Sequential transitions of bathtub vortex flow

The bathtub vortex has been found to autonomously arise owing to instability of a symmetric flow in a rectangular vessel when water is drained. We consider a model flow through a vessel with a rectangular horizontal cross section and a drain hole at the center of the bottom to investigate the physical mechanism for generation of swirling fluid motion like the bathtub vortex and the sequential transitions of the flow by numerical simulations and the linear stability analyses. The water surface is assumed to be flat even after instability. If the flow becomes unstable under this assumption, it assures that the surface deformation is irrelevant to the instability. It is emphasized that our interest is not limited to the real bathtub vortex but directed to occurrence of a large vortex in a flow having two reflectional symmetries. The configuration of the vessel has the double plane symmetry (DPS), which allows the flow have the same DPS at small Reynolds numbers. It is found that the instabilities and hence transitions occur accompanying symmetry-breaking of the flow field. Namely, the DPS flow experiences instability to yield vortical motion above a critical Reynolds number, losing the DPS but retaining the p-rotational (twofold rotational) symmetry around the center axis. The vortical flow also becomes unstable at a higher Reynolds number, makes a transition, and loses the p-rotational symmetry, but still keeps the time-translation symmetry, i.e., steadiness. The steadiness is broken at an even higher Reynolds number, owing to instability caused by an oscillatory mode of disturbance. The first and second transitions of the flow are identified as pitchfork bifurcations, and the third transition is identified as a Hopf bifurcation.