Pipe Flow: Subcritical Transition to Turbulence
Directed percolation universality · puff dynamics · Re threshold
Subcritical transition: Pipe (Hagen-Poiseuille) flow is linearly stable for all Re, yet transitions to turbulence subcritically around Re ≈ 2300. Turbulence appears in localized puffs that can decay or split. Decay is memoryless (exponential), splitting sets a critical Re.
Directed percolation (DP): Avila et al. (2011) showed that the onset obeys DP universality — the turbulent fraction ρ vanishes as ρ ~ (Re − Rec)^β with β ≈ 0.28 (DP exponent). Each pipe segment acts as a site that can be active (turbulent) or quiescent, with activity spreading downstream — the hallmark of DP. Rec ≈ 2040 for the balance of splitting vs decay rates.