Optimal control in wall bounded flows

Optimal control of transition in channel flow and boundary layer flow is attempted. First the optimization problem is stated and the corresponding adjoint equations used to compute the gradient of the objective function are derived for both the channel flow and boundary layer flow problems. Implementation and numerical issues are discussed. The governing equations are the incompressible Navier-Stokes equations with appropriate boundary conditions for the two cases. The boundary condition on the wall normal velocity at the walls of the channel, or at the single wall in the boundary layer case, is used as control and is determined in the iterative optimization procedure. The objective function used for the optimization problem contains the perturbation energy and a regularization term containing the control. The optimization problem is formulated using the primitive variables - velocity and pressure - and is then rewritten in a formulation containing only the wall normal velocity and the wall normal vorticity. An existing solver for the incompressible Navier-Stokes equations using this formulation can then also be used to solve the associated adjoint problem. The implementation is straightforward using this formulation and the efficiency of the original solver is maintained. To test the performance of the solver of the optimization problem, it is applied on different stages of the oblique transition scenario in the channel flow case. In a parallel Falkner-Skan-Cooke flow successful control of an inviscid instability is reported, and in the spatial Blasius flow the energy growth of a Tollmien-Schlichting wave is efficiently inhibited.