Optimal mixing in recirculation zones

Coarse-scale mixing in a recirculation zone is described with a simple vortex model. Time-dependent forcing is employed to change the vortex motion and mixing properties. An optimal mixing problem is defined in which the flux across the recirculation region shall be maximized under the side-constraints of bounded vortex motion and bounded actuation. Concepts of control theory and chaotic advection are used to achieve this goal. In particular, controllability is proven with a transformation into flat coordinates. Thus, a feedforward law for the optimal trajectory and a feedback law for its stabilization are derived. Observability of the vortex motion is indicated by a dynamic observer. Mixing in the optimized flow is studied using Poincaré maps. The low-frequency modulations to vortex motion are shown to substantially increase mixing in the average. Generalizations of the mathematical framework for mixing optimization are suggested for a larger class of models and flows.