Time Spectral Method for Periodic and Quasi-Periodic Unsteady Computations on Unstructured Meshes

Abstract

For flows with strong periodic content, time-spectral methods can be used to obtain time-accurate solutions at substantially reduced cost compared to traditional time-implicit meth- ods which operate directly in the time domain. However, these methods are only applicable in the presence of fully periodic flows, which represents a severe restriction for many aerospace en- gineering problems. This paper presents an extension of the time-spectral approach for problems that include a slow transient in addition to strong periodic behavior, suitable for applications such as transient turbofan simulation or maneuvering rotorcraft calculations. The formulation is based on a collocation method which makes use of a combination of spectral and polynomial basis func- tions and results in the requirement of solving coupled time instances within a period, similar to the time spectral approach, although multiple successive periods must be solved to capture the transient behavior. The implementation allows for two levels of parallelism, one in the spatial dimension, and an- other in the time-spectral dimension, and is implemented in a modular fashion which minimizes the modifications required to an existing steady-state solver. For dynamically deforming mesh cases, a formulation which preserves discrete conservation as determined by the Geometric Con- servation Law is derived and implemented. A fully implicit approach which takes into account the coupling between the various time instances is implemented and shown to preserve the baseline steady-state multigrid convergence rate as the number of time instances is increased. Accuracy and efficiency are demonstrated for periodic and non-periodic problems by comparing the performance of the method with a traditional time-stepping approach using a simple two-dimensional pitching airfoil problem, a three-dimensional pitching wing problem, and a more realistic transitioning rotor problem.