In this paper, we present an analysis of the unsteady aerodynamic response of cascades due to incident gusts (the forced response problem) or blade vibration (the flutter problem) when the cascade is part of a multistage fan, compressor, or turbine. Most current unsteady aerodynamic models assume the cascade to be isolated in an infinitely long duct. This assumption, however, neglects the potentially important influence of neighboring blade rows. We present an elegant and computationally efficient method to model these neighboring blade row effects. In the present method, we model the unsteady aerodynamic response due to so-called spinning modes (pressure and vorticity waves), with each mode corresponding to a different circumferential wave number and frequency. Then, for each mode, we compute the reflection and transmission coefficients for each blade row. These coefficients can be obtained from any of the currently available unsteady linearized aerodynamic models of isolated cascades. A set of linear equations is then constructed that couples together the various spinning modes, and the linear equations are solved via LU decomposition. Numerical results are presented for both the gust response and blade vibration problems. To validate our model, we compare our results to other analytical models, and to a multistage vortex lattice model. We show that the effect of neighboring blade rows on the aerodynamic damping of vibrating cascades is significant, but nevertheless can be modeled with a small number of modes.

1.
Buffum, D. H., 1993, “Blade Row Interaction Effects on Flutter and Forced Response,” Paper No. AIAA-93-2084; also NASA TM 106438.
2.
Giles
M. B.
,
1988
, “
Calculation of Unsteady Wake/Rotor Interaction
,”
Journal of Propulsion and Power
, Vol.
4
, No.
4
, pp.
356
362
.
3.
Hall
K. C.
,
1993
, “
Deforming Grid Variational Principle for Unsteady Small Disturbance Flows in Cascades
,”
AIAA Journal
, Vol.
31
, No.
5
, pp.
891
900
.
4.
Hall
K. C.
, and
Clark
W. S.
,
1993
, “
Linearized Euler Predictions of Unsteady Aerodynamic Loads in Cascades
,”
AIAA Journal
, Vol.
31
, No.
3
, pp.
540
550
.
5.
Hall
K. C.
, and
Crawley
E. F.
,
1989
, “
Calculation of Unsteady Flows in Turbomachinery Using the Linearized Euler Equations
,”
AIAA Journal
, Vol.
27
, No.
6
, pp.
777
787
.
6.
Hall
K. C.
, and
Lorence
C. B.
,
1993
, “
Calculation of Three-Dimensional Unsteady Flows in Turbomachinery Using the Linearized Harmonic Euler Equations
,”
ASME JOURNAL OF TURBOMACHINERY
, Vol.
115
, pp.
800
809
.
7.
Hanson, D. B., 1992, “Unsteady Coupled Cascade Theory Applied to the Rotor/Stator Interaction Noise Problem,” DGLR/AIAA Paper No. 92-02-084.
8.
Hanson, D. B., 1993, “Mode Trapping in Coupled 2D Cascades-Acoustic and Aerodynamic Results” AIAA Paper No. 93-4417.
9.
Hanson, D. B., 1994, private communication.
10.
Kaji
S.
, and
Okazaki
T.
,
1970
, “
Generation of Sound by Rotor-Stator Interaction
,”
J. Sound and Vibration
, Vol.
13
, No.
3
, pp.
281
307
.
11.
Kemp
N. H.
, and
Sears
W. R.
,
1955
, “
The Unsteady Forces Due to Viscous Wakes in Turbomachines
,”
Journal of the Aeronautical Sciences
, Vol.
22
, No.
7
, pp.
478
483
.
12.
Manwaring
S. R.
, and
Wisler
D. C.
,
1993
, “
Unsteady Aerodynamics and Gust Response in Compressors and Turbines
,”
ASME JOURNAL OF TURBOMACHINERY
, Vol.
115
, pp.
724
740
.
13.
Rai
M. M.
,
1989
a, “
Three-Dimensional Navier–Stokes Simulations of Turbine Rotor-Stator Interaction: Part I—Methodology
,”
Journal of Propulsion and Power
, Vol.
5
, No.
3
, pp.
307
311
.
14.
Rai
M. M.
,
1989
b, “
Three-Dimensional Navier–Stokes Simulations of Turbine Rotor-Stator Interaction: Part II—Results
,”
Journal of Propulsion and Power
, Vol.
5
, No.
3
, pp.
312
319
.
15.
Smith, S. N., 1972, “Discrete Frequency Sound Generation in Axial Flow Turbomachines,” Aeronautical Research Council Reports and Memoranda, R. & M. No. 3709.
16.
Verdon, J. M., 1987, “Linearized Unsteady Aerodynamic Theory,” AGARD Manual on Aeroelasticity in Axial Flow Turbomachines, Vol. 1, Unsteady Turbo-machinery Aerodynamics (AGARD-AG-298), M. F. Platzer and F. O. Carta, eds., Neuilly sur Seine, France, Chap. 2.
17.
Whitehead, D. S., 1987, “Classical Two-Dimensional Methods,” AGARD Manual on Aeroelasticity in Axial Flow Turbomachines, Vol. 1, Unsteady Turbo-machinery Aerodynamics (AGARD-AG-298), M. F. Platzer and F. O. Carta, eds., Neuilly sur Seine, France, Chap. 3.
18.
Whitehead, D. S., and Grant, R. J., 1981, “Force and Moment Coefficients of High Deflection Cascades,” Proc. 2nd International Symposium on Aeroelasticity in Turbomachines, P. Suter, ed., Juris-Verlag, Zurich, pp. 85–127.
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