Directed percolation and puff crystallization near the transition to pipe turbulence

Both theory and recent experiments in a quasi-one-dimensional Couette cell suggest that the onset of turbulence is a non-equilibrium phase transition in the directed percolation (DP) universality class. However, it is not experimentally clear if this universality class applies to pipe flow, where single-puff time scales vary with Reynolds number in a super-exponential way instead of the expected power-law scaling. To see how puff interactions contribute to the critical behavior, we develop stochastic models of puff dynamics by inputting the interaction function measured in pipe experiments, and calculate the phase diagram and critical phenomena. In agreement with renormalization group predictions, we find strong evidence for critical scaling of the turbulent fraction in the DP universality class, with complex crossovers due to finite size effects and the presence of a crystal-like spatio-temporal pattern which results from the repulsion between puffs.