*2.1. Context*

The approach to be developed is not new in the field of transitional flows. For example, studying plane channel flow, Sano and Tamai [21] introduced a plain 2D-DP model dedicated to support their experimental results, with a simple spatial shift implementing advection and a uniformly turbulent state upstream corresponding to their setup. Earlier, a similarly conceptual model was examined by Allhoff and Eckhardt [20], who introduced a PCA with two parameters accounting for persistence and lateral spreading appropriate for the symmetries of plane Couette flow, developed its mean-field treatment, and performed simulations to illustrate the spreading of spots and decay of turbulence in agreement with expectations. In a similar spirit but introducing more physical input, Kreilos et al. [22] analyzed the development of turbulent spots in boundary layers as a function of the residual turbulence level upstreams, separating a deterministic transport step from a stochastic growth/decay step with probabilities extracted from a numerical experiment, gaining insight into the statistics of boundary layer receptivity.

Following the lines of research suggested by those works, we developed a 2D model designed to interpret the decay of channel in the LTB regimes from two-sided to one-sided at decreasing Re, just qualitatively proposing a plausible variation of probabilities introduced as functions of Re. In our approach, the elementary agents are the LTBs themselves either propagating to the left or to the right of the stream-wise direction. To them we attach variables analogous to spins in magnetic phase transitions problems. Even if in computations, numerical values *S* = ±1 will be used, for descriptive and graphical convenience we shall associate them with colors—specifically: blue (*B*) and red (*R*) for right- and left-propagating LTBs, respectively. Laminar sites will be denoted using the empty-set symbol ∅, will have value 0, and will be graphically left blank. These agents will be seated at the nodes of a square lattice with coordinates (*i*, *j*), i.e., *S*(*i*,*j*) with *S* → {*R*, *B*, ∅} at the given site. As seen in Figure 5a, we place the stream-wise direction along the first diagonal of the lattice so that the LTBs will move along the horizontal and vertical axes; see Figure 5b.

**Figure 5.** (**a**) Cellular automata lattice with the two types of active states, *B* and *R*; the state at an empty node is denoted ∅ and left blank. (**b**) Left: the two possible kinds of propagation from an initial position marked with the "∗". Right: collision configuration to the point marked with the "∗". (**c**) Labeling of the von Neumann neighborhood used to account for the dynamics.

A strong assumption is that an LTB as a whole corresponds to a single active state, while the discretization of space coordinates (*i*, *<sup>j</sup>*) <sup>∈</sup> <sup>Z</sup>2, and time *<sup>t</sup>* <sup>∈</sup> <sup>N</sup> tacitly refers to an appropriate rescaling of time and space. Furthermore, interactions are taken as local, with configurations limited to nearest neighbors in each space direction. Accordingly, the dynamics at a site (*i*, *j*) only depend on the configuration of its von Neumann neighborhood V(*i*,*j*) := {(*i*, *j*),(*i* ± 1, *j*),(*i*, *j* ± 1)}, Figure 5c, while evolution is driven by a random process. We now turn to the definition of rules that mimic the actual continuous space-time, subcritical and chaotic, Navier–Stokes dynamics governing the LTBs' propagation, decay, splitting, and collisions, via educated guesses from the scrutiny of simulation results, in particular those in the supplementary material attached to [14].
