Instabilities, Symmetry Breaking and Mode Interactions in an Enclosed Swirling Flow
Abstract
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. The basic
state loses stability via a supercritical Hopf bifurcation as the
rotation rate is increased. For a certain range of cylinder aspect
ratio, the Hopf bifurcation does not break the SO(2) symmetry
(axisymmetry), while outside the range it does, leading to rotating
wave solutions. These two bifurcation scenarios are explored, using an
efficient and accurate numerical scheme for the three-dimensional
Navier-Stokes equations in primitive variables in a cylinder.
