Near-optimal source placement in forced convection

We consider the problem of optimal source placement, given the velocity field and sink distribution in forced convection. We use the ratio of scalar concentration variance in presence of advection compared to the purely diffusive case as our cost function. Using a semidefinite relaxation of the original optimization problem, we obtain near-optimal source distributions for several classes of velocity fields in 2D. We compare the numerical results with bounds obtained by analysis of space-time averaged advection-diffusion equation. The dependence of optimal variance reduction on the Lagrangian properties of the velocity field and shape of sink distribution are made explicit in this analysis. This work extends the earlier work in (Thiffeault, J. L., \& Pavliotis, G. A. (2008). Optimizing the source distribution in fluid mixing. Physica D: Nonlinear Phenomena, 237(7), 918-929), and provides a systematic framework that can be extended to more realistic models of forced and mixed convection.