**5. Discussion and Concluding Remarks**

Coming long after a conjecture by Pomeau [1], empirical evidence is growing that the ultimate stage of decay of wall-bounded turbulent flows towards the laminar regime follows a directed-percolation scenario. The evidence comes from laboratory experiments and direct numerical simulation of the Navier–Stokes equations but this support is still far from a theoretical justification. The recognition of the globally subcritical character of nontrivial states away from laminar flow and the elucidation of the structure of coherent structures involved in these nontrivial states [29] were first steps in this direction. The next ones would be the elucidation of special phase space trajectories from sustained localized turbulence accounting for the decay to laminar regime, on one side, and to proliferation via splitting, on the other side, using specific algorithms for the detection of rare events and the determination of transition rates that can be attached to them (see [30] for an illustrative example and references). These are heavy, and possibly not much rewarding, tasks but it would be nice to be able to attach numbers to specific events such as the splittings illustrated in Figure 3 or Figure 4. We have chosen to short-circuit such studies through analogical modeling, by which seemed more appropriate to make further progress regarding the thermodynamic limit and associated universality issues. One should though consider this practice as providing hints and not a demonstration that the results will apply to the case under study.

In the present paper, the problem has been considered from this last viewpoint, assuming that the ultimate decay stages were amenable to the most abstract level of implementation in terms of probabilistic cellular automata [2], following [20,22]. We focussed on the specific case of channel flow that offers a particularly rich transitional range. Its upper part displays regular non-intermittent laminar–turbulent patterns that can better be described using the tools of pattern-forming theory [15,25,31]. The lower transitional range is characterized by their spatiotemporally intermittent disaggregation, to which the considered type of modeling is particularly relevant. The analogy alluded to above has, however, been severely constrained to fit the empirical observations. The main assumptions were the introduction of two types of active agents attached to each kind of localized turbulent bands propagating in one of the two possible orientations with respect to the stream-wise direction. Interactions were assumed local so that the probabilistic cellular automata evolved simple nearest neighbors von Neumann neighborhoods (Figures 5–7). Scrutiny of simulation results lead to the introduction of a certain number of probabilities governing the fate of single-occupancy neighborhoods. Multiple-occupancy was treated as a combination of single-occupancy configurations supposedly independent, reducing the number of parameters to be introduced and drastically simplifying the interactions (at any rate impractical to estimate in detail). A clear-cut physical interpretation was, however, given to each parameter in the set reduced to four, accounting for every possible stochastic event affecting the agents, namely, propagation, decay, and splitting, either longitudinal or transversal. A mean-field study of the model, neglecting the nontrivial effects of stochastic fluctuations, reproduced the empirical bifurcation diagram of channel flow at a qualitative level (Figure 14). Transitions have been studied quantitatively by numerical simulation of the stochastic model considering variations of these parameters as putative functions of the Reynolds number Re, highlighting three situations:

In the two first cases, the parameter *p* <sup>4</sup> associated with transversal splitting, i.e., the nucleation of a daughter with orientation opposite of its mother, was switched off, as inferred from observations for Re 800, where the single-sided regime is well established. The coarsening observed when starting from two-sided initial conditions was faithfully reproduced (Figure 8) and decay seen to follow the directed-percolation expectations. The specific conclusion was that, when parameter *p*<sup>2</sup> is no-zero, with *p*<sup>2</sup> attached to longitudinal but upstream-shifted splitting, the scenario is typical of a 2D system with a high level of confidence, whereas when it is strictly zero, i.e., the daughter strictly aligned with the mother, the decay is 1D. A cross-over is observed when *p*<sup>2</sup> is reduced, that manifests itself as a transient reminiscent of 1D behavior, the longest the closest *p*<sup>2</sup> is to zero. Simulations of channel flow have shown that exponent *β* controlling the ultimate decay of the turbulent fraction was that of 1D directed percolation [16]. Since parameter *p*<sup>2</sup> is attached to the slight upstream trajectory shift experienced by a daughter upon splitting from its mother, this observation strongly suggests that the trajectory shift is mostly irrelevant and that localized turbulent bands propagate along independent tracks so that the end result is just a mean over the direction complementary to their propagation direction.

The last situation we have considered corresponds to *p* <sup>4</sup> = 0, with transversal splitting on. This parameter measures the frequency of transversal splitting and is expected to increase with Re. Accordingly, the system can change from one-sided when *p* <sup>4</sup> is zero or small, to two-sided when it is large. The transition has indeed been observed and mean-field predictions were well observed far from the transition point. Unfortunately, while the effect of fluctuations close to that point was obvious, strong size effects have forbidden us to approach it and evaluate critical corrections. This is the subject of on-going work within the framework of finite-size scaling theory [2,27,28]. This follow-up should allow us to establish the universality class to which this transition belongs. Here, the left-right symmetry of localized turbulent bands with respect to the stream-wise direction is reminiscent of the up-down symmetry of magnetic systems at thermodynamic equilibrium, which may lead to conjecture the relevance of the 2D Ising class [26]. This class appears also applicable to coupled map lattices with the same up-down symmetry when updated asynchronously, one site after the other, close to randomization by thermal fluctuations. In contrast, another universality class is obtained with synchronous update [27]. Here, the situation is unclear: on the one hand, configurations are treated as a whole in a simulation step, which tips the scales in favor of a synchronous

update model (in line with what is expected for a problem primitively formulated in terms of partial differential equations); on the other hand, spatial correlations generated by the deterministic dynamics governing the coupled map lattices are weakened by the independence of random drawings at the local scale, which can be viewed as a source of asynchrony in the probabilistic cellular automata. In its application to the symmetry-breaking bifurcation in channel flow, this uncertainty is, however, only of conceptual importance in view of size effects: owing to the large and unknown time-scale rescaling that allowed us to pass from flow structures to local agents in the model and to the narrowness of the region where critical corrections are expected, the mean-field interpretation developed in [14] appears amply sufficient.

In the three cases that were considered in detail (specific cuts in the parameter space), the transitions remained continuous. However, this may not always be the case since there are known example of similar systems displaying transitions akin to first-order ones [24]. Even while keeping the same general frame, a plethora of circumstances of physical interest can be mimicked: propagation can be made more stochastic by decreasing *p*5, splitting rules not observed in channel flow can be considered, e.g., with *p*<sup>3</sup> or *p*<sup>4</sup> different from zero, etc., though it seems hard to anticipate situations where the universal features pointed out here would not hold. In contrast, when dealing with highly populated configurations, even in the simple nearest-neighbor von Neumann setting, rules can be made more complicated by introducing the neighborhood's degree of occupation. This introduction might help us to account also for the upper part of the transitional range of wall-bounded flows characterized by the emergence of regular patterns in the same stochastic framework [11]. The construction of the present model is, of course, fully adapted to the study of universality in the framework of the theory of critical phenomena in statistical physics, especially directed percolation. Still, we are confident that the kind of approach illustrated here brings a valuable contribution to the understanding of the transition to turbulence, by rationalizing its key ingredients in an easily accessible way.

**Author Contributions:** Conceptualization, P.M., M.S.; methodology, P.M.; software, P.M., M.S.; validation, M.S.; formal analysis, P.M.; investigation, P.M.; resources, M.S.; data curation, M.S., P.M.; writing—original draft preparation, P.M.; writing—review and editing, P.M., M.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** P.M. wants to thank H. Chaté (CEA-Saclay, Gif-sur-Yvette, France) for pointing out a model that could help one uncover the universal contents of the symmetry-breaking bifurcation on the same footing as the decay near Reg.

**Conflicts of Interest:** The authors declare no conflict of interest.
