Stochastic theory for pattern formation and front propagation in transitional pipe turbulence.

The onset of turbulence in a pipe occurs through a subcritical transition. Once turbulent patches ("puffs") have been nucleated by some external perturbation, they decay; but above a threshold flow velocity, puffs split, leading to a nonzero turbulent fraction in the pipe at long times. Recent theoretical and experimental work has shown that this transition can be understood as a nonequilibrium phase transition in the universality class of directed percolation. At higher flow velocity, the turbulence spreads into the laminar state through front propagation, creating an expanding region of turbulence known as a "slug," which may exhibit either one or two sharp fronts depending on the flow velocity. It is an open question as to whether the phenomena associated with puff interactions and the slug phase can be understood within the statistical mechanical model framework that predicts the directed percolation transition. Here, we present a stochastic model for the decay, splitting, and propagation of turbulent patches in a background laminar state that accounts for the full range of behavior in the transitional regime. We show that activator-inhibitor (predator-prey) dynamics, coupled to the streamwise shear flow, recapitulates experimental profiles for puffs and slugs, as well as previous simulation results for the spatial structure of the energy flow, the kinematics of puff splitting and the transition to and between the two slug phases. Our work shows that nonequilibrium statistical mechanics can provide a detailed understanding not only of the laminar-turbulent transition, but also the pattern formation phenomena arising in the turbulent state.