Phase diagram of transitional pipe flow turbulence from a three trophic-level stochastic predator-prey model*

In transitional shear flows, transient turbulent local spots appear in the system with perturbation, and they decay or spread in a random fashion. Various patterns are formed by these transient turbulent spots before the flow becomes universally turbulent. For transitional pipe flow, a sequence of phases is observed experimentally as Reynolds number (Re) increases. Those phases are characterized by decaying local turbulent spots (puffs) for Re < 2040, splitting and propagating turbulent spots (puffs) for 2040 < Re < 2250, asymmetric expansion of turbulent regions (weak slugs) for 2250 < Re < 4500, and symmetric expansion of turbulent regions (strong slugs) for Re > 4500. An earlier stochastic model for puff decaying and splitting focused on the dynamics and fluctuations within a single puff and did not include stream-wise interactions arising through shear. Here, we extend the earlier model and include the neglected stream-wise interactions. Our extended model recapitulates the full phase diagram of the transition, including weak and strong slug behavior. The model is not restricted to one dimension and is extendable to other transitional shear flows. It can also be shown to reduce to excitable media dynamics in special cases.