Dynamics of axially localized states in Taylor-Couette flows

We present numerical simulations of the flow confined in a wide gap Taylor-Couette system, with a rotating inner cylinder and variable length-to-gap aspect ratio. A complex experimental bifurcation scenario differing from the classical Ruelle-Takens route to chaos has been experimentally reported in this geometry. The wavy vortex flow becomes quasiperiodic due to an axisymmetric very low frequency mode. This mode plays a key role in the dynamics of the system, leading to the occurrence of chaos via a period-doubling scenario. Further increasing the rotation of the inner cylinder results in the appearance of a new flow pattern which is characterized by large amplitude oscillations localized in some of the vortex pairs. The purpose of this paper is to study numerically the dynamics of these axially localized states, paying special attention to the transition to chaos. Frequency analysis from time series simultaneously recorded at several points has been applied in order to identify the flow transitions taking place. It has been found that the very low frequency mode is essential to explain the behavior associated with the different transitions towards chaos including localized states.

Figure 1

Convergence of the spectral coefficients (5) of the axial velocity using the infinity norm. The flow corresponds to a weakly chaotic state found at $\mathrm{Re}=1300$ and $\mathrm{\Gamma }=9$, computed with $L=36$ Chebyshev radial points, $M=16$ Fourier modes, and $N=448$ Chebyshev axial points. (a) Convergence in $r$ and $\theta$ and (b) in $z$.

Figure 2

Axisymmetric five-jet (10 vortices) TVF state at $\mathrm{Re}=400$ and $\mathrm{\Gamma }=9$ on a meridional plane $(r,z)\in [1,2]\times [-\mathrm{\Gamma }/2,\mathrm{\Gamma }/2]$. (a) Streamlines. There are 10 positive (red/dark gray) and negative (yellow/light gray) $\psi$ contours in $[-28.45,28.45]$. (b) Contours of angular momentum $rv$. There are 20 contours in $[0,400]$.

Figure 3

Small jet mode at $\mathrm{Re}=700$ and $\mathrm{\Gamma }=9$. Contours of angular momentum $rv$, (a) on a cylindrical surface $(\theta ,z)\in [0,2\pi ]\times [-\mathrm{\Gamma }/2,\mathrm{\Gamma }/2]$ at $r=1.5$ and (b) on a meridional plane $(r,z)\in [1,2]\times [-\mathrm{\Gamma }/2,\mathrm{\Gamma }/2]$ at $\theta =\pi$. (c) Overlapped axial velocity profiles $w$ over the line $r=1.8,\theta =\pi ,z\in [-\mathrm{\Gamma }/2,\mathrm{\Gamma }/2]$, corresponding to 20 different time steps.

Figure 4

Large jet mode at $\mathrm{Re}=720$ and $\mathrm{\Gamma }=9.25$. Contours of angular momentum $rv$, (a) on a cylindrical surface $(\theta ,z)\in [0,2\pi ]\times [-\mathrm{\Gamma }/2,\mathrm{\Gamma }/2]$ at $r=1.5$ and (b) on a meridional plane $(r,z)\in [1,2]\times [-\mathrm{\Gamma }/2,\mathrm{\Gamma }/2]$ at $\theta =\pi$. (c) Overlapped axial velocity profiles $w$ over the line $r=1.8,\theta =\pi ,z\in [-\mathrm{\Gamma }/2,\mathrm{\Gamma }/2]$, corresponding to 40 different time steps.

Figure 5

VLF state at $\mathrm{Re}=780$ and $\mathrm{\Gamma }=9$. (a) Contours of angular momentum on a meridional plane $(r,z)\in [1,2]\times [-\mathrm{\Gamma }/2,\mathrm{\Gamma }/2]$ and $\theta =\pi$. (b) The inverse symmetric of (a) $(\theta \to \theta +\pi$ and $z\to -z)$. There are 20 contours in $[0,780]$.

Figure 6

Schematic of the period-doubling route to chaos exhibited by the VLF as $\mathrm{Re}$ is increased, for $\mathrm{\Gamma }=9$. The solutions at the $\mathrm{Re}$ indicated are described in Fig. 7.

Figure 7

Power spectral density PSD, phase portrait, and Poincare section illustrating the sequence of period-doubling bifurcations sketched in Fig. 6 at $\mathrm{\Gamma }=9$. (a) VLF at $\mathrm{Re}=770$, near the bifurcation; (b) first period doubling $P2$ of VLF at $\mathrm{Re}=782$; (c) first chaotic solution at $\mathrm{Re}=790$; (d) state with period $P2$ but different structure at $\mathrm{Re}=810$; (e) second period doubling $P4$ of VLF at $\mathrm{Re}=820$; (f) chaotic flow at $\mathrm{Re}=840$.

Figure 8

Three-dimensional contour of the angular momentum $\left(rv=-450\right)$ and axial velocity profiles $w$ recorded at $r=1.8,\theta =\pi$, and $z\in [-\mathrm{\Gamma }/2,\mathrm{\Gamma }/2]$ for 40 different time steps and overlapped on the same plot. (a) ${\mathrm{ALS}}_{10}$ at $\mathrm{Re}=920$ and $\mathrm{\Gamma }=9$, (b) ${\mathrm{ALS}}_4$ at $\mathrm{Re}=960$ and $\mathrm{\Gamma }=9$, (c) ${\mathrm{ALS}}_8$ at $\mathrm{Re}=960$ and $\mathrm{\Gamma }=9.1$, and (d) ${\mathrm{ALS}}_2$ at $\mathrm{Re}=960$ and $\mathrm{\Gamma }=9.3$. The nomenclature ${\mathrm{ALS}}_{\text{number}}$ is explained in the text.

Figure 9

(a) Experimental transition curves between TVF, SJ, LJ, and ALS as indicated, for $N=8$ (thick lines, blue online) and $N=12$ (thin lines, black online); data from [12, Figs. 5 and 7]. Straight lines with arrow (red online) correspond to the transitions shown in Figs. 6 and 11. (b) Distribution in the parameter space of the numerically computed flow patterns for $\mathrm{Re}>850$. The solid lines represent the experimental stability boundaries for the case of $N=8$ shown in (a).

Figure 10

Contours of angular momentum $rv$ on a cylindrical surface $(\theta ,z)\in [0,2\pi ]\times [-\mathrm{\Gamma }/2,\mathrm{\Gamma }/2]$ at $r=1.5$ for (a) ${\mathrm{ALS}}_{10}$ at $\mathrm{Re}=920$ and $\mathrm{\Gamma }=9.25$; and (b) ${\mathrm{ALS}}_4$ at $\mathrm{Re}=1260$ and $\mathrm{\Gamma }=9.25$; (c) and (d) are, respectively, the power spectral density $\left(\mathrm{PSD}\right)$ and phase portrait for ${\mathrm{ALS}}_{10}$ at $\mathrm{Re}=920$ and $\mathrm{\Gamma }=9.25$.

Figure 11

Schematic of the transitions to chaos exhibited by the ${\mathrm{ALS}}_{10}$ (a) as $\mathrm{\Gamma }$ is decreased, at $\mathrm{Re}=920$, and (b) as $\mathrm{Re}$ is increased, at $\mathrm{\Gamma }=9.25$. The solutions (a) at the $\mathrm{\Gamma }$ indicated are described in Fig. 12, and those in (b) at the $\mathrm{Re}$ indicated are described in Fig. 13. The initial state, indicated as a solid disk, is the ${\mathrm{ALS}}_{10}$ state at $\mathrm{\Gamma }=9.25$ and $\mathrm{Re}=920$.

Figure 12

Power spectral density, phase portrait, and Poincaré section illustrating the sequence of states in the route to chaos when $\mathrm{Re}=920$ and $\mathrm{\Gamma }$ is decreased. The initial state is the ${\mathrm{ALS}}_{10}$ at $\mathrm{Re}=920$. (a) State resulting from the first transition at $\mathrm{\Gamma }=8.83$; (b) state resulting from the second transition at $\mathrm{\Gamma }=8.80$; (c) weakly chaotic flow at $\mathrm{\Gamma }=8.78$.

Figure 13

Power spectral density, phase portrait, and Poincaré section illustrating the route to chaos when $\mathrm{Re}$ is increased and $\mathrm{\Gamma }=9.25$ is fixed. The initial state is the ${\mathrm{ALS}}_{10}$ at $\mathrm{Re}=920$. (a) State resulting from the first transition at $\mathrm{Re}=952$; (b) intermediate state before the transition to the ${\mathrm{ALS}}_4$ at $\mathrm{Re}=980$; (c) ${\mathrm{ALS}}_4$ with a low frequency mode at $\mathrm{Re}=1030$; (d) ${\mathrm{ALS}}_4$ as rotating wave at $\mathrm{Re}=1260$. (e) Emergence of the low frequency mode at $\mathrm{Re}=1280$. (f) Chaotic flow at $\mathrm{Re}=1300$.