Complex dynamics in a short annular container with rotating bottom and timer cylinder

The nonlinear dynamics of the flow in a short annulus driven by the rotation of the inner cylinder and bottom endwall is considered. The shortness of the annulus enhances the role of mode competition. For aspect ratios greater than about 3, the flow dynamics are dominated by a centrifugal instability as the rotating inner cylinder imparts angular momentum to the adjacent fluid, resulting in a three-cell state; the cells are analoguous to Taylor-Couette vortices. For aspect ratios less than about 2.8, the dynamics are dominated by the boundary layer on the bottom rotating endwall that is turned by the stationary outer cylinder to produce an internal shear layer that is azimuthally unstable via Hopf bifurcations. For intermediate aspect ratios, the competition between these instability mechanisms leads to very complicated dynamics, including homoclinic and heteroclinic phenomena. The dynamics are organized by a codimension-two fold-Hopf bifurcation, where modes due to both instability mechanisms bifurcate simultaneously. The dynamics are explored using a three-dimensional Navier-Stokes solver, which is also implemented in a number of invariant subspaces in order to follow some unstable solution branches and obtain a fairly complete bifurcation diagram of the mode competitions.