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

TL;DR

This paper investigates the RZIF property in counter-rotating Taylor-Couette flow, showing that spirals and ribbons satisfy this property and exploring the dynamics of heteroclinic orbits connecting vortex states.

Contribution

It demonstrates that both spirals and ribbons in counter-rotating Taylor-Couette flow satisfy the RZIF property and analyzes the transition to heteroclinic orbits at higher Reynolds numbers.

Findings

Spirals and ribbons satisfy the RZIF property.

Heteroclinic orbits connect saddle vortex states.

Transition involves non-axisymmetric and axisymmetric orbits.

Abstract

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 non-axisymmetric, with excursions that resemble the ribbons, while the other remains axisymmetric.

Equations

$${\partial }_t\mathbf{U}$$

$$\nabla \cdot \mathbf{U}$$

$$\mathbf{U}$$

$${\partial }_t\mathbf{u}$$

$$\nabla \cdot \mathbf{u}$$

$$\mathbf{u}$$

$$u^{\mathrm{SPI}\pm }(r,\theta ,z,t)$$

$$u^{\mathrm{RIB}}(r,\theta ,z,t)$$

$${\partial }_t\mathbf{U}$$

$$\mathbf{U}=\overline{\mathbf{U}}+\sum _{n\ne 0}{\mathbf{u}}_n{e}^{in\omega t}$$

$$i\omega {\mathbf{u}}_1$$

Full text

I Introduction

A number of time-periodic flows have been found to have the following property: when a linear stability analysis is carried out about the temporal mean (rather than the usual steady state from which they bifurcate), an eigenvalue emerges whose imaginary part reproduces the frequency of the nonlinear oscillation and whose real part is near zero. This property was first discovered by Barkley Barkley (2006) and extensively studied Pier (2002); Noack et al. (2003); Mittal (2008); Mantič-Lugo et al. (2014, 2015); Sipp and Lebedev (2007) for the cylinder wake (for which it is strongly verified) and then for an open cavity Sipp and Lebedev (2007); Meliga (2017); Bengana et al. (for which it holds quite well, though not exactly). In these two-dimensional flows, a horizontal velocity imposed at infinity is deviated as it encounters objects (a cylinder) or openings (a cavity) and, for sufficiently high Reynolds number, undergoes a Hopf bifurcation.

In Turton, Tuckerman & Barkley Turton et al. (2015), this property was named RZIF – a mnemonic for the fact that the eigenvalue obtained by linearizing about the mean flow has a Real part of Zero and an Imaginary part which is the nonlinear Frequency – and was studied for a quite different kind of flow. Thermosolutal convection is driven by a vertical density gradient in a fluid mixture between two horizontal plates, which in turn is caused by imposed temperature and concentration differences at the two bounding plates. When the temperature and concentration have opposing effects on the density, instability is manifested as a Hopf bifurcation. In a two-dimensional domain with horizontally periodic boundary conditions, this is a classic case of the breaking of $O(2)$ symmetry (translations and reflection) and leads to the simultaneous formation of traveling and standing wave branches, at most one of which can be stable Knobloch (1986). Turton et al. Turton et al. (2015) discovered that the traveling waves in thermosolutal convection provide another clear realisation of the RZIF property, while the standing waves provide an equally clear counterexample: the real part of the eigenvalue obtained by linearizing about the mean part of the standing waves is far from zero and its imaginary part is far from the nonlinear frequency.

We have sought to continue this investigation on another pair of hydrodynamic traveling and standing waves. In Taylor-Couette flow, when only the inner cylinder rotates, a steady bifurcation leads to the classic Taylor vortices. In counter-rotating Taylor-Couette flow, however, these vortices are inclined to the rotation axis, and hence non-axisymmetric. A Hopf bifurcation then gives rise to branches of the well-known spirals Krueger et al. (1966); Langford et al. (1988); Antonijoan et al. (1998) and the somewhat lesser-known ribbons Demay and Iooss (1984); Tagg et al. (1989); Chossat and Iooss (2012); Pinter et al. (2006, 2008a, 2008b); Hoffmann et al. (2009); Altmeyer and Hoffmann (2010); Deguchi and Altmeyer (2013), which, like standing waves, are an equal superposition of spirals moving axially upwards and downwards. As in Knobloch (1986); Turton et al. (2015), this is a consequence of the breaking of $O(2)$ symmetry, here in the axial direction, when end effects are disregarded. Figure 1 shows the appearance of these two types of flows.

Counter-rotating Taylor-Couette flow has an additional symmetry when compared to the two-dimensional thermosolutal cases studied in Knobloch (1986); Turton et al. (2015). Along with the $O(2)$ axial $z$ symmetry, the Taylor-Couette configuration has $SO(2)$ symmetry, consisting of rotations (not reflection) in the azimuthal $\theta$ direction. The tilted spirals or azimuthally tiled ribbons are not axisymmetric, and hence they break $SO(2)$ symmetry. In such cases, bifurcations always lead to states which rotate in the azimuthal direction; see e.g. Ecke et al. (1992). Thus, although ribbons are standing waves in the axial direction, i.e. equal superpositions of axially-upwards and axially-downwards traveling waves, they rotate in the azimuthal direction, as do the spirals.

As the rotation velocity of the outer cylinder is varied, the ribbon branch is succeeded by two types of heteroclinic orbits, both of which connect saddle states consisting of two pairs of axisymmetric vortices. One heteroclinic orbit is non-axisymmetric, with excursions that resemble the ribbons, while the other remains axisymmetric.

II Methods

To investigate this problem numerically, we have used the spectral finite-difference code of Willis Dessup et al. (2018) to solve the Navier-Stokes equations in a cylindrical annulus,

[TABLE]

where the length scale is $d\equiv r_{\rm out}-r_{\rm in}$ and the time scale is $d^{2}/\nu$, where $\nu$ is the kinematic viscosity. The radius ratio is $\eta\equiv r_{\rm in}/r_{\rm out}$, the two Reynolds numbers are defined as $Re_{j}\equiv r_{j}\Omega_{j}d/\nu$ where $j={\rm in}$, out, and the numerical resolution is $(N_{r},N_{\theta},N_{z})=(33,16,16)$.

We use the parameters in Pinter et al. Pinter et al. (2006, 2008a, 2008b), namely radius ratio $\eta\equiv r_{\rm in}/r_{\rm out}=0.5$, inner Reynolds number $Re_{\rm in}=240$ and azimuthal wavenumber $M_{0}=2$ (azimuthal wavelength $2\pi/M_{0}=\pi$). We have chosen the axial wavelength $\lambda_{z}=1.2$, slightly less than the value studied intensively by Pinter et al. (2006, 2008a, 2008b) to avoid a secondary bifurcation they describe near the onset of spirals and ribbons for $\lambda_{z}=1.3$. Because counter-rotating Taylor-Couette flow is centrifugally unstable only near the inner cylinder, here for $r\in[1,1.24]$, the preferred axial wavelength is correspondingly less than that which would be expected if the vortices filled the gap. Because $Re_{\rm in}$ remains fixed and we will vary $Re_{\rm out}$, we will simplify the notation by using $Re$ instead of $Re_{\rm out}$ to designate the outer Reynolds number.

We have modified the Taylor-Couette code Dessup et al. (2018) in order to timestep the equations linearized about a velocity field $\mathbf{U}$:

[TABLE]

Time-integration of the linearized equations (2) about a steady solution $\mathbf{U}$ converges to the leading eigenvector, i.e. that whose corresponding eigenvalue has the most positive or least negative real part or growth rate. It is a means of carrying out the power method on the exponential of the right-hand-side of (2). Integration of (2) does not generally lead to a steady or periodic state, since the nonlinear terms which would otherwise saturate the amplitude is absent. Instead, it leads to states whose amplitude increases or decreases, but with a constant growth rate and frequency, in the case of a complex eigenvalue.

We first use the linearized code to compute the Hopf bifurcation threshold from laminar Couette flow. For various timesteps ${\Delta t}$, we calculate the growth rate $\sigma$ as a function of $Re$ and then use linear extrapolation or interpolation to find $Re_{c}$ such that $\sigma(Re_{c})=0$. (This procedure circumvents critical slowing down and so finds bifurcation thresholds faster and more accurately than could be done with nonlinear simulations.) The bifurcation thresholds calculated using ${\Delta t}=10^{-4}$, $10^{-5}$, and $10^{-6}$ are, respectively, $Re_{c}=-595.23$, $Re_{c}=-586.65$, $Re_{c}=-585.42$, the last of which we consider to be the converged value. (For greater legibility and simplicity, in what follows we cite its truncated value $-585$.)

The branches of spirals and ribbons illustrated in figure 1 bifurcate at $Re\approx-585$ towards increasing $Re$. Their functional form is:

[TABLE]

The spirals are traveling waves: they depend on $z$, $\theta$, and $t$ only via the single combination $z/\lambda_{z}+M_{0}\theta-\omega_{\rm SPI}t$. The ribbons have fixed axial nodal lines and are reflection symmetric; the axis of symmetry is $z=\lambda_{z}/2$ in equation (6) and in figure 1. The ribbons are traveling waves in $\theta$: they depend on $\theta$ and $t$ via $M_{0}\theta-\omega_{\rm RIB}t$. Many other complex and esthetic symmetry-breaking flows have been observed in numerical simulations of this configuration Pinter et al. (2006, 2008b, 2008a); Hoffmann et al. (2009); Altmeyer and Hoffmann (2010); Deguchi and Altmeyer (2013), including cross spirals, oscillating cross spirals, mixed cross spirals and mixed ribbons.

We will use numerical simulations of the nonlinear and linearized equations (1) and (2) to investigate the RZIF phenomenon mentioned in the introduction, in which linearization about the mean flow of a periodic solution yields an eigenvalue whose Real part is Zero and whose Imaginary part is the nonlinear Frequency. To do so, we will calculate the spirals and ribbons by time integrating the nonlinear equations (1) until an asymptotic state is reached. Because the spirals and ribbons are both azimuthally traveling waves, their temporal means are identical with their azimuthal means which are, in turn, the axisymmetric $(m=0)$ components of (6). We will then find the eigenvalues and eigenvectors about the resulting mean flows by integrating equations (2).

III RZIF analysis

We begin by showing the bifurcation diagram for spirals and ribbons. Their common threshold is $Re\approx-585$. On figure 2a, we represent the square root of kinetic energy of the two branches. In this parameter range, the spirals are unstable to ribbons, but can be computed by including only $k=m$ modes or, like Pinter et al. (2006), by setting the $m=0$ component of $u_{r}$ to zero. The ribbons themselves become unstable for $Re\gtrsim-530$. Figure 2b shows the frequency of the nonlinear spirals and ribbons, as well as those obtained by linearizing about laminar Couette flow (dashed line). Linearization about the laminar flow leads to the same set of eigenvalues for spirals or ribbons, since the eigenvectors corresponding to upgoing and downgoing spirals are related by symmetry, with their superposition leading to ribbons. For a supercritical Hopf bifurcation, these necessarily match the frequency of the nonlinear branches at the threshold. As $Re$ is increased, the frequencies of the nonlinear spirals deviate greatly from those derived from the laminar flow, but the frequencies of the nonlinear ribbons remain close. This was also true of the standing waves of thermosolutal convection Turton et al. (2015), which inspired this study: the frequencies of the nonlinear standing waves were much closer to those obtained by linearization about the laminar (conductive) flow than were the frequencies of the nonlinear traveling waves.

Figure 4 shows the temporal means of the nonlinear spiral and ribbon states shown in figure 1 at $Re=-550$. Since spirals are traveling waves in both the $\theta$ and the $z$ direction, their temporal means are equivalent to their spatial means and have no dependence on either direction. Ribbons are traveling waves in the $\theta$ direction and so their means have no $\theta$ dependence. We set $\mathbf{U}$ in the linearized equations (2) to the mean flows of spirals or ribbons shown in figure 4 and initialize $\mathbf{u}$ with fields of the spiral or ribbon form, respectively. We then integrate until the growth rate and the frequency are constant. The resulting field $\mathbf{u}(t)$ then cycles between the real and imaginary parts of the eigenvector which here are related by a spatial shift. Figure 4 shows samples of these eigenvectors. These greatly resemble the deviations of the corresponding nonlinear flows from laminar Couette flow shown in figure 1 and resemble even more the usual eigenvectors of spiral and ribbon form resulting from linearization about laminar Couette flow.

Figure 5 presents the main result of this section, namely that both the spirals and the ribbons satisfy the RZIF property. The growth rates or real parts of eigenvalues are shown in figure 5a and the frequencies or imaginary parts are shown in figure 5b. The solid curves show the frequencies obtained from nonlinear simulations. Linearization about the mean flows of spirals (red crosses) or of ribbons (blue stars) lead to eigenvalues whose imaginary parts agree with the nonlinear frequencies. The corresponding growth rates are seen to be near zero, i.e. the mean flows are nearly marginally stable. (This has no relation to the marginal stability of the full spirals or ribbons to shifted versions of themselves, which arises from their $\theta$ or $z$ dependence in a homogeneous domain.) Thus, both the spirals and the ribbons satisfy the RZIF property. This contrasts with the thermosolutal case, in which the traveling waves satisfied the RZIF property, while the standing waves did not.

Additional eigenvalues are shown in figure 6, which are not related to the basic RZIF property. Both correspond to spiral eigenvectors and have positive growth rates. These show the frequency and growth rate of spiral eigenvectors superposed on the mean flow of spirals of opposite helicity (dark green circles) or of ribbons (blue circle). The positive growth rate for the spirals of opposite chirality could be related to the linear instability of the spirals to ribbons; such a transition would take place via the addition of spirals of opposite chirality until equal amplitudes are reached. However, for these parameter values, ribbons show no tendency to make a transition to spirals, and yet the growth rates from the mean flow of ribbons to spirals are also positive (although less so than the growth rates from the mean flow of spirals). More understanding of the meaning of linearization about mean flows would be necessary to interpret these eigenvalues.

The RZIF property is implied by a near-monochromatic spectrum, as shown in Turton et al. (2015) as follows. Consider the evolution equation

[TABLE]

where $\mathcal{L}$ is linear and $\mathcal{N}(\cdot,\cdot)$ is a quadratic nonlinearity. Let

[TABLE]

(with $\mathbf{u}_{-n}=\mathbf{u}_{n}^{\ast}$) be the temporal Fourier decomposition of a periodic solution to (7) with mean $\mathbf{\overline{U}}$ and frequency $\omega$. The $n=1$ component of (7) is

[TABLE]

If, as is often the case, $||u_{n}||\sim\epsilon^{|n|}$, then $\mathcal{N}_{1}=O(\epsilon^{3})$ may be neglected and RZIF is satisfied: the linear operator $\mathcal{L}_{\bar{U}}$ in (9) has the pure imaginary eigenvalue $i\omega$, corresponding to the frequency of the periodic solution. Hence the RZIF property is satisfied for near-monochromatic oscillations in any system with a quadratic nonlinearity.

It remains to verify whether these periodic solutions are near-monochromatic. Figure 7 shows the temporal Fourier spectra of the azimuthal velocity for the spirals and the ribbons. For the thermosolutal case of Turton et al. (2015), the ratio $||u_{2}||/||u_{1}||$ was found to be much higher for the standing waves (greater than about $10^{-1}$) than for the traveling wave branch (between $10^{-3}$ and about $10^{-1}$), explaining the satisfaction of RZIF for the traveling waves and not for the standing waves. In contrast, for this counter-rotating Taylor-Couette case, figure 7 shows that the temporal spectra for spirals and ribbons are quite similar. The ratio $||u_{2}||/||u_{1}||$ is consistently less than $10^{-2}$ for both flows over the range of our investigation. This is consistent with the fact that RZIF is satisfied for both flows.

IV Heteroclinic orbits

Pinter et al. Pinter et al. (2006, 2008a, 2008b) observed that the ribbon branch became unstable and was succeeded by oscillating cross-spirals as $Re$ was increased. We observe this as well, near $Re\approx-525$. As $Re$ is further increased, however, the oscillating cross-spirals are themselves succeeded by a near-heteroclinic orbit, in which the system spends the overwhelming majority of its time in one of two saddles containing two pairs of axisymmetric vortices. The two saddles differ by a quarter of an axial period (one vortex). The system passes from one to the other via rapid non-axisymmetric excursions resembling the ribbon states. Snapshots of the phases of an excursion taken from a heteroclinic cycle at $Re=-502$ are shown in figure 8.

Figure 10(a,b,c) shows timeseries from this heteroclinic orbit. Time spent at the two saddles are seen as two plateaus. The approaches to and departures from the saddles are oscillatory; this appears clearly in the enlarged timeseries of figure 10b, as well as in the spirals of the phase portrait of figure 10a. In the phase portrait, the two saddles are distinguished, and the spiraling approaches to them are approximately perpendicular to one another. In the timeseries of figure 10c, the energy in the non-axisymmetric modes is seen to increase and to decrease logarithmically.

As Re is further increased, this regime is succeeded by another type of heteroclinic orbit, visualized in figure 11. Timeseries for this cycle are shown in figure 10(d,e,f) for $Re=−490$. The two plateaus are the same as for the previous heteroclinic cycle, but the approaches to and departures from the saddles are not oscillatory. These approaches and departures differ from excursion to excursion, as highlighted in the enlargements in figure 10e. The difference between the excursions is also seen in the phase portraits in figure 10b, which begins with a transient oscillatory phase. The black dots in the phase portrait are equally spaced in time and so accumulate on the saddles. Trajectories are seen approaching and leaving the saddles along different paths at different angles. Turning to the logarithmic energy timeseries of figure 10(f), after an initial transient (corresponding to the oscillatory transient in the phase portrait), the energy in the non-axisymmetric modes becomes and remains very small. Instead, it is the energy in the odd axial wavenumber modes which tracks the heteroclinic cycle. Since the saddles contain two stacked pairs of vortices, their axial Fourier spectrum contain only even multiples of the fundamental wavenumber $2\pi/\lambda_{z}$. Excursions are manifested by the logarithmic increase and decrease of the energy contained in odd multiples of $2\pi/\lambda_{z}$ in the axial spectrum. The variation from cycle to cycle is also seen in the logarithmic timeseries.

The axisymmetric heteroclinic cycle is very similar to that observed in the 1:2 mode-interaction Armbruster et al. (1988); Nore et al. (2003). An erratic alternation between a few different types of excursions is also seen in the cycles of Nore et al. (2003). As explained in these references, this cycle is a consequence of the interaction of axial modes with wavenumbers $2\pi/\lambda_{z}$ and $4\pi/\lambda_{z}$. Furthermore, the heteroclinic orbits can be characterized quantitatively by calculating the rates of approach and departure from the saddles which anchor the heteroclinic orbits. These are the two leading eigenvalues (in the axisymmetric subspace) of the saddles and are seen as the slopes of the black curve in the logarithmic timeseries in figure 10f. We plot these as a function of $Re$ in figure 13b. The existence of the heteroclinic cycle requires eigenvalues of opposite signs; thus, the axisymmetric cycle exists for $Re>-502$, where one of the leading eigenvalues becomes positive while the other remains negative. Stability of the cycle requires that the rate of departure be no greater than the rate of approach, i.e. the absolute value of the negative eigenvalue must exceed the positive eigenvalue. This occurs for $Re<-482$. Hence the axisymmetric cycle exists and is stable within this axisymmetric subspace in the range $Re\in[-502,-482]$.

The theory in Armbruster et al. (1988); Nore et al. (2003) does not concern oscillatory heteroclinic cycles or complex eigenmodes. However, assuming that the analysis of the oscillatory cycles is similar to that of the non-oscillatory cycles, we plot the slopes (or real parts of the eigenvalues) seen in the logarithmic timeseries of figure 10c as a function of $Re$. We also plot the frequencies associated with these eigenvalues in figure 13a, associated with the oscillations in the timeseries in figure 10a,b and the spiraling in and out in the phase portraits of figure 10a. Using the same criteria as for the non-oscillatory cycles, figure 13a shows that the oscillatory heteroclinic cycle exists and is stable within the subspace of these eigendirections for $Re\in[-522,-476]$.

The theory in Nore et al. (2003); Armbruster et al. (1988) also does not address the relative stability of two kinds of heteroclinic cycles. The transition that we observe when $Re$ is reduced from $-502$ to $-490$ implies that the non-axisymmetric heteroclinic cycles is unstable to the axisymmetric one at this value of $Re$. In fact, we observe non-axisymmetric heteroclinic cycles for $Re<-496$ and axisymmetric cycles for $Re>-496$, separated by the point at which axisymmetric and non-axisymmetric departure rates cross, as seen in figure 13b. This supports the idea that the cycle whose departure rate from the saddles is smaller is unstable to the cycle whose departure rate is greater (Ashwin and Chossat, 1998). Combining all of the criteria, the stability range of the non-axisymmetric cycle is $Re\in[-522,-496]$, while that of the axisymmetric cycle is $Re\in[-496,-482]$.

V Discussion

The problem of characterizing the nonlinear frequency of a periodic flow away from the threshold has led to one proposed solution: when a linear stability analysis is carried out about the temporal mean, an eigenvalue is obtained whose Real part is Zero and whose Imaginary part is the nonlinear Frequency. This RZIF property has heretofore been studied for the cylinder wake, an open cavity, and thermosolutal convection. It is strongly satisfied for the cylinder wake and the traveling waves of thermosolutal convection with oppositely directed density gradients, weakly satisfied for the open cavity, and not at all satisfied for the standing waves of thermosolutal convection. Although the RZIF property is a natural outcome of a near-monochromatic temporal spectrum, it is not entirely clear when this occurs.

We have investigated RZIF in another configuration, counter-rotating Taylor-Couette flow, which, like the thermosolutal convective case, has a Hopf bifurcation leading to branches of traveling and standing waves, here manifested as spirals and ribbons. In the thermosolutal case, the traveling waves display the RZIF property and have a temporal Fourier spectrum which is highly peaked, while the standing waves do not. Here, both the spirals and the ribbons display the RZIF property and both have temporal spectra which are equally peaked. However, the ribbons are standing waves only in the axial direction and are traveling waves in the azimuthal direction. This may be the cause of the difference between the Taylor-Couette and thermosolutal cases. The search for another counter-example to RZIF, especially in a purely hydrodynamic flow without additional fields, remains open.

During the course of our investigation, we discovered two heteroclinic cycles, one non-axisymmetric and the other axisymmetric. Both cycles are anchored by the same saddles: axisymmetric states containing two axially stacked pairs of vortices. The axisymmetric heteroclinic cycle is a manifestation of the classic 1:2 mode interaction, here the interaction between axisymmetric states with one and two pairs of vortices. The non-axisymmetric heteroclinic cycle is not of this type, since the approach to and from the saddles is oscillatory via transients resembling ribbons. However, it is anchored by the same axisymmetric saddles. The existence of two qualitatively different heteroclinic cycles connecting the same saddles is an unusual and intriguing feature. These cycles will be studied in more detail in a later investigation.

Acknowledgements.

This work was performed in part using high performance computing resources provided by the Institut du Developpement et des Ressources en Informatique Scientifique (IDRIS) of the Centre National de la Recherche Scientifique (CNRS), coordinated by GENCI (Grand Equipement National de Calcul Intensif) through grant A0042A01119.