Origin of the super-exponential scaling with Reynolds number of single turbulent puff lifetime in pipes

In transitional Couette and 2D Waleffe flows, careful measurements and simulations provide strong evidence of power-law scaling of turbulent fraction and a diverging turbulent lifetime near the critical Reynolds number. These results are consistent with the laminar-turbulent transition being a non-equilibrium phase transition in the directed percolation (DP) universality class. In pipe flow, however, the lifetime of single turbulent puffs does not seem to diverge at a critical Reynolds number, even though the interaction between small-scale turbulence and a zonal flow coherent structure implies that the transition should follow DP. We resolve this paradox by arguing that near the transition, as the correlation length of turbulent fluctuations grows to the pipe diameter, the fluctuations in a single puff would follow a universal finite-size scaling distribution. We derive this distribution from DP and show that its asymptotics closely match extreme value distributions, giving rise to a super-exponential dependence of lifetime on Reynolds number, in agreement with observations on single puffs very close to the laminar-turbulent transition.