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 methods 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 engineering 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 functions 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 another 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 Conservation 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.