*Article* **Transitional Channel Flow: A Minimal Stochastic Model**

**Paul Manneville 1,\* and Masaki Shimizu <sup>2</sup>**


Received: 2 November 2020; Accepted: 24 November 2020; Published: 29 November 2020 -

**Abstract:** In line with Pomeau's conjecture about the relevance of directed percolation (DP) to turbulence onset/decay in wall-bounded flows, we propose a minimal stochastic model dedicated to the interpretation of the spatially intermittent regimes observed in channel flow before its return to laminar flow. Numerical simulations show that a regime with bands obliquely drifting in two stream-wise symmetrical directions bifurcates into an asymmetrical regime, before ultimately decaying to laminar flow. The model is expressed in terms of a probabilistic cellular automaton of evolving von Neumann neighborhoods with probabilities educed from a close examination of simulation results. It implements band propagation and the two main local processes: longitudinal splitting involving bands with the same orientation, and transversal splitting giving birth to a daughter band with an orientation opposite to that of its mother. The ultimate decay stage observed to display one-dimensional DP properties in a two-dimensional geometry is interpreted as resulting from the irrelevance of lateral spreading in the single-orientation regime. The model also reproduces the bifurcation restoring the symmetry upon variation of the probability attached to transversal splitting, which opens the way to a study of the critical properties of that bifurcation, in analogy with thermodynamic phase transitions.

**Keywords:** transition to/from turbulence; wall-bounded shear flow; plane Poiseuille flow; spatiotemporal intermittency; directed percolation; critical phenomena
