The flow in a cylinder with a rotating endwall has continued to
attract much attention since Vogel (1968) first observed the vortex
breakdown of the central core vortex that forms. Recent experiments
have observed a multiplicity of unsteady states that coexist over a
range of the governing parameters. In spite of numerous numerical and
experimental studies, there continues to be considerable controversy
with fundamental aspects of this flow, particularly with regards to
symmetry breaking. Also, it is not well understood where these
oscillatory states originate from, how they are interrelated, nor how
they are related to the steady, axisymmetric basic state.
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In the aspect ratio (height/radius) range 1.6 < $\Lambda$ < 2.8, the
primary bifurcation is to an axisymmetric time-periodic flow (a limit
cycle). We have developed a suite of numerical techniques, exploiting
the biharmonic formulation of the problem in the axisymmetric case,
that allows us to compute the nonlinear time evolution, the basic
state, and its linear stability in a consistent and efficient
manner. We show that the basic state undergoes a succession of Hopf
bifurcations and the corresponding eigenvalues and eigenvectors of
these excited modes describe most of the characteristics of the
observed time-dependent states.
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The primary bifurcation is non-axisymmetric, to pure rotating wave, in
the ranges $\Lambda$ <1.6 and $\Lambda$ > 2.8. An efficient and
accurate numerical scheme is presented for the three-dimensional
Navier-Stokes equations in primitive variables in a cylinder. Using
these code, primary and secondary bifurcations breaking the SO(2)
symmetry are analyzed.
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We have located a double Hopf bifurcation, where an axisymmetric limit
cycle and a rotating wave bifurcate simultaneously. This codimension-2
bifurcation is very rich, allowing for several different scenarios. By
a comprehensive two-parameter exploration about this point we have
identified precisely to which scenario this case corresponds. The mode
interaction generates an unstable two-torus modulate rotating wave
solution and gives a wedge-shaped region in parameter space where the
two periodic solutions are both stable.
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For aspect ratios around three, experimental observations suggest that
the first mode of instability is a precession of the central vortex
core, whereas recent linear stability analysis suggest a Hopf
bifurcation to a rotating wave at lower rotation rates. This apparent
discrepancy is resolved with the aid of the 3D Navier-Stokes
solver. The primary bifurcation to an m=4 traveling wave, detected by
the linear stability analysis, is located away from the axis, and a
secondary bifurcation to a modulated rotating wave with dominant modes
m=1 and 4, is seen mainly on the axis as a precessing vortex breakdown
bubble. Experiments and the linear stability analysis detected
different aspects of the same flow, that take place in different
spatial locations.