Models for turbulent plane Couette flow using the proper orthogonal decomposition

We model turbulent plane Couette flow (PCF) by expanding the velocity field as a sum of optimal modes calculated via the proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier–Stokes equations onto these modes. For a minimal truncation including only the most energetic modes having no streamwise variation, we show under quite general conditions the existence of linearly stable nontrivial fixed points, corresponding to a state in which the mean flow is coupled to streamwise vortices and their associated streaks. When the two next most energetic modes, still lacking streamwise variations, are included, chaos and heteroclinic cycles associated with the fixed points are found. The attractors involve repeated visits near unstable fixed points and periodic orbits corresponding to steady and periodically varying vortices, and account for a self-sustaining process in which vortices interact with the mean flow. The models considered in this paper can also serve as a foundation for more sophisticated ordinary differential equation models for turbulent PCF, including those which include modes with streamwise variations.