Transitional channel flow: A minimal stochastic model

TL;DR

This paper introduces a minimal stochastic cellular automaton model to interpret the intermittent regimes and bifurcations observed in transitional channel flow, linking turbulence onset to directed percolation phenomena.

Contribution

It develops a probabilistic cellular automaton capturing band dynamics and bifurcations in transitional flow, connecting turbulence behavior to phase transition concepts.

Findings

Model reproduces band propagation and bifurcation phenomena.

Decays to laminar flow exhibit directed percolation properties.

Bifurcation analysis suggests phase transition analogy.

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 evolving von Neumann neighbourhoods 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 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.