Mean field model for turbulence transition in plane Poiseuille flow

etc15 Tracking Number 312

In the pipe flow model of Dwight Barkley the main idea is to model pipe flow as an excitable, bistable medium. Using a one-dimensional FitzHugh-Nagumo-type reaction-advection-diffusion system with two variables the model captures qualitatively a surprising number of features of the turbulence transition in pipe flow. Motivated by this success, we here describe a derivation of a set of two 1+1-dimensional coupled differential equations for the closely related system of plane Poiseuille flow from the Navier-Stokes equation. The model contains terms for the production of turbulent kinetic energy, its transfer between the modes and its dissipation by viscous terms. The model shows a bifurcation to a non-trivial state and reflects some of the complex dynamics observed in direct numerical simulations.