Time-Spectral Method for Overlapping Meshes

Abstract

Overset and Cartesian solvers have traditionally been employed to efficiently resolve unsteady flows with conventional time-marching methods. Temporal pseudospectral schemes have demonstrated the ability to dramatically reduce the computational effort required to simulate the important subclass of time-periodic phenomena. Incorporation of the Time-Spectral method within these approaches is desirable, but direct application is infeasible. Relative motion in an overset framework introduces the dynamic blanking of spatial nodes which move interior to the impermeable boundaries of solid bodies; the solution at these nodes is therefore undefined over specific intervals of time. This proves problematic for the the conventional Time-Spectral approach, as it expands the temporal variation at every node as an infinitely-supported Fourier series. An extension of the Time-Spectral method is outlined that treats the solution at dynamically-blanked nodes in an alternative manner. The standard Fourier differentiation operator and differentiation operators derived from barycentric rational interpolants are applied in tandem providing a hybrid Time-Spectral scheme capable of consistently resolving relative motion. The proposed scheme is applied to the cases of high-amplitude and high-frequency plunging airfoils, and the results compared against corresponding simulations obtained with a traditional time marching scheme. The results demonstrate that the hybrid scheme mirrors the performance of the conventional Time-Spectral method, and monotonically converges to corresponding high-resolution, time-accurate simulations with increasing temporal modes.