Adjoint-based optimal control of incompressible flows with convective-like energy-stable open boundary conditions

•The adjoint system for the incompressible Navier-Stokes equations with stabilized open boundary conditions is derived.•The primal and adjoint systems are solved using the least-squares finite element method.•Example problems are solved, including the drag reduction of a cylinder in a severely truncated domain. This paper describes an open-loop optimal control strategy for flow problems governed by the incompressible Navier-Stokes equations with convective-like energy-stable boundary conditions. A quasi-Newton optimization procedure is employed, and the required objective sensitivities are computed using the continuous adjoint method. The adjoint equations for the corresponding system are derived and discussed. Both the primal and adjoint systems are solved using the least-squares finite element method. This choice circumvents the LBB condition and leads to symmetric positive-definite algebraic systems. Numerical examples are provided to show the stability and accuracy of the proposed approach.