*3.4. Dynamics of Localized Turbulent Patches*

In this last subsection, we address the issue of the influence of azimuthal confinement/extension on the lower transition threshold *Reg*, as the estimations from Figure 3 suggest. In Ref. [36], a similar trend was noted (from measurements in shorter and narrower domains). The mechanism suggested in this former work addressed the presence of oblique stripes rather than their influence on the value of *Reg*. It was thereafter realized that the phenomenon governing the value of *Reg*, and by extension all statistics of the turbulent fraction, is the way different coherent structures interact together dynamically rather than the shape of such individual structures (although that shape certainly influences the interactions). In analogy with pipe [16,42,43] and channel [44,45], the finite turbulent fraction is the result of a dynamical competition between the proliferation of coherent structures and their tendency to decay in number. The transitional range where *Ft* > 0 is dominated by the splitting of coherent structures, whereas instantaneous relaminarizations become rare. We hence focus on the

dynamics of splitting events in two different computational domains, namely those with *L<sup>θ</sup>* = 32*π* and 128*π*. Figure 10 contains zooms on the radial velocity plotted for different values of *y* = *cst* surfaces (a different value for each row) and for different times (different columns). In Figure 10, the value of *L<sup>θ</sup>* is fixed to 32*π*, but the circumference in terms of *rθ*/*h* varies according to *r*. The global dynamics of these flows can also been scrutinized in the videos made available as Supplementary Materials. The comparison of different values of *y* is useful to confirm that, for all parameters, the spots remain coherent over the gap even during splitting events.

Lateral splitting events are considered in each of these figures and videos. Because of the different advection velocities in the azimuthal direction, spanwise collisions can occur. During spanwise collisions, usually one of the two spots disappears (see also Ref. [21] for similar observations in pCf). This tends to reduce the turbulent fraction while the other surviving spot is still active. In the presence of a short enough spanwise periodicity, a spot collides with itself rather than with a different neighbor. In such periodic domains, the local relaminarization of one spot is equivalent to the extinction of an infinity of identical spots. Hence, the turbulent fraction decreases more than in large domains where individual spots behave more like independent entities. We thus expect more turbulence to proliferate more for larger *Lθ*. As a consequence, the critical Reynolds number *Reg*, for which the rate of proliferation balances the probability to relaminarize locally, is lowered when *L<sup>θ</sup>* is increased, consistently with the thresholds reported in Figure 3 and Table 2. This effect is more marked at lower *η*.

**Figure 10.** Snapshots of splitting and self-colliding events in aCf for *Rew* = 262.5 with *L<sup>θ</sup>* = 32*π* and *η* = 0.1. Radial velocity in a frame moving with bulk velocity *um*. Here, *t* = 0 is an arbitrary time instant after reaching equilibrium. Top row, *y*<sup>∗</sup> = *y*/*h* ≈ 0.9; center row, *y*<sup>∗</sup> ≈ 0.5; lower row, *y*<sup>∗</sup> ≈ 0.1.

## **4. Conclusions**

The present DNS study deals with the statistical aspects of the intermittent transitional regime of aCf, with an emphasis on the low values of the radius ratio *η* close to 0.1. It is an extension of the simulations reported recently by Ref. [36]. The paper compares two computational situations, respectively the case of a realistic geometry and the one where the azimuthal extent is larger than the original value of 2*π*. In Ref. [36], this parametric trick was introduced in an explicit attempt to decouple the effects of wall curvature effects from the effects of azimuthal confinement induced by the geometry. The main conclusion for large *η* was that the reported absence of oblique laminar-turbulent patterns was due to azimuthal confinement, since they could re-appear for *L<sup>θ</sup>* > 2*π*. In the present article, the same trick is introduced for *η* = 0.1; however, larger values of *L<sup>θ</sup>* have been tried up to 128*π* (i.e., 64 times the original value). The oblique patterns do not reappear and a new percolating regime takes place with shorter spatial correlations. The statistical analysis of the STI is convergent as *L<sup>θ</sup>* grows, and is consistent with (1 + 1)-D DP. This updates the results of Ref. [36] where (2 + 1)-D DP was suggested from fits with *L<sup>θ</sup>* = 16*π*. The present results suggest now that the *L<sup>θ</sup>* = 16*π* algebraic statistics was still far from the true thermodynamic limit, while *L<sup>θ</sup>* = 128*π* seems to yield more decent results.

To our knowledge, there has been only poor evidence for the cross-over from exponential to algebraic scaling in the shear flow literature, as far as well-resolved simulations of the Navier–Stokes equations are concerned [2]. An exception is the work by Shi et al. [46] in a tilted periodic domain of pCf, which again is not a fully realistic numerical domain. It is interesting to speculate how much the present results can teach us something about a fully realistic system such as cylindrical pipe flow. Naive homotopy of the turbulent regimes is ruled out because of the singularity near the centerline. Instead, we can compare the rate at which these two effectively one-dimensional percolating systems tend towards their own thermodynamic limit. This issue was raised recently in the experimental study by Mukund and Hof [19]. There, despite pipes as long as 3000 diameters, no critical regime (with power-law statistics) was identified, only classical STI as reported in Refs. [47,48]. This issue was attributed to the narrowness of the critical range, and to a clustering property of puffs which delays the convergence to the thermodynamic limit. Here, in aCf with *η* = 0.1, the situation is different but depends on this artificial parameter *Lθ*. To our surprise, power-law statistics of the turbulent fraction as well as of the laminar gap distributions do appear in our simulations as *Rew* is reduced. All cases shown in Figure 8 suggest a cross-over from turbulent to power-law behavior as *Rew* is within ≈ 1% of the critical point. For *L<sup>θ</sup>* = 2*π* or around, the turbulent fraction curve still suggests an unconverged power-law. For *L<sup>θ</sup>* = 32*π* or 128*π*, power-law statistics of *Ft* are fully consistent with one-dimensional DP appear. This occurs despite a value of *Lx* of only 512*h*, i.e., much less than the pipe flow case and even less if one counts in outer pipe diameters. A possible interpretation is that azimuthal extension, by modifying the interaction with neighboring spots, can suppress the tendency to form clusters, and hence converge faster towards the thermodynamic limit. This is consistent with lower transition thresholds in *Rew* as well. One is left wondering if a similar approach to cylindrical pipe flow could also easily yield the percolation exponents from simulation measurements.

We conclude by noting that artificially modifying both the shape of turbulent patches and their interaction, as done here using azimuthal extension, is more than an esoteric thought experiment or an exotic parameter study. It is used here as a legitimate strategy in order to untangle complex phenomena, e.g., to decouple confinement from curvature effects. As demonstrated in our recent work using a simple modeling approach [49,50], wall roughness can have similar effects on transitional flows and change the way turbulence invades laminar flows. We expect similar strategies of artificial domain extension to be relevant to such cases too.

**Supplementary Materials:** Video S1: Time evolution of turbulent fraction *Ft*(*t*) and of fluctuating velocity fields visualized at mid-gap, for *η* = 0.1 with an artificially extended azimuthal domain size of *L<sup>θ</sup>* = 128*π*. On the right column, contours show *x*-*θ* distributions of the radial velocity fluctuation *u <sup>r</sup>* normalized by the inner-cylinder velocity *Uw*. Top (orange box and curve in the graph) : above the global critical Reynolds number *Reg*. Middle (red) : near *Reg*. Bottom (black) : below *Reg*. A supporting video article is available at https://doi.org/10.5281/zenodo.3985963.

**Author Contributions:** Conceptualization and methodology, T.T. and Y.D.; simulations and acquisition, K.T.; post-processing, K.T. and T.T.; writing Y.D. and T.T.; funding acquisition, T.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by Grant-in-Aid for JSPS (Japan Society for the Promotion of Science) Fellowship JP16H06066 and JP19H02071.

**Acknowledgments:** Numerical simulations were performed on SX-ACE supercomputers at the Cybermedia Center of Osaka University and the Cyberscience Center of Tohoku University.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
