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Research Papers: Gas Turbines: Structures and Dynamics

Solution Stabilization and Convergence Acceleration for the Harmonic Balance Equation System

[+] Author and Article Information
Ding Xi Wang

School of Power and Energy,
Northwestern Polytechnical University,
127 Youyixi Road,
Xi'an 710072, China
e-mail: dingxi_wang@nwpu.edu.cn

Xiuquan Huang

School of Power and Energy,
Northwestern Polytechnical University,
127 Youyixi Road,
Xi'an 710072, China
e-mail: xiuquan_huang@nwpu.edu.cn

1Corresponding author.

Contributed by the Structures and Dynamics Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received January 16, 2017; final manuscript received January 25, 2017; published online April 11, 2017. Editor: David Wisler.

J. Eng. Gas Turbines Power 139(9), 092503 (Apr 11, 2017) (9 pages) Paper No: GTP-17-1020; doi: 10.1115/1.4035912 History: Received January 16, 2017; Revised January 25, 2017

This paper presents an efficient approach for stabilizing solution and accelerating convergence of a harmonic balance equation system for an efficient analysis of turbomachinery unsteady flows due to flutter and blade row interaction. The proposed approach combines the Runge–Kutta method with the lower upper symmetric Gauss Seidel (LU-SGS) method and the block Jacobi method. The LU-SGS method, different from its original application as an implicit time marching scheme, is used as an implicit residual smoother with under-relaxation, allowing big Courant–Friedrichs–Lewy (CFL) numbers (in the order of hundreds), leading to significant convergence speedup. The block Jacobi method is introduced to implicitly integrate the time spectral source term of a harmonic balance equation system, in order to reduce the complexity of the direct implicit time integration by the LU-SGS method. The implicit treatment of the time spectral source term thus greatly augments the stability region of a harmonic balance equation system in the case of grid-reduced frequency well above ten. Validation of the harmonic balance flow solver was carried out using linear cascade test data. Flutter analysis of a transonic rotor and blade row interaction analyses for a transonic compressor stage were presented to demonstrate the stabilization and acceleration effect by the combination of the LU-SGS and the block Jacobi methods. The influence of the number of Jacobi iterations on solution stabilization is also investigated, showing that two Jacobi iterations are sufficient for stability purpose, which is much more efficient than existing methods of its kind in the open literature.

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Figures

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Fig. 1

Comparison of (a) the energy equation residual convergence, (b) the worksum convergence, and (c) the CPU time cost between the IRS and LU-SGS methods as a residual smoother for the linear cascade flutter analysis

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Fig. 2

Static pressure coefficient comparison at three span locations

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Fig. 3

The first harmonic static pressure coefficient comparison at three span locations

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Fig. 4

The first harmonic static pressure phase angle comparison at three span locations

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Fig. 5

Comparison of (a) the energy equation residual convergence, (b) the worksum convergence, and (c) the CPU time cost between the IRS and LU-SGS methods as a residual smoother for a transonic compressor rotor flutter analysis

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Fig. 6

Blade-to-blade view at the midspan (the two passages were constructed for visualization only, and one passage only was used in the analysis) and meridional view of the computational domain

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Fig. 7

Comparison of (a) the energy equation residual convergence, (b) the polytropic efficiency convergence, and (c) the CPU time cost between different analyses

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Fig. 8

Reconstructed instantaneous entropy at 50% span of the NASA stage 35

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