Abstract
This paper presents an investigation into the numerical stability of various implicit solution methods for an efficient solution of harmonic balance equations for turbomachinery unsteady flows. Those implicit solution methods were proposed to enhance stability and accelerate the convergence of harmonic balance solutions by implicitly integrating the time spectral source term of a harmonic balance equation. They include the block Jacobi method (BJ), the Jacobi iteration method (JI), and their variants, i.e. the modified block Jacobi method (MBJ) and the modified Jacobi iteration method (MJI). These implicit treatments are typically combined with the lower upper symmetric Gauss-Seidel method (LU-SGS) as a preconditioner of a Runge-Kutta scheme. In this study, the von Neumann analysis is applied to evaluate the stability and damping properties of all these methods. The findings reveal that the LU-SGS/BJ and LU-SGS/MJI schemes can allow larger Courant numbers, in the order of hundreds, leading to a significant convergence speedup, while the LU-SGS/MBJ and LU-SGS/JI schemes fail to stabilize the solution, resulting in a Courant number below 10 as the grid-reduced frequency increases. The stabilization effect of the number of Jacobi iterations is also investigated. It is found that the minimum allowable relaxation factor does not change monotonically with the number of Jacobi iterations. Typically 2–4 Jacobi iterations are suggested for the stability and computational efficiency consideration, while any value larger than 4 is not recommended. The stability analysis results are numerically verified by solving the harmonic balance equation system for two cases. One is the inviscid flow over a two-dimensional bump with a pressure disturbance at the outlet. The other is the turbulent flow in a three-dimensional transonic compressor stage.
