Numerical Solution of a Flow-Control Problem: Vorticity Reduction by Dynamic Boundary Action

In order to laminarize an unsteady, internal flow, the vorticity field is minimized, in a least-squares sense, using an optimal-control approach. The flow model is the Navier--Stokes equation for a viscous incompressible fluid, and the flow is controlled by suction and blowing on a part of the boundary. A quasi-Newton method is used for the minimization of a quadratic objective function involving a measure of the vorticity and a regularization term. The Navier--Stokes equations are approximated using a finite-difference scheme in time and finite-element approximations in space. Accurate expressions for the gradient of the discrete objective function are needed to obtain a satisfactory convergence rate of the minimization algorithm. Therefore, first-order necessary conditions for a minimizer of the objective function are derived in the fully discrete case. A memory-saving device is discussed without which problems of any realistic size, especially in three space dimensions, would remain computationally intractable. The feasibility of the optimal-control approach for flow-control problems is demonstrated by numerical experiments for a two-dimensional flow in a rectangular cavity at a Reynolds number high enough for nonlinear effects to be important.