Mesh adaptation criteria for unsteady periodic flows using a discrete adjoint time-spectral formulation

This papers develops and implements a discrete adjoint formulation for a periodic, time-spectral discretization of the equations of fluid flow. Time-spectral flow solution methodologies have been proposed in order to significantly shorten the computational time for the solution of unsteady (periodic) flows with a narrow frequency spectrum. As such, these methodologies are only applicable to periodic flows encountered, for example, in turbomachinery, rotorcraft, and fixed-wing aeroelastic applications among other areas. For these classes of problems, the time-spectral discretization permits the solution of the corresponding adjoint equations and facilitates both design and error estimation / correction applications. Details of the discrete adjoint solver and its implementations are provided as well as its validation (through the calculation of sensitivities of unsteady flows) and some preliminary applications to mesh refinement / distribution in unsteady periodic flows. The procedure is found to be very robust and it is currently being extended to fundamental turbomachinery test cases.