Figure 1

(Color online) Schematic illustrations of the vortex structures discussed in this paper. The colored (gray shaded) regions denote radial outflow, $u>0$, and the white ones indicate radial inflow, $u<0$, respectively, in the plane of an unrolled cylinder surface in the annulus. The inner squares cover the azimuthal period $2\pi$ in horizontal direction and one axial wavelength $\lambda$ in vertical direction. For better visibility the structures are periodically continued slightly beyond these limits. The wSPI and SPI with azimuthal wave number $M=1$ are shown in their left-winding variety.Reuse & Permissions

Figure 2

(Color online) Geometry of the Taylor-Couette system. Only the bottom lid serving as one of the axial end walls is indicated (dark gray).Reuse & Permissions

Figure 3

(Color online) Schematic bifurcation diagram of vortex flow amplitudes versus control parameter, e.g., $R_2$ along the horizontal axis. Stable (unstable) solutions subject to axially periodic boundary conditions are displayed as solid (dashed) lines. TVF, SPI, and RIB (ribbon state) denote nonhysteretic solution branches that bifurcate out of the CCF state. The solutions for wTVF and wSPI are indicated by thick arrows. Thin arrows indicate the transients corresponding to the “jump” bifurcation mentioned in [7].Reuse & Permissions

Figure 4

(Color online) Numerically obtained bifurcation diagrams of TVF, SPI, wTVF, wSPI, and ribbons (RIB) versus $R_1$ for rbc (a,c) and for pbc (b,d). The axial wave numbers $k$ of these structures are indicated in the legend of (a). Solid (dashed) lines with filled (open) symbols refer to stable (unstable) states. Shown are the moduli $\left|u_{m,k}\right|$ of the dominant axial Fourier amplitudes of the azimuthal modes $u_m\left(z,t\right)$ [Eq. (4)] of the radial flow at midgap and the corresponding frequencies $\left|{\omega }_{m,k}\right|$; see text for further explanations. The indices R and L correspond to right- and left-handed spiral modes. The short arrows below the abscissa of (a) denote $R_1$-values of snapshots in Fig. 5. The long arrows in the figures indicate transients to final stable states. The partitions of the $R_1$-interval labeled by unprimed and primed letters correspond to the stability regions for the pbc case [18] in (b,d) and the rbc case in (a,c), respectively. They are listed in table with s denoting stable, u unstable, and—nonexistent.Reuse & Permissions

Figure 5

(Color online) Spatiotemporal changes of the vortex structure in the transition $\text{TVF}\to \text{wTVF}\to \text{wSPI}$ (from right to left). (a) Numerically obtained snapshots of isosurfaces of the azimuthal vorticity ${\partial }_{z}u-{\partial }_{r}w=\pm 40$ [red (dark gray): $+40$, green (light gray): $-40$]. The whole $2\pi$-cylinder is displayed to present the whole structure in one single 3D plot. In the $r-z$–plane shown at the right side of each cylinder for a fixed $\phi$, red (dark gray) implies positive vorticity and green (light gray) negative vorticity. The Reynolds numbers for the snapshots [$R_1=106$ (No. 1), 109 (No. 2), 110 (No. 3), 111 (No. 4), 115 (No. 5)] are marked by arrows below the abscissa of Fig. 4a. All snapshots in (a) represent “stationary” (i.e., fully relaxed, nontransient) structures. (b) Experimental flow visualization after an instantaneous jump from $R_1=115$ down to 109. The space-time plots result from successively recording the luminosity at a fixed $\phi$–position along a narrow axial stripe with a CCD-camera in front of the cylinder. The shown records cover about 13 diffusion times after the jump. All plots in (a) and (b) cover the whole system length $\Gamma =12$.Reuse & Permissions

Figure 6

(Color online) Numerically obtained bifurcation diagrams of TVF and wSPI versus $R_1$ for rbc. See caption of Fig. 4a for an explanation of the details. The short arrows below the abscissa identify the $R_1$-values of the snapshots in Fig. 7a. The long arrow indicates the direction of the transition $\text{wSPI}\to \text{TVF}$ as discussed in Sec. 4B. Region ${\text{P}}_2$ includes the behavior of the amplitudes during the transformation of wSPI into wTVF.Reuse & Permissions

Figure 7

(Color online) Spatiotemporal changes of the vortex structure in the transition $\text{wSPI}\to \text{TVF}$ (from left to right). (a) Numerically obtained snapshots of isosurfaces of the azimuthal vorticity ${\partial }_{z}u-{\partial }_{r}w=\pm 40$ [red (dark gray): $+40$, green (light gray): $-40$]. The whole $2\pi$-cylinders is depicted to present the whole structure in one single 3D plot. In the $r-z$–plane shown at the right side of each cylinder for a fixed $\phi$, red (dark gray) implies positive vorticity and green (light gray) negative vorticity. The Reynolds numbers for the ten snapshots are marked by arrows below the abscissa of Fig. 6. The snapshots five to nine in (a) show transient flow structures with a downwards propagating defect that are described in more detail in the text. The other snapshots in (a) represent “stationary” (i.e., fully relaxed, nontransient) structures. (b) Experimental flow visualization after an instantaneous jump from $R_1=107$ up to 120. The space-time plots result from successively recording the luminosity at a fixed $\phi$ position along a narrow axial stripe with a CCD camera in front of the cylinder. The shown records cover about 11 diffusion times after the jump. All plots in (a) and (b) cover the whole system length $\Gamma =12$.Reuse & Permissions

Figure 8

(Color online) Schematic bifurcation diagram of flow amplitudes. It summarizes results for the finite system shown in Figs. 4a, 6. Stable (unstable) solutions are displayed as solid (dashed) lines. TVF solution branches are shown exemplarily for two possible wave numbers, $k_1$ and $k_2$. In the transitions from TVF to wTVF they are preserved. Eventually wTVF (lines labeled wTVF) undergo complex transients (thin vertical arrows) to wSPI with a uniquely selected wave number, say, $k_0$. The broad dashed arrow refers to the transition from wSPI to TVF that involves among others a propagating defect (see text for more details). Thereby, the TVF wave number closest to $k_0$, here $k_1$, is selected.Reuse & Permissions