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Symmetry breaking and instability mechanisms in medium depth torsionally driven open cylinder flows

Published online by Cambridge University Press:  14 February 2011

STUART J. COGAN ,
KRIS RYAN  and
GREGORY J. SHEARD
STUART J. COGAN*
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne 3800, Australia
KRIS RYAN
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne 3800, Australia
GREGORY J. SHEARD
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne 3800, Australia
*
†Email address for correspondence: stuart.cogan@eng.monash.edu.au
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Abstract

A numerical investigation was conducted into the different flow states, and bifurcations leading to changes of state, found in open cylinders of medium to moderate depth driven by a constant rotation of the vessel base. A combination of linear stability analysis, for cylinders of numerous height-to-radius aspect ratios (H/R), and nonlinear stability analysis and three-dimensional simulations for a cylinder of aspect ratio 1.5, has been employed. Attention is focused on the breaking of SO(2) symmetry. A comprehensive map of transition Reynolds numbers as a function of aspect ratio is presented by combining a detailed stability analysis with the limited existing data from the literature. For all aspect ratios considered, the primary instabilities are identified as symmetry-breaking Hopf bifurcations, occurring at Reynolds numbers well below those of the previously reported axisymmetric Hopf transitions. It is revealed that instability modes with azimuthal wavenumbers m = 1, 3 and 4 are the most unstable in the range 1.0 < H/R < 4, and that numerous double Hopf bifurcation points exist. Critical Reynolds numbers generally increase with cylinder aspect ratio, though a decrease in stability occurs between aspect ratios 1.5 and 2.0, where a local minimum in critical Reynolds number occurs. For H/R = 1.5, a detailed characterisation of instability modes is given. It is hypothesized that the primary instability leading to transition from steady axisymmetric flow to unsteady three-dimensional flow is related to deformation of shear layers that are present in the flow, in particular at the interfacial region between the vortex breakdown bubble and the primary recirculation.

JFM classification

Instability: Instability Vortex Flows: Vortex breakdown
Type
Papers
Information
Journal of Fluid Mechanics , Volume 672 , 10 April 2011 , pp. 521 - 544
Copyright
Copyright © Cambridge University Press 2011

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