Directed percolation model for turbulence transition in shear flows

We analyze a 1 + 1-dimensional directed percolation system as a model for the spatio-temporal aspects of the turbulence transition in pipe flow and other shear flows. Space and time are discrete, and the model is characterized by two parameters: one describes the probability to remain turbulent in the next step and the other characterizes the spreading of turbulence to the neighboring cells. The transition to a persistent turbulence is evident in mean field arguments, but the actual critical values and exponents are considerably renormalized by fluctuations. Extensive numerical tests show that the model falls into the universality class of one-dimensional (1D) directed percolation. We also discuss the spreading of localized perturbations and an extension to 2D systems.