Session: 34-05 Harmonic balance methods

Paper Number: 125758

125758 - Numerical Stability Analysis of Implicit Solution Methods for Harmonic Balance Equations 

This paper presents an investigation into the numerical stability of various implicit solution methods for efficiently solving harmonic balance equations for turbomachinery unsteady flows. A series of methods have been proposed to enhance stability and accelerate convergence of harmonic balance solutions by implicitly integrating the time spectral source term of a harmonic balance equation. These methods include the block Jacobi method(BJ), the Jacobi iteration method(JI), and their variants such as the modified block Jacobi method(MBJ) and the modified Jacobi iteration method(MJI). All the modified versions have their Courant numbers dropped for their Jacobi iterations. 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 determine 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 effect of the number of Jacobi iterations is also examined, showing that two or three Jacobi iterations are sufficient for stabilization. An interesting finding is the adverse effect of using more than three Jacobi iterations. It may reduce the stability of the LU-SGS/BJ scheme for high grid-reduced frequencies, apart from increasing computational cost. 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.

Presenting Author: Yuxuan Zhang Northwestern Polytechnical University

Presenting Author Biography: My name is Yuxuan Zhang. I'm a third-year graduate student in the School of Power and Energy, Northwestern Polytechnical University. My supervisor is Professor Dingxi Wang . My research interest is numerical scheme stability of the harmonic balance method.

Authors:

Yuxuan Zhang Northwestern Polytechnical University
Dingxi Wang Northwesten Polytechnical University
Sen Zhang Northwestern Polytechnical University
Yuze Zhu Northwestern Polytechnical University