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Abstract. Theoretical and practical issues arising in optimal boundary control of the unsteady two-dimensional compressible Navier-Stokes equations are discussed. Assuming a sufficiently smooth state, formal adjoint and gradient equations are derived. For a vortex rebound model problem wall normal suction and blowing is used to minimize cost functionals of interest, here the kinetic energy at the final time.

A discontinuous Galerkin (DG) method is formulated, implemented, and tested for simulation of compressible turbulent flows. The method is applied to a range of test problems including steady and unsteady flow over a circular cylinder, inviscid flow over an inclined ellipse, and fully developed turbulent flow in a planar channel. In all cases, local hp-refinement is utilized to obtain high quality solutions with fewer degrees of freedom than traditional numerical methods. The formulation and validation cases presented here lay the foundation for future applications of DG for simulation of compressible turbulent flows in complex geometries.

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 on a subset of an otherwise 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 at inflow versus outflow boundaries. Another important issue is the derivation of adjoint equations which are required to compute 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 results for three 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.