This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis on its impact on the scaling of the work-per-cycle curves, using an asymptotic approach. A perturbation analysis of the linearized Navier–Stokes equations for real modes at low reduced frequency is presented and some conclusions are drawn. The first important result is that the loading of the airfoil plays an essential role in the trends of the phase and modulus of the unsteady pressure caused by the vibration of the airfoil. For lightly loaded airfoils, the unsteady pressure and the influence coefficients (ICs) scale linearly with the reduced frequency whereas the phase departs from π/2 and changes linearly with the reduced frequency. As a consequence, the work-per-cycle scales linearly with the reduced frequency for any interblade phase angle (IBPA), and it is independent of its sign. For highly loaded airfoils, the unsteady pressure modulus is fairly constant exhibiting only a small correction with the reduced frequency, while the phase departs from zero and varies linearly with it. In this case, only the mean value of the work-per-cycle scales linearly with the reduced frequency. This behavior is independent of the geometry of the airfoil and the mode shape in first-order approximation in the reduced frequency. For symmetric cascades, the work-per-cycle scales linearly with the reduced frequency irrespective of whether the airfoil is loaded or not. These conclusions have been numerically confirmed in Part II of the paper.

References

References
1.
Corral
,
R.
,
Gallardo
,
J. M.
, and
Vasco
,
C.
,
2007
, “
Aeroelastic Stability of Welded-in-Pair Low Pressure Turbine Rotor Blades: A Comparative Study Using Linear Methods
,”
ASME J. Turbomach.
,
129
(
1
), pp.
72
83
.
2.
Corral
,
R.
,
Gallardo
,
J. M.
, and
Martel
,
C.
,
2009
, “
A Conceptual Flutter Analysis of a Packet of Vanes Using a Mass-Spring Model
,”
ASME J. Turbomach.
,
131
(
2
), p.
021016
.
3.
Buffum
,
D.
, and
Fleeter
,
S.
,
1990
, “
The Aerodynamics of an Oscillating Cascade in a Compressible Flow Field
,”
ASME J. Turbomach.
,
112
(
4
), pp.
759
767
.
4.
He
,
L.
,
1998
, “
Unsteady Flow in Oscillating Turbine Cascades: Part 2—Computational Study
,”
ASME J. Turbomach.
,
120
(
2
), pp.
269
275
.
5.
Nowinski
,
M.
, and
Panovsky
,
J.
,
2000
, “
Flutter Mechanisms in Low Pressure Turbine Blades
,”
ASME J. Eng. Gas Turbines Power
,
122
(
1
), pp.
89
98
.
6.
Theodorsen
,
T.
,
1935
, “
General Theory of Aerodynamic Instability and the Mechanism of Flutter
,” National Advisory Committee for Aeronautics, Langley Aeronautical Laboratory, Langley Field, VA, Report No.
NACA
-TR-496.
7.
Moore
,
F.
,
1951
, “
Unsteady Laminar Boundary Layer Flow
,” National Advisory Committee for Aeronautics, Lewis Flight Propulsion Laboratory, Cleveland, OH, Report No.
NACA
-TN-2471.
8.
Bölcs
,
A.
, and
Fransson
,
T. H.
,
1986
, “
Aeroelasticity in Turbomachines: Comparison of Theoretical and Experimental Cascade Results
,” Laboratoire de Thermique Appliquee et de Turbomachines, EPFL, Lausanne, Switzerland, Technical Report No. 13.
9.
Vega
,
A.
, and
Corral
,
R.
,
2013
, “
Physics of Vibrating Airfoils at Low Reduced Frequency
,”
ASME
Paper No. GT2013-94906.
10.
Waite
,
J.
, and
Kielb
,
R.
,
2014
, “
Physical Understanding and Sensitivities of Low Pressure Turbine Flutter
,”
ASME J. Turbomach.
,
137
(
1
), p.
012502
.
11.
Vega
,
A.
, and
Corral
,
R.
,
2015
, “
The Low Reduced Frequency Limit of Vibrating Airfoils—Part II: Numerical Experiments
,”
ASME J. Turbomach.
(in press).
12.
Sbardella
,
L.
, and
Imregun
,
M.
,
2001
, “
Linearized Unsteady Viscous Turbomachinery Flows Using Hybrid Grids
,”
ASME J. Turbomach.
,
123
(
3
), pp.
568
582
.
13.
Corral
,
R.
,
Escribano
,
A.
,
Gisbert
,
F.
,
Serrano
,
A.
, and
Vasco
,
C.
,
2003
, “
Validation of a Linear Multigrid Accelerated Unstructured Navier–Stokes Solver for the Computation of Turbine Blades on Hybrid Grids
,”
AIAA
Paper No. 2003-3326.
14.
Verdon
,
J.
, and
Caspar
,
J.
,
1984
, “
A Linearized Unsteady Aerodynamic Analysis for Transonic Cascades
,”
J. Fluid Mech.
,
149
, pp.
403
429
.
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