Solution Stabilization and Convergence Acceleration for the Harmonic Balance Equation System

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Ding Xi Wang ◽  
Xiuquan Huang
Keyword(s):  
Balance Equation ◽  
Harmonic Balance ◽  
Source Term ◽  
Equation System ◽  
Jacobi Method ◽  
Implicit Time ◽  
Transonic Compressor ◽  
Blade Row ◽  
Jacobi Iterations ◽  
Blade Row Interaction

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.

Stabilizing and Accelerating Solution of Harmonic Balance Equation System Using the LU-SGS and Block Jacobi Methods

2016 ◽  
Author(s):  
Xiuquan Huang ◽  
Ding Xi Wang
Keyword(s):  
Balance Equation ◽  
Harmonic Balance ◽  
Equation System ◽  
Jacobi Method ◽  
Implicit Time ◽  
Solution Stability ◽  
Reduced Frequency ◽  
Solution Convergence ◽  
Time Marching ◽  
Jacobi Iterations

The paper presents the combination of the Lower Upper Symmetric Gauss Seidel (LU-SGS) method and the block Jacobi method for stabilizing and accelerating the solution of a harmonic balance equation system. First the baseline LU-SGS procedure for a harmonic balance equation system with explicit discretization of the time spectral source term is derived. The LU-SGS method, different from its original application as an implicit time marching scheme, is used as an implicit residual smoother, allowing big CFL numbers (in order of 1000s), within the Runge-Kutta explicit time marching loops. Then the block Jacobi method is introduced to augment solution stability for the solution of a harmonic balance equation system under situation where grid reduced frequency is in the order of 10. Results are presented to show the effectiveness of the combination of LU-SGS and the block Jacobi method on the solution stabilization and acceleration. The influence of the number of Jacobi iterations on solution convergence is also investigated.

A Numerical Investigation of the Interaction Between Rotor Bow Waves and Upstream Wake Generators in a Transonic Compressor Stage

2003 ◽  
Author(s):  
Gregory Bloch ◽  
James Loellbach ◽  
Chunill Hah
Keyword(s):  
Numerical Investigation ◽  
Rotor Blade ◽  
Similar Case ◽  
Axial Compressor ◽  
Navier Stokes ◽  
Transonic Compressor ◽  
Blade Row ◽  
Transonic Axial Compressor ◽  
Main Effects ◽  
Blade Row Interaction

A numerical investigation of unsteady blade row interaction in a transonic axial compressor was performed. The compressor consists of an upstream wake generator (WG) blade row followed by a rotor blade row. Blade row interaction consists of two main effects: the downstream influence on the rotor flowfield of wakes and unsteady vortices shed from the wake generator, and the upstream influence on the wake generator of the rotor bow shock waves. An unsteady, two-dimensional, Navier-Stokes simulation was performed at the 75% span location of the compressor. Results from the numerical simulation are compared to previously reported numerical results and to experimental measurements from a similar case.

Stator-Rotor Interactions in a Transonic Compressor: Part 2 — Description of a Loss Producing Mechanism

2002 ◽  
Author(s):  
Steven E. Gorrell ◽  
Theodore H. Okiishi ◽  
William W. Copenhaver
Keyword(s):  
Pressure Wave ◽  
Pressure Ratio ◽  
Bow Shock ◽  
Trailing Edge ◽  
Transonic Compressor ◽  
Blade Row ◽  
Wave Forms ◽  
Stator Blade ◽  
Blade Row Interaction ◽  
Additional Loss

A previously unidentified loss producing mechanism resulting from the interaction of a transonic rotor blade-row with an upstream stator blade-row is described. This additional loss occurs only when the two blade rows are spaced closer together axially. Time-accurate simulations of the flow and high-response static pressure measurements acquired on the stator blade surface reveal important aspects of the fluid dynamics of the production of this additional loss. At close spacing the rotor bow shock is chopped by the stator trailing edge. The chopped bow shock becomes a pressure wave on the upper surface of the stator that is nearly normal to the flow and that propagates upstream. In the reference frame relative to this pressure wave, the flow is supersonic and thus a moving shock wave that produces an entropy rise and loss is experienced. The effect of this outcome of blade-row interaction is to lower the efficiency, pressure ratio, and mass flow rate observed as blade-row axial spacing is reduced from far to close. The magnitude of loss production is affected by the strength of the bow shock and how much it turns as it interacts with the trailing edge of the stator. At far spacing the rotor bow shock degenerates into a bow wave before it interacts with the stator trailing edge and no significant pressure wave forms on the stator upper surface. For this condition, no additional loss is produced.

Stator-Rotor Interactions in a Transonic Compressor—Part 2: Description of a Loss-Producing Mechanism

2003 ◽  
Vol 125 (2) ◽  
pp. 336-345 ◽  
Author(s):  
Steven E. Gorrell ◽  
Theodore H. Okiishi ◽  
William W. Copenhaver
Keyword(s):  
Pressure Wave ◽  
Pressure Ratio ◽  
Bow Shock ◽  
Trailing Edge ◽  
Transonic Compressor ◽  
Blade Row ◽  
Wave Forms ◽  
Stator Blade ◽  
Blade Row Interaction ◽  
Additional Loss

A previously unidentified loss producing mechanism resulting from the interaction of a transonic rotor blade row with an upstream stator blade row is described. This additional loss occurs only when the two blade rows are spaced closer together axially. Time-accurate simulations of the flow and high-response static pressure measurements acquired on the stator blade surface reveal important aspects of the fluid dynamics of the production of this additional loss. At close spacing the rotor bow shock is chopped by the stator trailing edge. The chopped bow shock becomes a pressure wave on the upper surface of the stator that is nearly normal to the flow and that propagates upstream. In the reference frame relative to this pressure wave, the flow is supersonic and thus a moving shock wave that produces an entropy rise and loss is experienced. The effect of this outcome of blade-row interaction is to lower the efficiency, pressure ratio, and mass flow rate observed as blade-row axial spacing is reduced from far to close. The magnitude of loss production is affected by the strength of the bow shock and how much it turns as it interacts with the trailing edge of the stator. At far spacing the rotor bow shock degenerates into a bow wave before it interacts with the stator trailing edge and no significant pressure wave forms on the stator upper surface. For this condition, no additional loss is produced.

Harmonic Balance Analysis of Blade Row Interactions in a Transonic Compressor

2010 ◽  
Vol 26 (2) ◽  
pp. 335-343 ◽  
Author(s):  
Kivanc Ekici ◽  
Kenneth C. Hall ◽  
Robert E. Kielb
Keyword(s):  
Harmonic Balance ◽  
Transonic Compressor ◽  
Harmonic Balance Analysis ◽  
Blade Row ◽  
Balance Analysis

scholarly journals Blade Row Interaction Effects on the Performance of a Moderately Loaded NASA Transonic Compressor Stage

2002 ◽  
Author(s):  
Dale E. Van Zante ◽  
Wai-Ming To ◽  
Jen-Ping Chen
Keyword(s):  
Experimental Data ◽  
Velocity Field ◽  
Interaction Effects ◽  
Operating Conditions ◽  
Laser Doppler Velocimeter ◽  
Compressor Stage ◽  
Peak Efficiency ◽  
Transonic Compressor ◽  
Blade Row ◽  
Blade Row Interaction

Blade row interaction effects on loss generation in compressors have received increased attention as compressor work-per-stage and blade loading have increased. Two dimensional Laser Doppler Velocimeter measurements of the velocity field in a NASA transonic compressor stage show the magnitude of interactions in the velocity field at the peak efficiency and near stall operating conditions. The experimental data are presented along with an assessment of the velocity field interactions. In the present study the experimental data are used to confirm the fidelity of a three-dimensional, time-accurate, Navier Stokes calculation of the stage using the MSU-TURBO code at the peak efficiency and near stall operating conditions. The simulations are used to quantify the loss generation associated with interaction phenomena. At the design point the stator pressure field has minimal effect of the rotor performance. The rotor wakes do have an impact on loss production in the stator passage at both operating conditions. A method for determining the potential importance of blade row interactions on performance is presented.

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