NASA / TM--2001-210693

Aeroelastic Stability Computations

for Turbomachinery

R. Srivastava, M.A. Bakhle, and T.G Keith, Jr.

University of Toledo, Toledo, Ohio

G.L. Stefko

Glenn Research Center, Cleveland, Ohio

Prepared for the

International Forum on Aeroelasticity and Structural Dynamics

cosponsored by the CEAS, AIAA, and AIAE

Madrid, Spain, June 5-7, 2001

National Aeronautics and

Space Administration

Glenn Research Center

June 2001
Acknowledgments

This work was supported by the Advanced Subsonic Technology and the Ultra Efficient Engine

Technology Projects through a grant from NASA Glenn Research Center.

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AEROELASTIC STABILITY COMPUTATIONS

FOR TURBOMACHINERY

R. Srivastava, M.A. Bakhle, and T.G. Keith, Jr.

University of Toledo

Toledo, Ohio

G.L. Stefko

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio

Abstract. This paper describes an aeroelastic analysis program for turbomachines. Unsteady

Navier-Stokes equations are solved on dynamically deforming, body fitted, grid to obtain the

aeroelastic characteristics. Blade structural response is modeled using a modal representation of

the blade and the work-per-cycle method is used to evaluate the stability characteristics. Nonzero inter-blade phase angle is modeled using phase-lagged boundary conditions. Results

obtained showed good correlation with existing experimental, analytical and numerical results.

Numerical analysis also showed that given the computational resources available today,

engineering solutions with good accuracy are possible using higher fidelity analyses.

1 INTRODUCTION

Aeroelastic problems contribute significantly to the development and maintenance costs of

aircraft engines. In turbomachinery, both flutter and synchronous vibration can cause blade

failures. Flutter problems, usually detected during the design phase, cause significant program

delays and cost overruns. Numerical methods are being developed to help design flutter free

turbomachinery components.

Several methods have been developed for the prediction of aeroelastic characteristics of

turbomachines with various degrees of fidelity. Aeroelastic analyses based on energy exchange

between vibrating blades and the surrounding fluid, have been reported for turbomachines using

semi-analytical methods (Lane and Friedman, _ Smith2), linearized methods (Hall and Clark, 3

Verdon4), Euler methods (He, 5 Gerolymos and Vallet,* Bakhle, et al. 7) and viscous methods

(Giles and Haimes, 8 Siden, 9 He and Denton l°). A limited number of coupled aeroelastic analyses

of turbomachine configurations have also been reported. 11-13Williams, Cho, and Dalton _t used a

linear panel method to solve the eigenvalue problem. Gerolymos, t2 and Srivastava and Reddy 13

solved the coupled aeroelastic equations based on an inviscid aerodynamic analysis. The

majority of the methods are either analytical, applicable to two-dimensional geometry and/or

ignore the viscous effects.

The objective of the present effort is to report an aeroelastic analysis program based on NavierStokes equations and its application to turbomachinery components. The results obtained are

compared to those obtained from a two-dimensional linear method which solves the linear,

compressible, potential flow equation, 2 to those obtained from a three-dimensional linearized

Euler analysis, 14 and to existing experimental data from a linear turbine blade cascade 15 and a

transonic fan configuration. 16

NASA/TM--2001-210693 1
THE TURBO-AE CODE

The aeroelastic solver TURBO-AE is briefly described in this section. The solver can model

multiple blade rows undergoing harmonic oscillations with arbitrary inter-blade phase angles

(IBPAs). It is based on a Navier-Stokes unsteady aerodynamic solver for internal flow

calculations of axial flow turbomachinery components TURBO. tT,_s Viscous effects are modeled

using the Reynolds-averaged Navier-Stokes equations. The Baldwin-Lomax or two-equation

k-eturbulence model is used for closure. The aerodynamic equations are solved using a finite

volume scheme. Flux vector splitting is used to evaluate the flux Jacobians on the left-hand side.

The right-hand side fluxes are discretized using the high order Total Variation Diminishing

(TVD) scheme based on Roe's flux difference splitting. Newton subiterations are used at each

time-step to maintain accuracy. Symmetric Gauss-Sidel iterations are applied to the discretized

equations for improved convergence.

The aeroelastic characteristics of the rotor are obtained by calculating the energy exchange

between the vibrating blade and its surrounding fluid. If work on the blade is positive, this

indicates instability. The aeroelastic analysis is carried out by first obtaining the "steady"

aerodynamic solution for a given operating condition. The blades are then forced into a

prescribed harmonic motion (specified mode, frequency, and IBPA) to calculate the unsteady

aerodynamic response and work-per-cycle. The blade motion is simulated using a dynamic grid

deformation technique. Phase-lagged boundary conditions are used to calculate the non-zero

IBPA vibrations, which eliminates the need to model multiple blade passages.

3 RESULTS AND DISCUSSION

Results were obtained for a flat plate helical fan, a linear cascade of turbine blades and a

transonic fan configuration. These results are compared with existing analytical, numerical or

experimental data.

Flat Plate Helical Fan: The helical fan geometry consists of 24 unswept, untapered,

zero thickness, twisted flat plate blades enclosed within a rigid cylindrical duct of infinite length

with no tip gap. The fan's operating condition was set at a relative inflow Mach number of

0.7 and zero incidence at midspan with a 0.495 axial Mach number.

Linear Turbine Cascade: A cascade of turbine blades with large flow turning (112 deg) was

tested for several expansion ratios, exit Mach numbers and IBPAs.15 The experimental facility

consisted of a cascade comprising of five airfoil sections in a stator configuration. In

TURBO-AE the linear cascade was modeled as an annular cascade with 96 blades, with a

0.98 hub-tip-ratio, and a tip radius of 21.4625 inches. These parameters were selected to match

the solidity of the test cascade at midchord. The analysis was carried out for the measured

condition of 2.731 expansion ratio with an exit flow Mach number of 1.25.

Transonic Fan: A scale model of an experimental transonic fan designated Quiet High Speed Fan

(QHSF), with 22 blades, design rpm of 15444, relative tip Mach number of 1.4, and mass flow of

98.89 Kg/s was tested in a rig. The fan fluttered at part speed in the first bending mode. 16 Results

are presented for 90% speed at several back pressures and IBPAs.

NASA/TM--2001-210693 2
3.1 Fiat Plate Helical Fan

Results obtained for the fiat plate helical fan configuration and an inviscid analysis using

TURBO-AE are presented in this section. The advantage of this configuration and flow

condition is that the flow is subsonic and well behaved, and a large hub to tip ratio allows

modeling a 2-D flowfield that can be closely approximated at the midspan of the blade. This

allows a comparison with results obtained from an analytical solution using linear theory. 2

Montgomery and Verdon js using a linearized Euler analysis have also analyzed this particular

configuration. Since this configuration is a flat plate with zero thickness at zero incidence, there

is no steady load on the blade. The unsteady results for two different configurations -0 deg IBPA

for pitching motion and 180 deg IBPA for plunging motion are shown in Figs. 1 and 2. The

variation of the unsteady pressure difference with chord at midspan for pitching motion is

compared with linear theory 2 and linearized Euler analysis _s in Fig. 1 and for the plunging

motion in Fig. 2.

10

8

eS

._-_.._,_.,_ _ __ .._,_o_ _._ ,o_o-_ _ " ,_,_._._'

_" - .... TUR BO-AE

-- Linearized Euler

...... L near TheorY/

-10

xt¢

Figure l. Unsteady pressure difference variation with chord at midspan for 0 deg IBPA pitching oscillations

Excellent agreement is realized among the three methods. Figure 3 shows the variation of the

unsteady pressure difference with chord for several phase angles for the pitching motion. The

variation for the plunging motion is shown in Figure 4. These results are for the first harmonic of

the unsteady blade surface pressure difference at the midspan location. The results obtained from

TURBO-AE are compared with results obtained from the linear theory 2 in these figures. Good

agreement is obtained over most of the range except in the neighborhood of acoustic resonance

(cut-on and cut-off regions). Acoustic resonance occurs at phase angles of 107.3 and 330.6 deg.

The phase angles between these resonances are associated with subresonant t9 (cut-off)

conditions in which all disturbances attenuate away from the cascade. No disturbances propagate

upstream or downstream under subresonant conditions. The phase angles between 0 and 107.3

deg. and between 330.6 and 360 deg. are associated with super-resonant (cut-on) conditions in

which at least one disturbance propagates in either the far upstream or downstream direction.

NASA/TM--2001-210693 3
5,

i

@

[

1

-5

-10

-15

_._:_'-"_" '_ "_' Imag.

_ '

Legend i

TURBO-AE

Linearized Euler

Linear Theory

xlc

Figure 2. Unsteady pressure difference variation with chord at midspan tbr 180 deg IBPA plunging oscillations

(a) 0 deg. Pitching

] i

t

(c) 90 deg. Pitching

• TI RF_, kF,_ .d

(e) 180 deg. Pitching (g} 270 deg. Pitching

i"_ F : 14_%t}7:1::"_

....F'z-. F. ,', ............... 7{

% _,-'7 _,'Z" ,,,'-7 ,77 _,

• ¢

_ x:,  ̧

(b) 45 deg. Pitching (d} 135 deg. Pitching (f) 225 deg. Pitching

' :_7t......

(h) 315 deg. Pitching

Figure 3. Unsteady pressure difference (first harmonic) for pitching motion

N ASA/TM--2001-210693 4
i ll.....

(a) 0 deg. Plunging

.l,i

(c) 90 deg. Plunging (e) 180 deg. Plunging (g) 270 deg. Plunging

(b) 45 deg. Plunging (d) 135 deg. Plunging (t) 225 deg. Plunging

x L

(hi 315 deg. plunging

Figure 4. Unsteady pressure difference (first harmonic) for plunging motion

Figure 4 shows the comparison with linear theory 2 for plunging blade motion. The overall level

of agreement with linear theory is good. Once again deviations are observed close to the acoustic

resonances, as for pitching. The level of agreement was found to be better for pitching motion

than for plunging motion.

3.2 Linear Turbine Cascade

Results for the linear turbine cascade configuration are presented next• A 2.713 expansion ratio

provided a test case in which the cascade inlet velocity was subsonic (0.52 Mach), while the exit

velocity was supersonic (1.25 Mach). Predictions for the steady blade surface pressure

distribution, obtained from TURBO-AE, are compared to measurements in Fig. 5. Excellent

agreement with the measured values is obtained over most of the blade surface. However, in the

leading edge region on the pressure surface and the aft section of the suction surface small

differences are apparent. In the leading edge region, a separation bubble was observed, which is

not captured very accurately in the analysis. An expansion fan originating from the trailing edge

of the adjacent blade impinges on the aft 30 percent of the suction surface. Because of relatively

coarse grid in the blade passage the expansion fan impacts the suction surface a little upstream

from that measured.

The unsteady pressure calculated for the cascade is shown in Figs. 6 and 7 for a forced blade

motion of 180 deg IBPA and 340 Hz vibration frequency. Comparison of calculated and

measured pressure amplitude variation with chord is shown in Fig. 6. The phase comparison is

shown in Fig. 7. Good comparison is seen for the unsteady pressure variation with chord. The

phase comparison, which is more important in flutter prediction, shows better agreement than

pressure amplitude. The unsteady pressure measurements were sparse on the suction surface

since a transducer located at the 30% chord location was not working. In the trailing edge region

of the suction surface the code missed the measured increase in unsteady pressure magnitude aft

of the point where the trailing edge expansion fan intersects the blade.

NASA/TM--2001-210693 5
1

. I

o.8

0.7

a.

_ 0.6

_ 0.5

_ 0.4

0.3

0.2 0.25 0.5 0.75 1

Axial Chord, X/C

Figure 5. Comparison of steady blade loading for the turbine cascade at an expansion ratio of 2.713

0.15

"0

'_ 0.12

E

<

0.09

2

a. 0.06

-o

'_ 0.03

C

0 o

-- Calculated

• Measured, Pressure

• Measured, Suction

I I I

0.25 0.5 0.75

Axial Chord, X/C

Figure 6. Comparison of unsteady pressure response amplitude for the turbine cascade vibrating in a torsional mode

at 340 Hz, 180° IBPA, and 2.713 expansion ratio

NASA/TM--200I-210693 6
®

"O

v

efl.

>.

"o

e,.

180

9O

-90

-180

-270

-360 0

-- Calculated

• Measured, Pressure

• Measured, Suction

• •

0.25 0.5 0.75

Axial Chord, X/C

Figure 7. Comparison of unsteady pressure response phase for the turbine cascade vibrating in a torsional mode at

340 Hz, 180 ° IBPA, and 2.713 expansion ratio

3.3 Transonic Fan

The analysis was next applied to calculate the aeroelastic characteristics of the QHSF. The

results presented here are for the 90% speed line. Experiments showed blade flutter in the first

natural mode (natural frequency 351 Hz) for 32.73 IBPA (2 Nodal Diameter forward travelling

wave). 16The calculated time h/story of mass flow is shown in Fig. 8 for flow with and without

blade vibration for back pressure of 16.2 pounds per square inch (psi). The "steady" flow results

or results for no blade vibrations show a time dependency, although the mean mass flow is

constant. A closer examination of the flowfield revealed flow separation and subsequent

shedding of the vortex in the vicinity of the hub in the aft section of the blade. Increasing the

back pressure lengthened the region of separation and also moved it further upstream. Also

shown on this figure is the variation of mass flow after the blades are forced to undergo

prescribed vibration of 180 deg IBPA at 351 Hz. Because of the mean flow unsteadiness, one

needs to ensure that the blade vibration amplitude is large enough so as to minimize the impact

of mean flow unsteadiness on the unsteadiness due to blade vibrations. It can be seen from Fig. 8

that for the vibration amplitude chosen for the analysis, the unsteadiness due to blade vibration is

at least an order of magnitude larger than the mean flow unsteadiness. It can also be seen that the

flow converges to periodicity in roughly 10 blade vibration cycles.

Figure 9 shows the aerodynamic damping calculated for several different back pressures and

IBPAs. For lower back pressures, the least stable IBPA was found to be 0 deg. However, as the

back pressure was raised moving the fan operating condition towards the stall line, the 32.73 deg

IBPA (2 nodal diameter forward travelling wave) became least stable. Flutter was observed in

the wind tunnel for 32.73 deg IBPA. The calculated variation of aerodynamic damping with back

pressure is shown in Fig. 10 for 32.73 deg IBPA. As may be seen in the figure, as the back

pressure is increased, the aerodynamic damping decreases rapidly, dropping to approximately

0.2% of critical damping for the back pressure of 16.4 psi. For the grid used, and because of

computational constraints, the highest back pressure analyzed was 16.4 psi. Increasing the back

pressure above 16.4 psi for 90% speed resulted in stalled flow with large separation on the

NASA/TM--2001-210693 7
suction surface in the mid section of the blade. This separation was found to be shock induced

and prevented a steady mean flow from which the blade vibration analysis could be carried out.

These results indicate that the 32.73 IBPA is the least stable IBPA for first natural mode, as

observed in experiments, and that the aerodynamic damping is rapidly decreasing with

increasing back pressure. It can be easily extrapolated from Fig. 10 that further increase in the

back pressure would result in a negative aerodynamic damping indicating flutter.

89.00

87.00'

85.00

83.00

Min

Blade Vibrations started

after 2300 iterations

Blade Vibration

81.00

500 1000 1500 2000 2500 3000 3500

Iteration

Figure 8. Variation of mass flow for the transonic fan, with and without blade vibration

2.5

I I I I I I i

-32.73 -16.36 0.00 16.36 32.73 49.09 65.46

Inter-blade phase angle (deg)

Figure 9. Variation of aerodynamic damping with back pressure and IBPA for the transonic fan with forced blade

vibration in the first natural mode at the natural frequency of 351 Hz

NASA/TM--2001-210693 8
1.75

1.5

A

"'" 1.25

Q.

I= 1-

m

E o.75-

_ 0.5

<

0.25

£:_.5

I • 20os ,,,,,cyo,, I

r I I [ I [ i I | I I I h

14.5 15.5 16.5

Back Pressure (PSI)

Figure 10. Variation of aerodynamic damping for 16.4 PSI back pressure. Forced vibration in the first natural mode

at 351 Hz and 32 deg IBPA

For the back pressure of 16.4 psi, the experiment showed flutter, whereas the analysis predicted

the fan to be marginally stable. In the analysis, the 100% speed blade shape and blade modal

properties were used at the 90% speed. Changes in rotational speed impact the geometry by

affecting tip-gap, blade natural frequency, and blade shape or more specifically twist

distribution. These factors may have contributed to the over prediction of aerodynamic damping.

It has been reported that tip-gap may have significant impact on stability, z° The analyses used a

uniform value of tip-gap which is based on measurements made during testing. However, the

axial distribution of tip-gap and the accuracy of the measurements are not known. Further, a

sensitivity analysis did not show a strong influence of natural frequency on aerodynamic

damping.

Analysis was also performed at the second natural vibration mode, which was stable in the

experiment. The aerodynamic damping calculated was much higher than those for the first mode,

indicating the second mode to be more stable than the first mode. These results clearly show that

the analysis successfully calculates the natural mode and IBPA of instability identifying the

flutter characteristics of the QHSF.

CONCLUDING REMARKS

An aeroelastic analysis program based on the Navier-Stokes equation has been reported and

applied to several turbomachinery components in this study. The calculated unsteady

aerodynamic pressure variations are compared with existing numerical, analytical and

experimental results for a fan and a turbine configuration. Good comparisons between analytical

and numerical results were found for a fan-type geometry, with the exception being the

neighborhood of acoustic resonances. Comparisons with the more challenging case of a turbine

N ASA/TM--2001-210693 9
configuration was also good in general. The unsteady aerodynamic characteristics obtained lbr

the turbine configuration compared well with measured data. The grid used was not able to

capture all the details of flow separation resulting in some differences in the calculated pressure.

The important characteristics of phase variation, which determines the flutter characteristics,

were captured well. Increasing the grid density and using smaller time-steps in time march can

further improve the details of the calculated results.

Calculated variations of aerodynamic damping with back pressure and IBPA is presented for a

transonic fan at a condition where flutter occurred during testing. Although a negative

aerodynamic damping was not calculated, the analysis yielded some good results and trends. The

analysis correctly predicted the mode and IBPA of flutter. Also, the trend for calculated

aerodynamic damping clearly indicated a negative aerodynamic damping would result with

further increasing the back pressure. Unfortunately, the analysis showed a stalled flow field

would emerge for any further increase of the back pressure. Although flow separation was

captured along with vortex shedding, deep blade stall posed a numerical problem.

The results presented here required roughly 4 to 5 days on a SGI Octane per analysis, these

computational times are within the realm of engineering analyses. Moreover, these results

indicate that the aeroelastic analysis program presented here is accurate and fast enough to be

used in engineering applications for aeroelastic analysis of modem turbomachinery components.

5 REFERENCES

[t] Lane, F. and Friedman, M., "Theoretical Investigation of Subsonic Oscillating Blade-Row

Aerodynamics," NACA TN 4136. 1958.

[2] Smith, S.N., "Discrete Frequency Sound Generation in Axial Flow Turbomachines,"

R&M 3709, British Aeronautical Research Council, London, England, UK, 1972.

[3] Hall, K.C., Clark, W.S., "Linearized Euler Predictions of Unsteady Aerodynamic Loads in

Cascades," A1AA Journal, vol. 31, pp. 540-550, 1993.

[4] Verdon, J.M., "Unsteady Aerodynamic Methods for Turbomachinery Aeroelastic and

Aeroacoustic Applications," AIAA Journal, vol. 31, no. 2, pp. 235-250, 1993.

[5] He, L., "An Euler Solution for Unsteady Flows Around Oscillating Blades," ASME

Journal OfTltrbomachinery, vol. 112, no. 4, pp. 714-722, 1989.

[6] Gerolymos, G.A., and Vallet, I., "Validation of 3-D Euler Methods for Vibrating Cascade

Aerodynamics," ASME Paper 94-GT-294, 1994.

[7] Bakhle, M.A., Srivastava, R., Keith, T.G. Jr., Stefko, G.L., "A 3D Euler/Navier-Stokes

Aeroelastic Code for Propulsion Applications," AIAA Paper No. 97-2749, 1997.

[8] Giles, M., Haimes, R., "Validation of a Numerical Method for Unsteady Flow

Calculations," ASME Paper 91-GT-271, 1991.

[9] Siden, L.D.G., "Numerical Simulation of Unsteady Viscous Compressible Flows Applied

to a Blade Flutter Analysis," ASME Paper No. 91-GT-203, 1991.

N ASA/TM--2001-210693 10
[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

He, L. and Denton, J.D., "'Three Dimensional Time Marching Inviscid and Viscous

Solutions for Unsteady Flows Around Vibrating Blades," ASME Journal of

Turbomachinet3,, vol. 116, pp. 469-476, 1994.

Williams, M.H., Cho, J., and Dalton, W.N., "Unsteady Aerodynamic Analysis of Ducted

Fans," Journal o[Propulsion and Power, vol. 7, no. 5, pp. 800-804, 1991.

Gerolymos, G.A, "Coupled 3-D Aeroelastic Stability Analysis of Bladed Disks," ASME

Paper No. 92-GT- 171, 1992.

Srivastava, R. and Reddy, T.S.R., "Comparative Study of Coupled-Mode Flutter-Analysis

Methods for Fan Configurations," Journal o[" Propulsion and Power, vol. 15, no. 3,

pp. 447-453, 1999.

Montgomery, M.D. and Verdon, J. M., "A Three-Dimensional Linearized Unsteady Euler

Analysis for Turbomachinery Blade Rows," NASA CR-4770, 1997.

Rothrock, M.D., Jay, R.L., and Riffel, R.E., "Time Variant Aerodynamics of High-Turning

Blade Elements," ASME Paper No. 81-GT-123, 1981.

Fite, B.E., Private Communications, NASA Glenn Research Center, January 2001.

Chen, J.P., "Unsteady Three-Dimensional Thin-Layer Navier-Stokes Solutions for

Turbomachinery in Transonic Flow," Ph. D. Dissertation, Mississippi State University,

1991.

Janus, J.M., "Advanced 3-D CFD Algorithm for Turbomachinery," Ph. D. Dissertation,

Mississippi State University, Mississippi, 1989.

Verdon, J.M., "The Unsteady Flow in the Far Field of an Isolated Blade Row," Journal of

Fluids and Structures, vol. 3, no.2, pp. 123-149, 1989.

Srivastava, R. and Reddy, T.S.R., "Aeroelastic Analysis of Ducted Rotors," Proceedings of

International Symposium on Computational Fluid Dynamics in Aeropropulsion, San

Francisco, AD-vol. 49, pp. 1-9, November 1995.

NASA/TM--2001-210693 11
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Aeroelastic Stability Computations for Turbomachinery

6. AUTHOR(S)

R. Srivastava, M.A. Bakhle, T.G Keith. Jr., and G.L. Stefko

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John H. Glenn Research Center at Lewis Field

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NASA TM--2001-210693

Prepared for the International Forum on Aeroelasticity and Structural Dynamics cosponsored by the CEAS, AIAA, and

AIAE, Madrid, Spain, June 5-7, 2001. R. Srivastava, M.A. Bakhle, and T.G Keith, Jr., University of Toledo, Toledo,

Ohio 43606; and G.L. Stet_,_o. NASA Glenn Research Center. Responsible person, G.L. Stefko, organization code 5930,

216--433-3920.

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13. ABSTRACT (Maximum 200 words)

This paper describes an aeroelastic analysis program for turbomachines. Unsteady Navier-Stokes equations are solved

on dynamically deforming, body fitted, grid to obtain the aeroelastic characteristics. Blade structural response is

modeled using a modal representation of the blade and the work-per-cycle method is used to evaluate the stability

characteristics. Nonzero interblade phase angle is modeled using phase-lagged boundary conditions. Results obtained

showed good correlation with existing experimental, analytical, and numerical results. Numerical analysis also showed

that given the computational resources available today, engineering solutions with good accuracy are possible using

higher fidelity analyses.

14. SUBJECT TERMS

Turbomachinery; Flutter; Analysis; Navier-Stokes; Three-dimensional

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