Time-Spectral Method for Overlapping Meshes

Abstract

Overset and Cartesian solvers typically employ conventional time-marching schemes to simulate unsteady flows. Temporal pseudospectral schemes have demonstrated the ability to dramatically reduce the computational effort required to resolve the important subclass of time-periodic phenomena. Incorporating the time-spectral method within these approaches is desirable, but direct application is infeasible. Relative motion introduces dynamic blanking of spatial nodes which move interior to solid bodies; the solution at such nodes is therefore undefined over specific intervals of time. This proves problematic for the conventional time-spectral method because it expands the temporal variation at every node as an infinitely supported Fourier series. An extension of the time-spectral method is presented where dynamically blanked nodes are handled in an alternative manner; the solution through intervals of consecutively unblanked time samples are represented with barycentric rational interpolants. The Fourier- and rational interpolant-based differentiation operators are applied in tandem, providing a hybrid time-spectral scheme capable of consistently resolving relative motion on overlapping meshes. The hybrid scheme is applied to relevant cases in two and three spatial dimensions and the results demonstrate that the hybrid scheme mirrors the performance of the conventional time-spectral method and monotonically converges to analogous high-resolution, time-accurate simulations with increasing temporal modes.