Subcritical transition: puff splitting and decay in Hagen-Poiseuille flow
Puffs: 0 | Time: 0
Pipe flow transition is subcritical — Hagen-Poiseuille flow is linearly stable at all Reynolds numbers.
Turbulence appears via localized "puffs" that advect downstream, split (Re↑), or decay (Re↓).
The critical Re ≈ 2040 marks the cross-over where splitting rate equals decay rate.
Space-time diagram: horizontal = pipe position, vertical = time. Red = turbulent, blue = laminar.