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Received Nov 26, 2017; Accepted Feb 15, 2018
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1. Introduction
Challenges in helicopter rotor flow analysis are several. Blade wake interactions need to be solved accurately requiring higher mesh resolution in the wake-trailing regions. Moving boundaries of the rotating blades need to be handled dynamically in time [1–5]. A typical local, finite difference method for the time integration of the flow governing equations of the URANS (unsteady Reynolds-averaged Navier-Stokes equations), such as BDF2 (2-step backward differentiation formula) or BDF3 (3-step backward differentiation formula), of the dual-time-stepping method [6, 7] requires a small time step for solution stability. Although the pseudo-time integration for the inner iteration and a multigrid technique can make the solution converge faster, the time-marching computation is still expensive even with highly parallel computation as it needs to fully resolve unsteady transient flows before flows reach a periodic steady-state. On the other hand, a spectral method for time integration has been proposed [8–16] as an alternative to a time-marching, dual-time stepping method with much reduced computational cost at equivalent solution accuracy. With the flow solution approximation by a discrete Fourier series, a time derivative term of the governing equations reduces to a spectral derivative term and removes time dependence of the flow solutions. The time-spectral derivative is the multiplication of the spectral derivative matrix and solution vectors at all time instances. The flow solutions advance in a pseudo-time domain with periodic steady-state remaining throughout the iterations until flows converge. Variants of the time-spectral method have been developed by various researchers. Hall et al. [8] originally developed a harmonic balance method (HBM) to solve internal flows of a two-dimensional compressor and use a discrete Fourier series for the solution approximation. The improved HBM by Thomas et al. [9, 10] also approximates flow variables by a discrete Fourier series and represents the time derivative term of the governing equations in Fourier coefficients but solves the FFT-transformed governing equations directly in the time domain by taking the inverse transform of the frequency domain governing equations. This approach is more convenient as the flow solutions are integrated...