Modeling and control of minimal flow unit turbulence in plane Couette flow

We model turbulent plane Couette flow for a minimal flow unit (the smallest domain in which turbulence can be sustained) 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. We consider a 6 mode (11-dimensional) model. When losses to neglected modes are ignored, the model captures some aspects of the turbulent state very well, but fails to accurately reproduce the velocity field dynamics. However, when energy-transfer to neglected modes is included, there is strong agreement with results from direct numerical simulations. This model provides empirical evidence that the "backbone" for minimal flow unit turbulence is a periodic orbit. We then calculate the phase response curve for this periodic orbit, which describes the phase-shift of the oscillation as a function of the phase of an impulsive perturbation to the mean flow. We see that, depending on its timing, such a perturbation can either advance or retard the time at which subsequent streak breakdowns occur. This suggests how streak breakdown can be controlled through appropriately timed perturbations.