Manneville Paul, Shimizu Masaki
LadHyX, École Polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau, France.
Graduate School of Engineering Science, Osaka University, Toyonaka 560-0043, Japan.
Entropy (Basel). 2020 Nov 29;22(12):1348. doi: 10.3390/e22121348.
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.
根据波莫关于有向渗流（DP）与壁面流动中湍流起始/衰减相关性的猜想，我们提出了一个最小随机模型，用于解释在通道流恢复层流之前观察到的空间间歇性状态。数值模拟表明，在最终衰减为层流之前，一种具有在两个流向对称方向上倾斜漂移带的状态会分叉为一种不对称状态。该模型以演化的冯·诺依曼邻域的概率元胞自动机来表示，其概率是通过对模拟结果的仔细研究得出的。它实现了带的传播以及两个主要的局部过程：涉及具有相同取向带的纵向分裂，以及产生与其母带取向相反的子带的横向分裂。在二维几何中观察到的最终衰减阶段表现出一维DP特性，这被解释为是由于在单取向状态下横向扩展不相关所致。该模型还再现了随着赋予横向分裂的概率变化而恢复对称性的分叉，这为类似于热力学相变那样研究该分叉的临界特性开辟了道路。
