For flows with strong periodic content, time-spectral methods can be used to obtaintime-accurate solutions at substantially reduced cost compared to traditionaltime-implicit methods which operate directly in the time domain. However, these methodsare only applicable in the presence of fully periodic flows, which represents a severerestriction for many aerospace engineering problems. This paper presents an extension ofthe time-spectral approach for problems that include a slow transient in addition tostrong periodic behavior, suitable for applications such as transient turbofan simulationor maneuvering rotorcraft calculations. The formulation is based on a collocation methodwhich makes use of a combination of spectral and polynomial basis functions and results inthe requirement of solving coupled time instances within a period, similar to the timespectral approach, although multiple successive periods must be solved to capture thetransient behavior.

The implementation allows for two levels of parallelism, one in the spatial dimension,and another in the time-spectral dimension, and is implemented in a modular fashion whichminimizes the modifications required to an existing steady-state solver. For dynamicallydeforming mesh cases, a formulation which preserves discrete conservation as determined bythe Geometric Conservation Law is derived and implemented. A fully implicit approach whichtakes into account the coupling between the various time instances is implemented andshown to preserve the baseline steady-state multigrid convergence rate as the number oftime instances is increased. Accuracy and efficiency are demonstrated for periodic andnon-periodic problems by comparing the performance of the method with a traditionaltime-stepping approach using a simple two-dimensional pitching airfoil problem, athree-dimensional pitching wing problem, and a more realistic transitioning rotor problem.