The topology and the dynamics of Vogel–Escudier flow, which is the flow inside a circular cylinder with a top rotating lid, are presented in this paper. A three-dimensional direct numerical simulation of the Navier–Stokes equations in cylindrical coordinates is used to investigate the flow. Various combinations of Reynolds number and aspect ratio are studied and classified based on the flow topology. The flow is found to exhibit steady axisymmetric, unsteady axisymmetric, rotating azimuthal waves, and weak turbulence regimes. The perturbations found in the system are characteristically different for various flow regimes and are used for the classification of flow. The presence of several modes at high Reynolds number suggests a weak turbulence state, and a Taylor–Görtler type instability wave is found in the sidewall boundary layer.

In this paper, we investigate mixing in flows dominated by bubble-type vortex breakdown. Three-dimensional Navier–Stokes equations in cylindrical polar coordinates are solved to simulate flow in a cylindrical container. We have found that in steady regime of the flow, the vortex breakdown bubble is axisymmetric and apparent non-axisymmetric features observed in experiments are artifacts of imperfections in experimental set-ups. We also find that the heteroclinic manifold joining hyperbolic points of the vortex breakdown bubble is stable in the absence of any perturbation and no chaotic advection was found within vortex breakdown bubble. This makes the vortex breakdown bubble impermeable to outer fluid and hence, the vortex breakdown bubble inhibits mixing. We conclude that symmetry is a barrier to mixing.

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Taylor–Couette flow is a canonical flow to study Taylor–Görtler (TG) instability or centrifugal instability and the associated vortices. TG instability has been traditionally associated with flow over curved surfaces or geometries. In the computational study, we confirm the presence of TG-like near-wall vortical structures in two lid-driven flow systems, the Vogel–Escudier (VE) and the lid-driven cavity (LDC) flows. The VE flow is generated inside a circular cylinder by a rotating lid (top lid in the present study), while the LDC flow is generated inside a square or rectangular cavity by the linear movement of the lid. We look at the emergence of these vortical structures through reconstructed phase space diagrams and find that the TG-like vortices are seen in the chaotic regimes in both flows. In the VE flow, these vortices are seen when the side-wall boundary layer instability sets in at large R e . The VE flow is observed to go to a chaotic state in a sequence of events from a steady state at low R e . In contrast to VE flows, in the LDC flow with no curved boundaries, TG-like vortices are seen at the emergence of unsteadiness when the flow exhibits a limit cycle. The LDC flow is observed to have transitioned to chaos from the steady state through a periodic oscillatory state. Various aspect ratio cavities are examined in both flows for the presence of TG-like vortices. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical transactions paper (Part 2)’.