Wave-number dependence of the transitions between traveling and standing vortex waves and their mixed states in the Taylor-Couette system

End wall effects on the transitions between Taylor vortices and spiral vortices

We present numerical simulations as well as experimental results concerning transitions between Taylor vortices and spiral vortices in the Taylor-Couette system with rigid, nonrotating lids at the cylinder ends. These transitions are performed by wavy structures appearing via a secondary bifurcation out of Taylor vortices and spirals, respectively. In the presence of these axial end walls, pure spiral solutions do not occur as for axially periodic boundary conditions but are substituted by primary bifurcating, stable wavy spiral structures. Similarly to the periodic system, we found a transition from Taylor vortices to wavy spirals mediated by so-called wavy Taylor vortices and, on the other hand, a transition from wavy spirals to Taylor vortices triggered by a propagating defect. We furthermore observed and investigated the primary bifurcation of wavy spirals out of the basic circular Couette flow with Ekman vortices at the cylinder ends.

Bifurcation of standing waves into a pair of oppositely traveling waves with oscillating amplitudes caused by a three-mode interaction

A flow state consisting of two oppositely traveling waves (TWs) with oscillating amplitudes has been found in the counter-rotating Taylor-Couette system by full numerical simulations. This structure bifurcates out of axially standing waves that are nonlinear superpositions of left- and right-handed spiral vortex waves with equal time-independent amplitudes. Beyond a critical driving, the two spiral TW modes start to oscillate in counterphase due to a Hopf bifurcation. The trigger for this bifurcation is provided by a nonlinearly excited mode of different symmetry than the spiral TWs. A three-mode coupled amplitude equation model is presented that captures this bifurcation scenario. The mode-coupling between two symmetry degenerate critical modes and a nonlinearly excited one that is contained in the model can be expected to occur in other structure-forming systems as well.