Ch. Hoffmann

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.

Transitions between Taylor vortices and spirals via wavy Taylor vortices and wavy spirals

We present numerical simulations of closed wavy Taylor vortices and of helicoidal wavy spirals in the Taylor-Couette system. These wavy structures appearing via a secondary bifurcation out of Taylor vortexflow and out of spiral vortex flow, respectively, mediate transitions between Taylor and spiral vortices and vice versa. Structure, dynamics, stability and bifurcation behaviour are investigated in quantitative detail as a function of Reynolds numbers and wave numbers for counter-rotating as well as corotating cylinders. These results are obtained by solving the Navier-Stokes equations subject to axial periodicity for a radius ratio = 0.5 with a combination of a finite differences method and a Galerkin method.

Spiral vortices traveling between two rotating defects in the Taylor-Couette system

Numerical calculations of vortex flows in Taylor-Couette systems with counter rotating cylinders are presented. The full, time-dependent Navier-Stokes equations are solved with a combination of a finite difference and a Galerkin method. Annular gaps of radius ratio eta=0.5 and of several heights are simulated. They are closed by nonrotating lids that produce localized Ekman vortices in their vicinity and that prevent axial phase propagation of spiral vortices. The existence and spatiotemporal properties of rotating defects, modulated Ekman vortices, and the spiral vortex structures in the bulk are elucidated in quantitative detail.

Spiral vortices and Taylor vortices in the annulus between rotating cylinders and the effect of an axial flow

We present numerical simulations of vortices that appear via primary bifurcations out of the unstructured circular Couette flow in the Taylor-Couette system with counter rotating as well as with corotating cylinders. The full, time dependent Navier Stokes equations are solved with a combination of a finite difference and a Galerkin method for a fixed axial periodicity length of the vortex patterns and for a finite system of aspect ratio 12 with rigid nonrotating ends in a setup with radius ratio eta=0.5. Differences in structure, dynamics, symmetry properties, bifurcation, and stability behavior between spiral vortices with azimuthal wave numbers M=+/-1 and M=0 Taylor vortices are elucidated and compared in quantitative detail. Simulations in axially periodic systems and in finite systems with stationary rigid ends are compared with experimental spiral data. In a second part of the paper we determine how the above listed properties of the M=-1, 0, and 1 vortex structures are changed by an externally imposed axial through flow with Reynolds numbers in the range -40< or =Re< or =40. Among other things we investigate when left handed or right handed spirals or toroidally closed vortices are preferred.

Spiral and Taylor vortex fronts and pulses in axial through flow.

The influence of an axial through flow on the spatiotemporal growth behavior of different vortex structures in the Taylor-Couette system with radius ratio eta=0.5 is determined. The Navier-Stokes equations (NSE) linearized around the basic Couette-Poiseuille flow are solved numerically with a shooting method in a wide range of through flow strengths Re and different rates of co-rotating and counter-rotating cylinders for toroidally closed vortices with azimuthal wave number m=0 and for spiral vortex flow with m=+/-1. For each of these three different vortex varieties we have investigated (i) axially extended vortex structures, (ii) axially localized vortex pulses, and (iii) vortex fronts. The complex dispersion relations of the linearized NSE for vortex modes with the three different m are evaluated for real axial wave numbers for (i) and over the plane of complex axial wave numbers for (ii) and (iii). We have also determined the Ginzburg-Landau amplitude equation (GLE) approximation in order to analyze its predictions for the vortex structures (ii) and (iii). Critical bifurcation thresholds for extended vortex structures are evaluated. The boundaries between absolute and convective instability of the basic state for vortex pulses are determined with a saddle-point analysis of the dispersion relations. Fit parameters for power-law expansions of the boundaries up to Re4 are listed in two tables. Finally, the linearly selected front behavior of growing vortex structures is investigated using saddle-point analyses of the dispersion relations of NSE and GLE. For the two front intensity profiles (increasing in positive or negative axial direction) we have determined front velocities, axial growth rates, and the wave numbers and frequencies of the unfolding vortex patterns with azimuthal wave numbers m=0,+/-1, respectively.

Secondary bifurcation of mixed-cross-spirals connecting travelling wave solutions

We present numerical results of secondarily forward bifurcating, stationary flow states that mediate transitions between travelling helical waves (spirals). These so-called mixed-cross-spirals (MCSs) can be seen as nonlinear superpositions of spiral solutions with different helicity and pitch. Thereby, the contribution of the respective spiral to the entire MCS varies continuously with the control parameters. Furthermore, MCSs connect the bifurcation branches of the involved spirals, even when both spiral pitches differ. This makes them interesting for pattern-forming systems in general. The bifurcation scenarios of MCSs differ from the well-studied cross-spiral-mediated transitions between mirror-symmetric left- and right-winding spirals, even in the case of MCS solutions that start and end in the same spiral branch. The structure, spatiotemporal dynamics, bifurcation behaviour and stability of MCSs are elucidated in detail for the axially periodic Taylor-Couette system as a prototypical reference for pattern formation. The results are obtained by solving the full Navier-Stokes equations with a combination code of a finite differences and a Galerkin method.

Direction reversal of a rotating wave in Taylor-Couette flow

In Taylor-Couette systems, waves, e.g. spirals and wavy vortex flow, typically rotate in the same direction as the azimuthal mean flow of the basic flow which is mainly determined by the rotation of the inner cylinder. In a combined experimental and numerical study we analysed a rotating wave of a one-vortex state in small-aspect-ratio Taylor-Couette flow which propagates either progradely or retrogradely in the inertial (laboratory) frame, i.e. in the same or opposite direction as the inner cylinder. The direction reversal from prograde to retrograde can occur at a distinct parameter value where the propagation speed vanishes. Owing to small imperfections of the rotational invariance, the curves of vanishing rotation speed can broaden to ribbons caused by coupling between the end plates and the rotating wave. The bifurcation event underlying the direction reversal is of higher codimension and is unfolded experimentally by three control parameters, i.e. the Reynolds number, the aspect ratio, and the rotation rate of the end plates.

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

Previous numerical investigations of the stability and bifurcation properties of different nonlinear combination structures of spiral vortices in a counter-rotating Taylor-Couette system that were done for fixed axial wavelengths are supplemented by exploring the dependence of the vortex phenomena waves on their wavelength. This yields information about the experimental and numerical accessibility of the various bifurcation scenarios. Also backward bifurcating standing waves with oscillating amplitudes of the constituent traveling waves are found.

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.

Controlling the stability transfer between oppositely traveling waves and standing waves by inversion-symmetry-breaking perturbations

The effect of an externally applied flow on symmetry degenerated waves propagating into opposite directions and standing waves that exchange stability with the traveling waves via mixed states is analyzed. Wave structures that consist of spiral vortices in the counter rotating Taylor-Couette system are investigated by full numerical simulations and explained quantitatively by amplitude equations containing quintic coupling terms. The latter are appropriate to describe the influence of inversion-symmetry-breaking perturbations on many oscillatory instabilities with O(2) symmetry.