Vortex breakdown in a cylinder with a rotating bottom and a flat stress-free surface

Vortex breakdown and transition to time-dependent regimes are investigated in a cylinder (H/R = 4) with a rotating disk and a free-surface. The aim of this study is to show how, by changing upstream conditions it is possible to alter on the flow, particularly the vortex breakdown process. The understanding of such effects on vortex breakdown is very useful in the development of a control strategy in order to intensify or remove the phenomenon. The flow dynamics are explored through numerical solution of the three-dimensional Navier-Stokes equations based on high-order spectral approximations. The use of a flat, stress-free model for the air/water interface is shown to be entirely satisfactory at least for moderate Reynolds numbers. A particular interest of these results is to show how the bubble related to the vortex breakdown becomes attached to the free-surface and grows in diameter as the Reynolds number is increased, Re >= 2900. Such a phenomenon removes the cylindrical vortex core upstream of the breakdown which is usually included in classical theories based on idealized models of vortex flows. The flow is shown to be unstable to three-dimensional perturbations for sufficiently large rotation rates. The bifurcated state takes the form of a k = 3 rotating wave at Re = 3000. The existence of the free-surface promotes the onset of periodicity, with a critical Reynolds number about 15% lower than in the case with a rigid cover. Moreover, the successive bifurcations occur over a much shorter range of Reynolds numbers and lead rapidly to a multi-frequency regime with more than five different frequencies. In the unsteady regime, the vortex breakdown is characterized by an elongated, asymmetric recirculation zone, attached to the free-surface and precessing around the axis of the container. By increasing the rotation, the circular stagnation line on the free-surface takes a more irregular form and starts to move around the axis of the cylinder in the same sense as the rotating disk. Finally, our results show that the vertical boundary layer controls both the vortex breakdown process and the transition to unsteadiness. (c) 2006 Elsevier Inc. All rights reserved.