Investigation of the Sigma Approximation Technique for the Solution of the Time Spectral Equation System

The time spectral method (TSM) is commonly utilized to analyze periodic unsteady flows within turbomachines. However, the time spectral solutions in regions with large temporal and spatial gradients, such as wakes, shocks, and boundary layers, can be plagued with unphysical oscillations, known as the Gibbs phenomenon. This paper presents an investigation into the sigma approximation technique, which is capable of significantly attenuating the Gibbs phenomenon by dampening high-frequency nonlinear components in time spectral solutions. Central to this technique are three sigma parameters: the exponent, the cutoff number, and the number of harmonics, which collectively determine the damping effects for each frequency. However, the optimal combination of them remains to be ascertained considering the tradeoff between the accuracy of solutions and the reduction of unphysical oscillations. In this study, a two-row compressor configuration is first selected, and numerical simulations are performed to evaluate the efficacy of this technique under near-peak efficiency and near-stall operating conditions. It is found that Gibbs-type unphysical oscillations can be effectively mitigated without notably compromising solution accuracy through an optimal combination of sigma parameters. The NASA Stage 35 case study further verifies the effectiveness of the proposed optimal sigma parameter combination in preserving the accuracy of time spectral solutions based on the data from test rig. Moreover, the von Neumann analysis reveals that the sigma approximation technique can enhance the numerical stability of the time spectral equation system, thereby allowing the use of aggressive numerical parameters to accelerate convergence. The stability analysis results are verified by a two-dimensional bump under two different flow conditions and further assessed using the two-row compressor.