3.1 Changes in the rolls with Reynolds number

In order to analyse the properties of the large-scale structures, we have performed several simulations fixing the cylinder radius ratio to $\unicode[STIX]{x1D702}=0.909$ , and increasing the driving by rotating the inner cylinder only and varying the Reynolds number $Re_{s}$ between $10^{2}$ and $3.6\times 10^{4}$ , thus covering the full range between linear instability and turbulent rolls. Indeed, for all the cases, the axially pinned structures exist and fill up the entire domain. These Taylor rolls are known to redistribute angular momentum and have a visible large-scale pattern (Lathrop et al. Reference Lathrop, Fineberg and Swinney1992; Ostilla-Mónico et al. Reference Ostilla-Mónico, Verzicco, Grossmann and Lohse2016), as shown in the first two rows of figure 1: if we look at the temporally and azimuthally averaged azimuthal velocity and azimuthal vorticity, indeed, we can see that the signature of the rolls is preserved for all cases. Following Busse (Reference Busse2012), Taylor rolls were analysed with the terminology of thermal convection by Ostilla-Mónico et al. (Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014a ), where it was shown that at both cylinder walls, there existed three types of areas: plume ejection regions, where the velocity was largely in the positive wall-normal direction, and turbulent plume-like structures would leave the cylinder, plume impact regions, where the velocity was largely in the negative wall-normal direction, and plume-like structures from the other cylinder would impact the wall, and wind-shear regions where the velocity was largely parallel to the wall, and plumes would be generated. In Ostilla et al. (Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013), Ostilla-Mónico et al. (Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014a ), the plumes were related to herringbone structures, or hairpin vortices, which can also be seen in figure 2(a).

Figure 1.

Temporally and azimuthally averaged azimuthal velocity (in the rotating reference frame) $\hat{u} _{\unicode[STIX]{x1D703}}$ (a–d) and normalized averaged azimuthal vorticity $\hat{\unicode[STIX]{x1D714}}_{\unicode[STIX]{x1D703}}/\text{max}|\hat{\unicode[STIX]{x1D714}}_{\unicode[STIX]{x1D703}}|$ (e–h) for inner cylinder rotation and several shear Reynolds numbers (from left to right, $Re_{s}=1.96\times 10^{2}$ , $1.37\times 10^{3}$ , $4.97\times 10^{3}$ , $3.61\times 10^{4}$ ) at $\unicode[STIX]{x1D702}=0.909$ and $\unicode[STIX]{x1D6E4}=2.33$ . Contours levels for vorticity are shown for values between $-0.001$ and 0.001, to highlight the roll structures. The panels on the bottom row show two quantities, azimuthal velocity (i) and azimuthal vorticity (j) at the mid-gap $r=d/2$ for Reynolds numbers $Re_{s}=1.96\times 10^{2}$ (black dash-dot), $1.37\times 10^{3}$ (grey dash-dot), $4.97\times 10^{3}$ (black dashed), $3.61\times 10^{4}$ (grey dashed) and $1\times 10^{5}$ (black solid).

Figure 2.

(a) Instantaneous visualization of the azimuthal (streamwise) velocity at $r^{+}\approx 13$ , for $Re_{s}=3.4\times 10^{4}$ , $\unicode[STIX]{x1D702}=0.909$ and $\unicode[STIX]{x1D6E4}=2.33$ . (b) Same velocity field as on the left under a low-pass filter. The characteristic undulatory streamwise instability of the streaks of the self-sustained process can be clearly seen.

In figure 1, the two quantities show different behaviours: the mean azimuthal velocity preserves a similar pattern during all the transition, revealing the Taylor rolls and the two zones where the fluid detaches from the inner/outer boundary layer and impacts the opposite wall. Alternatively, the intensity of the azimuthal (streamwise) vorticity concentrates near the wall. While we could expect the axial (spanwise) vorticity to concentrate near the walls with increasing Reynolds number due to boundary layer thinning, the behaviour of the azimuthal vorticity is more complicated. If there were no secondary flow, one could expect the mean azimuthal vorticity to average out to zero. In contrast, if secondary flows were present, as for example in the case of square-duct flow (Pirozzoli et al. Reference Pirozzoli, Modesti, Orlandi and Grasso2018), azimuthal vorticity would be largely concentrated not only in the boundary layers but also in the core of the secondary structures. For Taylor–Couette flow with inner cylinder rotation, the azimuthal vorticity in the boundary layer becomes much higher than that of the core of the secondary flow, i.e. the Taylor roll. For this to happen, the mean flow velocity of the secondary motion must be relatively constant with Reynolds number, and the boundary layer of the secondary motion must become thinner. This results in a relative ‘emptying’ process of the bulk of the rolls. It is possible to see this process in the last row of figure 1, where in the right panel we have checked the magnitude of the mean streamwise vorticity in the core of the rolls, i.e. at the mid-gap, along the spanwise coordinate. We can clearly see that, while at low Reynolds number of $Re=1.96\times 10^{2}$ the peak is located in the very core of the roll, already during the transition to ultimate regime the centre of the structure is ‘deflating’, and vorticity starts to concentrate at the borders of the structure. This condition is enhanced at the highest Reynolds numbers $Re=3.61\times 10^{4}$ and $Re=1\times 10^{5}$ , where the relatively constant value of azimuthal vorticity drops. This fact shows that other processes are taking place. As a consequence rolls become relatively more quiescent and inactive even if some circulation remains in the secondary flow cores.

This matches previous intuition that as the Reynolds number increases and the flow reaches the ultimate regime, the rolls undergo a series of transitions. In Ostilla-Mónico et al. (Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014a ), when the ultimate regime was reached for pure inner cylinder rotation at $\unicode[STIX]{x1D702}=0.714$ , the rolls simply vanished. For $\unicode[STIX]{x1D702}=0.909$ , even if they persist, between the right-most upper panels at $Re_{s}=3.61\times 10^{4}$ , corresponding to the so-called ultimate regime, and the panels at $Re_{s}=4.97\times 10^{3}$ , the azimuthal velocity in the bulk can be seen to increase even if strong axial signatures are still present. The left panel of the last row of figure 1, representing the magnitude of the mean streamwise velocity in the core of the rolls, i.e. at the mid-gap, along the spanwise coordinate, is a confirmation of this fact: even if, as we have already said, the profile of each velocity preserves the concentrated shape due to fluid detaching/impacting pattern, there is a velocity value jump in correspondence with the transition to the ultimate regime, between $Re_{s}=4.97\times 10^{3}$ and $Re_{s}=3.61\times 10^{4}$ , that somehow matches the drop of azimuthal vorticity in the core seen in the right panel.

This calls into question the assumption that turbulent Taylor rolls before the ultimate regime such as those in Andereck et al. (Reference Andereck, Liu and Swinney1986) and those in the ultimate regime such as those in Huisman et al. (Reference Huisman, van der Veen, Sun and Lohse2014) are generated by the same mechanisms. The continuity between the left and right upper panels becomes less apparent upon close examination, and it appears that fundamental changes happen as the rolls are becoming relatively empty of vorticity.

3.2 The Taylor roll in the context of the self-sustained process

It thus seems useful to analyse the turbulent Taylor rolls in the ultimate regime not in the context of centrifugal instabilities, but instead in the framework of shear flows. For this aim we introduce the self-sustained process of shear flows. Following the argument of Waleffe (Reference Waleffe1997), we know that a self-sustained process is composed of three main phases:

1. (i) the redistribution of mean shear stress by streamwise rolls to create streaks;

2. (ii) the wake-like instability of the streaks;

3. (iii) the regeneration of the streamwise rolls from the nonlinear development of the streak instability.

In the previous section, we saw that Taylor rolls are essentially large-scale streamwise rolls which redistribute mean shear stress. Dessup et al. (Reference Dessup, Tuckerman, Wesfreid, Barkley and Willis2018) showed that already at low Reynolds numbers, the axisymmetric Taylor roll was part (i) of the self-sustained process, and once the transition to wavy Taylor vortices took place, all three parts of the process were active. With increasing Reynolds numbers, we expect the centrifugal (linear) instabilities to become bypassed by the shear instabilities (Faisst & Eckhardt Reference Faisst and Eckhardt2000; Brauckmann & Eckhardt Reference Brauckmann and Eckhardt2017). Thus for high Reynolds numbers we can expect aspects of the Taylor rolls to be related to parts (i) and (iii) of the SSP. We expect large-scale streaky structures to also exist for Taylor–Couette flow even in the turbulent regime. These structures would then feed the Taylor rolls and axially pinned Taylor rolls would simply be a fixed instance of the above self-sustained process.

A streaky flow would contain strong spanwise inflections. In the near-wall region of turbulent flows this will lead to two possible patterns, a typical staggered row of vortices and a less frequent horseshoe structure. To detect the existence of these spanwise variations, we have checked the instantaneous azimuthal velocity near the wall for pure inner cylinder rotation, and the results are illustrated in figure 2. After applying a low-pass filter to the fields, the staggered row structure of streaks clearly comes up. This provides an initial indication that a Taylor roll in the ultimate regime is an axially pinned process reminiscent of the self-sustained process of shear flows.

A further assessment of the similarity between the two processes can be made by studying the temporal evolution of streaks and rolls. There are several definitions used in the literature to quantify the energy of streaks and rolls. First, we following the definition given by Dessup et al. (Reference Dessup, Tuckerman, Wesfreid, Barkley and Willis2018) of streaks and rolls in Taylor–Couette flow where their energy is calculated from the energy in axisymmetric modes.

For this, first of all we separate the field $u$ from its mean  $U$ , averaged with respect to the streamwise coordinate  $\unicode[STIX]{x1D703}$ :

(3.1) $$\begin{eqnarray}u(t,\unicode[STIX]{x1D703},r,z)=U(t,r,z)+u^{\prime }(t,\unicode[STIX]{x1D703},r,z),\end{eqnarray}$$

and we take a further step by analysing two derived quantities, the energy in the rolls:

(3.2) $$\begin{eqnarray}E_{r}(t,\unicode[STIX]{x1D703},r,z)={\textstyle \frac{1}{2}}\int _{r,z}[U_{r}^{2}(t,r,z)+U_{z}^{2}(t,r,z)]r\,\text{d}r\,\text{d}z,\end{eqnarray}$$

and the energy in the streaks as defined by Dessup et al. (Reference Dessup, Tuckerman, Wesfreid, Barkley and Willis2018), which are defined through the axial anomaly, computed taking out the $\unicode[STIX]{x1D703}$ and $z$ average of the field from  $U$ ,

(3.3) $$\begin{eqnarray}E_{s}(t)={\textstyle \frac{1}{2}}\int _{r,z}[U_{\unicode[STIX]{x1D703}}(t,r,z)-\langle U_{\unicode[STIX]{x1D703}}(t,r,z)\rangle _{z}]^{2}r\,\text{d}r\,\text{d}z.\end{eqnarray}$$

This is shown in figure 3(a), where the energy of both structures can be seen to oscillate at time scales of $O(u_{\unicode[STIX]{x1D70F}}/d)$ . The energy in the rolls can be seen to consistently lag behind the energy in the streaks by a small time, and this is probably due to the fact that they are regenerated through something analogous to step (iii) of the SSP cycle.

Figure 3.

(a) Temporal evolution of the kinetic energy associated with streaks $E_{s}$ (solid) and rolls $E_{r}$ (dashed) normalized by their mean value. (b) Temporal evolution of the normalized modal root-mean-square (r.m.s.) velocity in the Fourier space associated with the $M(0,\unicode[STIX]{x1D6FD})$ mode (solid) and the $M(0,0)$ mode (dashed). Both plots shown are for pure inner cylinder rotation with $\unicode[STIX]{x1D702}=0.909$ and $Re_{s}=3.4\times 10^{4}$ .

A second method to quantify the energy of streaks and rolls, and the link between the rolls and the SSP process is done following Hamilton et al. (Reference Hamilton, Kim and Waleffe1995). Here, the study of streaks and rolls is done through the modal root-mean-square (r.m.s.) velocity (or the square root of the ‘kinetic energy’), in the Fourier space, given by:

(3.4) $$\begin{eqnarray}M(k_{\unicode[STIX]{x1D703}}=\unicode[STIX]{x1D6FC}m,k_{z}=\unicode[STIX]{x1D6FD}n)=\left[\int _{r_{i}}^{r_{o}}(\widehat{u_{\unicode[STIX]{x1D703}}^{2}}(m\unicode[STIX]{x1D6FC},r,n\unicode[STIX]{x1D6FD})+\widehat{u_{r}^{2}}(m\unicode[STIX]{x1D6FC},r,n\unicode[STIX]{x1D6FD})+\widehat{u_{z}^{2}}(m\unicode[STIX]{x1D6FC},r,n\unicode[STIX]{x1D6FD}))r\,\text{d}r\right]^{1/2},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}=2\unicode[STIX]{x03C0}/L_{x}$ and $\unicode[STIX]{x1D6FD}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{z}$ are the fundamental streamwise and spanwise wavenumbers, $L_{x}$ corresponding to the streamwise domain extent, and $\unicode[STIX]{x1D706}_{z}$ as defined in § 2.

In particular we focus on the two principal modes:

1. (i) $M(0,0)$ , the axially and streamwise-invariant mode, which represents the mean flow.

2. (ii) $M(0,\unicode[STIX]{x1D6FD})$ the streamwise independent, fundamental in $z$ mode, corresponding to both streaks and rolls, as both $u_{\unicode[STIX]{x1D703}}$ and $u_{r}/u_{z}$ add to the total energy in  $M$ .

In figure 3(b) we see that the two energies oscillate at time scales of $O(u_{\unicode[STIX]{x1D70F}}/d)$ , and the period of the two quantities are almost anti-correlated. This anti-correlation is different from the time lag seen in the left panel and is consistent with a breakdown–regeneration structure that resembles the one described in Hamilton et al. (Reference Hamilton, Kim and Waleffe1995). Energy is constantly being redistributed to the mean flow into streaks and rolls.

We note that Hamilton et al. (Reference Hamilton, Kim and Waleffe1995) analysed $M(\unicode[STIX]{x1D6FC},0)$ , which is invariant in the axial direction, and fundamental in  $\unicode[STIX]{x1D703}$ . Hamilton et al. (Reference Hamilton, Kim and Waleffe1995) states that this mode is responsible for roll regeneration (Hamilton et al. Reference Hamilton, Kim and Waleffe1995, § 6), but in our simulations it evolves at faster time scales and we do not see any positive or negative correlation between this mode and the two shown in figure 3(b). This indicates that we are not exactly seeing the SSP as studied by Hamilton et al. (Reference Hamilton, Kim and Waleffe1995), but instead observe a cycle which has properties reminiscent of it. As a final remark, both the plots of figure 3 are realized at $Re=3.4\times 10^{4}$ , corresponding, as we already said, to the so called ‘ultimate regime’, but we have observed almost identical behaviour of the energies and modal velocities at smaller Reynolds number ( $Re_{s}=5\times 10^{3}$ , during the transition to this regime).

Additional evidence for a general similarity between the rolls and the SSP is given by a previous study performed by Ostilla-Mónico et al. (Reference Ostilla-Mónico, Zhu, Spandan, Verzicco and Lohse2017) on decaying Taylor–Couette turbulence. In channel flow, the self-sustaining process regenerates itself with slow time scales ( $100tU/d$ ). Once forcing is removed, the vortices would regenerate themselves increasingly slowly but with the same intensity, until the turbulence dies out (Jiménez & Moin Reference Jiménez and Moin1991; Hamilton et al. Reference Hamilton, Kim and Waleffe1995). In the decaying Taylor–Couette simulations of Ostilla-Mónico et al. (Reference Ostilla-Mónico, Zhu, Spandan, Verzicco and Lohse2017), similar phenomena could be observed. The cylinders were made stress free at $tU/d=0$ , thus removing the forcing from the rolls. The first life stage of decay, attributed in the manuscript to the decay of the rolls lasting for a time scale of $tU/d\approx 15{-}40$ . The energy of the azimuthal (streamwise) velocity first decayed up to $tU/d\approx 10$ , while the energy of the radial (wall-normal) and axial (spanwise) components did not start decaying up to $tU/d\approx 10$ and had a monotonic decay. This matches the findings of Jiménez & Moin (Reference Jiménez and Moin1991), Hamilton et al. (Reference Hamilton, Kim and Waleffe1995): the streamwise rolls are regenerated in full force in the first life stage of decay until they completely die down and the flow enters the second life stage of decay.

3.3 The pinning of Taylor rolls

If the Taylor roll is reminiscent of a self-sustained process, which is axially pinned, what is causing the pinning? Is it the instabilities caused by curvature, or can rotation alone cause the pinning of the rolls? To answer this question we have simulated several narrow-gap radius ratios up to the limit of $\unicode[STIX]{x1D702}\rightarrow 1$ , i.e. the plane Couette limit. By varying the curvature and keeping the other control parameters constant we can control for its effect. We set $Re_{s}=3.4\times 10^{4}$ , in the ultimate regime (Ostilla-Mónico et al. Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014b ). The non-dimensional rotation parameter $R_{\unicode[STIX]{x1D6FA}}$ is set to $R_{\unicode[STIX]{x1D6FA}}=0.1$ , so we are considering weak anti-cyclonic rotation very close to the value for pure inner cylinder rotation at $\unicode[STIX]{x1D702}=0.909$ , which is $R_{\unicode[STIX]{x1D6FA}}=0.0909$ . Analysing the instantaneous and averaged azimuthal or streamwise velocity fields we find that large-scale structure analogous to Taylor rolls are present for all cases. The average and instantaneous flow fields look very similar, so we expect them to be sustained by the same pattern of streaks that we have described previously. These facts are illustrated in figure 4(a,b).

Figure 4.

(a) Mean azimuthal velocity for Taylor–Couette flow at $\unicode[STIX]{x1D702}=0.85$ and $R_{\unicode[STIX]{x1D6FA}}=0.1$ , $Re_{s}=3.4\times 10^{4}$ . (b) Mean streamwise velocity for plane Couette flow ( $\unicode[STIX]{x1D702}\rightarrow 1$ ) at $R_{\unicode[STIX]{x1D6FA}}=0.1$ , $Re_{s}=3.4\times 10^{4}$ . Spanwise-pinned large-scale structures analogous to the turbulent Taylor rolls can be seen. (c) Strength of the Fourier component associated with the turbulent Taylor roll at the mid-gap as a function of curvature for $R_{\unicode[STIX]{x1D6FA}}=0.1$ , $Re_{s}=3.4\times 10^{4}$ .

Moreover we have checked the magnitude of the rolls as curvature vanishes by looking at the magnitude of the Fourier component associated with them in the $u_{r}$ – $u_{\unicode[STIX]{x1D703}}$ mean at the mid-gap: the axisymmetric mode in the streamwise direction ( $k_{x}=0$ ) and the axially/spanwise mode associated with the roll wavelength ( $k_{z}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{z}$ ). This is shown in figure 4(c), where we can clearly see that the variations of this quantity are around 6 % between $\unicode[STIX]{x1D702}=0.909$ and $\unicode[STIX]{x1D702}=1$ . This matches the intuition coming from the Navier–Stokes equations that the strength of forces due to curvature is measured by the $R_{c}=d/\sqrt{r_{i}r_{o}}$ parameter (Brauckmann, Salewski & Eckhardt Reference Brauckmann, Salewski and Eckhardt2015), which corresponds to $R_{c}\approx 0.095$ for $\unicode[STIX]{x1D702}=0.909$ . This is further evidence that the nature of the rolls are similar in all the systems, and that their pinning is caused by rotation, and not curvature. Indeed, the Coriolis forces arising from solid-body rotation, reflected non-dimensionally as  $R_{\unicode[STIX]{x1D6FA}}$ , seem crucial in controlling the regions of parameter space in which rolls form.

We note that while we use the word ‘pinning’, this is a simplification. Coriolis forces do not break the vertical translation symmetry of the problem. Instead, their presence modifies the rolls and adds a feature that makes them more resistant to spanwise translation when compared to the $R_{\unicode[STIX]{x1D6FA}}=0$ case. Large-scale structures are known to exist in non-rotating plane Couette flow from the work of Tsukahara, Kawamura & Shingai (Reference Tsukahara, Kawamura and Shingai2006). These were isolated in the simulations of Avsarkisov et al. (Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014), and found to consist of counter-rotating pairs of rolls with high vorticity in their boundaries. This pattern is similar to the structure of the Taylor rolls seen in figure 1. It is not curvature, but the addition of a mild anti-cyclonic rotation $R_{\unicode[STIX]{x1D6FA}}=0.1$ which breaks the spanwise wandering of these structures, resulting in the fixed rolls. Proof of this is seen not only in our simulations, where the rolls are barely modified as the curvature vanishes, and correspond to the structures of Avsarkisov et al. (Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014), but also in the large-box rotating plane Couette simulations of Tobias & Marston (Reference Tobias and Marston2017) which show identical signatures for RPCF.

While in principle any rotating reference frame could be chosen to express a Coriolis force, obtaining a different expression for  $R_{\unicode[STIX]{x1D6FA}}$ , the choice of this particular one in which the velocities at the cylinders are equal and opposite is the correct way to reproduce the transition between TC flow and RPCF (Brauckmann et al. Reference Brauckmann, Salewski and Eckhardt2015). As the curvature increases, TC flow driven by pure inner cylinder rotation can be seen as TC flow driven by counter-rotating cylinders with an increasing solid-body rotation. The corresponding Coriolis force increases with increasing  $\unicode[STIX]{x1D702}$ , i.e. $R_{\unicode[STIX]{x1D6FA}}(Re_{o})=1-\unicode[STIX]{x1D702}$ . With this Coriolis parameter  $R_{\unicode[STIX]{x1D6FA}}$ , we can revisit older data to show that mild anti-cyclonic rotation, and not centrifugal instabilities, play a defining role for moderate values of curvature. By recalculating the parameters in terms of  $R_{\unicode[STIX]{x1D6FA}}$ , we see that the $\unicode[STIX]{x1D702}=0.714$ ( $R_{c}\approx 0.33$ ) experiments of Huisman et al. (Reference Huisman, van der Veen, Sun and Lohse2014) show the signature of rolls between $R_{\unicode[STIX]{x1D6FA}}\approx 0.1$ and $R_{\unicode[STIX]{x1D6FA}}\approx 0.2$ when the cylinder counter-rotation is expressed as a Coriolis force, corresponding to the values discussed above. In the $\unicode[STIX]{x1D702}=0.5$ ( $R_{c}\approx 0.71$ ) experiments of van der Veen et al. (Reference van der Veen, Huisman, Merbold, Harlander, Egbers, Lohse and Sun2016), a fixed roll structure for counter-rotating cylinders was seen at $\unicode[STIX]{x1D714}_{o}/\unicode[STIX]{x1D714}_{i}=-0.2$ , which corresponds to $R_{\unicode[STIX]{x1D6FA}}=0.25$ , but not for pure inner cylinder rotation corresponding to $R_{\unicode[STIX]{x1D6FA}}=0.5$ . This also serves to rationalize why the rolls vanish for pure inner cylinder rotation at $\unicode[STIX]{x1D702}=0.5$ ( $R_{\unicode[STIX]{x1D6FA}}=0.5$ ) and 0.714 ( $R_{\unicode[STIX]{x1D6FA}}=0.286$ ), but not for 0.909 ( $R_{\unicode[STIX]{x1D6FA}}=0.091$ ) as seen in Ostilla-Mónico et al. (Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014b ). Mild anti-cyclonic Coriolis forces and not centrifugal forces are the primary cause of the pinning of structures in the ultimate regime.

3.4 The effect of the domain size

From the previous sections, it seems apparent that mild anti-cyclonic rotation causes some self-organization in shear flows by pinning the self-sustained process in the spanwise (axial) direction. However, while the pinning seemed natural for experiments, due to the presence of end plates, it is still unclear what the effect of box size is on simulations. As mentioned in the introduction, it has been found by numerical simulation of Lee & Moser (Reference Lee and Moser2018) that for plane Couette flow, a large computational box of $20\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}\times 2\unicode[STIX]{x1D6FF}\times 5\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}$ , where $\unicode[STIX]{x1D6FF}$ is half the distance between the walls, causes the pinning of flow structures. We note that this pinning was not observed either in the DNS simulations in a domain of size $20\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}\times 2\unicode[STIX]{x1D6FF}\times 6\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}$ by Avsarkisov et al. (Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014), or the $18\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}\times 2\unicode[STIX]{x1D6FF}\times 8\unicode[STIX]{x03C0}$ domain of Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014).

To probe the pinning of rolls, we decompose the velocity in a similar manner as in § 3.2 into mean flow and fluctuations. We return to the definition of streamwise-averaged velocities $U(t,r,z)$ from (3.1), and we take a further step, as in Moser & Moin (Reference Moser and Moin1987) by analysing two derived quantities. First, we look at the axisymmetric azimuthal vorticity, which provides a measure of the circulation of the rolls:

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}=\frac{\unicode[STIX]{x2202}U_{r}}{\unicode[STIX]{x2202}z}-\frac{\unicode[STIX]{x2202}U_{z}}{\unicode[STIX]{x2202}r}.\end{eqnarray}$$

and second, we look at the axisymmetric axial anomaly, computed taking out the $\unicode[STIX]{x1D703}$ and $z$ average of the field from  $U$ , that was used by Dessup et al. (Reference Dessup, Tuckerman, Wesfreid, Barkley and Willis2018) (3.3) to identify streaks:

(3.6) $$\begin{eqnarray}U_{s}(t,r,z)=U(t,r,z)-\langle U(t,r,z)\rangle _{z}.\end{eqnarray}$$

We first compare two TC systems, with $R_{\unicode[STIX]{x1D6FA}}=0.0909$ , corresponding to pure inner cylinder rotation, with strong turbulent rolls, the other with $R_{\unicode[STIX]{x1D6FA}}=-0.1$ , corresponding to pure outer cylinder rotation, with fixed $Re_{s}=3.4\times 10^{4}$ and $\unicode[STIX]{x1D702}=0.909$ . As we previously discussed, large-scale axially pinned rolls are present only with a mild anti-cyclonic rotation, and so are a characteristic of the first system only, while the second does not present this feature at all. This is also reflected on the streak fields  $U_{s}$ , as we have found also there large-scale axially pinned structure, only for inner cylinder rotation, while they are completely missing in the other case. This is displayed in figure 5(a,b).

Figure 5.

(a,b) Mean azimuthal velocity anomaly ( $U_{s}$ ) for outer cylinder rotation (a) and inner cylinder rotation (b) at $Re_{s}=3.4\times 10^{4}$ , $\unicode[STIX]{x1D702}=0.909$ . (c) Normalized energy of the streaks and rolls as a function of the streamwise domain length at the mid-gap $L_{x}/d=2\unicode[STIX]{x03C0}(r_{i}+r_{o})/(2dN)$ for the same values of $Re_{s}$ and $\unicode[STIX]{x1D702}$ for inner cylinder rotation (ICR) and outer cylinder rotation (OCR). The dot-dashed line indicates the expected $A\sim L_{x}^{-1}$ behaviour for a structureless flow where the energy comes from aliasing.

Moreover if we look closer at the central panel, we can see that the axial signature of the rolls has a well-defined shape and an influence also in the bulk, analogous to what was seen for the azimuthal velocity in § 3.1. This is further proof that different mechanisms are taking place, and that Taylor rolls are part of an axially pinned process reminiscent of the SSP: the Coriolis forces are also pinning the streak fields, which in turn reinforces the rolls, keeping them fixed.

We now compare the effect of the azimuthal domain size on the two cases, to further prove that the rolls are not simply a product of aliasing, i.e. of large wavelengths mapped onto the streamwise invariant modes, caused by small domain extents. Even if recent studies have found that the computational domain (or box) size does affect the outline of the rolls (Ostilla-Mónico et al. Reference Ostilla-Mónico, Verzicco and Lohse2015, Reference Ostilla-Mónico, Lohse and Verzicco2016), since the axial extent of the box plays a crucial role on the correlations and spectra of the velocity field, no size of box was able to unpin the rolls. Moreover larger boxes in the azimuthal extent allow for azimuthal wave-like patterns in the Taylor rolls to develop, which affects the statistics in the bulk region, but does not unpin them anyway. The question remains of how large is ‘large enough’? Here, we attempt to rigorously quantify the effect of box size to understand whether the presence of Taylor rolls is a product of numerics, since we are also imposing artificial rotational symmetry in the azimuthal direction. We can do this by systematically varying the azimuthal (streamwise) extent of the computational box. If the rolls are a product of aliasing, then, the energy of the rolls and streaks should decrease with box size as ${\sim}1/N$ , as they are essentially a mean ( $k_{x}=0$ ). On the other hand, if they are physical, the energy should plateau to a constant with box size.

We again use the comparison between a pure inner rotating cylinder system and a pure outer rotating one, as the latter, as we pointed out before, has a lack of roll structures. The simulations are performed with the same parameters, and varying the imposed rotational symmetry of the system with the values $N=5,10,20$ . In figure 5(c) we show the streak and roll energies for both systems: for inner cylinder rotation we can see that even if the azimuthal extent of the domain change, the amplitude of rolls and streaks is almost constant; on the other hand for pure outer cylinder rotation the amplitude goes down with box size as predicted for a mean mode.

As the last point we still want to understand if whether Taylor rolls are effectively fixed in the axial direction for all times or if they move about their position as a random process. For this reason, we decompose the flow in low-wavenumber roll modes and high-wavenumber fluctuations, in the spirit of the generalized quasi-linear approximation (Tobias & Marston Reference Tobias and Marston2017). We have already mentioned that the energy of the rolls is mainly contained in the axisymmetric ( $k_{x}=0$ ) mode, and the axial mode associated with the base wavelength of the roll ( $k_{z}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{z}$ ). By looking at the phase of the Fourier transform of the streamwise velocity $u_{\unicode[STIX]{x1D703}}$ at the mid-gap we can probe the extent of pinning. For the roll to move the phase of this mode, which we will denote from here as  $\unicode[STIX]{x1D6FC}$ , has to shift with time.

In the figure 6(a) we show a visualization of the roll movement, with the predicted peak from the phase of the Fourier component superimposed. The method tracks relatively well the position of the minimum and maximum velocity. We note that an implicit assumption we make when using the Fourier mode, is that each of the two rolls behaves in a symmetric manner, which might not be strictly the case.

Figure 6.

(a) Space–time (Hovmöller) diagram showing the streamwise-averaged azimuthal velocity at mid-gap for $N=20$ at $Re_{s}=3.4\times 10^{4}$ , $\unicode[STIX]{x1D6E4}=2.33$ and $\unicode[STIX]{x1D702}=0.909$ . The position of the maximum and minimum velocities of the Taylor roll Fourier mode is shown with a dashed line. (b) Phase ( $\unicode[STIX]{x1D6FC}$ ) of the Fourier mode associated with the Taylor roll for three different azimuthal (streamwise) domain lengths. (c) Temporal auto-correlation of the phases for the three domains. (d) Probability distribution function (p.d.f.) for the phase velocity ( $\text{d}\unicode[STIX]{x1D6FC}/\text{d}t$ ) for the three simulations. (e) Standard deviation of the phase velocity as a function of streamwise domain extent $L_{x}=20\unicode[STIX]{x03C0}/N$ . The dashed line shows the proposed $1/\sqrt{L_{x}}$ behaviour. (f) The three probability distributions of (d) normalized by their standard deviation with a Gaussian distribution of zero mean, and standard deviation of one overlaid.

In (b) we show the time evolution of the phase for pure inner cylinder rotation cases at $Re_{s}=3.4\times 10^{4}$ and $\unicode[STIX]{x1D702}=0.909$ , with a varying imposed rotational symmetry with the values of $N=5,10,20$ . The movement of the rolls can be seen to become smaller for increasing box size. We also show the temporal auto-correlation of the roll phase in figure 6(c), where it can be seen that the spatial changes decorrelate in time, with the decorrelation time increasing as the streamwise extent of the domain increases. Thus the drifts become both longer in time and shorter in space with increasing domain size.

Furthermore, we can measure the amplitude of the phase change (i.e. the speed of the drift of the roll) $\text{d}\unicode[STIX]{x1D6FC}/\text{d}t$ , and calculate the probability distribution function (p.d.f.) of this quantity. This is shown in (d). The drifts are approximately symmetric, as could be expected by the symmetries in the system. In the bottom right panel, we show the three curves normalized by their standard deviation and compared to a standard Gaussian distribution. The three curves can be seen to approximate the Gaussian distribution with their standard deviation, i.e. the size of the spatial drifts, decreasing with increasing domain size.

It thus seems that the spatial drifts not only become shorter, but they also become more correlated with increasing domain size. This can be reasoned as follows: all nonlinear interactions are triadic interactions of low- or high-wavenumber modes. A force that could shift the phase has to come from the interaction of two high-wavenumber fluctuations, that is, from a triadic interaction of the form $\widehat{u}^{\prime }(k_{1}),\widehat{u}^{\prime }(k_{2})\rightarrow \widehat{U}(k_{1}+k_{2})$ . If we now assume that fluctuations are random, and have zero mean, using the central limit theorem we expect that the mean of all triadic interactions affecting the roll behave like a Gaussian. We also expect that an increase in the computational box, which adds more modes to the averaging, makes the expected value of the fluctuations approach zero. In figure 6(e) we show the standard deviation of the drift, which approaches zero even faster than $1/\sqrt{L_{x}}$ , that is what would be expected from a simple random process. This is an indication that the phenomenon is more complicated than what could be predicted from this simple analysis, but anyway the pinning process holds and works stronger than our expectation.

Overall, the results verify reasonably well our expectations. We have shown that rolls are moving, but this movement depends on the box size: the larger the domain the smaller the spatial variations. Computational effects cannot be completely avoided. Indeed, boxes which are very large might pin structures by the averaging out fluctuations over long spatial extents.