Self-sustaining turbulence in a Restricted Nonlinear (RNL) Model of plane Couette flow

In this work we develop a restricted non-linear model (RNL) of plane Couette flow based on stochastic structural stability theory (S3T). The S3T system consists of a coupled set of equations for the evolution of a streamwise averaged mean flow forced by the ensemble averaged Reynolds stresses computed from the perturbations. The RNL model calculates the evolution of a single member of the perturbation ensemble interacting with the time varying streamwise averaged mean flow. Simulations of the RNL exhibit self-sustaining turbulent behavior that closely resembles DNS. S3T based analysis of this system shows that this self-sustaining activity arises due to the coupling from the mean flow to the perturbations, in other words the fact that the perturbations depend parametrically on the current state of the streamwise averaged mean flow. Elimination of this interaction reduces the system to the so-called 2D/3C model, which is asymptotically stable and consequently, does not exhibit turbulent behavior in the absence of forcing. Studies of the RNL confirm that the turbulence intensity decreases as the coupling strength is reduced, and that its behavior collapses to that of the 2D/3C model at a non-zero threshold.