Transitional Channel Flow: A Minimal Stochastic Model

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.

Aging and memory of transitional turbulence

“…We propose that once individual stripes continuously grow also turbulence as a whole will eventually cease to decay and hence the critical point for channel flow can be estimated to be Re c ≈ 650. In agreement with recent numerical studies [41][42][43][44] our experiments also confirm that the critical point is significantly lower than that reported by [33].…”

“…1). Finally as shown by a number of recent studies [41][42][43][44], the actual critical point where turbulent stripes first become sustained in channel flow is substantially lower than the critical point proposed by [33] and consequently these latter experiments were carried out far from the onset of sustained turbulence.…”

“…A model study, attempted to remedy the shortcomings of these channel experiments by suggesting that at the actual (i.e. much lower) critical point the transition may fall into the 1+1D DP universality class and crosses over to 2+1 D DP further from critical [42,44]. On the other hand, a numerical study of nonlocalized stripes in periodic domains [39] reported that the characteristic time scales for the decay and the nucleation of new stripes exceed 10 6 advective time units at the critical point.…”

Size-dependent transient nature of localized turbulence in transitional channel flow

“…Continuing efforts have been taken to prove that channel flow transition also belongs to the DP universality class (Sano & Tamai 2016; Shimizu & Manneville 2019; Manneville & Shimizu 2020), thus unifying all canonical shear flows as a DP-type phase transition. However, this is a difficult task given the complex dynamics exhibited by channel flow turbulence as presented here and elsewhere.…”

The phase transition to turbulence in spatially extended shear flows

“…The dynamics of the process are governed by the decay and splitting rates of puffs and in the proximity of the critical point on average a puff travels 10 7 diameters downstream before either event occurs [12]. While various numerical studies determined directed percolation exponents for models of varying complexity(including direct numerical simulations) [14,[17][18][19][20][21][22], measurements in actual fluid flows have to resolve both, the large spatial and temporal scales.…”