Statistical mechanics of puff-splitting in the transition to pipe turbulence

Very close to the laminar-turbulent transition in pipe flow, localized regions of turbulence known as puffs proliferate and interact, leading to spatiotemporal intermittency. We develop a one-dimensional stochastic model of puff dynamics, taking into account puff decay, propagation and splitting, as well as short-range interactions whereby a puff experiences lifetime suppression and velocity pushing by a sufficiently close upstream puff. The universality class for the resulting laminar-turbulence transition in pipe flow is (1+1)-dimensional directed percolation, with puff-puff interactions irrelevant at the renormalization group fixed point. Continuum and discrete numerical simulations are in agreement with the theory, after taking into account finite-size and crossover effects.