Optimal Control of Aeroacoustic Flows: Transpiration Boundary Control

We consider the optimal boundary control of aeroacoustic noise governed by the two-dimensional unsteady compressible Euler equations. The control is the time and space varying wall-normal velocity (transpiration velocity) on a subset of a solid wall. The objective functional to be minimized is a measure of acoustic amplitude. Optimal transpiration boundary control of aeroacoustic noise introduces challenges beyond those encountered in direct aeroacoustic simulations or in many other optimization problems governed by compressible Euler equations. One nontrivial issue that arises in our optimal control problem is the formulation and implementation of transpiration boundary conditions. Since we allow suction and blowing on the boundary, portions of the boundary may change from inflow to outflow, or vice versa, and different numbers of boundary conditions must be imposed depending on whether a boundary portion is an inflow or an outflow boundary. Another important issue is the derivation of adjoint equations, which are needed for the computation of the gradient of the objective function with respect to the control. Among other things, this is influenced by the choice of boundary conditions for the compressible Euler equations. This paper describes our approaches to meet these challenges and presents first results for two model problems. These problems are designed to validate our transpiration boundary conditions and their implementation, study the accuracy of gradient computations, and assess the performance of the computed controls.