**1. Context**

How laminar flow becomes turbulent, or the reverse, when the shearing rate changes, is a problem of great conceptual interest and practical importance. This special issue is focused on the case when the transition is characterized by the fluctuating coexistence of domains either laminar or turbulent in physical space at a given Reynolds number Re (control parameter), a regime called spatiotemporal intermittency, relevant to wall-bounded flows in particular. Several years ago, Y. Pomeau [1] placed that problem in the realm of statistical physics by proposing its approach in terms of a non-equilibrium phase transition called directed percolation (DP). This process displays specific statistical properties defining a universality class liable to characterize systems with two competing local states, one active, the other absorbing, with remarkably simple dynamical rules: any active site may contaminate a neighbor and/or decay into the absorbing state, and an absorbing state cannot give rise to any activity [2]. The coexistence is regulated by the contamination probability, and a critical point can be defined above which the mixture of active and absorbing states is sustained and below which the active state recedes, leaving room for a globally absorbing state. The fraction of active sites is a measure of the global status of the system. The subcritical context typical of wall-bounded flows, initially pointed out by Pomeau, seems an interesting testbed for universality [3,4]. Here, turbulence plays the role of the

active state and laminar flow, being linearly stable, represents the absorbing state. DP has indeed been shown relevant to simple shear between parallel plates (Couette flow) [5] and its stress-free version (Waleffe flow) [6]. The most recent contributions to the field can be found in [7]. In this paper we will be interested in plane channel flow (also called plane Poiseuille flow), the flow driven by a pressure gradient between two parallel plane plates, which is not fully understood despite recent advances.

In this context, universal properties are notably difficult to extract from experiments, since they relate to the thermodynamic limits of asymptotically large systems in the long time limit, whereas what plays the role of microscopic scales involves already macroscopic agents, e.g., roll structures in convection or turbulent streaks in open flows, and the turnover time associated with such structures. However, universality focuses on quantitative aspects of systems sharing the same qualitative characteristics, in particular symmetries and the effective space dimension *D* in which these systems evolve. Delicate questions can thus be attacked by modeling attempts that implement these traits appropriately. This approach involves simplifications from the primitive equations governing the problem, here the Navier–Stokes equations, to low-order differential models implementing the building blocks of the dynamics [8], to coupled map lattices (CML) in which the evolution is rendered by maps and space is discretized [9,10], to cellular automata for which local state variables are also discretized, and ultimately to probabilistic cellular automata (PCA), where the evolution rule itself becomes stochastic [11]. The absence of a rigorous theoretical method supporting the passage from one modeling level to the next, such as multi-scale expansions or Galerkin approximations, makes the simplification rely on careful empirical observations of the case under study, which somehow comes and limits the breadth of the conclusions drawn.

#### *1.1. Physical Context: Plane Channel Flow*

Of interest here, the transitional range of plane channel flow displays a remarkable series of steps at decreasing Re from large values where a regime of featureless turbulence prevails. It has been the subject of numerous studies and references to them can be found in the article by Kashyap, Duguet, and Dauchot in this special issue [12]; see also [13]. Our own observations based on numerical simulations are described in [14,15] and summarized in Figure 1.

**Figure 1.** Bifurcation diagram of plane channel flow after [14]. Reg ≈ 700. Transversal splitting sets in at Re ∼ 800 (event A). The extrapolated 2D-DP threshold is ReDP 984. The "one-sided → two-sided" transition takes place at Re2 1011. localized turbulent bands (LTBs) exist up to Re ≈ 1200 (event B), beyond which a continuous laminar–turbulent oblique pattern prevails up to the threshold for featureless turbulence Ret ≈ 3900.

The Reynolds number used to characterize the flow regime is defined as Re = *U*c*h*/*ν*, where 2*h* is the gap between the plates, *U*c is the mid-gap stream-wise speed of a supposedly laminar flow under the considered pressure gradient, and *ν* the kinematic viscosity. This definition using *U*<sup>c</sup> is appropriate for our numerical simulations under constant pressure-gradient driving. Other definitions involve the friction velocity *Uτ*, or the stream-wise speed averaged over the gap *U*b. They are related either empirically, vis., *<sup>U</sup>*<sup>b</sup> vs. *<sup>U</sup>*c, or theoretically, vis., Re*<sup>τ</sup>* <sup>=</sup> <sup>√</sup>2Re to be used in particular for connecting to the work presented in [12], and some other articles. See [14] for details. Below a first threshold Ret, featureless turbulence leaves room for a laminar–turbulent, oblique, patterned regime (upper transitional range) that next turns into a sparse arrangement of localized turbulent bands (LTBs) propagating obliquely along two directions symmetrical with respect to the general stream-wise flow direction, experiencing collisions and splittings ("two-sided" lower transitional regime). Event B in

Figure 1 corresponds to the opening of laminar gaps along the intertwined band arrangement observed in the tight laminar–turbulent network regime, and the simultaneous prevalence of downstream active heads (DAHs) driving the LTBs. Upon decreasing Re further, a symmetry-breaking bifurcation takes place at a second threshold Re2, below which a single LTB orientation prevails. Figure 2 displays snapshots of the flow illustrating these last two stages.

**Figure 2.** Illustration of the different regimes featuring the wall-normal velocity component at the mid-gap; turbulent/laminar flow is pink/white, after data in Figure 1 of [14]. The domain size is 250 × 500 (span-wise × stream-wise). The flow is from left to right. **Left**: Strongly intermittent loose continuous LTB network at Re = 1200 (∼event B). **Centre**: Two-sided regime at Re = 1050 (Re - Re2). **Right**: One-sided regime at Re = 850. Downstream active heads (DAHs) are easily identified in the two right-most panels; a single one is visible in the upper left corner of the left image, marking the transition between sustained regular patterns and loose intermittent ones. Images here and in Figures 2 and 3 are adapted from snapshots taken out of the supplementary material of reference [14].

A significant result in [14] was that the decrease of turbulence intensity with Re below event *B* followed expectations for directed percolation in two dimensions but that, controlled by the decreasing probability of transversal splitting, the bifurcation at Re2 prevented the flow to reach the corresponding threshold. The latter could nevertheless be extrapolated to a value ReDP < Re2. The ultimate decay stage takes place at Reynolds numbers below the point whereat transversal splitting ceases to operate. Figure 3 illustrates an extremely rare occurrence of transversal splitting at a Reynolds number roughly corresponding to event A in Figure 1.

**Figure 3.** First observed occurrence of transversal splitting during a simulation at Re = 800 for *t* ∈ (17100:100:17500). The stream-wise direction is horizontal and the flow is from left to right.

At lower Re, deprived of the possibility to nucleate daughters' LTBs of opposite propagation orientation, LTBs are forcibly maintained in the "one-sided" regime that eventually decays below a third threshold Reg, marking the global stability of the laminar flow. Corresponding flow patterns are illustrated in Figure 4, the right panel of which displays the surprising result that the turbulent fraction decreases as a power law with an exponent *β* of the order of that for directed percolation in one dimension, despite the fact that the flow develops in two dimensions [16].

The objective of the present work is the design of a minimal PCA model for these two last stages that is applicable to flow states for Re below event B, incorporates the anisotropy features visible in Figures 2–4, and accounts for the specific role transversal splitting above event A, in view of providing clues to their statistical properties in relation to dimensionality and universality issues.

**Figure 4.** (**Left**): One-sided flow at Re = 725, 750; same representation as in Figure 2. The domain size is now 500 × 1000, the stream-wise direction is vertical, and the flow upwards. (**Right**): Used as a proxy for the turbulent fraction, *Ey* = *V*−<sup>1</sup> *u*<sup>2</sup> *<sup>y</sup>* d*V* is displayed as a function of 1/Re; inset: same data raised at power 1/*β* with *β* = 0.28 suggesting decay according to the DP scenario in 1D, adapted from [16].
