Spirals and ribbons in counter-rotating Taylor-Couette flow: Frequencies from mean flows and heteroclinic orbits

A number of time-periodic flows have been found to have a property called RZIF: when a linear stability analysis is carried out about the temporal mean (rather than the usual steady state), an eigenvalue is obtained whose real part is zero and whose imaginary part is the nonlinear frequency. For two-dimensional thermosolutal convection, a Hopf bifurcation leads to traveling waves which satisfy the RZIF property and standing waves which do not. We have investigated this property numerically for counter-rotating Couette-Taylor flow, in which a Hopf bifurcation gives rise to branches of upwards and downwards traveling spirals and ribbons which are an equal superposition of the two. In the regime that we have studied, we find that both spirals and ribbons satisfy the RZIF property. As the outer Reynolds number is increased, the ribbon branch is succeeded by two types of heteroclinic orbits, both of which connect saddle states containing two axially stacked pairs of axisymmetric vortices. One heteroclinic orbit is nonaxisymmetric, with excursions that resemble the ribbons, while the other remains axisymmetric.

Figure 1

Nonlinear spirals (a,b) and ribbons (c,d) at $\mathrm{Re}=-550$. The deviation from laminar Couette flow is shown. Spirals of the opposite helicity exist and have identical properties but are not shown. (a,c) $(\theta ,z)$ velocity (arrows) with $u_r$ (colors) at $r=1.26$, near the maximum of the deviation from laminar Couette flow. (b,d) $(r,z)$ velocity (arrows) with $u_{\theta }$ (colors).

Figure 2

(a) Bifurcation diagram for spirals and ribbons. Square root of kinetic energy of the deviation from laminar Couette flow is shown as a function of $\mathrm{Re}$. The ribbons are stable where shown, while the spirals are calculated by using a spiral initial condition. (b) Frequencies of spiral and ribbon states as a function of $\mathrm{Re}$. The dashed line shows the frequencies obtained by linearizing about laminar Couette flow.

Figure 3

Mean flow of spirals (a,b) and ribbons (c,d) at $\mathrm{Re}=-550$. The deviation from laminar Couette flow is shown. (a,c) $(\theta ,z)$ velocity (arrows) with $u_r$ (colors) at $r=1.26$, near the maximum of the deviation from laminar Couette flow. (b,d) $(r,z)$ velocity (arrows) with $u_{\theta }$ (colors). Since the spirals are traveling waves in the $\theta$ and $z$ directions, their means have no dependence on either. The ribbons are traveling waves in $\theta$ and hence their mean is axisymmetric.

Figure 4

Eigenvectors resulting from linearization about mean flow of spirals (a,b) and ribbons (c,d) at $\mathrm{Re}=-550$. (a) $(\theta ,z)$ velocity (arrows) with $u_r$ (colors) at $r=1.26$. (b) $(r,z)$ velocity (arrows) with $u_{\theta }$ (colors). These eigenvectors resemble the deviation of nonlinear spirals and ribbons from the laminar Couette flow as well as the eigenvectors of spiral and ribbon form obtained from linearizing about the laminar flow.

Figure 5

Growth rates (a) and frequencies (b). Results from nonlinear numerical simulations (black curves) and from Navier-Stokes equations linearized about the laminar flow (dashed lines), about the mean flow of spirals (red crosses), and about the mean flow of ribbons (blue stars). Satisfaction of the RZIF property for the frequencies is demonstrated by the agreement between the red crosses and the black curve for spirals and between the blue stars and the black curve for ribbons. Satisfaction of the RZIF property for the growth rates is demonstrated by the fact that the blue stars and the red crosses are near zero.

Figure 6

Same as Fig. 5 with additional eigenvalues. The hollow circles are eigenvalues corresponding to spiral eigenvectors. The dark green circles (upper set) are obtained by linearizing around a mean spiral flow whose orientation is opposite to that of the eigenvector while the blue ones (lower set) are obtained by linearizing around the mean flow of a ribbon.

Figure 7

Spectra of spirals and ribbon states. (a) Full spectrum for states at $\mathrm{Re}=-550$, normalized by amplitude of fundamental frequency. (b) Ratio of second harmonic to fundamental frequency.

Figure 8

Nonaxisymmetric heteroclinic cycle at $\mathrm{Re}=-502$. Colors show the radial velocity in a $(\theta ,z)$ slice at $r=1.26$. The first (top left) and last (bottom right) visualizations show the saddles which anchor the heteroclinic cycles. These contain two pairs of axisymmetric vortices and differ by an axial phase shift of a single vortex. The visualizations between the two illustrate the excursion or rapid transition between the two saddles, via states which resemble ribbons.

Figure 9

Time series for (a,b,c) oscillatory and nonaxisymmetric heteroclinic cycle at $\text{Re}=-502$ and for (d,e,f) nonoscillatory and axisymmetric heteroclinic cycle at $\text{Re}=-490$. (a,d) Complete time series alternating between two plateaus. (b,e) Enlargements of time series. (b) Enlargement of a single period from a nonaxisymmetric cycle, with further enlargement around the excursion. The differences between the growth/decay rates and frequencies during the spiraling approach and departure are clear. (e) Enlargements near the peaks of the axisymmetric nonoscillatory excursions, highlighting their variation. (c,f) Logarithmic plot of energy, which grows and decays exponentially. The nonaxisymmetric energy is shown in black and, for the axisymmetric heteroclinic cycle in (f), the energy corresponding to axisymmetric but axially antisymmetric modes is shown in red. Both types of heteroclinic cycles have the same plateaus, up to numerical roundoff.

Figure 10

Phase portraits for (a) oscillatory heteroclinic cycle at $\mathrm{Re}=-502$ and for (b) nonoscillatory heteroclinic cycle at $\mathrm{Re}=-490$ using $u_r(r=1.2,\theta =0,z=0),u_r(r=1.2,\theta =\pi /3,z={\lambda }_z/3)$, and $u_{\theta }(r=1.2,\theta =2\pi /3,z=2{\lambda }_z/3)$ corresponding to time series in Fig. 9. The trajectory in the oscillatory heteroclinic cycle spirals inwards along one curved surface and outwards along a perpendicular curved surface. The nonoscillatory phase portrait begins with a spiraling transient, then follows different paths into and out of the saddles. The color coding (blue and red) of these paths matches that in the time series, where can be seen subtle differences between the excursions. The saddles for the two cycles are the same; note the different scale for $u_r^{\left(2\right)}$ between (a) and (b).

Figure 11

Axisymmetric heteroclinic cycle at $\mathrm{Re}=-490$. Colors show the radial velocity in a $(\theta ,z)$ slice at $r=1.26$. The first (top left) and last (bottom right) visualizations show the saddles which anchor the heteroclinic cycles. These contain two pairs of axisymmetric vortices and differ by an axial phase shift of a single vortex. For this heteroclinic cycle, the states remain axisymmetric throughout the rapid transition between the two saddles.

Figure 12

Rate of approach to and departure from saddles as a function of $\mathrm{Re}$. These are the slopes of the logarithmic plots of figure 9 and also the real parts of the leading eigenvalues. a) Nonaxisymmetric oscillatory heteroclinic cycle. The cycle exists when the eigenvalues are of opposite sign, here for $\mathrm{Re}<-476$. The cycle is stable within the space of these eigenmodes when the rate of approach exceeds the rate of departure, here for $\mathrm{Re}>-522$. b) Axisymmetric nonoscillatory heteroclinic cycle. The cycle exists when the eigenvalues are of opposite sign, here for $\mathrm{Re}>-502$. The cycle is stable within the space of these eigenmodes when the rate of approach exceeds the rate of departure, here for $\mathrm{Re}<-482$.

Figure 13

(a) Imaginary parts of eigenvalues involved in nonaxisymmetric oscillatory heteroclinic cycle. (b) Real parts of eigenvalues involved in both heteroclinic cycles, axisymmetric and nonaxisymmetric. Nonaxisymmetric cycles are observed for $\mathrm{Re}<-496$, when the largest eigenvalue (departure rate, dark green circles) corresponds to a nonaxisymmetric eigenpair. Axisymmetric cycles are observed for $\mathrm{Re}>-496$, when the largest eigenvalue (blue stars) corresponds to an axisymmetric eigenvector. The other endpoints of the stability ranges, shown as the black rightwards and red leftwards arrows, involve the next-to-largest eigenvalues (rates of approach, red diamonds and black squares); see Fig. 12 and its caption. Summarizing, the stability range of the nonaxisymmetric cycle is $\mathrm{Re}\in [-522,-496]$, while that of the axisymmetric cycle is $\mathrm{Re}\in [-496,-482]$, which matches our observations.