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.

References

References
1.
Hall
,
K.
,
Thomas
,
J.
, and
Clark
,
W.
,
2002
, “
Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique
,”
AIAA J.
,
40
(
5
), pp.
879
886
.
2.
Gopinath
,
A.
,
van der Weide
,
E.
,
Alonso
,
J.
,
Jameson
,
A.
,
Ekici
,
K.
, and
Hall
,
K.
,
2007
, “
Three-Dimensional Unsteady Multi-Stage Turbomachinery Simulations Using the Harmonic Balance Technique
,”
AIAA
Paper No. 2007-892.
3.
van der Weide
,
E.
,
Gopinath
,
A. K.
, and
Jameson
,
A.
,
2005
, “
Turbomachinery Applications With the Time Spectral Method
,”
AIAA
Paper No. 2005-4905.
4.
Ekici
,
K.
,
Hall
,
K. C.
, and
Dowell
,
E. H.
,
2008
, “
Computationally Fast Harmonic Balance Methods for Unsteady Aerodynamic Predictions of Helicopter Rotors
,”
ASME J. Turbomach.
,
227
(
12
), pp.
6206
6225
.
5.
Frey
,
C.
,
Ashcroft
,
G.
,
Kersken
,
H.-P.
, and
Voigt
,
C.
,
2014
, “
A Harmonic Balance Technique for Multistage Turbomachinery Applications
,”
ASME
Paper No. GT2014-25230.
6.
Yang
,
Z.
, and
Mavriplis
,
D. J.
,
2006
, “
Time Spectral Method for Quasi-Periodic Unsteady Computation on Unstructured Meshes
,”
AIAA
Paper No. 2010-5034.
7.
Du
,
P.
, and
Ning
,
F.
,
2014
, “
Application of the Harmonic Balance Method in Simulating Almost Periodic Turbomachinery Flows
,”
ASME
Paper No. GT2014-25457.
8.
Hall
,
K.
,
Ekicib
,
K.
,
Thomas
,
J. P.
, and
Dowell
,
E. H.
,
2013
, “
Harmonic Balance Methods Applied to Computational Fluid Dynamics Problems
,”
Int. J. Comput. Fluid Dyn.
,
27
(
2
), pp.
52
67
.
9.
Swanson
,
R.
,
Turkel
,
E.
, and
Rossow
,
C.-C.
,
2007
, “
Convergence Acceleration of Runge-Kutta Schemes for Solving the Navier-Stokes Equations
,”
J. Comput. Phys.
,
224
(
1
), pp.
365
388
.
10.
Yoon
,
S.
, and
Jameson
,
A.
,
1988
, “
Lower-Upper Symmetric-Gauss-Seidel Method for the Euler and Navier-Stokes Equations
,”
AIAA J.
,
26
(
9
), pp.
1025
1026
.
11.
Su
,
X.
, and
Yuan
,
X.
,
2010
, “
Implicit Solution of Time Spectral Method for Periodic Unsteady Flows
,”
Int. J. Numer. Methods Fluids
,
63
(
7
), pp.
860
876
.
12.
Ma
,
C.
,
Su
,
X.
,
Gou
,
J.
, and
Yuan
,
X.
,
2014
, “
Runge-Kutta/Implicit Scheme for the Solution of Time Spectral Method
,”
ASME
Paper No. GT2014-26474.
13.
Sicot
,
F.
,
Puigt
,
G.
, and
Montagnac
,
M.
,
2008
, “
Block-Jacobi Implicit Algorithms for the Time Spectral Method
,”
AIAA J.
,
46
(
12
), pp.
3080
3089
.
14.
Crespo
,
J.
,
Corral
,
R.
, and
Pueblas
,
J.
,
2016
, “
An Implicit Harmonic Balance Method in Graphics Processing Units for Oscillating Blades
,”
ASME J. Turbomach.
,
138
(
3
), p.
031001
.
15.
Im
,
D.-K.
,
Choi
,
S.
, and
Kwon
,
J. H.
,
2015
, “
Unsteady Rotor Flow Analysis Using a Diagonally Implicit Harmonic Balance Method and an Overset Mesh Topology
,”
Int. J. Comput. Fluid Dyn.
,
29
(
1
), pp.
82
89
.
16.
Weiss
,
J. M.
,
Subramanian
,
V.
, and
Hall
,
K. C.
,
2011
, “
Simulation of Unsteady Turbomachinery Flows Using an Implicitly Coupled Nonlinear Harmonic Balance Method
,”
ASME
Paper No. GT2011-46367.
17.
Alton James
,
L. I.
,
2013
, “
A Block-Jacobi Time-Spectral Method for Incompressible Flow
,” Ph.D. thesis, University of Michigan, Ann Arbor, MI.
18.
Yoon
,
S.
, and
Jameson
,
A.
,
1987
, “
An LU-SSOR Scheme for the Euler and Navier-Stokes Equations
,”
AIAA
Paper No. 87-0600.
19.
Spalart
,
P.
, and
Allmaras
,
S.
,
1992
, “
A One-Equation Turbulence Model for Aerodynamic Flows
,”
AIAA
Paper No. 92-0439.
20.
Jameson
,
A.
,
Schmidt
,
W.
, and
Turkel
,
E.
,
1981
, “
Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes
,”
AIAA
Paper No. 81-1259.
21.
Arnone
,
A.
,
1993
, “
Viscous Analysis of Three-Dimensional Rotor Flow Using a Multigrid Method
,” NASA Lewis Research Center, Cleveland, OH, Report No.
NASA-TM-106266
.
22.
Sicota
,
F.
,
Guédeney
,
T.
, and
Dufour
,
G.
,
2013
, “
Time-Domain Harmonic Balance Method for Aerodynamic and Aeroelastic Simulations of Turbomachinery Flows
,”
Int. J. Comput. Fluid Dyn.
,
27
(
2
), pp.
68
78
.
23.
Guédeney
,
T.
,
Gomar
,
A.
,
Gallard
,
F.
,
Sicot
,
F.
,
Dufour
,
G.
, and
Puigt
,
G.
,
2013
, “
Non-Uniform Time Sampling for Multiple-Frequency Harmonic Balance Computations
,”
J. Comput. Phys.
,
236
, pp.
317
345
.
24.
Reid
,
L.
, and
Moore
,
R. D.
,
1978
, “
Performance of Single-Stage Axial-Flow Transonic Compressor With Rotor and Stator Aspect Ratios of 1.19 and 1.26, Respectively, and With Design Pressure Ratio of 1.82
,” NASA Lewis Research Center, Cleveland, OH, Report No.
NASA-TP-200
.
You do not currently have access to this content.