Adjoint-consistency of a spacetime discontinuous Galerkin discretization for the incompressible Navier-Stokes equations is considered and the spacetime continuous and discrete adjoint problems are derived. The spacetime discretization is used to compute primal and discrete adjoint solutions to problems involving an incompressible fluid moving through a channel with oscillating walls. We consider both the case of a prescribed inflow velocity that is parabolic and a pressure driven flow at Reynolds numbers of 100 and 250. In all cases, separation and reconnection is observed, with recirculation regions forming at one quarter wave period and repeating every half wave period. The adjoint solution is shown to be smooth at surfaces over which the cost function is evaluated, indicating the discretization is adjoint consistent.
展开▼
