STEADY-STATES AND OSCILLATORY INSTABILITY OF SWIRLING FLOW IN A CYLINDER WITH ROTATING TOP AND BOTTOM

In this study we present a numerical investigation of steady states, o nset of oscillatory instability, and slightly supercritical oscillator y states of an axisymmetric swirling flow of a Newtonian incompressibl e fluid in a cylinder, with independently rotating top and bottom. The first part of the study is devoted to the influence of co- and counte r-rotation of the bottom on the steady vortex breakdown, which takes p lace in the well-known problem of flow in a cylinder with a rotating t op. It is shown that weak counter-rotation of the bottom may suppress the vortex breakdown. Stronger counter-rotation may induce a stable st eady vortex breakdown at relatively large Reynolds numbers for which a vortex breakdown does not appear in the case of the stationary bottom . Weak corotation may promote the vortex breakdown at lower Reynolds n umbers than in the cylinder with the stationary bottom. Stronger corot ation leads to the detachment of the recirculation zone from the axis and the formation of an additional vortex ring. The second part of the study is devoted to the investigation of the onset of oscillatory ins tability of steady flows. It is shown that the oscillatory instability sets in due to a Hopf bifurcation. The critical Reynolds number and t he critical frequency of oscillations were calculated as a function of the rotation ratio (xi=Omega(bottom)/Omega(top)) for a fixed value of the aspect ratio gamma (height/radius) of the cylinder gamma=1.5. The stability analysis showed that there are several most unstable linear modes of the perturbation that become successively dominant with a co ntinuous change of xi. It is shown that the oscillatory instability ma y lead to an appearance and coexistence of more than one oscillating s eparation vortex bubble. (C) 1996 American Institute of Physics.