Mechanical and Product Design Engineering, Swinburne University of Technology, VIC 3122, Australia
Patrice Meunier
Affiliation:
IRPHE, CNRS, Aix–Marseille Université, Centrale Marseille, 49 Rue Joliot-Curie, 13013 Marseille, France
*
†Email address for correspondence: thomas.albrecht@monash.edu
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Abstract

Contained rotating flows subject to precessional forcing are well known to exhibit rapid and energetic transitions to disorder. Triadic resonance of inertial modes has been previously proposed as an instability mechanism in such flows, and that idea was developed into a successful model for predicting instability in a cylindrical container when departures from solid-body rotation are sufficiently small. Using direct numerical simulation and dynamic mode decomposition, we analyse instabilities of precessing cylinder flows whose three-dimensional basic states, steady in the gimbal frame of reference, may depart substantially from solid-body rotation. In the gimbal frame, the instability can be interpreted as resulting from a supercritical Hopf bifurcation that results in a limit-cycle flow. In the cylinder frame of reference, the basic state is a rotating wave with azimuthal wavenumber $m=1$, and the instability satisfies triadic-resonance conditions with the instability mode maintaining a fixed orientation with respect to the basic state. Thus, we are able to demonstrate the existence of two alternative but congruent explanations for the instability. Additionally, we show that basic states may depart substantially from solid-body rotation even with modest cylinder tilt angles, and growth rates for instabilities may be sufficiently large that nonlinear saturation to disordered states can occur within approximately ten cylinder revolutions, in agreement with experimental observations.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Transition to turbulence Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 841 , 25 April 2018 , R3
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Copyright
© 2018 Cambridge University Press 

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Albrecht et al. supplementary movie 1

Kalliroscopic flow visualizations of a `Type A collapse' at Re = 21120 from the experiments of Manasseh (1992). The movie commences after establishment of solid-body rotation in the cylinder, followed by tilting through α = 3°. At early times, the initial bright area that we associate with a vortex core is thought to arise from locally large spatial gradients of flake orientation and finite light sheet thickness.

Download Albrecht et al. supplementary movie 1(Video)
Video 27.2 MB

Albrecht et al. supplementary movie 2

100 snapshots of ± isosurfaces of axial vorticity as seen from the gimbal frame of reference, normalised such that Em=2 = 1 to remove exponential growth, and extracted from the interval 31.890 ≤ t/T1 ≤ 34.733 (cf. figure 3). The wave envelope remains steady with respect to the gimbal frame (and the basic state), but there is a dominant frequency associated with progress of the wave; this is the phase frequency of the wave in the gimbal frame (cf. figure 5,i>a).

Download Albrecht et al. supplementary movie 2(Video)
Video 1.6 MB

Albrecht et al. supplementary movie 3

As for movie 2, but visualised in the cylinder frame of reference. The whole wave as well as the basic state revolve at circular frequency Ω1 with respect to the observer.

Download Albrecht et al. supplementary movie 3(Video)
Video 3.9 MB