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Mean Field Model for Turbulence Transition in Plane Poiseuille Flow

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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.

Author(s):

Michael Rath    
Philipps-Universität Marburg
Germany

Bruno Eckhardt    
Philipps-Universität Marburg
Germany