Experimental Investigation of Three-Dimensional Mechanisms ...7902/FULLTEXT01.pdf · Experimental...

Experimental Investigation of Three-Dimensional Mechanisms in

Low-Pressure Turbine Flutter

Damian Vogt

Doctoral Thesis 2005

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av teknisk doktorsexamen i energiteknik, fredagen den 27e maj 2005, kl. 10.00 i salen M3, Brinellvägen 64, Kungliga Tekniska Högskolan, Stockholm. Avhandlingen försvaras på engelska. TRITA-KTV-2005-01 ISSN 1100/7990 ISRN KTH-KRV-R-05-01-SE ISBN 91-7178-034-3 © 2005 Damian Vogt

Doctoral Thesis / Damian Vogt Page 1

Abstract The continuous trend in gas turbine design towards lighter, more powerful and more reliable engines on one side and use of alternative fuels on the other side renders flutter problems as one of the paramount challenges in engine design. Flutter denotes a self-excited and self-sustained aeroelastic instability phenomenon that can lead to material fatigue and eventually damage of structure in a short period of time unless properly damped. The design for flutter safety involves the prediction of unsteady aerodynamics as well as structural dynamics that is mostly based on in-house developed numerical tools. While high confidence has been gained on the structural side unanticipated flutter occurrences during engine design, testing and operation evidence a need for enhanced validation of aerodynamic models despite the degree of sophistication attained. The continuous development of these models can only be based on the deepened understanding of underlying physical mechanisms from test data. As a matter of fact most flutter test cases treat the turbomachine flow in two-dimensional manner indicating that the problem is solved as plane representation at a certain radius rather than representing the complex annular geometry of a real engine. Such considerations do consequently not capture effects that are due to variations in the third dimension, i.e. in radial direction. In this light the present thesis has been formulated to study three-dimensional effects during flutter in the annular environment of a low-pressure turbine blade row and to describe the importance on prediction of flutter stability. The work has been conceived as compound experimental and computational work employing a new annular sector cascade test facility. The aeroelastic response phenomenon is studied in the influence coefficient domain having one blade oscillating in various three-dimensional rigid-body modes and measuring the unsteady response on several blades and at various radial positions. On the computational side a state-of-the-art industrial numerical prediction tool has been used that allowed for two-dimensional and three-dimensional linearized unsteady Euler analyses. The results suggest that considerable three-dimensional effects are present, which are harming prediction accuracy for flutter stability when employing a two-dimensional plane model. These effects are mainly apparent as radial gradient in unsteady response magnitude from tip to hub indicating that the sections closer to the hub experience higher aeroelastic response than their equivalent plane representatives. Other effects are due to turbomachinery-typical three-dimensional flow features such as hub endwall and tip leakage vortices, which considerably affect aeroelastic prediction accuracy. Both effects are of the same order of magnitude as effects of design parameters such as reduced frequency, flow velocity level and incidence. Although the overall behavior is captured fairly well when using two-dimensional simulations notable improvement has been demonstrated when modeling fully three-dimensional and including tip clearance. Keywords: turbomachinery, flutter, aeroelastic instability, aeroelastic testing, CFD, linearized unsteady numerical method, 3D aerodynamic effects

Page 2 Doctoral Thesis / Damian Vogt


Preface The thesis is based on the following papers: 1 Vogt, D.M., Fransson, T.H., 2000

“Aerodynamic Influence Coefficients on an Oscillating Turbine Blade in Three-Dimensional High Speed Flow” Paper presented at the 15th Symposium on Measuring Techniques in Transonic and Supersonic Flows in Cascades and Turbomachines, Florence, Italy

2 Vogt, D.M., Fransson, T.H., 2002 “A New Turbine Cascade for Aeromechanical Testing” Paper presented at the 16th Symposium on Measuring Techniques in Transonic and Supersonic Flows in Cascades and Turbomachines, Cambridge, UK

3 Fransson, T.H., Vogt, D.M., 2003

“A New Facility for Investigating Flutter in Axial Flow Turbomachines” Paper presented at the 8th National Turbine Engine High Cycle Fatigue (HCF) Conference, Monterey, California, USA

4 Vogt, D.M., Fransson, T.H., 2004a

“Effect of Blade Mode Shape on the Aeroelastic Stability of a LPT Cascade” Paper presented at the 9th National Turbine Engine High Cycle Fatigue (HCF) Conference, Pinehurst, North Carolina, USA

5 Vogt, D.M., Fransson, T.H., 2004b

“A Technique for Using Recessed-Mounted Pressure Transducers to Measure Unsteady Pressure” Paper presented at the 17th Symposium on Measuring Techniques in Transonic and Supersonic Flows in Cascades and Turbomachines, Stockholm, Sweden

6 Mårtensson, H., Vogt, D.M., Fransson, T.H., 2005

“Assessment of a 3D Linear Euler Flutter Prediction Tool using Sector Cascade Test Data” ASME Paper GT2005-68453

7 Vogt, D.M., Mårtensson, H., Fransson, T.H., 2005 “Validation of a Three-Dimensional Flutter Prediction Tool” Paper submitted to the NATO Symposium on Evaluation, Control and Prevention of High Cycle Fatigue in Gas Turbine Engines for Land, Sea and Air Vehicles, Seville, Spain

The involvement of Prof. Torsten Fransson and Mr. Hans Mårtensson in the above publications consisted in problem formulation and discussion of results. For all publications the underlying material was part of the work elaborated in this thesis.



Acknowledgements I would like to express my gratitude to my supervisor Prof. Torsten Fransson at the Chair of Heat and Power Technology at the Royal Institute of Technology, Stockholm, who made this work possible and established a stimulating environment. To Hans Mårtensson at Volvo Aero, Trollhättan, I would to like to express my gratitude for giving the right inputs at the right time and providing valuable feedback. Besides that he was also one of the few persons who genuinely appreciated my old Volvo cars. Special thanks goes to Karl-Erik Andersson, retired designer at former ABB STAL, Finspång, who assisted during the design of the test facility. Staying with the facility I would like to thank the technicians in the lab, Rolf Bornhed, Stellan Hedberg, Christer Blomqvist, Bernt Jansson and last but not least Jan-Inge Ringström, who not only provided first class technical support but also contributed to my adaptation to the Swedish culture. The work would not have been possible without a considerable amount of manufacturing jobs that were placed outside the lab at various workshops in Sweden. The jobs were rarely standard and required often special diligence. Many thanks to all involved technicians for performing excellent work and for being open to sometimes uncommon ideas. Financial support for the present work has been provided through the Swedish Gas Turbine Center (GTC) and the EU funded project DAIGTS (contract number ENK5-CT2000-00065), which is greatly acknowledged. Special thanks goes to Lic. Eng. Sven Gunnar Sundkvist, director of GTC, as well as Dr. Andrew Minchener, technical monitor of DAIGTS, who followed the respective parts with great interest. I would like to thank my colleagues Björn Laumert, Jürgen Jacoby, Kai Freudenreich, Markus Jöcker, Mikkel Myhre, Olivier Bron and all the ones that I have not mentioned in persona for fruitful discussions, mutual motivation and just good fun time. Special thanks is directed to all my friends from back home for being good friends. Invaluable thanks goes to my family at home, who gave me continuous support and besides that provided me with the necessary “culinary fuel”. Special thanks goes also to Anna’s family here in Sweden for taking me on board in the various senses of the word and introducing me to Swedish habits. Finally I would like to express my sincere gratitude to my dearest Anna, who supported me with love, patience and motivation and for giving birth to our son Leo, the most wonderful little boy in the world.



Content 1 Introduction...................................................................................................... 23 2 Background ..................................................................................................... 25

2.1 Description of the Flutter Phenomenon....................................................... 25 2.2 Review of Flutter Analysis Methods............................................................ 37 2.3 Review of Flutter Testing Methods.............................................................. 40

2.3.1 Free Flutter Testing........................................................................... 40 2.3.2 Controlled Flutter Testing.................................................................. 42

3 Fundamental Concepts.................................................................................... 49 3.1 Introduction................................................................................................. 49 3.2 Determination of Flutter Stability................................................................. 50 3.3 Aerodynamic Influence Coefficients............................................................ 53

4 Objectives and Approach................................................................................. 57 5 Experimental Investigation of Aerodynamic Influence Coefficients................... 59

5.1 Description of Test Setup ........................................................................... 59 5.1.1 Test Object ....................................................................................... 59 5.1.2 Test Facility....................................................................................... 61 5.1.3 Controlled Blade Oscillation .............................................................. 65 5.1.4 Conventions...................................................................................... 68 5.1.5 Measurement Setup.......................................................................... 73 5.1.6 Data Acquisition and Data Reduction Procedure............................... 79

5.2 Validation of Test Setup.............................................................................. 80 5.2.1 Steady-State Aerodynamic Performance .......................................... 80 5.2.2 Blade Oscillation ............................................................................... 93 5.2.3 Unsteady Performance ..................................................................... 94

6 Numerical Prediction of Aerodynamic Influence Coefficients ........................... 99 6.1 Description of Numerical Model .................................................................. 99 6.2 Validation of Numerical Method ................................................................ 102

6.2.1 Mesh Convergence......................................................................... 102 6.2.2 Effect of Numerical Approximation .................................................. 105 6.2.3 Effect of Finite Cascade on Influence Coefficient Technique........... 106

7 Investigation Strategy .................................................................................... 109 7.1 Flutter Testing .......................................................................................... 109 7.2 Unsteady CFD Simulations....................................................................... 110

8 Results .......................................................................................................... 111 8.1 Steady-State Test Data ............................................................................ 111 8.2 Flutter Test Data....................................................................................... 119

8.2.1 Aeroelastic Response at Different Modes ....................................... 119 8.2.2 Effect of Reduced Frequency on Aeroelastic Response.................. 126 8.2.3 Effect of Flow Velocity Level on Aeroelastic Response ................... 130 8.2.4 Effect of Flow Incidence on Aeroelastic Response.......................... 133

8.3 Three-Dimensional Effects of Aeroelastic Response ................................ 136


8.4 Correlation of CFD Results to Test Data................................................... 144 9 Discussion ..................................................................................................... 159

9.1 Quantification of Three-Dimensional Mechanisms during Flutter .............. 161 9.2 Assessment of CFD Prediction Accuracy of Aeroelastic Stability .............. 168

10 Summary ..................................................................................................... 173 10.1 Conclusions.............................................................................................. 173 10.2 Recommendations and Future Work ........................................................ 176

11 References .................................................................................................. 179 Appendix I: Blade Profile Description..................................................................... 187

Profile Denotations ............................................................................................ 187 Profile Data........................................................................................................ 188


Nomenclature Latin Symbols

a complex torsion component A matrix containing modal unsteady aerodynamic forces c blade chord

pc specific heat at constant pressure

Apc ,ˆ normalized unsteady pressure coefficient, refdynAp pApc ,, ˆˆ ⋅= ; in

case of three-dimensional consideration the angular oscillation amplitude in deg is taken as normalization basis; in case of two-dimensional plane motion the normalization is based on equivalent translational amplitude in mm for the bending modes and rotational amplitude in rad for the torsion mode

pC averaged static pressure coefficient, refsref

refss

pp

pppC

,,0

,

−−

=

0pC averaged total pressure coefficient, refsref

refs

pp

pppC

,,0

,00 −

−=

vc specific heat at constant volume

df infinitesimal force component

d diameter dm infinitesimal moment component ds infinitesimal arcwise surface component, per unit span e total energy per unit volume

ζer

torsion direction (radial)

F force, force vector HGF ,, fluxes

G damping matrix

hr

complex mode shape vector

h complex bending component

i imaginary unit, 1−=i k reduced frequency, based on full chord ufck π2=

ik aerodynamic probe calibration coefficient i

K stiffness matrix l nodal diameter

mn, blade indices m mass M mass matrix M Mach number

isoM isentropic Mach number,

2

11

0 11

2

−

−

=

−γ

γ

γ siso p

pM


nr

normal vector to surface element N number of blades p pressure

p mean pressure

p~ time-varying perturbation pressure

p complex pressure amplitude

cp pitch

Q conserved variables vector

Q modal displacement vector

r radius, radial coordinate rr

distance from center of torsion to force realization point

rt mode shape displacement ratio, LTrt δδ=

s span

zytxt SSSS ,,, metric terms

t time, time of flight T oscillation period u flow velocity

wvu ,, Cartesian velocity components

cycleW work per cycle; positive if the fluid is transferring work to the

structure unstable situation zyx ,, Cartesian coordinates

X displacement vector X& first derivative of displacement vector (velocity) X&& second derivative displacement vector (acceleration) Y radius ratio, hubshr rrY =

Greek Symbols

α yaw flow angle β pitch flow angle

γ ratio of specific heats, vp cc=γ

δ displacement ζ torsion orthogonal mode coordinate

η circumferential orthogonal mode coordinate

ϑ angular coordinate in cylindrical coordinate system λ wavelength µ mass ratio

ξ axial orthogonal mode coordinate

π number “pi” 3.1415927… ρ density

σ interblade phase angle; forward traveling wave Nl⋅= πσ 2 ,

backwards traveling wave ( ) NlN −= πσ 2

τ pseudo time


ϕ mode shape

ω rotational frequency, fπω 2=

Π pressure ratio Ξ stability parameter φ potential

hp→φ phase angle of response with respect to excitation (motion); the

phase angle is per definition positive if the response is leading the excitation

Subscripts

0 total 1 cascade inlet 2 cascade outlet 2* outlet plenum ae aerodynamic amp amplitude (magnitude of complex quantity) avg average ax axial circ_avg circumferential average damping related to damping disturbance related to disturbance dyn dynamic EA ensemble-averaged end end of separation bubble hub hub i inner ic influence coefficient max maximum phase phase of complex quantity pitch_avg pitchwise average pseudo pseudo value ref reference (r*theta) unwrapped circumferential direction s static sec secondary shr shroud start start of separation bubble twm traveling wave mode

ζ torsion direction

η circumferential bending direction

θ circumferential component

ξ axial bending direction


Superscripts

* normalized value T trailing edge L leading edge ° degree

Abbreviations

2D two-dimensional 3D three-dimensional A/D analog to digital AGARD Advisory Group for Aerospace Research and Development AIAA American Institute of Aeronautics and Astronautics arc normalized arcwise coordinate; negative branch on suction side,

positive on pressure side ARC Aeronautical Research Council ASME The American Society of Mechanical Engineers avg average CFD computational fluid dynamics COT center of torsion CUED Cambridge University Engineering Department DC direct current deg degree EPFL École Polytechnique Fédérale de Lausanne GPIB general purpose interface bus HCF high cycle fatigue HPT Heat and Power Technology FEM finite element method IBPA interblade phase angle IFASD International Forum on Aeroelasticity and Structural Dynamics Im imaginary part of complex number INFC influence coefficient ISABE International Symposium on Airbreathing Engines ISUAAAT International Symposium on Aerodynamics, Aeroacoustics and

Aeroelasticity of Turbomachines JSME The Japan Society of Mechanical Engineers K Kelvin kg kilogram KTH Kungliga Tekniska Högskolan (Royal Institute of Technology) LE leading edge LPT low-pressure turbine Mag magnitude MIT Massachusetts Institute of Technology MP measurement point MW mega Watt NASA National Aeronautics and Space Administration ONERA Office National d’Etudes et de Recherches Aérospatiales OP operating point PC personal computer


PS pressure side PSI Pressure System Inc. PT Platinum resistance thermometer PVC polyvinyl chloride rad radian Re real part of complex number rev revolution span spanwise coordinate; hub at span=0, tip at span=1 SS suction side tc tip clearance TE trailing edge TWM traveling wave mode UTRC United Technologies Research Center VAC Volvo Aero Corporation VDC direct current voltage VKI von Karman Institute

Denotations of operating points

First letter (denoting velocity level) L low subsonic M medium subsonic H high subsonic

Second letter (denoting inflow incidence) 1 nominal, zero incidence 2 off-design 1, medium negative incidence 3 off-design 2, high negative incidence



List of Tables Table 5-1. Set of target design properties of test object .......................................... 59 Table 5-2. Set of blade profile parameters .............................................................. 60 Table 5-3. Probe traverse parameters..................................................................... 75 Table 6-1. CFD Mesh parameters......................................................................... 102 Table 7-1. Overview of measured test conditions (passage-averaged values) ...... 109 Table 8-1. Impact of mode shape on relative change in throat size (amplitudes

indicated at midspan) ..................................................................................... 124



List of Figures Figure 2-1. Collar’s triangle of forces ...................................................................... 25 Figure 2-2. Graphical interpretation of mass ratio and reduced frequency .............. 26 Figure 2-3. Cross-section of a modern aero engine turbomachine (Volvo RM12);

flutter susceptible areas indicated, area of current work highlighted; picture courtesy of Volvo Aero ..................................................................................... 27

Figure 2-4. Influence phenomenon during blade row flutter in turbomachines ........ 28 Figure 2-5. Nodal diameters of a disk, traveling wave mode shape and

corresponding instantaneous blade row geometry............................................ 29 Figure 2-6. Campbell diagram indicating the occurrence of flutter (arrows); adapted

from Fransson (1999)....................................................................................... 29 Figure 2-7. Unshrouded and shrouded bladed disk assemblies.............................. 30 Figure 2-8. Blade first-order eigenmodes................................................................ 30 Figure 2-9. Graphical representation of critical reduced frequency versus torsion axis

location for a 2D section of a cascade (Panovsky-Kielb method); from Chernysheva et al. (2003) ................................................................................ 31

Figure 2-10. Flutter stability versus mode shape displacement ratio for assessing the effect of mode shape on stability; from Peng and Vahdati (2002) ..................... 32

Figure 2-11. Possible vibration modes for blade packages; from Ewins (1988)....... 33 Figure 2-12. Effect of phase angle between bending and torsion mode on the

aerodynamic work performed; adapted from Försching (1991)......................... 33 Figure 2-13. Effect of multistage coupling on flutter stability (2D simulations); from

Hall et al. (2003)............................................................................................... 36 Figure 2-14. Free flutter test setup for single mode testing; Urban et al. (2000)...... 41 Figure 2-15. Free-flutter test setup for variable mode testing; Kirschner et al. (1976)

......................................................................................................................... 41 Figure 2-16. Annular non-rotating cascade for traveling wave mode and influence

coefficient testing.............................................................................................. 43 Figure 2-17. Purdue 3-stage experimental compressor; Frey and Fleeter (1999) ... 43 Figure 2-18. UTRC Oscillating Cascade Wind Tunnel (OCWT); Carta (1983)......... 44 Figure 2-19. NASA Lewis Transonic Oscillating Cascade; Buffum and Fleeter (1991)

......................................................................................................................... 45 Figure 2-20. Controlled flutter testing by aerodynamic excitation; Crawley (1981) .. 46 Figure 2-21. Example of type of blade oscillation device......................................... 47 Figure 3-1. System of orthogonal modes ................................................................ 50 Figure 3-2. Indexing of blades in cascade............................................................... 53 Figure 3-3. Effect of interblade phase angle on traveling wave mode response;

superposition of influence coefficients from blades -1, 0 and +1....................... 54 Figure 3-4. Schematic influence of blade pairs on blade row aeroelastic stability ... 55 Figure 3-5. Characteristic variation of stability versus interblade phase angle (S-

curve) ............................................................................................................... 56 Figure 5-1. Test object............................................................................................ 60 Figure 5-2. Measurement setup.............................................................................. 61 Figure 5-3. Test facility ........................................................................................... 63 Figure 5-4. Inlet and outlet flexible sidewalls........................................................... 63 Figure 5-5. Test section; upstream lateral sidewalls removed................................. 64 Figure 5-6. Blade oscillation principle ..................................................................... 65 Figure 5-7. Kinematics of oscillation actuator; bending mode shown ...................... 66


Figure 5-8. Oscillation actuator device (opened) and oscillating blade.................... 67 Figure 5-9. Blade oscillation actuator...................................................................... 67 Figure 5-10. Blade indexing and cascade coordinates............................................ 68 Figure 5-11. Test rig coordinate system.................................................................. 69 Figure 5-12. Local on-blade coordinate system ...................................................... 70 Figure 5-13. Definition of flow angles...................................................................... 71 Figure 5-14. Definition of blade oscillation .............................................................. 72 Figure 5-15. Head of aerodynamic 4-hole probe, calibration coefficients and

calibration surfaces .......................................................................................... 74 Figure 5-16. Distribution of static pressure taps...................................................... 75 Figure 5-17. Distribution of unsteady pressure measurement points on non-

oscillating blades (recessed-mounted transducers) .......................................... 76 Figure 5-18. Distribution of unsteady pressure measurement points on oscillating

blade (recessed-mounted transducers) ............................................................ 77 Figure 5-19. Oscillating and non-oscillating blades used for fast-response pressure

measurements ................................................................................................. 77 Figure 5-20. Dynamic calibration procedure and transfer characteristic .................. 78 Figure 5-21. Cascade flow field and periodicity assessment traverses ................... 81 Figure 5-22. Inlet flow field characteristics; low subsonic ........................................ 82 Figure 5-23. Inlet flow field characteristics; medium subsonic................................. 83 Figure 5-24. Inlet flow field characteristics; high subsonic....................................... 84 Figure 5-25. Outlet flow field characteristics; low subsonic ..................................... 85 Figure 5-26. Outlet flow field characteristics; medium subsonic .............................. 86 Figure 5-27. Outlet flow field characteristics; high subsonic.................................... 87 Figure 5-28. Inlet flow field periodicity data at different spanwise positions; low

subsonic........................................................................................................... 88 Figure 5-29. Outlet flow field periodicity data at different spanwise positions; low

subsonic........................................................................................................... 89 Figure 5-30. Outlet flow field periodicity data at midspan for different velocity levels

......................................................................................................................... 90 Figure 5-31. Blade loading periodicity data at different spanwise positions; low

subsonic........................................................................................................... 91 Figure 5-32. Blade loading periodicity data at midspan for different velocity levels . 92 Figure 5-33. Blade oscillation data.......................................................................... 93 Figure 5-34. Power spectra of wind tunnel acoustics; blade oscillating at 44Hz...... 94 Figure 5-35. Aerodynamic response on blade -1 measured with oscillating blade at

two different indices.......................................................................................... 95 Figure 5-36. Aerodynamic response on blade +1 measured with oscillating blade at

two different indices.......................................................................................... 96 Figure 6-1. Simulation of flutter in the traveling wave mode domain ..................... 101 Figure 6-2. Coarse, medium and fine meshes used; midspan shown ................... 103 Figure 6-3. Medium meshes with different node distribution; midspan shown....... 103 Figure 6-4. Steady loading and unsteady response on blade 0 at midspan and close

to hub for different meshes............................................................................. 104 Figure 6-5. Comparison of unsteady response on blades +1 through -1 using

different numerical schemes (TWM simulation) .............................................. 105 Figure 6-6. Test section and single passage traveling wave mode model............. 106 Figure 6-7. Comparison of unsteady response on blades +2 through -2 from TWM

and INFC simulation....................................................................................... 107 Figure 7-1. Computational meshes used for 3D simulations ................................. 110


Figure 8-1. Effect of flow velocity level on steady blade loading; low subsonic to high subsonic; operating point 1 (zero incidence)................................................... 112

Figure 8-2. Termination lines of pressure side separation bubble from steady loading data; low subsonic velocity level ..................................................................... 113

Figure 8-3. Effect of off-design operation on steady blade loading; low subsonic; operating points L1 through L3....................................................................... 114

Figure 8-4. Blade surface flow visualization results and corresponding steady-state blade loading data; operating point L1............................................................ 115



Figure 8-7. Unsteady response on blades +2 through -2 at midspan; operating point L1; axial bending, k=0.1 ................................................................................. 120

Figure 8-8. Unsteady response on blades +2 through -2 at midspan; operating point L1; circumferential bending, k=0.1.................................................................. 122

Figure 8-9. Unsteady response on blades +2 through -2 at midspan; operating point L1; torsion, k=0.1............................................................................................ 123

Figure 8-10. Comparison of product of blade loading and its second derivative to aeroelastic response data .............................................................................. 125

Figure 8-11. Effect of reduced frequency on unsteady response on blades +1, 0, -1; operating point L1; axial bending................................................................... 126

Figure 8-12. Effect of reduced frequency on unsteady response on blades +1, 0, -1; operating point L1; circumferential bending ................................................... 127

Figure 8-13. Effect of reduced frequency on unsteady response on blades +1, 0, -1; operating point L1; torsion ............................................................................. 128

Figure 8-14. Effect of flow velocity level on unsteady response on blades +1, 0, -1; operating points L1, M1 and H1; axial bending.............................................. 130

Figure 8-15. Effect of flow velocity level on unsteady response on blades +1, 0, -1; operating points L1, M1 and H1; circumferential bending .............................. 131

Figure 8-16. Effect of flow velocity level on unsteady response on blades +1, 0, -1; operating points L1, M1 and H1; torsion ........................................................ 132

Figure 8-17. Effect of flow incidence on unsteady response on blades +1, 0, -1; operating points L1, L2 and L3; axial bending ................................................ 133

Figure 8-18. Effect of flow incidence on unsteady response on blades +1, 0, -1; operating points L1, L2 and L3; circumferential bending................................. 134

Figure 8-19. Effect of flow incidence on unsteady response on blades +1, 0, -1; operating points L1, L2 and L3; torsion........................................................... 135

Figure 8-20. Spanwise variation of aeroelastic response on blade +1; operating point L1................................................................................................................... 137

Figure 8-21. Spanwise variation of aeroelastic response on blade -1; operating point L1................................................................................................................... 138

Figure 8-22. Blade surface flow visualization results and corresponding unsteady response amplitudes on primary surfaces; operating point L1; axial bending . 140

Figure 8-23. Blade surface flow visualization results and corresponding unsteady response amplitudes on primary surfaces; operating point L1; torsion............ 141

Figure 8-24. Spanwise variation of aeroelastic response on blade +1; operating points L2 and L3, axial bending ...................................................................... 142

Figure 8-25. Blade surface flow visualization results and corresponding unsteady response amplitudes; operating points L2 and L3; axial bending.................... 143


Figure 8-26. Comparison of predicted and measured steady loading; nominal low subsonic (L1) and high subsonic (H1) ............................................................ 144

Figure 8-27. Comparison of measured and predicted unsteady response on blades +2 through -2 at midspan; operating point L1; axial bending, k=0.1................ 145

Figure 8-28. Comparison of measured and predicted unsteady response on blades +2 through -2 at midspan; operating point L1; circumferential bending, k=0.1 147

Figure 8-29. Comparison of measured and predicted unsteady response on blades +2 through -2 at midspan; operating point L1; torsion, k=0.1 .......................... 148

Figure 8-30. Comparison of measured and predicted unsteady response on blade -1 at different reduced frequencies; operating point L1; axial bending ................ 149

Figure 8-31. Effect of model detailing on prediction of steady-state flow field; operating point L1........................................................................................... 150

Figure 8-32. Effect of model detailing on prediction of aeroelastic response on blades +1 through -1; operating point L1; axial bending; k=0.1....................... 151

Figure 8-33. Effect of model detailing on prediction of aeroelastic response on blades +1 through -1; operating point L1; torsion; k=0.1 ................................. 152

Figure 8-34. Comparison of measured and predicted unsteady response on blade -1 close to hub and close to tip; operating point L1; axial bending; k=0.1 ........... 153

Figure 8-35. Comparison of measured and predicted unsteady response on blade -1 close to hub and close to tip; operating point L1; torsion; k=0.1...................... 154

Figure 8-36. Comparison of predicted and measured steady loading at various incidence; low subsonic.................................................................................. 155

Figure 8-37. Comparison of measured and predicted unsteady response on blades +1 through -1 at high negative incidence; operating point L3; axial bending ... 156

Figure 8-38. Comparison of measured and predicted unsteady response on blade 0 at high negative incidence; operating point L3; circumferential bending and torsion ............................................................................................................ 157

Figure 9-1. Fragmenting of profile into arcwise regions......................................... 159 Figure 9-2. Torsion mode representation of rigid-body modes .............................. 160 Figure 9-3. Spanwise distribution of stability contribution of blade -1; IBPA=0deg; low

subsonic L1; k=0.1 ......................................................................................... 161 Figure 9-4. Variation of stability contribution of blade -1 with interblade phase angle;

low subsonic L1; k=0.1; axial bending ............................................................ 163 Figure 9-5. Variation of stability contribution of blade +1; IBPA=90deg; low subsonic;

k=0.1; axial bending ....................................................................................... 164 Figure 9-6. Graphical quantification of three-dimensional effects with respect to

effects of reduced frequency and flow incidence ............................................ 165 Figure 9-7. Variation of stability contribution in span, reduced frequency and

operating point for blade pair ±1; low subsonic; axial bending ........................ 166 Figure 9-8. Variation of stability contribution in span, reduced frequency and

operating point for blade pair ±1; low subsonic; torsion .................................. 167 Figure 9-9. Comparison of measured and predicted arcwise stability contribution at

midspan on blade -1; IBPA=90deg; operating point L1; axial bending ............ 168 Figure 9-10. Comparison of measured and predicted arcwise stability contribution at

midspan on blade -1; IBPA=90deg; operating point L1; circumferential bending....................................................................................................................... 169

Figure 9-11. Comparison of measured and predicted arcwise stability contribution at midspan on blade -1; IBPA=90deg; operating point L1; torsion ...................... 169

Figure 9-12. Comparison of measured and predicted stability plot at midspan; IBPA=0deg; shaded areas mark stable regions.............................................. 170


Figure 9-13. Comparison of measured and predicted stability plot at midspan; IBPA=90deg; shaded areas mark stable regions............................................ 172

Figure 10-1. Proposition for real and locally deforming test objects ...................... 177 Figure 10-2. Predicted steady-state total pressure distribution from inviscid and

viscous models............................................................................................... 178



1 Introduction Flutter denotes a self-excited and self-sustained aeroelastic instability phenomenon that involves vibration of a structure when exposed to a fluid flow. In turbomachines flutter is prone to occur in the parts of the engine where long and slim blades are exposed to aggressive blade loading such as in the fore part of compressors or the aft part in turbines. Unless properly damped, it can result in material fatigue and eventually to damage of engine components in a short period of time. Depending on the extent of the instability, failure may be caused due to excessive stress or high cycle fatigue (HCF) in case of limit cycle oscillations. In the present work the focus has been put on flutter in low-pressure turbines (LPT). Safety and economical aspects rather than efficiency concerns therefore drive the assessment of the flutter phenomenon. The relevance of the problem has been indicated by El-Aini (1997) stating that although 90% of potential HCF occurrences are uncovered during engine development the remaining 10% stand for one third of the total engine development costs. Field experience as the one presented by Sieg (2000) has shown that during the last decades as much as 46% of fighter aircrafts were not mission-capable in certain periods due to high cycle fatigue related mishaps. The design for aeroelastic stability is therefore one of the paramount tasks in engine design. Aeroelastic stability denominates the ability of a system to resettle back to a stable situation upon stochastic excitation rather than escalating and leading to occurrence of high cycle fatigue. The goal for design engineers is to ensure aeroelastic stability over as wide part of the operating range as possible and by this guarantee flutter-free operation of the turbomachine. Designing for aeroelastic stability involves the prediction of unsteady aerodynamic forces that are due to blade oscillation and their reciprocal influence on the oscillation mode. Traditionally this process has involved empirical correlations that have been based on experience from existing engines and tuned such as to allow for predictive assessment of new designs. Such methods have long been applied with good confidence within families of similar engines but have failed, as soon as similarity of a new design no longer could be guaranteed. Analytical methods for determining aeroelastic stability have successfully been employed for lightly loaded components such as fans and low-pressure compressor stages but have failed for highly loaded turbine blades. Detailed assessment of the unsteady aerodynamic flow field during flutter for various blade geometries has been made possible with the advances in computational power during the last decades and the emerge of computational fluid dynamics (CFD). In conjunction with modern finite element (FEM) analysis methods for predicting structural modes design tools have been made available that opened up for an intimate treatment of the aeroelastic design problem. These advanced aeroelastic design methods allowed for designing new generations of flutter-safe engines by eliminating most of the known flutter occurrences. Consequently increased confidence in flutter prediction methods has led to pushing engines to higher power densities and more aggressive blade loading. Flutter related mishaps in the recent engine generation however indicate that the prediction tools


have been used beyond their range of validity and that noticeable limitations exist. The problem is especially apparent when analyzing highly loaded compressors but is also encountered in low-pressure turbines. As a matter of fact flutter problems are mostly treated in two-dimensional manner for the sake of simplicity indicating that the blade-to-blade geometry is modeled at a certain section rather than the generally complex three-dimensional geometry of turbomachine blading. It is however believed that part of the encountered limitations in predicting aeroelastic stability is due to three-dimensional effects that are present in real engines but not captured by the models. Although more advanced numerical tools have recently made it possible to treat the aeroelastic problems in three-dimensional environment the validation is still progressing due to the lack of experimental data. The present work contributes by investigating three-dimensional aerodynamic effects on the aerodynamic damping during flutter based on a compound experimental and numerical approach.


2 Background

2.1 Description of the Flutter Phenomenon

Flutter denotes an unstable vibration phenomenon of a structure that is exposed to fluid flow. In a condition of instability initially small vibrations at the verge of flutter induce unsteady aerodynamic forces that feed energy into the structure leading to rapidly growing in magnitude by each cycle. Normally the escalation of the instability cannot be prevented leading to excessive structural oscillations and material failure. The difference between flutter and resonant vibrations in turbomachines is that the flow unsteadiness leading to structural oscillation is induced by the motion of the structure itself rather than an external source. The existence of flutter is yielding from a balance between the unsteady forcing by the fluid, the inertial and damping forces of the structure respectively and the elastic forces of the structure. Flutter is occurring if this balance attains unstable condition meaning that the fluid is feeding energy into the structure leading to larger oscillation amplitude and consequently to even larger aerodynamic forcing. From a phenomenological point of view the assessment of the flutter involves steady and unsteady aerodynamics as well as structural dynamics and is embraced by the science of aeroelasticity. A graphical interpretation of the phenomenological interaction leading to flutter has been given by Collar (1946) as depicted in Figure 2-1.

Figure 2-1. Collar’s triangle of forces Structures, which are long, slim and exposed to high aerodynamic loading are prone to flutter as the unsteady aerodynamic forces gain in magnitude relative to the structural forces. Within the aeronautic field flutter was first observed on wings as airplanes reached higher velocities in the first half of last century. Soon it was recognized that the ratio between wing mass and the mass of surrounding air inside a circle with radius half chord has a noticeable influence. This ratio, referred to as mass ratio, is given by


20

4

c

m

πρµ = , Eq. 2-1

where m is the mass per unit blade span, 0ρ the air density and c the blade chord.

As the mass ratio decreases flutter susceptibility increases. In turbomachines the mass ratio attains comparatively large values and thus does not gain the same importance. Meldahl (1946) has found that flutter in turbomachine blade rows occurs above certain flow velocities and that it is rather the ratio of flow velocity, blade chord and oscillation frequency that dominates flutter stability. The ratio is known as reduced frequency and relates the time of flight for a fluid particle needed to travel across blade chord to the oscillation period as is

u

fc

T

tk

π2== , Eq. 2-2

where f is the oscillation frequency, c the blade chord and u the flow velocity.

Another though equivalent interpretation of the reduced frequency is that it relates to the blade chord to the wavelength drawn out by a sinusoidal oscillation as given by

λc

k = , where f

uu

πωλ

2== Eq. 2-3

Small values of reduced frequency indicate that the time of flight is short compared to the oscillation period, in other words that the flow is able to settle to changed conditions and thus has a quasi-steady character. For each setup a value of critical reduced frequency can be found below which flutter can occur. For a certain oscillation frequency this indicates that the value of critical reduced frequency is approached as flow velocity increases. For turbomachine blades critical reduced frequencies have been reported in the range between 0.1 and 1.0. A graphical interpretation of mass ratio and reduced frequency is included in Figure 2-2.

0 20 40 60 80 100 120 140 160 180 200

20

10

0

10

20

30

Mass ratio Reduced frequency

Figure 2-2. Graphical interpretation of mass ratio and reduced frequency

ρ0 m

c c

λ


The occurrence of flutter in turbomachines is almost exclusively limited to blades in the fore and aft part of the engine (Srinivasan, 1997) such as fans, low-pressure compressor (LPC) and low-pressure turbine (LPT). Figure 2-3 shows a typical aero engine turbomachine denoting the areas susceptible to flutter. The area of interest of the present work (LPT) is highlighted in the figure.

Figure 2-3. Cross-section of a modern aero engine turbomachine (Volvo RM12); flutter susceptible areas indicated, area of current work highlighted; picture courtesy

of Volvo Aero Given the blade row environment in turbomachines the phenomenon of flutter usually involves an arrangement of blades and for that reason a blade row rather than a single blade is regarded. The motion of each single blade is influencing instantaneously the flow field in its direct neighborhood inducing a response on itself and on its direct neighbors as depicted in Figure 2-4. This phenomenon is referred to as aerodynamic coupling. In one of the early studies Bellenot and Lalive d‘Epinay (1950) have recognized that an arrangement of blades might become aeroelastically unstable although a comparable isolated blade would not flutter. Several efforts have been dedicated to the coupling phenomena with the aim to understand the effect of coupling on cascade aeroelastic stability. Triebstein (1976), Kirschner et al. (1976) and Carta and St.Hilaire (1980) can be cited as early studies addressing the coupling phenomena in blade rows. All investigations concluded that the aerodynamic response on the blades in an oscillating blade row is influenced to a large degree by coupling effects. Széchényi (1985) has indicated that the aptitude of a single blade to flutter was of the same importance as coupling effects in a cascade. In a systematic categorization of unsteady flow phenomena in turbomachines Greitzer et al. (1994) classify the phenomenon of flutter being one order of magnitude larger in extent than the blade pitch.

Fan Low-pressure turbine (LPT)


−100 −50 0 50 100 150

−100

−50

0

50

100

+3

+2

+1

0

−1

−2

−3

Figure 2-4. Influence phenomenon during blade row flutter in turbomachines The coupling influence is thereby largely affected by the relative motion between two adjacent blades. In a traditional approach as for example given by Crawley (1988) the oscillatory motion of a tuned blade row during flutter can be characterized as a traveling wave mode indicating that all blades oscillate in the same mode, amplitude and frequency but at a certain phase lag between two adjacent blades. The phase lag between two adjacent blades is referred to as interblade phase angle and can take discrete values that yield from the kinematical constraint to fulfill full cycle periodicity as is

N

l⋅= πσ 2, Nl ,...3,2,1= Eq. 2-4

The order of the traveling wave is given by parameter l also referred to as nodal diameter. For each nodal diameter pattern a pair of traveling waves is induced, a

forward traveling wave with N

l⋅= πσ 2 and a backwards traveling wave with

−=

N

lNπσ 2 . An example of nodal diameter patterns and corresponding cascade

geometries is depicted in Figure 2-5.

stator stator rotor


−100 0 100

−300

−200

−100

0

100

200

300

400

−100 0 100

−300

−200

−100

0

100

200

300

400

Nodal diameter pattern Blade row Nodal diameter pattern Blade row

Nodal diameter 1=l Nodal diameter 3=l

Figure 2-5. Nodal diameters of a disk, traveling wave mode shape and corresponding instantaneous blade row geometry

The excitation of structural modes in the turbomachine environment is traditionally assessed by means of a Campbell diagram as the one shown in shown in Figure 2-6. The diagram depicts structural vibration characteristics and engine order excitation lines versus rotational speed and allows recognizing potential vibration problems at crossings between these lines. As flutter occurrences are not bound to engine order lines and therefore can occur at any rotational speed the predictive assessment of flutter gets one additional unknown dimension.

Figure 2-6. Campbell diagram indicating the occurrence of flutter (arrows); adapted from Fransson (1999)

The vibration properties of blade rows depend largely on the structural setup of the assembly. From a general perspective a turbomachine blade row can mechanically be modeled as a disk with blades fixed to it. Depending on whether the blades are interconnected to each other above the root the assembly is referred to as unshrouded bladed disk or shrouded bladed disk as shown in Figure 2-7. The location of the shroud may vary from part span to full span, i.e. tip shroud.

+ - +

+

+ -

-

-

Flutter occurrences

Synchronous vibrations


Unshrouded Shrouded

Figure 2-7. Unshrouded and shrouded bladed disk assemblies. Unshrouded bladed disk assemblies result in blades being freestanding. The oscillation modes of such assemblies can be separated into blade-dominated or disk-dominated modes. The former summarizes the modes that are either blade global modes such as bending or torsion or local blade modes as for example corner modes or stripe modes. In disk-dominated modes the individual blades play a subordinate role and are rather to be seen as passive element attached to a rotor disk. To assess blade dominated modes the single blades can in a simplified manner be modeled as beams. The three mode shapes with lowest frequencies, which are of greatest interest in the present context, are accordingly two bending modes and one torsion (twisting) mode defined by the position of the elastic axes. With relation to the blade these modes are referred to as flap, edgewise bending and torsion (twisting) as depicted in Figure 2-8. All modes feature a certain eigenfrequency at rest, which change with the rotational speed of the turbomachine due to centrifugal forces. Other modes include stripe mode (in-plane oscillation) as well as local deformation modes such as corner modes.

0.3

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

−0.02 −0.01 0 0.01 0.02

0

0.01

0.02

0.03

0.04

0.39

0.4

0.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

−0.01 0 0.01 0.02 0.030

0.01

0.02

0.03

0.04

0.05

0.3

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

−0.01 0 0.01 0.02 0.03

0

0.01

0.02

0.03

0.04

Flap Edgewise bending Torsion

Figure 2-8. Blade first-order eigenmodes Several researchers have addressed the mode shape as major influence for flutter stability. Bendiksen and Friedmann (1982) have analytically studied a cascade oscillating in bending and torsion modes over a large range of flow regime and showed that bending and torsion stabilities develop differently. Systematic studies on the influence of mode shape have been carried out by Panovsky and Kielb (2000), Nowinski and Panovsky (2000) and Tchernycheva et al. (2001) employing a


graphical method referred to as Panovsky-Kielb method that assesses the aeroelastic stability on a two-dimensional section of a cascade. The method assumes pure rigid-body modes and is based on representing any possible modes as torsion modes with respective center of torsion resulting in so-called stability maps. An example of such map is included in Figure 2-9 specifying the critical reduced frequency below which flutter could occur for a given center of torsion.

Figure 2-9. Graphical representation of critical reduced frequency versus torsion axis location for a 2D section of a cascade (Panovsky-Kielb method); from Chernysheva

et al. (2003) The Panovsky-Kielb plots illustrate efficiently the effect of mode shape on flutter stability; as can be seen in Figure 2-9 there are regions of high gradients around the blade suggesting that a small change in torsion axis location does have a detrimental effect on the magnitude of critical reduced frequency. It can be noted as interesting fact that stability maps are of similar nature for different types of turbine geometries as resulted from a study performed by Tchernycheva et al. (2001). Kielb et al. (2003) therefore draw the conclusion that standard maps for turbine mode shape stability can be used in the preliminary aeroelastic design thus omitting time-consuming unsteady aeroelastic analyses in a first stage and performing structural dynamic analyses only. Peng and Vahdati (2002) have employed a ratio of mode shape displacement of blade leading and trailing edge as included in Figure 2-10 such as to assess the effects of mode shape on stability. The result indicates that although the method is different in its roots from the above-presented graphical tool it leads to similar conclusion of the mode shape affecting greatly flutter stability.

Reference blade Example: flutter

is predicted below k=0.4 if the reference blade is oscillating around this center of torsion (motion indicated) Reduced frequency k here based on semi-chord


Figure 2-10. Flutter stability versus mode shape displacement ratio for assessing the effect of mode shape on stability; from Peng and Vahdati (2002)

Further kinematical constraints might be imposed due to tying blades together to blade packages such as to increase structural stiffness. This method is commonly employed for suppressing flutter both due to an increased mechanical stiffness of the assembly and due to restricting the coupling of the blades to few interblade phase angles. The measures can extend over the entire circumference as in the case of shrouded rotors or part-span shrouds or to cyclic symmetric sectors as in the case of sectored vane assemblies. Ewins (1988) has given a schematic description of blade assembly modes as shown in Figure 2-11.


Figure 2-11. Possible vibration modes for blade packages; from Ewins (1988) Chernysheva et al. (2003) conclude from a theoretical analysis that although sectoring of blades to blade packages might increase stability the existence of unstable regions might not be suppressed completely. Corral et al. (2003) have performed a computational analysis on pairs of low-pressure turbine blades. Their main finding is that aeroelastic stability of welded pairs of blades is greater than for single airfoils especially for torsion and circumferential bending modes, which indicated that sectoring of blades is an efficient mean to increase flutter stability. Structural coupling of blades might lead to the occurrence of coupled modes that combine two modes under a certain phase angle. In this case the phase angle between the involved modes detrimentally affects aeroelastic stability. Bendiksen and Friedmann (1982) have analyzed the problem analytically concluding that the coupled bending and torsion flutter affects the flutter boundary significantly. A tendency of the two modes to coalesce could however not be observed as flutter has been approached. Försching (1991) has elucidated the effect of phase angle between bending and torsion mode on the work performed from a quasi-steady consideration; for a phase difference of zero degree the total work examined by the fluid equals to zero. Positive work results for a phase angle of 90deg, see Figure 2-12.

Figure 2-12. Effect of phase angle between bending and torsion mode on the aerodynamic work performed; adapted from Försching (1991)

negative work

positive work

positive work

negative work

Direction of oscillation

Direction of force

positive work

positive work

Phase angle 0deg total work zero

Phase angle 90deg total work positive


In addition to reduced frequency and mode shape it has been observed that flow incidence has a dominant effect on aeroelastic stability. On one hand it affects directly the mean loading of the blades and has identical effects in turbomachine blade rows as it has for isolated airfoils. Generally it has been shown from several investigations that higher angles of attack lead to reduced stability. Carta and St.Hilaire (1980) have performed systematic studies on the influence of incidence, reduced frequency and interblade phase angle on a linear research compressor cascade finding that critical reduced frequency decreases with increasing angle of attack. They stated however clearly that this was not due to an incidence-induced separation. Széchényi (1985) has shown the effect of reduced stability at high angles of attack from tests in a compressor cascade. Above a certain angle of attack though the flow may separate around the leading edge of the blade profile leading to the aeroelastic stability being dominated by the separated flow behavior. Buffum et al. (1998) have investigated a compressor cascade at high mean incidence and shown that the character of damping largely depends on the state of flow around the leading edge, which in turn was directly influenced by flow incidence; a high angle of attack led to separated flow that had a destabilizing effect whereas attached flow acted stabilizing in this region. Peng and Vahdati (2002) have drawn the conclusion that high incidence leads to destabilizing from an analytical study of a compressor at near stall and near choke. Ellenberger and Gallus (1999) have drawn the conclusion from a compressor cascade experiment on torsional flutter that although a shock-induced separation was present the shock rather than the separation bubble dominated the stability character. He (1996) investigated the effect of separated flow experimentally in a cascade of low-pressure turbine blades at various incidence angles with one blade oscillating in torsion mode and found that a large separation bubble on the pressure acted destabilizing. However it is interesting to notice that the region downstream of the reattachment point acted stabilizing, which balanced the negative effect of the separation bubble. Queune and He (2000) have performed flutter testing on a steam turbine profile in bending mode and at massive part-span separation. The tests yielded that the separated region featured reduced stability compared to attached flow.


Closer attention shall at this position be drawn to the space adjacent to the regarded blade row. Due to the oscillation of the blades a varying pressure field is established up- and downstream of the blade row that can induce acoustic resonance in the adjacent ducts. Whitehead (1973) draws the distinction between three regions of acoustic resonance flutter:

1) subcritical flutter where acoustic waves can not propagate in the duct 2) acoustic resonance flutter where a pair of waves are just at the verge to

propagate 3) supercritical flutter where at least one pair of waves can propagate.

Acoustic resonances can only exist over a certain range of interblade phase angles in which the traveling wave mode pattern matches the resonance pattern and therefore suggest the distinct influence of higher blade indices. The practical implication is that the acoustic resonance adds one degree of freedom to the aeroelastic system, which might be effective for extraordinary coupling of blades during flutter. Acoustic resonance flutter is almost exclusively of relevance in empty ducts adjacent to the regarded blade row as adjacent blade rows tend to suppress resonant behavior (Whitehead, 1973). Wu et al. (2003) have recently underlined the relevance of aeroacoustic flutter for a high bypass-ratio jet engine where the aeroacoustic properties of the inlet duct triggered flutter of the fan, which was observed as sharp and local drop in the flutter stability margin referred to as “flutter bite”. Up to this point the phenomenon has exclusively been discussed in the extent of a single blade row. In the turbomachine environment the adjacent blade rows are affecting the flutter properties in various ways. On one side they are inducing flow disturbances that are overlaid to the unsteadiness induced by the oscillating blade row and by this diffusing the line between flutter and forced response. Frey and Fleeter (1999) have addressed the phenomenon of combined gust and flutter in a low-speed rotating compressor facility with the blades oscillated in torsion mode and the inflow having been subjected to a 2/rev inlet distortion. The influence of gust on flutter stability has been found being dependent on the phase lag between the two mechanisms leading to either a constructive or destructive constellation meaning that flutter stability is decreased in the latter case.


Another effect from adjacent blade row is the influence on the aerodynamic coupling as a neighboring blade row has the ability to reflect pressure waves and thus change the coupling response. Hall et al. (2003) have modeled the coupling effect by means of spinning modes, i.e. a set of pressure and vorticity waves that propagate between blade rows. Reflection and transmission properties from adjacent blade rows were thereby computed from isolated blade row analyses. Figure 2-13 depicts the effect of multistage coupling by comparing the results of an isolated blade row analysis to a multistage analysis using two-dimensional models. Large differences can therein be recognized in the range of interblade phase angles of 90deg to 180deg. Using the same approach Chuang (2004) could show that the flutter boundary was shifted by 5% in flow margin.

Figure 2-13. Effect of multistage coupling on flutter stability (2D simulations); from Hall et al. (2003)

The effect of aeroelastic mistuning has early been identified as favorable and as potential method for preventing flutter. Széchényi (1985) has found that mistuning can be an effective method to shrink the stability locus loop of aerodynamic damping. A limitation however is that the method cannot be used to change the position of the loop relative to the axis; if a single-blade flutter is prone to occur it is not possible to be remedied by mistuning. Nowinski and Panovsky (2000) have assessed the method of mistuning in an annular cascade flutter experiment and concluded that alternate mistuning represents the most stabilizing pattern. Silkowski et al. (2001) have made a similar observation of that the destabilizing influence of blade pair ±1 can be broken down by arranging blades with alternating varying natural frequency around the circumference.


2.2 Review of Flutter Analysis Methods

Analysis methods for predicting flutter stability are employed to prevent flutter occurrences prior to operation of new engine or engine components. In detail the methods used involve the analysis of structure dynamics as well as unsteady aerodynamics and the evaluation of results with respect to aeroelastic instability. From an analytical point of view the aeroelastic problem can be described by the aeroelastic equation as given by

[ ] [ ] [ ] )(tFXKXGXM ae=++ &&& Eq. 2-5

The left-hand side of the above equation reflects the structural part where M being the mass, G the structural damping and K the stiffness matrices. The right-hand side includes the unsteady aerodynamic forces that are due to the blade motion and contains the aerodynamic damping. Generally there are three different approaches used in the prediction of flutter stability. The first approach foresees an implicit analytical description of the structural and the aerodynamic part and is also referred to as reduced-order model. A simplified analytical model is thereby usually used to describe the structural part. Rather than resolving the flow around the airfoils a representation of the unsteady aerodynamic force only is employed. The aeroelastic system is then fully defined and can be reduced to an eigenvalue problem yielding dynamic behavior and occurrences of flutter instability over a parameter range of interest. The approach has its limitations in that the analytical description of the unsteady aerodynamic force is heavily simplified and therefore cannot be used with confidence for highly loaded geometries. Imregun (1995) has presented a method of using frequency response functions to determine flutter stability. Whereas a lumped parameter model was used to describe the structural part linearized unsteady aerodynamic theories presented by Smith (1972) for subsonic flow and Nagashima and Whitehead (1976) for supersonic flow have been implemented for representing the unsteady aerodynamic force. The technique was employed on a 12-bladed disk of flat plates yielding frequency response functions over a range of frequencies. The strength of the method lies in the inherent coupling of structure and flow and comparatively small computational times. Copeland and Rey (2004) have used an actuator disk model to represent a fan stage and could show that the overall aeroelastic model was tunable to experimental data. Whereas this approach could be used to assess the influences of changes from a known setup such as for example the assessment of mistuning, it did however not allow predictive application. The second approach foresees a separate i.e. decoupled treatment of the structural and the aerodynamic part. The coupling is effectuated by aerodynamic forces for the structural part and motion of the structure for the aerodynamic part. The dynamic behavior of the aeroelastic system yields from modal comparisons of the structural and the aerodynamic part. This approach has the great advantage that practically any method can be used for predicting the structural and the unsteady aerodynamic part. As the critical mode shapes of the setup are not known a priori a method must


be used to predict the aerodynamic damping properties for a number of possible structural modes. Rather than performing an immense number of unsteady aerodynamic predictions simplifying assumptions can be made such as linearity and non-deforming blade sections, which allows spanning of a mode space by three orthogonal modes. The evaluation of stability is then yielding from solving the modal equations, i.e. detecting unstable modal coalescences of the structural and aerodynamic system. The drawback of this method however is that the effects of aerodynamic damping on the structural modes are not taken into consideration unless the process is performed in iterative manner. On the aerodynamic side the methods can be separated depending on the flow model used and the employed temporal discretization. The different flow models cover analytical methods, linear potential methods assuming inviscid, isentropic and irrotational flow or discrete models solving either the inviscid Euler or the viscous Navier-Stokes equations on a computational mesh. Analytical models as the ones described by Whitehead (1987) are assuming flat plates and thus do not account for steady blade loading. The linear potential method (Verdon and Caspar, 1982) allows studying effects of airfoil geometry on loading and flow structure even in the transonic regime provided the shocks are weak and the condition of isentropic flow is not harmed. The discrete Euler method can be used for the same conditions as potential models but have the advantage that it can deal with rotational and non-isentropic flow, i.e. the limitation of weak shocks is no longer valid. It resolves the flow field in detail but does not reflect shear layers such as boundary layers or separated flow bubbles correctly due to absence of viscosity. The most accurate description of the flow is thus given by the Navier-Stokes model taking into account viscosity. This model is applicable to the same flows as can be treated by an Euler analysis. From the temporal discretization point of view the unsteady equations can either be solved time-marching or in a linearized manner. In the first case the characteristic equations are solved at each node of the computational mesh at each time step. Convergence is achieved if a time-periodic flow is established. Non-linear time-marching models have for example been presented by Fransson and Pandolfi (1986), Giles (1988) and Whitfield et al. (1987) employing Euler equations and Huff (1987) and Rai (1989) for Navier-Stokes equations. The linearized approach assumes small perturbations of flow variables around a steady mean value. The steady flow solution can thereby be determined from a steady non-linear flow analysis. Thereafter the perturbation equations are solved on this mean value until a steady perturbation amplitude is reached. Such models have for example been presented by Hall and Crawley (1989), Holmes and Chuang (1991), Lindström and Mårtensson (2001) and Petrie-Repar (2003). Whereas unsteady non-linear viscous models allow for the most accurate representation of unsteady flow they are associated with high computational costs in the order of magnitude of several days, which decreases applicability especially in the industrial environment. On the other hand it has been pointed out above how different flow phenomena such as flow separation act on flutter stability. The decision for what model to use for a certain analysis must therefore be justified by the targeted application.


The third and last approach treats the structure and the flow as one continuum and foresees the solving the structural and the aerodynamic part in a fully coupled and time-marching manner. The boundary conditions of force for the structural part and motion for the aerodynamic part are used at the structure/flow interface. The aeroelastic equation is solved at each time step for a number of cycles and the aeroelastic stability is determined from the temporal development of characteristic variables. Divergence of the variables indicates an aeroelastic unstable situation whereas convergence indicates stability. The prominent mode shape during flutter is directly yielded from such analysis. Applications of fully coupled models have for example been presented by Vahdati and Imregun (1995) and McBean et al. (2002). Fully coupled models represent the most accurate model for the dynamic analysis of an aeroelastic system but feature the highest computational costs.


2.3 Review of Flutter Testing Methods

Flutter testing can be performed on a component basis or in the frame of engine tests and finds its application in research as well as engine commissioning. Component tests feature inherently lower complexity compared to full engine tests and by this open up for a more intimate analysis of the flutter phenomenon. On the other hand such tests are performed in an idealized environment and therefore often do not model all the effects that are influencing flutter. In a cascade flutter test the effects of adjacent blade rows are for example not present. The section gives a review of different testing techniques and overview of the most prominent examples. The occurrence of flutter is an instability phenomenon involving an oscillation system comprising fluid flow and structures as outlined above. Flutter testing can either be aimed at detecting eventual instability of the system or at determining the transfer function of the system over a range of oscillation parameters. In the first case the test performed are of free-flutter type indicating that outer parameters such as flow velocity or angle of attack are changed until onset of flutter can be observed. The second case represents a forced oscillation testing method during which the system is excited in a controlled manner and the response is monitored such as to determine the transfer function. These two types of flutter testing are reviewed below.

2.3.1 Free Flutter Testing

Free flutter testing stems from a “build it – test it” approach with the aim to detect if flutter occurs at a certain set of flow conditions. The object of investigation is thereby exposed to a well-defined fluid flow and measurements are taken such as to detect unstable oscillatory motion and eventually also aerodynamic response characteristics. Free flutter tests can be performed in real engines and test frames that are accurate models (annular rotating rigs) or on a more generic basis in simpler cascades. The former types of tests are rare due to the inherently higher complexity. If the scope of the tests is to assess the aerodynamic damping characteristics it is tried to increase aerodynamic forcing compared to structural damping, which can be achieved by suspending blades elastically or increasing blade aspect ratio. In certain cases the spring constant of the suspension and the mass of the blade are made variable such as to measure inversely aerodynamic damping. For monitoring the motion of the blades traditionally strain gauges are employed although alternative motion capturing methods such as optical techniques have been reported. Some of the few full-scale flutter tests reported in open literature have been performed in the Compressor Research Facility (CRF) at Wright Patterson Air Force Base. Sanders et al. (2002) studied the flutter properties of a transonic low aspect ratio fan blisk and found that a blade passage shock acted most destabilizing. Additionally it was found that structural dynamics was the key driver for mistuning response. Manwaring et al. (1996) employed the same facility for the investigation of forced response due to inlet distortion in a 2-stage low aspect ratio fan. Bellenot and Lalive d’Epinay (1950) have employed a linear cascade consisting of five low-pressure compressor profiles with increased aspect ratio compared to their


counterparts in real engines such as to promote the influence of aerodynamic forcing over elastic forces. Detection of flutter as well as determination of flutter frequency was performed by means of a stroboscope. Flutter was detected above a certain flow velocity, which gave proof for the validity of reduced frequency approach as stability criteria. Urban et al. (2000) have used a seven bladed linear cascade of last stage profiles of steam turbines. The blades were elastically suspended as shown in Figure 2-14 such as to allow a torsional oscillation with the center of torsion being located upstream of the leading edge. Blade motion was monitored by means of strain gauges. In addition the blades were equipped with miniature pressure transducers such as to provide information on the unsteady loading during flutter.

Figure 2-14. Free flutter test setup for single mode testing; Urban et al. (2000) A setup with four elastically suspended blades has been used by Kirschner et al. (1976), see Figure 2-15. Friction in the blade suspension has been minimized with the aim to cancel out the damping contribution in the aeroelastic balance. The tests performed aimed at acquiring stability characteristics of damping and frequency versus reduced flow velocity (inverse of reduced frequency) for the cascade oscillating in bending mode, torsion modes with different center of torsion as well as combined modes. The cited investigation is one of the first ones addressing the importance of blade mode shape on the aerodynamic damping.

Test section Blade support

Figure 2-15. Free-flutter test setup for variable mode testing; Kirschner et al. (1976)


As a matter of fact free flutter tests generally have lower complexity compared to controlled flutter tests. There are however several drawbacks. Firstly flutter data can only be obtained at the least stable condition, which establishes naturally. Flutter data are therefore only available for the least stable interblade phase angle. Secondly no aerodynamic damping information can be gathered at no flutter condition and thus no conclusions can be drawn on how sensitive the system is. Thirdly, as most free flutter tests employ a cascade that restricts or promotes certain blade oscillation modes, see for example the study performed by Urban et al. (2000), test data are only available for the addressed modal dimension of the system. Hennings and Send (1998) have eliminated these drawbacks by employing a linear cascade of elastically suspended compressor airfoils undergoing torsional motion that could be operated in both free oscillation as well as excited mode by forcedly oscillating one blade. The method used consisted in reducing blade displacement data to eigenvectors and finally frequency response functions to describe aeroelastic stability.

2.3.2 Controlled Flutter Testing

Controlled flutter testing is used for determining aerodynamic damping characteristics of a setup. The flow is thereby considered as oscillation system that similar to a structural oscillation system features inherent dynamic characteristics. By exciting the system in a controlled manner and measuring the aerodynamic response information on aerodynamic damping and consequently on aeroelastic stability can be acquired. Most controlled flutter tests excite the system via the motion of the blades although it is generally possible to induce the excitation via aerodynamic disturbance forces. In the former case the regarded setup is exposed to fluid flow while the structure is oscillated in a controlled manner and aerodynamic response data is acquired. In accordance thereof the latter method foresees as well that the setup under investigation is exposed to fluid flow however the forcing is introduced by aerodynamic disturbance forces that are terminated abruptly such as to let the structure oscillate freely. Aerodynamic damping data can then be deduced from the oscillation properties of the structure. Motion-induced controlled flutter testing is widely used for testing. Under application of the linearized theory that has been lined out above there are the following two testing methods:

• Traveling wave mode testing: all blades in the cascade are oscillated at identical mode shape and at various interblade phase angles. The response is measured on one blade.

• Influence coefficient testing: only one blade is oscillated and the

response is measured on all the blades in cascade. The data are superimposed at different interblade phase angles according to the theory presented below such as to yield damping data in the traveling wave mode domain.


These two testing methods have different strengths and disadvantages. Traveling wave mode testing in an annular cascade is the one that most accurately represents the circumstances in real engines. In such setups pressure waves can freely propagate in circumferential direction, which allows the detection of acoustic resonances. The test setup gets however costly and complex firstly due to higher complexity of annular facilities and secondly due to the fact that all blades in the cascade have to be oscillated in a controlled manner. Bölcs and Fransson (1986) have employed an annular non-rotating cascade shown in Figure 2-16 where all blades or only single blades could be oscillated for the systematic investigation of flutter phenomenon in compressor and turbine cascades. Flow conditions could be varied from subsonic to transonic. Controlled oscillation of the blades is achieved by a spring type suspension of the blades that are submitted to electromagnetic excitation.

Elastic blade suspension

Kahl and Hennings (2000) Assembled blade carrier

Nowinski and Panovsky (2000)

Figure 2-16. Annular non-rotating cascade for traveling wave mode and influence coefficient testing

Another type of annular cascade has been used by Frey and Fleeter (1999) for the investigation of combined gust and flutter in low-speed tests. The facility comprises a 3-stage experimental compressor with blades that can be made oscillating in traveling wave mode at frequencies proportional to the rotational speed. Oscillation of the blades is achieved by a cam follower assembly as shown in Figure 2-17.

Figure 2-17. Purdue 3-stage experimental compressor; Frey and Fleeter (1999)


A simplification of annular facilities is achieved by traveling wave mode testing in linear cascades as the complexity of the setup decreases to an easier manageable level. Linear cascades are widely used due to the fact that data can readily be compared to two-dimensional theory without the need for taking into account complex three-dimensional mean flow phenomena. Bölcs and Fransson (1986) have compiled a set of 9 two-dimensional test cases, so-called standard configurations, for validation purposes. There are however two major drawbacks in the use of linear cascades. On the steady-state side passage-to-passage periodicity, which otherwise is inherently present in annular cascades, must be achieved by means of flow and geometric devices at least in the center passages to achieve acceptance of data. On the unsteady side the walls limiting the cascade induce deteriorating pressure wave reflections and prevent the establishment of acoustic resonance flutter. These drawbacks have been assessed by several studies. Carta (1983) has addressed the unsteady blade-to-blade periodicity in a cascade of 11 blades depicted in Figure 2-18 and has shown that although good unsteady periodicity could be achieved for low incidence conditions at low speed higher mean incidence led to deteriorated unsteady periodicity. In addition the presence of an acoustic resonance has been reported at a specific interblade phase angle.

Figure 2-18. UTRC Oscillating Cascade Wind Tunnel (OCWT); Carta (1983) In a series of investigations Buffum and Fleeter (1991) and Buffum and Fleeter (1994) have investigated the influence on wind tunnel walls on the unsteady performance during flutter testing in the NASA Lewis Transonic Oscillating Cascade shown in Figure 2-19. The facility features capabilities of oscillating all blades in traveling wave mode as well as single blade modes. Oscillation is achieved in a mechanical way by means of a cam follower assembly. To reduce acoustic reflections it has been suggested to treat the tunnel walls adequately, which could be achieved by means of specially absorbing material. The locations of acoustic treatments is indicated in Figure 2-19. Lepicovsky et al. (2000) have addressed the steady-state performance in a transonic linear compressor cascade and indicated the importance of wind tunnel wall geometry on steady flow field periodicity.


Figure 2-19. NASA Lewis Transonic Oscillating Cascade; Buffum and Fleeter (1991) The advantages in influence coefficient testing lies mainly in the lower complexity of the experimental setup, as only one blade needs to be oscillated. Furthermore the treatment of the phenomenon gets more intimate as the blade specific influence coefficients are determined directly. Several research groups have addressed the experimental validation of the linearized theory underlying influence coefficient testing. Hanamura et al. (1980) have found excellent agreement of lift and moment coefficients measured in an oscillating cascade in traveling wave mode and single blade mode oscillation in a water tunnel and proved the validity of the assumption for incompressible flow. Széchényi (1985) has presented data from tests in a linear cascade with one and three blades oscillating and has found that the data acquired with the two different methods correlate well. As the scope of the paper was not primarily directed towards the validation of influence coefficient technique no conclusive statements were made. One of the few conclusive investigations was the one by Bölcs and Fransson (1986) performed in an annular oscillating cascade. As data from the two different testing methodologies has shown very good agreement for small oscillation amplitudes and even in the transonic regime it was concluded that the influence coefficient was valid for testing aeroelastic stability. The major part of tests however is performed in linear facilities and it would be wrong to apply this result one-to-one to this type of setup. Buffum and Fleeter (1994) have addressed this issue in the NASA Lewis Transonic Oscillating Cascade where all the blades and only single blades could be made oscillating. They detected however acoustic modes that were due to reflections of pressure wave from the tunnel walls, which deteriorated the otherwise good correlation between traveling wave mode and influence coefficient data at certain interblade phase angle. Unless the aforementioned proper measures are taken validity of the testing technique was not guaranteed over the entire interblade phase angle range. From these investigations it can be concluded that the influence coefficient technique is validated for both annular and linear cascades at the investigated flow conditions and within the assumptions made, i.e. small perturbations. In case of circumferentially limited setups as are linear facilities wind tunnel acoustic modes must be taken into consideration and if necessary appropriate measures must be taken.


Aerodynamically induced controlled flutter testing can rather be seen as exception but shall shortly be revised here for the sake of completeness. Crawley (1981) has presented a flutter testing method in which the forcing is induced by aerodynamic disturbance forces rather than a controlled motion. Tests were performed in the MIT blowdown facility on a transonic compressor rotor with flexible blades. The disturbance was introduced as steady in the absolute frame of reference by upstream injectors the pattern of which resulted in excitation of a certain interblade phase angle. The facility and the injection device are shown in Figure 2-20. After speeding-up of the rotor gas was injected upstream of the rotor and then abruptly shut down to have the rotor oscillating freely. The structural response was measured by means of piezo-electric transducers and strain gauges.

Test Section Injection device

Figure 2-20. Controlled flutter testing by aerodynamic excitation; Crawley (1981) Blade oscillation systems used in motion induced controlled flutter testing vary widely in type and achievable oscillation parameters. The purpose of the systems is to induce a controlled oscillation of one or several airfoils at well-defined oscillation parameters as are oscillation mode, amplitude and frequency. In all controlled flutter testing setup the airfoils are oscillated as rigid bodies and not flexibly deforming as their counterparts in real engines. The induced modes are mostly two-dimensional modes such as rigid-body torsional motion or plunging (heaving). As one of the few investigations of blades oscillation in a three-dimensional mode Bell and He (2000) have performed tests in a one-bladed linear wind tunnel with the blade oscillating in hinging mode. By type the systems can be separated into mechanical, electromagnetic and hydraulic systems. Mechanical systems usually transform rotational movement from a speed-controlled electrical motor into an oscillating bending or torsion motion. The resulting mode yields from pure kinematical constraints. Mechanical oscillation systems are generally less complex than the other two types but are subject to wear and can feature a lower power density. Electromagnetic systems usually include an elastic suspension of the blade and an electromagnetic shaker such as to excite oscillation. The resulting mode yields from the degree of freedom given by the suspension or in case of several shakers are used from the operation of the shaker setup. Hydraulic systems feature the highest power density and feature mode control similar to mechanical systems. The control at high frequencies and loads is however


difficult, which leads to these systems being costly. Figure 2-21 shows an example for each type of oscillation device.

Mechanical NASA Lewis Oscillating Cascade Buffum and Fleeter (1991)

Electromagnetic EPFL Annular Cascade Bölcs and Fransson (1986)

Hydraulic ONERA Linear Cascade Széchényi (1985)

Figure 2-21. Example of type of blade oscillation device The different systems vary widely in size and power density depending on the flow regime of the wind tunnel. For low-speed flows it is sufficient to have a system oscillating at moderate frequencies (<50Hz) to achieve relevant reduced frequencies. High-speed flow tests in the high subsonic and transonic regime however require high-speed oscillation systems capable of exciting at several hundred Hertz with much increased power density.



3 Fundamental Concepts

3.1 Introduction

This chapter covers the fundamental concepts for the analysis of turbomachine flutter stability. The aeroelastic equation presented above reflects a balance between structural and aerodynamic forces as is

[ ] [ ] [ ] )(tFXKXGXM ae=++ &&& , Eq. 3-1

where [ ]M denotes the modal mass matrix, [ ]G the modal damping matrix and [ ]K

the modal stiffness matrix. X denotes the modal coordinate comprising the

bending and torsion part. The unsteady modal forcing by the fluid flow is included on the right-hand side containing the following elements

)()()( tFtFtF edisturbancdampingae += Eq. 3-2

dampingF represents the aerodynamic damping forces that result from the motion of

the airfoil in the fluid. The forces due to upstream and downstream disturbances are contained in edisturbancF . It is important to note that in cases where the structural

damping is inhomogeneous and not necessarily proportional to the temporal derivative of the modal coordinate the structural damping part [ ] XG & is moved to the

right-hand side of Eq. 3-1 (Crawley, 1988). From a flutter point-of-view only the forces induced by the motion of the airfoil are of interest and the right-hand side of Eq. 3-1 simplifies to )()( tFtF dampingae = .

A common way to solve the aeroelastic equation is to introduce a model coordinate system

[ ] tieQtX ⋅= ωϕ)( , Eq. 3-3

where ω denotes the frequency, Q the modal displacement and ϕ the mode shape. Eq. 3-1 can be reformulated to yield the following modal equation

[ ] [ ] [ ] [ ] [ ] 0*2 =

−++− QAKGiM

Tmmm ϕωω Eq. 3-4

[ ]A denotes therein a matrix containing the modal unsteady aerodynamic forces. Eq. 3-4 represents a complex eigenvalue problem the solution of which, i.e. the eigenvalues, is describing the stability of the system. In turbomachines the structural terms are comparatively large in comparison to the aerodynamic damping term due to high mass ratio. The practical implication of this is that the structural and the aerodynamic part can be treated decoupled, which greatly


simplifies the aeroelastic problem; the structural eigenmodes are in such approach determined assuming no-flow conditions (i.e. vacuum), while the character of the aerodynamic contribution with regard to stability is yielding from a purely unsteady aerodynamic analysis as lined out below.

3.2 Determination of Flutter Stability

Incipiently it shall be made clear that the term of “flutter stability” in the present context denotes the stabilizing character of the flow rather than the stability of the entire structure-flow system. The term is used in this meaning below. For the analysis of flutter stability a two-dimensional section of a turbo blade shall be considered. Furthermore it is assumed that the blade is part of a blade row oscillating in traveling wave mode as introduced above and that the blade profile is not deforming. The harmonic motion of the blade shall be described by a complex vector

h consisting of three orthogonal components ζηξ ahh ˆ,ˆ,ˆ as are two bending modes

and one torsion mode as sketched in Figure 3-1. In the present case the direction of the two fundamental bending modes coincide with axial and circumferential direction of the blade row respectively. The direction of the torsion axis is orthogonal two these two directions and collides here with the radial direction.

−30

−20

−10

0

10

20

Figure 3-1. System of orthogonal modes An arbitrary motion of the blade profile will lead to an unsteady pressure distribution around the blade profile that in turn will result in an unsteady force on the profile. The work per oscillation cycle exerted by the fluid on the profile determines whether the flow has a stabilizing or a destabilizing character; in case of the fluid examining work on the structure the situation is referred to as unstable. In such case the work is per definition positive. It has been shown by Verdon (1987) that in case of small perturbations the unsteady pressure due to harmonic motion can be represented as harmonic oscillation with respect to the blade motion around a steady mean value as

η: circumferential bending

ξ: axial bending

ζ: torsion


( )hpti

epyxptyxpyxptyxp →+⋅⋅+=+= φωˆ),(),,(~),(),,( Eq. 3-5

where )(xp is the steady mean pressure at a certain location, ),(~ txp the respective

time-varying perturbation part and p the complex pressure perturbation amplitude of

the harmonic oscillation. Consequently the unsteady aerodynamic force is also of

harmonic nature and can be written as Fr

. For convenience the unsteady pressure amplitude is normalized by a reference dynamic head and the oscillation amplitude such that a complex unsteady pressure coefficient is defined as

refdynAp pA

pc

,,

ˆˆ

⋅= Eq. 3-6

Here the reference dynamic head is taken upstream of the blade row as

101, srefdyn ppp −= . The oscillation amplitude is referring to the two-dimensional

motion of the profile and reflects a translational amplitude for the bending modes and a rotational amplitude for the torsion mode respectively. In the same sense as for the unsteady pressure a normalized force shall be defined as follows

refdynpA

Ff

,

ˆˆ

⋅=

rr

Eq. 3-7

The unsteady normalized force yields from integration around the blade profile given by

dsfdf ⋅= ∫ˆˆ rr

Eq. 3-8

The infinitesimal normalized force components df per surface element ds are given

by

dsncfd Ap ⋅⋅= ξξr

,ˆˆ Eq. 3-9

dsncfd Ap ⋅⋅= ηηr

,ˆˆ Eq. 3-10

dsecrmd Ap ⋅⋅×=⊥ζζ

rr)ˆ(ˆ , Eq. 3-11

where n

r is the local normal direction to the surface element. Note that the force

components are per unit span due to the two-dimensional extent of the analysis and that the third component reflects a moment rather than a force.


The work per oscillation cycle yields from integration of the product of force and motion as

dtehFdthFW ti

TTcycle

⋅∫∫ ⋅⋅=⋅⋅= ωˆˆ~~ rrrr Eq. 3-12

, where hr

denotes the complex motion of the blade. After integration this writes explicitly to

⋅⋅+

⋅⋅+⋅⋅⋅=

→

→→

)sin(

)sin()sin(

ςς

ηηξξ

φ

φφπ

ςς

ηηξξ

am

hfhfcycle

ma

fhfhW Eq. 3-13

The phase-related terms )sin(ξξ

φξ hff →⋅ , )sin(ηη

φη hff →⋅ and )sin(ςς

φς amm →⋅

respectively indicate that only the respective imaginary parts of the perturbation force and moment enter the work. This leads to the conclusion that if the response is lagging the excitation, the imaginary part provides a negative contribution, indicating the flow has a stabilizing character. Finally a normalized stability parameter as reported by Verdon (1987) is applied for the characterization of aeroelastic stability. The stability parameter is based on the negated work performed per cycle normalized by the oscillation amplitude and π . A positive stability parameter indicates that the fluid acts in a stabilizing manner. The stability parameter is given by

ζ

ζ

η

η

ξ

ξπππ a

W

h

W

h

W cycleper cycleper cycleper −+

−+

−=Ξ Eq. 3-14


3.3 Aerodynamic Influence Coefficients

For a system of blades the resulting aerodynamic response during flutter represents a superposition of the influences from several blades. Consider a cascade of N+1 blades as numbered in Figure 3-2 oscillating in traveling wave mode with all the blades having the same mode shape but differing in phase, which has been introduced as interblade phase angle above.

−20 0 20 40 60 80

−100

−50

0

50

100+N/2

...

+1

0

−1

...

−N/2

Figure 3-2. Indexing of blades in cascade For small perturbations the influences of the various blades superimpose linearly as has been shown by Hanamura et al. (1980) and Crawley (1988). Under such conditions the total unsteady response on a blade composes of the individual responses from itself and from the other blades lagged by the respective multiple of the interblade phase angles. The response in traveling wave mode at a certain point is given by

∑+=

−=

⋅−⋅=2

2

,,

,,

),(ˆ),(ˆ

Nn

Nn

nimnicAp

mtwmAp eyxcyxc σσ Eq. 3-15

where )(ˆ ,,

tc mtwmApσ is the complex pressure coefficient at point ),,( zyx , acting on

blade m with the cascade oscillating in traveling wave mode with interblade phase

angle σ and ),,(ˆ ,,

zyxc mnicAp is the pressure coefficient of the vibrating blade n , acting

on the non-vibrating reference blade m at point ),,( zyx . The coefficients mnicApc ,,

ˆ are

commonly referred to as local aerodynamic influence coefficients. Note that the


exponent in the above equation is negated due to the numbering of the blades in the cascade. Interpretation of Eq. 3-15 reveals that the interblade phase angle plays a dominant role on the resulting traveling wave mode response as it lags the respective influence coefficients when superposing. Its impact on the magnitude and phase of the resulting response becomes very descriptive when drawing the superposition in the complex plane as it is done in Figure 3-3. The complex pressure is therein represented by a vector with the length corresponding to the magnitude and the orientation to the phase with respect to the blade motion. For this graphical reflection only the reference blade and its two immediate neighbors are regarded. The influence coefficients are depicted on the left-hand side whereas the superposition is shown for two distinct interblade phase angles on the right hand side. In addition the loci of all possible combinations are included on the bottom right.

Real

Imag

Real

Imag

icpAc ,ˆ blade +1

∑ at deg0=σ

Real

Imag

Real

Imag

icpAc ,ˆ blade 0

∑ at deg90=σ

Real

Imag

Real

Imag

0 20 40

−100

−50

0

50

100+3

+2

+1

0

−1

−2

−3

icpAc ,ˆ blade -1

Loci of ∑ at

deg180...180 +−=σ

Figure 3-3. Effect of interblade phase angle on traveling wave mode response; superposition of influence coefficients from blades -1, 0 and +1


This consideration makes apparent that the contribution of the reference blade (index 0) enters the sum as constant value, i.e. the reference blade vector is not rotated regardless of the interblade phase angle. The contributions of blades +1 and -1 are rotated respectively by the interblade phase angle at their respective signs resulting in variations of superimposed influence.

−150 −100 −50 0 50 100 150−5

0

5

IBPA, deg

stab

ility

par

amet

er Ξ

, −

unstable

stable

Influence of blade 0

−150 −100 −50 0 50 100 150−5

0

5

IBPA, deg

stab

ility

par

amet

er Ξ

, −

unstable

stable

Influence of blades +1 and -1

−150 −100 −50 0 50 100 150−5

0

5

IBPA, deg

stab

ility

par

amet

er Ξ

, −

unstable

stable

0 20 40

−100

−50

0

50

100+3

+2

+1

0

−1

−2

−3

Influence of blades +2 and -2

Figure 3-4. Schematic influence of blade pairs on blade row aeroelastic stability


Finally the influence of interblade phase angle on flutter stability shall be analyzed more in detail at this position. In the previous section the relation between unsteady pressure of force coefficient respectively and a stability parameter has been elucidated and it followed that only the imaginary part of the force enters the work and thus affects stability. In the complex plane this equals to the value of the superimposed influence, in this case of the force coefficient, on the ordinate. The influence of the reference blade (index 0) on stability is therefore again of constant nature. The immediate neighbors (indices +1 and -1) give a harmonically varying contribution, which can be seen as first harmonic oscillation in interblade phase angle, whereas blades further away contribute their respective higher harmonic variation (2nd harmonic for blade pair ±2, 3rd harmonic for pair ±3, etc). This phenomenon is expressed graphically in Figure 3-4. A number of studies have addressed the extent of coupling during cascade flutter (Hanamura et al. (1980), Carta and St.Hilaire (1980), Széchényi (1985), Crawley (1988), Buffum and Fleeter (1990), Nowinski and Panovsky (2000)). It has commonly been found that the influence decreases rapidly with increasing distance from the reference blade and attains convergence after blade pair ±2. The influence of the direct neighbors is generally in the order of magnitude of the direct influence of the reference airfoil on itself indicating that although the single airfoil might not flutter there can be a condition at which the coupling influences overrule this stability and the entire setup gets aeroelastically unstable. The contribution from ±2 pair is generally of one order of magnitude less than the ones from blades 0 and ±1, which leads to a characteristic S-shape. The variation of stability parameter with respect to interblade phase angle is therefore commonly referred to as “S-curve”. A characteristic example of an S-curve is included in Figure 3-5.

−150 −100 −50 0 50 100 150−5

−4

−3

−2

−1

0

1

2

3

4

5

IBPA, deg

stab

ility

par

amet

er Ξ

, −

unstable

stable

Figure 3-5. Characteristic variation of stability versus interblade phase angle (S-curve)


4 Objectives and Approach Flutter prediction tools that are nowadays used in industry still depend on a large degree on empirical correlations and simplified models although unsteady CFD tools for predicting aerodynamic damping have matured to a degree where industrial applicability is provided. Limitations in the models used are however obvious given the fact of unanticipated flutter incidents still being observed during engine testing and commissioning. Further investigations of the physical mechanisms underlying aerodynamic damping are therefore needed with the aim to formulate more accurate and more reliable models. As a matter of fact the major part of test data underlying the validation of current flutter models are of two-dimensional resolution or have been acquired in two-dimensional mean flow. The set of standard configurations compiled by Bölcs and Fransson (1986) reflect the most comprehensive set of data for the understanding of the flutter phenomenon in two-dimensional environments. Three-dimensional data of aerodynamic damping are nowadays very limited in the open literature as mostly the focus is put on structural vibration data using strain gauges during testing. A strong need for comprehensive and systematic aerodynamic damping data in three-dimensional environment is therefore widely expressed throughout the aeroelastic research community. In the light of the above the objective of the current work has been formulated to describe three-dimensional mechanisms in low-pressure turbine flutter from experimental studies. For that purpose a new test facility has been developed that allowed the controlled investigation of flutter in three-dimensional mean flow and at three-dimensional mode shapes using the influence coefficient technique. The facility comprised an annular sector of low-pressure turbine blades one of which could be made oscillating in different rigid-body modes while the response was measured on several blades in the cascade. The facility has first been validated with regard to steady-state and unsteady performance. In order to provide a physical background of the investigated problem the effects of design parameters of the of blade mode shape, reduced frequency, velocity level and inflow incidence on aeroelastic stability have been investigated in detail prior to addressing three-dimensional effects. The set of unsteady data acquired has been used for correlation to results from CFD predictions. A non-commercial unsteady numerical prediction tool (VOLSOL) providing a linearized inviscid model has been employed. Firstly it has been shown that the numerical method converges with increased mesh density. The tool has then been used to predict aerodynamic damping properties at different degree of model detailing (2D, 3D, 3D with tip clearance). By comparing aerodynamic damping properties resolved over the blade surface three-dimensional effects were assessed in detail and conclusions were drawn on the degree of modeling required to accurately predict aeroelastic stability.



5 Experimental Investigation of Aerodynamic Influence Coefficients

5.1 Description of Test Setup

5.1.1 Test Object

The test object chosen for the present investigation is a blade row of twisted and highly loaded low-pressure (LP) turbine rotor profiles. The profile geometry is non-proprietary and has been designed for the present application with the intention to reflect typical aerodynamic features of aero engine low pressure turbines (LPT). To increase structural stiffness while reducing the inertial loads during controlled flutter testing the aspect ratio has been limited to around 2 although aspect ratios as large as 10 might be encountered in last stages of turbines. An unshrouded geometry has been chosen such as to allow for mechanically decoupled setup during controlled flutter testing as well as to include tip leakage flow. A summary of target properties employed during test object design is given below in Table 5-1.

Property Value Flow velocity level High subsonic Loading distribution Uniformly high, radial

variation due to annular geometry

Aspect ratio Around 2 Blade tip Unshrouded, nominally 1%

tip clearance Geometry Three-dimensional twisted

Table 5-1. Set of target design properties of test object The operation of the blade row has been chosen non-rotating such as to simplify the experimental setup. To conserve the three-dimensionality of the flow field as present in real engines the blade row has been tested in an annular setup yielding in a radial

pressure gradient proportional to r

u

dr

dp 2θρ−= . The impact of the absence of the

centrifugal field as well as of non-sheared inflow conditions as otherwise present in turbomachine stages has been assessed numerically and taken into consideration during the blade design process. Under conservation of engine Mach and Reynolds number the profile has been adapted to cold flow testing conditions indicating that it has been a secondary design goal to keep the loading distribution during test rig operation close to the real engine properties. Uniform inflow to the cascade is provided in the test facility conserving the flow direction at midspan. To allow arbitrary oscillation of the blade hub and tip sections have been transformed to constant radius, which has shown little influence as to judge from steady-state analyses. The impact of constant radius hub and tip sections on unsteady performance has however not been investigated as it was regarded lying outside the scope of the present work. In addition to the adaptation of the profile to test rig


conditions a scaling by a factor of 1.5 has been performed such as to enhance instrumentation possibilities. From CFD results it has been shown that the used blade profile shows similar behavior at test rig operation condition and real engine conditions. The set of blade profile parameters is given in Table 5-2. Figure 5-1 depicts the blade and profile sections at various heights as well as passage shape and passage width normalized by the local pitch at the respective spanwise height.

Parameter Symbol Unit Value Real chord (midspan) c mm 50 Axial chord (midspan) axc mm 45

Span s mm 97 Pitch (midspan) cp deg 4.5

Solidity (midspan) cp thetarc /*, - 0.68

Aspect ratio cs / - 1.94 Radius ratio Y - 1.25 Hub radius hubr mm 383

Shroud radius shrr mm 480

Table 5-2. Set of blade profile parameters

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005 10%50%90%

Isometric view Profile sections

0 0.5 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

PS

SS

normalized axial coordinate, −

norm

aliz

ed p

itch,

−

10%50%90%

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

normalized axial coordinate, −

norm

aliz

ed p

assa

ge w

itdh,

−

10%50%90%

Passage shape Passage width

Figure 5-1. Test object

cax=45mm

100m

m


A set of steady, i.e. non-oscillating, and oscillating blades was manufactured for the tests. The steady blades were milled from aluminum alloy (Al7075-T6) and connected to a stainless steel root. The oscillating blades were machined from titanium alloy (Ti6Al4V) and mounted by a swivel bearing to a stainless steel root. Both types of blades have been hand polished such as to achieve a smooth surface. Verification of geometry by a certified institute yielded a maximum geometry standard deviation of 0.03mm from nominal.

5.1.2 Test Facility

The tests have been carried out in a newly purpose-built test facility that was developed within the framework of the current study. The facility allows for controlled flutter testing in continuous mode and was conceptualized as exchangeable module that was connected to an existing large-scale air supply of 1MW power delivering 4.75kg/s cold air (303K) at 4bar maximum. As explained below the facility was discharged to ambient rather than to an exhaust system for suppressing acoustic resonances. Due to this the facility was operated at atmospheric outlet conditions. A sketch of the measurement setup is depicted in Figure 5-2.

Test module (not to scale)

Figure 5-2. Measurement setup

Inlet plenum Outlet plenum

Annular cascade

Variable annular inlet and outlet ducts

Flow


A functional description of the test setup is included below:

• Pressurized and pre-conditioned (cooled, dehumidified) air is entering a fully circular inlet plenum containing flow straightener and a setup of turbulence meshes to achieve a uniform turbulence level of approximately 0.5% and 2mm length scale

• The flow is then directed through a bellmouth and a variable annular sector channel to the test section (cascade). The variability allowed control of inflow angle

• The test section is made up of a cascade of nominally 7 freestanding LP blades with one full passage on either side (total of 8 passages). The end walls in the cascade region were shaped as pressure and suction side of the tested profile respectively, see Figure 5-2

• Downstream of the cascade the flow was discharging through an adjustable annular sector duct to a fully circular outlet plenum. The inclination of the outlet walls were adapted such as to achieve periodic flow in the cascade as described below

• Discharge to ambient through a double-walled circular silencer reducing noise. Prior to this the facility discharged to an exhaust system, which caused a strong acoustic resonance, see below

The variability of inlet annular sector duct was used to achieve different inflow directions. The direction could thereby be set continuously in the range -30deg to +30deg opening up for an incidence range during testing of -4 to +56deg. At the outlet the variability of the lateral duct walls has been used to achieve periodic flow conditions inside the cascade. The direction of the walls equaled approximately nominal outflow direction. By changing the inclination of the outer sidewall (outer with respect to the flow deviation) the flow inside the cascade could be trimmed to high degree of periodicity. The functionality of such system is well-known from linear cascade and has here for the first time been employed on an annular cascade. A theoretical treatment of the working principle has been included in Vogt and Fransson (2000) using full-scale viscous CFD predictions of the test section and the adjacent ducts. A picture of the test facility is shown in Figure 5-3.


Figure 5-3. Test facility To enable continuous variability of the inlet and outlet duct walls a new type of semi-flexible structure has been developed within the framework of this study. The structure contains a semi-flexible framework made of steel, which has been molded into polyurethane. Thanks to this the walls were smooth and featured capabilities of elongation, bowing and twisting while withstanding normal load (here aerodynamic load). Integrated flexible tubes on either side provided sealing. A picture of the inlet and outlet walls is included in Figure 5-4.

Inlet sidewalls and cascade Outlet sidewall

Figure 5-4. Inlet and outlet flexible sidewalls The test section included an annular sector cascade of 7 freestanding twisted LP profiles with one full passage on either side. All blades featured a nominal tip clearance of 1% blade height. As the cascade could be rotated to different positions inside a fully annular casing great care has been spent on achieving high accuracy in concentricity of blade carrier and casing. Combined roundness and concentricity of the casing itself has been measured by a certified institute to ±0.04mm yielding in a maximum error of 7% of the nominal tip clearance (1.0mm±0.07mm).

Inlet plenum

Outlet plenum

Test section Silencer

Controls

Flow conditioner

Flow

1m

100mm 50mm


One of the blades could be made oscillating in controlled modes as described below. All blades in the cascade could be exchanged without dismantling the facility by means of a blade charging and locking mechanism located underneath the hub. As all blades, i.e. oscillating, non-oscillating and instrumented blades, were identical practically no restrictions were imposed on composition of blades in the cascade. A picture of the test section is given in Figure 5-5.

Figure 5-5. Test section; upstream lateral sidewalls removed The test section featured capabilities for automated probe traverses up- and downstream (-20% cax, +120% cax) of the cascade in circumferential and radial direction as well as probe yawing. Circumferential traverses were achieved by rotating the cascade and having the probe fixed relative to the casing covering all passages of the sector. Radial traverse range was limited from 6% to 94% blade span avoiding data points closer than two probe head diameters to hub or casing wall.

100mm

Outlet sidewall

Oscillation actuator

Oscillating blade

Instrumentation

Inlet sidewall hinge base

Instrumented blade


5.1.3 Controlled Blade Oscillation

Controlled oscillation of one blade has been achieved by pivoting the blade at a short distance below the hub by means of a swivel bearing and actuating the blade from below the hub. As a result the blade was oscillated as rigid-body in torsion, bending or a combination of torsion and bending modes. Due to the kinematical setup the bending modes were of three-dimensional nature indicating that the local bending amplitude gradually increased from hub to tip. A sketch of the blade actuation principle included in Figure 5-6 elucidates the oscillation principle.

Casing

Hub

Actuator

Blade

Figure 5-6. Blade oscillation principle The position of the swivel bearing was approximately on the stacking line of the profile at a radial distance of 8.2% blade span underneath the hub. The movement of the blade actuation mechanism was either translational or torsional resulting in bending or torsion modes respectively. Direction of the bending axis could thereby be varied stepwise. For the present tests the following set of orthogonal modes were applied:

• Axial bending mode • Circumferential bending mode • Torsion mode

The actuation mechanism was of mechanical type and was developed within the framework of the present study. It employs two co-rotating circular eccentric cams that actuated a guided actuator disk in a sinusoidal oscillatory movement. By controlling the phase lag between the two cams the achieved mode could be varied continuously from pure bending over combined bending and torsion to pure torsion. As mentioned above for the present tests only pure being or torsion modes have been employed. Variation in bending direction was achieved by twisting the actuation mechanism accordingly. The operation principle of the actuator is depicted in Figure 5-7 showing four instantaneous positions during one oscillation period.


t/T 0

t/T 0.25

t/T 0.75

t/T 0.5

Figure 5-7. Kinematics of oscillation actuator; bending mode shown A speed-controlled DC servomotor and two planetary gear drives were used for driving the actuation mechanism. By controlling the position of the inner gear of one of the two planetary gears continuous variation in cam phase lag could be achieved. Two inductive one-per-rev indexing devices were used to monitor the motion of the two cams and for phase reference for the aerodynamic response. The actuation part itself was of compact sized (d=120mm, h=40mm) and designed for actuation at high amplitudes (7deg bending/torsion) at low frequencies or low amplitudes (0.5deg bending/torsion) up to 500Hz. For the tests presented herein moderate amplitudes of 0.9deg have been used at a maximum excitation frequency of 260Hz. To reduce friction compound air film and oil mist lubrication was used. The air film was generated on either side of the actuation disk by means of perforated plates and appropriate air distribution channels. In addition the actuator disk has been made of leaded bronze alloy featuring inherent friction reduction properties. The connection between the actuation disk and the actuation rod of the blade was made by means of a hexagonal swivel joint. To avoid deteriorating leakage flow at blade hub a flexible transition part molded of polyurethane has been included between the oscillating blade and the blade root. A picture of the actuation mechanism and the oscillating blade is shown in Figure 5-8.


Figure 5-8. Oscillation actuator device (opened) and oscillating blade The oscillation actuator was designed as independent device that was pivoted on the main axle of the test facility. It included the necessary features for driving, mode control, control of bending axis direction, lubrication and indexing and was interfaced by a number of connectors for power, controls and pressurized air. The actuator could be positioned underneath each blade and thus having any blade in the cascade oscillated in controlled modes individually. Figure 5-9 shows the oscillation actuator and its position while mounted on the blade carrier.

Oscillation actuator Actuator mounted on blade carrier

Figure 5-9. Blade oscillation actuator

100m

m

Actuation disk Cams

Actuation rod

Oscillating blade

Drive motor Actuation device Cascade

Actuator

Instrumentation

300mm

400mm

Control device for bending direction


5.1.4 Conventions

This section contains the following conventions:

• Convention of blade indexing and cascade coordinates • Convention of flow angles • Convention of blade oscillation and aerodynamic response

Convention of blade indexing and cascade coordinates The present tests were run with a cascade containing 7 freestanding blades with one full passage on either side. The blades were indexed –3 to +3 ascending in direction from pressure to suction side. The center blade featured index 0. Two coordinate systems were used in the present investigation, one referring to the cascade and the other to the blade surface locally. The cascade coordinate system was based on axial, pitchwise and spanwise directions. Figure 5-10 depicts the indexing of the blades and the cascade coordinate system in an unwrapped blade-to-blade view. Here the normalized coordinates are shown based on the following normalization bases

• Normalized axial distance: normalized by axial chord at midspan • Normalized pitchwise distance: normalized by unwrapped blade pitch at

respective radius

0 20 40

−100

−50

0

50

100+3

+2

+1

0

−1

−2

−3

0 50 100

−100

−50

0

50

100

centerline

pitch=+1

pitch=0

pitch=−1

1

0.5

0

−0.5

−1

normalized pitchw

ise coordinate

normalized axial coordinate

−1 −0.5 0 0.5 1 1.5 2 2.5

Blade indexing Cascade coordinates (center blades

shown)

Figure 5-10. Blade indexing and cascade coordinates


The origin of the normalized pitchwise distance has been bound to approximate stagnation streamlines, which allowed capturing a coherent phenomenological section of the flow at different axial positions with the origin coinciding with the blade leading and trailing edge respectively. This is indicated by constant pitch lines in Figure 5-10. The normalized cascade coordinate system has in turn been rooted in a global test rig coordinate system as depicted in Figure 5-11 that is defined as follows:

• The z-axis collides with the machine axis and is oriented in main flow direction (i.e. pointing from in- to outlet). Origin of the z-axis is located at the leading edge stagnation point (nominal inflow) at hub of blade 0

• The x-axis is leading through the leading edge stagnation point (nominal inflow) at hub of blade 0

• The y-axis completes the coordinate system according to the right-hand rule and pointes tangentially in direction of lower blade indices

• The polar angle ϑ of the respective cylindrical coordinate system is measured from the x-axis according to the right-hand rule, i.e. ascending in direction of negative blade indices

• The radial direction of the cylindrical coordinate system is pointing from the origin outwards

Figure 5-11. Test rig coordinate system

z-axis

y-axis

x-axis

+ϑ direction

Blade index 0

Reference point x=383, y=0, z=0

Origin

0 -1 -2 -3

+1 +2 +3


A local plane on-blade coordinate system was introduced to describe the blade surface defined as follows:

• The blade surface is spanned by an arcwise and a spanwise local coordinate

• The arcwise direction follows the blade surface at constant spanwise height around the blade. Origin is at the respective leading edge stagnation point (nominal inflow) at each spanwise height. Negative branch spans the suction side, positive the pressure side. The arcwise coordinate is normalized by the total local arc length.

• The spanwise direction follows the blade surface at constant arc. Origin is located at the hub with ascending coordinate towards blade tip. The spanwise coordinate is normalized by the total local channel height

−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.030

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

−0.5−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

−0.4 −0.2 0 0.2 0.4

0

0.2

0.4

0.6

0.8

1

SS normalized arcwise coordinate, − PS

norm

aliz

ed s

panw

ise

coor

dina

te, −

Arcwise coordinates at midspan Plane representation of blade surface

Figure 5-12. Local on-blade coordinate system

tip

hub

LE TE TE


Convention of Flow Angles The flow direction was characterized by a yaw and a pitch angle as follows:

• The yaw angle α is the angle of the flow direction in the cylindrical surface measured from axial. The yaw angle is positive in direction of positive polar angle ϑ.

• The pitch angle β is the angle of the flow direction in the axial-radial plane. The pitch angle is positive in the direction towards the blade tip, i.e. positive spanwise coordinate.

Definition of yaw angle Definition of pitch angle

Figure 5-13. Definition of flow angles

+β

−β

+ϑ direction

axial

+α −α

casing

hub

axial


Convention of Blade Oscillation and Aerodynamic Response The blade oscillation is characterized by a rigid-body rotation around an axis of rotation of the form

tieAt ⋅⋅= ωα )( Eq. 5-1

For bending modes the axis of rotation lies normal to the radial direction that points to the center of blade oscillation, i.e. the blade swivel bearing pivot point. For torsion modes the axis of rotation collides with the radial direction pointing to the center of blade oscillation. Expressed in global coordinates the center of blade oscillation is located at x=0.375m, y=-0.0038m and z=0.0181m. In terms of local blade coordinates the center of rotation is located at 40% cax. The directions of the bending axes of the three investigated orthogonal modes are included in Figure 5-14.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

Axial bending Circumferential bending

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

10%50%90%COT

Torsion

Figure 5-14. Definition of blade oscillation The aerodynamic response is referred to the blade oscillation (=excitation) as follows

)(,,

~ˆ)(~ αφω →+⋅

⋅= pctiApAp ectc Eq. 5-2

The definition of the phase shift between response and excitation αφ →pc~ is such that

the phase gets positive when the complex response is leading the complex excitation.

Axis of rotation

Direction of positive motion

Axis of rotation


Axis of rotation (out of page)



5.1.5 Measurement Setup

The instrumentation of the test facility included both steady-state and unsteady measurement devices. Steady-state measurements were performed for the determination of mean flow conditions both for validation of the test setup and for description of the mean flow field. Unsteady measurements were taken to assess the acoustic properties of the test facility, to monitor the blade oscillation parameters and to determine the response of the aerodynamic system upon excitation by controlled blade oscillation. All measurement devices were centrally controlled from an industrial master PC, which was part of the test rig control unit. The various systems were thereby interfaced using Ethernet, GPIB and serial communications. The steady-state measurement setup consisted of a monitoring part of global flow parameters and local cascade instrumentation. The global parameters of atmospheric pressure, mass flow, total inlet pressure, total inlet temperature and static outlet pressure were continuously monitored and logged for traceability purposes. The atmospheric pressure was monitored by means of a Solartron high-sensitive barometer with an accuracy of 0.01% (±11.5Pa). The mass flow was measured in the air supply line upstream of test module by means of a standard orifice at an accuracy of 2% in the operated flow range. The pressure levels up- and downstream of the orifice were measured by 350kPa modules of a 16-channel PSI9016 system with an accuracy of 0.04% full scale (±140Pa) with atmospheric reference. Inlet total pressure and outlet static pressure were measured by 100kPa modules of the same system at ±40Pa accuracy. A PT100 sensor connected in 4-wire circuit to a high-quality conditioning module yielded total temperature data at 0.1K accuracy. Inlet and outlet flow field were traversed using a 4-hole aerodynamic wedge probe with a wedge base of 3mm and 45deg wedge angle. Probe traverse data were used to validate the mean flow conditions and to provide boundary conditions for numerical simulations. The probe has been calibrated in an in-house probe calibration facility described by Vogt (2004) over a Mach number range of M=0.1…0.9 and ±20deg in pitch and yaw angle. The confident angle range of the probe has however been limited to ±15deg in both pitch and yaw direction as reasonable accuracy could not be guaranteed outside this range. Calibration data has shown good monotony and low scatter within a range of ±15deg in both pitch and yaw direction. Data points outside this range have been treated with reduced confidence. A sketch of the probe head and calibration surfaces for the different Mach numbers are included in Figure 5-15.


avg

ref

pp

ppk

−

−=

1

1,01

avg

refsref

pp

ppk

−

−=

1

,,02

avgpp

ppk

−−

=1

323

avg

avg

pp

ppk

−

−=

1

44

232 pp

pavg+

=

−20−10

010

20

−20

−100

1020

0

0.5

1

1.5

pitch angle, degyaw angle, deg

coef

ficie

nt, −

−20

−100

1020

−20

−100

1020

1

1.5

2

2.5

3


coef

ficie

nt, −

Coefficient k1 Coefficient k2

−20−10

010

20

−20

−100

1020−2

−1

0

1

2


coef

ficie

nt, −

−20

−100

1020

−20

−100

1020−2

−1.5

−1

−0.5

0

0.5


coef

ficie

nt, −

Coefficient k3 Coefficient k4

Figure 5-15. Head of aerodynamic 4-hole probe, calibration coefficients and calibration surfaces

Probe pressure data were sampled using 100kPa modules of a 16-channel PSI9016 system with an accuracy of 0.04% full scale (±40Pa). The measurements were taken relative to atmosphere with the atmospheric pressure being measured by means of


the aforementioned Solartron barometer. Table 5-3 gives an overview of probe traverse parameters.

Parameter Unit Value Circumferential traverse range

Passage index

-2…2

Radial traverse range - 0.06…0.94 Mach number accuracy % 0.1 Yaw angle accuracy deg 0.5 Pitch angle accuracy deg 0.5

Table 5-3. Probe traverse parameters The blade loading was mapped by total of 95 static taps distributed on different blades. Data has been acquired on 5 blades in the cascade (indices –2 through +2). The static pressure taps were of 0.4mm in diameter (0.8% chord) and were distributed evenly on 5 spanwise sections between 10% and 90% span with 19 taps on each section. The taps were connected by miniature stainless steel tubes (di=0.4mm) to the lower end of the blade root and thereafter by Vinyl tubes (di=1.0mm and di=1.6mm) to the measurement equipment. The stainless steel tubes were embedded in pre-milled grooves in the blade surface, covered with Epoxy resin and hand-ground and –polished to restore the original blade surface. On each blade three additional taps that were placed at identical locations to serve as reference when recombining test data from several runs. Repeatability in both cascade geometry and achievement of flow conditions has been found excellent, which allowed confident recombination of data from different runs. Figure 5-16 shows the distribution of static pressure taps on a section and over the blade span.

−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.030

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.440.4

0.3

0.2

0.12

0.06

0.02

0

−0.02−0.05

−0.08

−0.11

−0.15

−0.22

−0.29

−0.35

−0.41

−0.47−0.52

−0.4 −0.2 0 0.2 0.4

0

0.2

0.4

0.6

0.8

1

10%

30%

50%

70%

90%


norm

aliz

ed s

panw

ise

coor

dina

te, −

Arcwise distribution of pressure taps

(midspan shown) Mesh of pressure taps on blade surface

Figure 5-16. Distribution of static pressure taps Blade static pressures were sampled by means of a multi-channel PSI8400 system with atmospheric reference. Modules of 100kPa (15psi) featuring 0.05% (±50Pa) accuracy were used for the tests. The atmospheric pressure was taken from the aforementioned Solartron barometer. Inside the test section the instrumented blades were connected by means of miniature quick disconnect couplers (Scanivalve, 19 channels) such as to allow for repetitive and time-efficient exchanging of blades.


The dynamic measurement setup consisted of trigger devices to characterize blade oscillation and fast-response pressure instrumentation to measure the unsteady blade surface pressure. Two one-per-rev trigger signals from the two actuator cams were recorded and served both for describing the mode from determination of relative cam phase lag and for giving a phase reference for the unsteady pressure measurements. A third trigger signal was recorded featuring 100 pulses per blade oscillation period such as to provide a measure for verifying constancy of blade actuation frequency. The unsteady blade surface pressure was measured by means of recessed-mounted pressure transducers that were placed underneath the blade hub at typical distances of 50mm to 150mm from the measurement location. This type of setup was chosen such as to place the transducers away from the deterioration source of high acceleration and to have the possibility of reusing sensors for covering a larger number of measurement positions transducers were available. For the measurement of coupling coefficients on the non-oscillating neighbor blades the above described statically instrumented blades were employed, which yielded in 90 data points distributed on 5 spanwise positions for the coupling coefficients. Figure 5-17 shows the distribution of the static taps that were used for unsteady pressure measurements.

−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.030

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

−0.52−0.47

−0.35

−0.29

−0.22

−0.15

−0.11

−0.08−0.05

−0.02

0

0.02

0.06

0.12

0.2

0.3

0.410.45

−0.4 −0.2 0 0.2 0.4

0

0.2

0.4

0.6

0.8

1

10%

30%

50%

70%

90%


norm

aliz

ed s

panw

ise

coor

dina

te, −



Figure 5-17. Distribution of unsteady pressure measurement points on non-oscillating blades (recessed-mounted transducers)

For measuring the direct coefficients (blade index 0) one of the oscillating blades was equipped with 19 pressure taps (d=0.4mm, 0.8% chord) at midspan. The taps were connected by spark eroded miniature holes (di=0.9mm) to the lower end of the blade and transferred by means of miniature PVC tubes (di=0.8mm) to the lower end of the blade root. The PVC tubes were thereby molded into the flexible transition part and harnessed such as to avoid any deteriorating effect during blade oscillation. As the manufacturing process was not free of problems, five taps featured poor transfer characteristics and could not be used for measurements. The distribution of the usable measurement locations on the oscillating blade is shown in Figure 5-18.


−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.030

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

−0.52

−0.41

−0.29

−0.22

−0.15

−0.11

−0.08−0.05

−0.020.02

0.12

0.2

0.24

0.41

−0.4 −0.2 0 0.2 0.4

0

0.2

0.4

0.6

0.8

1

50%


norm

aliz

ed s

panw

ise

coor

dina

te, −



Figure 5-18. Distribution of unsteady pressure measurement points on oscillating blade (recessed-mounted transducers)

Figure 5-19 shows the oscillating and non-oscillating blades used for unsteady pressure measurements.

Oscillating blade Non-oscillating blade

Figure 5-19. Oscillating and non-oscillating blades used for fast-response pressure measurements

All taps used for unsteady pressure measurements were calibrated dynamically as described by Vogt and Fransson (2004b) such as to allow for reconstruction of the unsteady pressure signal at the tap location. The calibration procedure included the application of a reference unsteady pressure signal to a miniature cavity placed on top of the tap to be calibrated. The pressure signal was generated by an in-house built pressure pulse generator that allowed controlling pulse frequency, shape, amplitude and mean level described by Vogt (2001). The signals from the sensor to calibrate and from a reference sensor placed in the calibration cavity were acquired and treated such as to yield complex dynamic transfer properties in the frequency domain, i.e. magnitude ratio and phase. Dynamic calibration was performed in a range of 4Hz to 2kHz with respect to the maximum theoretical blade excitation frequency of 500Hz. Figure 5-20 shows the calibration setup and examples of transfer properties for taps at different spanwise heights. It is noticeable that the magnitude ratio decreases with increasing spanwise position, i.e. the attenuation increases. Additionally the taps at higher span feature increased phase lag.

100m

m

100m

m


0 100 200 300 400 5000

0.5

1

mag

rat

io, −

0 100 200 300 400 5000

50

100

150

200

phas

e, d

eg

frequency, Hz

10%50%90%

Calibration procedure Transfer characteristic at different

spanwise heights

Figure 5-20. Dynamic calibration procedure and transfer characteristic For the recessed mounted type of unsteady pressure measurements a total of 20 Kulite sensors of the type XCQ-062 and LQ-080 were used. The signals from the sensors were acquired by means of a digital high-speed data acquisition system (Kayser Threde KT8000) that also provided stabilized 10VDC sensor excitation. The system featured 32 channels with programmable amplifiers, 14bit A/D conversion for each channel and a maximum sampling rate for all 32 channels simultaneously of 200kHz. Each channel could be programmed individually such as to set gain and a low-pass filter with variable cut-off frequency. The tests were performed with a gain of 25, no low-pass filtering and at a sampling rate of 20kHz. The accuracies of the sensor for measuring the perturbation part of the pressure was determined to ±30Pa taking into account the static and dynamic transfer characteristic of the sensor. The resolution of the A/D-converter adds with ±30µV (gain 25), which corresponds to ±50Pa taking into account the transfer characteristic of the sensor. The transfer characteristic adds with an average of ±50Pa over the calibrated range, which results in a total measurement accuracy of ±130Pa for the fast-response measurement setup.


5.1.6 Data Acquisition and Data Reduction Procedure

Steady-state measurements were centrally acquired by the PC control unit of the test facility. For steady-state measurements each data point represents an average of 20 samples acquired at a rate of 10Hz. The data were reduced to average values before storing on a hard drive. Selected measurement locations were monitored in real time for assessment of measurement performance. Standard deviation values and variation were monitored for control reason. During traverse measurement a holding time of 2s has been passed prior to signal acquisition to allow settling pressure in the pressure measurement chain. The measured pressures were reduced to normalized pressure coefficient components as follows

refdynp p

xpxc

,

)()( = Eq. 5-3

The reference dynamic head refstatrefrefdyn ppp ,,0, −= was taken from total

pressure measurements in the settling chamber and static pressure measurements 40% upstream of the cascade. Unsteady measurements were mastered by the main control unit and remotely acquired by a built-in PC of the high-speed data acquisition system. To provide accurate representation of the time-dependent signals a sampling rate of 20kHz was chosen. The data reduction procedure was performed using the software package MATLAB and comprised the following steps:

1) Determination of the exact excitation, i.e. blade oscillation frequency from signal analysis of the oscillation actuator trigger signals

2) Ensemble-averaging of unsteady pressure data (typically 200 periods) with respect to the oscillation period and starting at a specified time point given by

∑=

+=N

nEA nTtxp

Ntxp

00 ),(

1),(~ Eq. 5-4

3) Normalizing of the unsteady pressure by inlet dynamic head and oscillation

amplitude such as to yield the unsteady pressure coefficient as was

refdyn

EAAp pA

txptxc

,,

),(~),(~

⋅= Eq. 5-5

4) Signal analysis of unsteady pressure coefficient such as to yield first

harmonic amplitude Apc ,ˆ and phase Acp →φ related to the oscillation of the

airfoil as

tieAt ⋅⋅= ωα )( Eq. 5-6 ( )Apcti

ApAp exctxc →+⋅⋅=

φω)(ˆ),(~

,, Eq. 5-7


5.2 Validation of Test Setup

For the present facility to be a valid test case for turbomachine flutter issues of steady and unsteady performance were verified. Some of the validation issues addressed the achievement of target requirements such as three-dimensional flow field and three-dimensional blade oscillation. On the other hand a number of validation issues were necessary to prove that turbomachine representative flow with regard to both steady and unsteady aspects could be achieved despite the limited extent of the test section.

5.2.1 Steady-State Aerodynamic Performance

The validation of the steady-state aerodynamic performance included two issues; firstly the achievement of three-dimensional flow features as originally targeted was verified. Secondly the achievement of blade-to-blade periodicity was addressed. For both issues blade surface pressure measurements as well as in- and outlet probe traverses were employed as measurement technique. The inlet and outlet flow fields were measured at 20% axial chord upstream and 20% downstream of the cascade at three operating points as were low subsonic (M2,avg=0.37), medium subsonic (M2,avg=0.62) and high subsonic (M2,avg=0.71) as described more in detail below. Probe traverses spanned the center five passages ranging from +2 to -2 and starting at mid passage (normalized pitch -0.5) respectively as depicted in Figure 5-21. To assess the characteristics of the flow field spanwise profiles as well as two-dimensional contour and vector plots of the secondary flow are adapted. The spanwise profiles represent circumferential mass flow averaged values as determined by

dAcpCpdAc

spC np

c

pn

avgcirc

c

c

⋅⋅⋅

= ∫∫

∗ ρρ

)(1

)(_ Eq. 5-8

The contour and flow vector plots were based on pitchwise periodically averaged values over N passages as given by

∑=

∗∗ ⋅+=N

ncccavgpitch spnpCp

NsppC

1_ ),(

1),( Eq. 5-9

The two-dimensional plots were plotted at blade 0 position as indicated in Figure 5-21 and ranged in normalized pitch from -0.5 to +0.5.


−20 0 20 40 60

−150

−100

−50

0

50

100−0.2 1.2

normalized arcwise coordinate

−2.5−1.5

−0.50.5

1.52.5

+2

+1

0

−1

−2norm

alized pitchwise coordinate

passage index

Figure 5-21. Cascade flow field and periodicity assessment traverses Inlet flow field data for the three operating points are included in Figure 5-22 through Figure 5-24 plotting circumferentially averaged distributions of static and total pressure coefficients, secondary flow angles and Mach numbers. Additionally Mach number contours and secondary flow vectors over one flow passage are included. For all three operating points it can be observed that the total pressure coefficient is constant over the main part of the span and decreases rather slightly above 85% span, which indicates the beginning of the boundary layer towards the casing. The yaw and pitch secondary flow angles change rapidly in this region indicating the appearance of secondary flow structures. Close to the hub no sign of decrease in total pressure coefficient can be seen down to 6% span, which leads to the conclusion that the hub boundary layer lies below this value. The distribution of the static pressure coefficient shows characteristically lower values around midspan with smooth increase towards both hub and casing side. In conjunction with the fairly constant total pressure distribution this increase reveals a lowered velocity level towards hub and casing as also visible in the circumferentially averaged Mach number distribution, which is believed being due to blockage effects probably caused by the hub and tip horseshoe vortices around the blade leading edge. The Mach number contours evidence this finding as distinctive regions of higher and lower velocity are visible; around mid span and mid passage a region of higher velocity indicates the location of the main flow passage. Close to the casing and close to the hub two regions of lower velocity can be observed that are located around normalized pitch zero, i.e. upstream of the blade leading edge. Note that the two regions have a slight connection in between, which visualizes the potential effect, i.e. the blocking of the flow upstream of the leading edge.


0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

pressure coefficient, −

norm

aliz

ed s

panw

ise

coor

dina

te, −

Cp0Cp

−10 −5 0 5 100

0.2

0.4

0.6

0.8

1

secondary flow angles, deg

norm

aliz

ed s

panw

ise

coor

dina

te, −

αsec

βsec

Circumferentially averaged pressure

coefficients Circumferentially averaged secondary flow

angles

0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

isentropic Mach number, −

norm

aliz

ed s

panw

ise

coor

dina

te, −

Miso

−20 −10 0 10

380

390

400

410

420

430

440

450

460

470

480

0

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.2

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.4no

rmal

ized

spa

nno

rmal

ized

spa

n

0.6

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.8

norm

aliz

ed s

pan

norm

aliz

ed s

pan

1

norm

aliz

ed s

pan

−0.4pitchpitch

0pitchpitch

0.4pitch

0.175

0.18

0.185

0.19

0.195

0.2

−30 −20 −10 0 10 20

390

400

410

420

430

440

450

460

470

480

0

0.2

0.4

0.6

0.8

1

−0.4pitchpitch

0pitchpitch

0.4pitch

Circumferentially averaged Mach number Mach number contours and secondary

flow vectors

Figure 5-22. Inlet flow field characteristics; low subsonic On the secondary flow vector plots it can be observed that the blade leading edge blocking leads to an upwash around pitch=0. At all three velocity levels the flow seems to be diverted towards the pressure side in the lower half of the span and towards the suction side in the upper half respectively. The presence of blade leading edge horseshow vortices is not obvious from the presented plots, which might be due to the upstream traverse plane being located too far away from the leading edge. Inlet flow field data at the two higher velocity levels included in Figure 5-23 and Figure 5-24 reveal similar features as observed for the low subsonic case, namely distinct presence of potential effect upstream of blade leading edge with the flow tending to be diverted towards the suction side in the upper half of the span and towards the pressure side in the lower half respectively. The circumferentially Mach number distributions show balanced behavior with minor decreases towards the casing and the hub endwalls.


0 0.5 1 1.50

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

Cp0Cp

−10 −5 0 5 100

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

αsec

βsec



angles

0.15 0.2 0.25 0.3 0.350

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

Miso

−20 −10 0 10

380

390

400

410

420

430

440

450

460

470

480

0

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.2

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.4

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.6

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.8

norm

aliz

ed s

pan

norm

aliz

ed s

pan

1

norm

aliz

ed s

pan

−0.4pitchpitch

0pitchpitch

0.4pitch

0.245

0.25

0.255

0.26

0.265

0.27

0.275

0.28

−30 −20 −10 0 10 20

390

400

410

420

430

440

450

460

470

480

0

0.2

0.4

0.6

0.8

1

−0.4pitchpitch

0pitchpitch

0.4pitch

Circumferentially averaged Mach number Mach number contours and secondary flow vectors

Figure 5-23. Inlet flow field characteristics; medium subsonic


0 0.5 1 1.50

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

Cp0Cp

−10 −5 0 5 100

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

αsec

βsec



angles

0.15 0.2 0.25 0.3 0.350

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

Miso

−20 −10 0 10

380

390

400

410

420

430

440

450

460

470

480

0

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.2

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.4

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.6

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.8

norm

aliz

ed s

pan

norm

aliz

ed s

pan

1

norm

aliz

ed s

pan

−0.4pitchpitch

0pitchpitch

0.4pitch

0.26

0.27

0.28

0.29

0.3

0.31

−30 −20 −10 0 10 20

390

400

410

420

430

440

450

460

470

480

0

0.2

0.4

0.6

0.8

1

−0.4pitchpitch

0pitchpitch

0.4pitch


Figure 5-24. Inlet flow field characteristics; high subsonic


Outlet flow field data are presented in Figure 5-25 through Figure 5-27 for all three velocity levels. The distribution of total pressure coefficient shows constant behavior over the main part of the span at the two lower velocity levels. At the high subsonic operating point minor deterioration of the total pressure coefficient can be observed in the region of span=0.6 to 0.8. At all three velocity levels a distinct decrease in total pressure coefficient can be observed above 85% span indicating the beginning of the casing boundary layer, which is also evidenced by a rather strong increase in yaw secondary flow angle in this region. Towards the hub the only evidence for the start of the boundary layer is visible in the pitch secondary flow angle distribution by a distinct decrease below 15% span. In addition to the spanwise distribution of the static pressure coefficient a linear approximation has been added such as to highlight the radial gradient in static pressure on a per span basis. For the low and medium subsonic case the static pressure coefficient distribution lies fairly well in line with the linear approximation indicating a constant radial pressure gradient over span. Similar to the total pressure distribution slight deviations are observed at the high subsonic case in the range of span=0.6 to 0.8. It is believed that these differences are due to different vortex activity downstream of the cascade depending on the velocity level. The radial pressure gradient amounts to ∆Cp=1.08 for the low subsonic, ∆Cp=1.51 for the medium and ∆Cp=1.73 for the high subsonic case and thereby agrees well with the targeted properties of three-dimensional flow.

−4 −2 0 20

0.2

0.4

0.6

0.8

1

∆Cp=1.08


norm

aliz

ed s

panw

ise

coor

dina

te, −

Cp0Cptrend

−5 0 5 10 150

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

αsec

βsec



angles

0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

Miso

−20 −10 0 10

380

390

400

410

420

430

440

450

460

470

480

0

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.2

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.4

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.6

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.8

norm

aliz

ed s

pan

norm

aliz

ed s

pan

1

norm

aliz

ed s

pan

−0.4pitchpitch

0pitchpitch

0.4pitch

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

−20 −10 0 10 20

390

400

410

420

430

440

450

460

470

480

0

0.2

0.4

0.6

0.8

1

−0.4pitchpitch

0pitchpitch

0.4pitch


Figure 5-25. Outlet flow field characteristics; low subsonic


The contour plots reveal distinct gradient in Mach number from casing to hub, which agrees well with the observed static pressure gradient. A local Mach number deficit being due to the wake downstream of the blade can be observed around pitch=0. Note that the wake region is inclined slightly to the right, which is due to the three-dimensional twisting of the blade profile, i.e. the trailing edge is not oriented purely radial but inclined slightly towards the pressure side. To the left of the wake region a prominent low Mach number region can be observed at all three operating points. It is believed that this region is due to the tip vortex that deploys into the passage flow on the suction side of the blade. The plots of secondary flow vectors reveal distinct downwash inside the wake region. This downwash is due to the radial pressure gradient acting on fluid particles with reduced momentum. Close to the casing a distinct upwash is observed. It is possible that this is an effect of the tip vortex. The tip vortex itself is faintly identifiable in the upper left quadrant of the vector plots. Generally it can be stated that the observed flow features agree well at the different velocity levels.

−4 −2 0 20

0.2

0.4

0.6

0.8

1

∆Cp=1.51


norm

aliz

ed s

panw

ise

coor

dina

te, −

Cp0Cptrend

−5 0 5 10 150

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

αsec

βsec



angles

0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

Miso

−20 −10 0 10

380

390

400

410

420

430

440

450

460

470

480

0

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.2

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.4

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.6

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.8

norm

aliz

ed s

pan

norm

aliz

ed s

pan

1

norm

aliz

ed s

pan

−0.4pitchpitch

0pitchpitch

0.4pitch

0.4

0.45

0.5

0.55

0.6

0.65

−20 −10 0 10 20

390

400

410

420

430

440

450

460

470

480

0

0.2

0.4

0.6

0.8

1

−0.4pitchpitch

0pitchpitch

0.4pitch


Figure 5-26. Outlet flow field characteristics; medium subsonic


−4 −2 0 20

0.2

0.4

0.6

0.8

1

∆Cp=1.73


norm

aliz

ed s

panw

ise

coor

dina

te, −

Cp0Cptrend

−5 0 5 10 150

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

αsec

βsec



angles

0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1


norm

aliz

ed s

panw

ise

coor

dina

te, −

Miso

−20 −10 0 10

380

390

400

410

420

430

440

450

460

470

480

0

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.2

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.4no

rmal

ized

spa

nno

rmal

ized

spa

n

0.6

norm

aliz

ed s

pan

norm

aliz

ed s

pan

0.8

norm

aliz

ed s

pan

norm

aliz

ed s

pan

1

norm

aliz

ed s

pan

−0.4pitchpitch

0pitchpitch

0.4pitch

0.45

0.5

0.55

0.6

0.65

0.7

0.75

−20 −10 0 10 20

390

400

410

420

430

440

450

460

470

480

0

0.2

0.4

0.6

0.8

1

−0.4pitchpitch

0pitchpitch

0.4pitch


Figure 5-27. Outlet flow field characteristics; high subsonic In the next step the flow field periodicity is assessed. This is done based on both flow field traverse as well as blade loading data. Pressure coefficient periodicity data from flow field traverses has been obtained by from differences between local pitchwise values and pitchwise averaged data as follows

avgpassavgpass

avgpitchn

pCpC

ppCpCpCp

__,0

_ )()(

−

−=∆

∗∗ Eq. 5-10

The difference has been normalized by the local passage averaged dynamic head and thus gives a percentage value of the flow field non-periodicity at a specific axial position. In the present case periodicity data has been acquired for the five center passages ranging from +2 to -2. Each passage is thereby represented by an individual difference distribution, i.e. five periodicity data distributions are shown in each plot. To refer periodicity data to the flow field the respective averaged pressure coefficient distributions are added. Inlet flow field periodicity data are shown in Figure 5-28 for the low subsonic case. The top left plot shows the average distribution at three spanwise positions (close to hub, midspan, close to casing). Periodicity data of total and static pressure coefficient are then included at these spanwise positions. Close to hub and at midspan the


periodicity distributions lie in general well below 5% in both total and static coefficient and thereby indicate fairly periodic flow. At 90% span passage +2 exhibits differences that exceed 10% of the normalized inlet dynamic head. This might be due to cross flows being present due to the proximity to the lateral end walls.

−0.4 −0.2 0 0.2 0.4 0.60.8

0.9

1

1.1

Cp0

, −

−0.4 −0.2 0 0.2 0.4 0.6−0.2

−0.1

0

0.1

0.2

Cp,

−

normalized pitchwise distance, −

10%50%90%

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp0

, −

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp,

−


+2+10−1−2

Average distribution Periodicity data at 10% span

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp0

, −

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp,

−


+2+10−1−2

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1∆C

p0, −

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp,

−


+2+10−1−2

Periodicity data at 50% span Periodicity data at 90% span

Figure 5-28. Inlet flow field periodicity data at different spanwise positions; low subsonic

Outlet periodicity data is included in Figure 5-29 for the low subsonic case and different spanwise heights and in Figure 5-30 for the two higher velocity levels at midspan only respectively. The averaged distribution of the total pressure coefficient shows a distinct dip at 90% span, which above has been associated to the tip vortex. Furthermore the wake is visible around pitch=0. Close to hub and at midspan the distributions show little difference and lie mostly well below ±5% non-periodicity. Exceptions are observed in the region of pitch -0.1 to +0.2, i.e. across the wake, which are partially explained by the limited resolution. Major differences are obvious in total pressure coefficient close to the casing. As has been shown above this region is dominated by viscous effects and therefore much reduced flow periodicity must be expected. Maximum non-periodicity in this region lies around 15%.


−0.4 −0.2 0 0.2 0.4 0.6−2

−1

0

1

Cp0

, −

−0.4 −0.2 0 0.2 0.4 0.6

−5

−4

−3

−2

Cp,

−


10%50%90%

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp0

, −

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp,

−


+2+10−1−2


−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp0

, −

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp,

−


+2+10−1−2

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp0

, −

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1∆C

p, −


+2+10−1−2


Figure 5-29. Outlet flow field periodicity data at different spanwise positions; low subsonic

Periodicity data for the two higher velocity levels shows that the flow increasingly looses its periodic character; whereas total pressure periodicity is still highly acceptable at M2=0.6 apart from the wake region it gets slightly worse at M2=0.8 indicating passage +2 as the odd-one-out due to a 5% increased total pressure level. Static pressure coefficient periodicity data show that larger differences exist in passages +2 and +1, i.e. the outer passages in the cascade, whereas the other three passages form a rather coherent group. At the medium subsonic velocity the maximum difference lies around ±7.5% for passage +2 whereas it increases to ±10% at the high subsonic level.


−0.4 −0.2 0 0.2 0.4 0.6−2

−1

0

1

Cp0

, −

−0.4 −0.2 0 0.2 0.4 0.6

−5

−4

−3

−2

Cp,

−


10%50%90%

−0.4 −0.2 0 0.2 0.4 0.6−2

−1

0

1

Cp0

, −

−0.4 −0.2 0 0.2 0.4 0.6

−5

−4

−3

−2

Cp,

−


10%50%90%

Average distribution, medium subsonic Average distribution, high subsonic

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp0

, −

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp,

−


+2+10−1−2

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1

∆Cp0

, −

−0.4 −0.2 0 0.2 0.4 0.6

−0.1

−0.05

0

0.05

0.1∆C

p, −


+2+10−1−2

Periodicity data at 50% span

medium subsonic Periodicity data at 50% span

high subsonic

Figure 5-30. Outlet flow field periodicity data at midspan for different velocity levels Finally the focus shall be directed to blade loading periodicity. In the same manner as flow field periodicity data the blade loading periodicity data represents blade-wise normalized differences from an averaged distribution. The normalized outlet dynamic head has thereby been taken as normalization basis. Figure 5-31 and Figure 5-32 include blade loading periodicity data on blades +2 through -2 as well as averaged distributions at different spanwise sections and velocity levels. Periodicity data at 10% span and low velocity level depicted in Figure 5-31 show that the blade loading is periodic within ±3% apart from a region around blade leading edge where the maximum differences are observed for the two extreme indices +2 and -2. At midspan high periodicity is observed apart from the region around the leading edge. As can be seen from the average distribution large gradients exist in this region, which lead to a high susceptibility of periodicity data. Periodicity lies also here approximately in a range of ±3%. At 90% span larger differences are observed both around the leading edge and aft suction side; whereas the non-periodicity amounts to ±15% around the leading edge it stays below ±4% on the rest of the blade.


−0.4 −0.2 0 0.2 0.4−5

−4

−3

−2

−1

0

1


Cp,

−

10%50%90%

−0.4 −0.2 0 0.2 0.4

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2


∆Cp,

−

+2+10−1−2


−0.4 −0.2 0 0.2 0.4−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2


∆Cp,

−

+2+10−1−2

−0.4 −0.2 0 0.2 0.4

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2


∆Cp,

−

+2+10−1−2


Figure 5-31. Blade loading periodicity data at different spanwise positions; low subsonic

The effect of velocity level on blade loading periodicity is shown at midspan only in Figure 5-32. It is obvious that the two outer blades +2 and -2 feature larger differences especially on the suction side. The inner blades +1 through -1 feature blade loading non-periodicity of below ±4% for the medium subsonic and ±5% for the high subsonic case.


−0.4 −0.2 0 0.2 0.4−5

−4

−3

−2

−1

0

1


Cp,

−

10%50%90%

−0.4 −0.2 0 0.2 0.4

−5

−4

−3

−2

−1

0

1


Cp,

−

10%50%90%

Average distribution, medium subsonic Average distribution, high subsonic

−0.4 −0.2 0 0.2 0.4−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2


∆Cp,

−

+2+10−1−2

−0.4 −0.2 0 0.2 0.4

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2


∆Cp,

−

+2+10−1−2

Periodicity data at 50% span

medium subsonic Periodicity data at 50% span

high subsonic

Figure 5-32. Blade loading periodicity data at midspan for different velocity levels The results of the steady-state performance of the facility can be summarized as follows:

• The inlet flow field shows uniform total and static pressure distributions under presence of potential effects; close to hub and casing increased blockage has been observed upstream of the blade leading edge, which is believed being due to endwall horseshoe vortices. Normalized non-periodicity of the inlet main flow field is below ±5%.

• The outlet flow field features characteristic structures as wake and tip vortex and is dominated by an approximately linear radial pressure gradient from casing to hub. The gradient increases from ∆Cp=1.08 for the low subsonic case to ∆Cp=1.73 for the high subsonic case and is one of the paramount features of the present test case. Normalized non-periodicity of the outlet main flow field ranges from ±5% to ±10% with increasing velocity level.

• Blade loading non-periodicity at midspan ranges from ±3% to ±5% with increasing velocity level for the three inner blades of the cascade indexed +1, 0 and -1. Non-periodicity increases for the outer blades +2 and -2 and attains values of ±10% at midspan. In general blade loading periodicity is lowered close to hub and close to tip.


5.2.2 Blade Oscillation

The oscillation of the blade has been verified at flow and no-flow conditions by means of a Laser vibrometer. The Laser vibrometer was used to take point-wise samples of the blade motion over several oscillation periods. Figure 5-33 shows blade oscillation data at 40Hz excitation frequency. From comparison of raw data and ensemble-averaged data it can be recognized that the oscillation is sinusoidal, highly periodic and constant in frequency. The spectrum of the oscillation evidences these observations by a very sharp peak at the excitation frequency.

0 1 2 3 4

−1.5

−1

−0.5

0

0.5

1

1.5

instantaneous time, t/T

norm

aliz

ed m

agni

tude

, −

raw dataensemble averaged

0 50 100 150 200 250 300

0

0.2

0.4

0.6

0.8

1

frequency, Hz

norm

aliz

ed m

agni

tude

, −

Signal (time domain) Spectrum (frequency domain)

Figure 5-33. Blade oscillation data Validation of blade oscillation amplitude at variable frequency has revealed that the resulting oscillation amplitude slightly increased with increasing frequency. It is believed that this is an effect of limited stiffness in the assembly of oscillation actuator and oscillating blade an increasing inertial load with increasing actuation frequency. The effect of flow velocity level has been found negligible. As no instantaneous measurements of the oscillation amplitude during flutter testing were performed due to limitations in the optical access to the measurement section, blade oscillation data have been acquired prior to and after test series to provide a reference amplitude. The variation of resulting blade oscillation amplitude with absolute frequency has thereafter been approximated linearly. To validate the resulting mode shape during oscillation successive point-wise scanning has been performed at no flow conditions using the aforementioned Laser vibrometer. Results have shown that the resulting oscillation shape corresponded to requirements, i.e. featured linear increasing amplitude from hub to tip for the bending modes and constant amplitude over span for the torsion mode. The performance at flow conditions has been checked by means of a stroboscope with the light frequency set close to blade oscillation frequency such as to slow down the movement optically. From these tests no degradation of the rigid-body modes was apparent in the range of tested reduced frequencies and flow operating points.


5.2.3 Unsteady Performance

The unsteady performance of the facility has been validated under consideration of two objectives. Firstly it was aimed at detecting any acoustic resonances in the test section that might be due to wind tunnel acoustic modes. The second objective was to validate the unsteady passage-to-passage periodicity and to detect deteriorating reflections from the test section sidewalls. For detection of wind tunnel acoustic resonances the facility was operated both at steady mean flow and in excited manner by oscillating the blade. Pressure power spectra were acquired at several positions inside the test section by means of the aforementioned fast-response pressure measurement instrumentation. The power spectra depicted in Figure 5-34 have been acquired with the blade oscillating in controlled mode at 44Hz such as to give a reference for the magnitude of noise. Originally the test facility was connected to a fixed mounted exhaust system in the laboratory. It was soon recognized that the exhaust system led to a comparatively strong acoustic wind tunnel resonance at 126Hz in the order of magnitude of the aerodynamic perturbations induced by the blade oscillation. To avoid deteriorating influence of this resonance during testing the facility has thereafter been modified such that it discharged into the laboratory through a silencer. After modification the previously observed acoustic resonance was no longer present and the maximum observed noise was well below the aerodynamic perturbations induced by the blade oscillation. All tests have consequently been performed using the silencer and discharging to ambient.

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

frequency, Hz

norm

aliz

ed m

agni

tude

, −

0 50 100 150 200 250 300

0

0.2

0.4

0.6

0.8

1

1.2

frequency, Hz

norm

aliz

ed m

agni

tude

, −

Discharging to air exhaust system Discharging to ambient through silencer

Figure 5-34. Power spectra of wind tunnel acoustics; blade oscillating at 44Hz Additionally the power spectra depicted in Figure 5-34 evidence that the aerodynamic perturbation features dominant harmonic behavior with negligible higher harmonic content. This led to the conclusion that the applied excitation, i.e. the blade oscillation, was such (in terms of shape and amplitude) that the assumption of linearity in aerodynamic response could be considered valid.

Acoustic resonance


Acoustic performance of the facility during blade oscillation has in addition been assessed by determining identical blade influence coefficients at various positions in the cascade. Coefficients on the blades -1 and +1 have been regarded in this study. In accordance to the studies performed by Buffum and Fleeter (1991) and Buffum and Fleeter (1994) the following indicators have been used to recognize acoustic resonances:

1) Perturbation distributions of constant amplitude featuring linearly varying phase might be due to propagating waves

2) Differences in perturbation response depending on the position of the oscillating and measuring blades indicate the presence of an acoustic mode and reflections of pressure waves from sidewalls

Figure 5-35 and Figure 5-36 show the unsteady response measured on blades +1 and -1 with one blade oscillating in axial bending mode at k=0.5. Note that the indices here are referring to the relative position with respect to the oscillating blade (index 0), i.e. index -1 denotes the blade adjacent to the suction side while index +1 refers to the pressure side neighbor. For the resent tests the oscillating blade was once placed at absolute index 0 in the cascade and once at index +1, i.e. one blade pitch closer to the outer wall. In both figures the average response is included on the top right while the unsteady periodicity is assessed in the bottom right plot. Note that the intention of this section is to assess unsteady flow periodicity rather than to describe the unsteady response phenomenon in detail. This is included below.

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200


Cp

phas

e, −

avg

0 20 40

−100

−50

0

50

100+3

+2

+1

0

−1

−2

−3

−0.4 −0.2 0 0.2 0.40

0.02

0.04

∆Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−20

−10

0

10

20


∆Cp

phas

e, −

osc 0osc +1

Figure 5-35. Aerodynamic response on blade -1 measured with oscillating blade at two different indices

meas

osc 0

meas

osc +1

meas

Oscillated blade

Measured blade


The unsteady response on blade -1 depicted in Figure 5-35 shows a sharp increase on the suction side up to arc=-0.11 whereupon the response decreases gradually towards the trailing edge. On the pressure side the response is constantly low. The phase is fairly constant on the major part of the suction side as well as on the fore part of pressure side and does not suggest the presence of an acoustic disturbance. Regarding unsteady periodicity it can be observed that the differences in response amplitude are small apart from a region around arc=-0.11 where they amount to ±3% of the local perturbation amplitude. The differences in phase are moderate around ±5deg in regions with high response magnitude and increase towards the trailing edge on the pressure side to values around ±20deg. Response data on blade +1 included in Figure 5-36 indicate that the response amplitude is of moderate level on the pressure side and low on the suction side. The amplitude level is thereby lower than on blade -1. Despite this fact the phase shows constant behavior on the pressure and aft suction side. An abrupt decrease on the fore suction side can be observed however does not have the characteristic of an acoustic disturbance. Unsteady periodicity data indicate that non-periodicity amounts to around ±5% of the local response amplitude on the pressure side while it features very low values on the suction side. Phase non-periodicity stays below ±10deg on most part of the blade.

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200


Cp

phas

e, −

avg

0 20 40

−100

−50

0

50

100+3

+2

+1

0

−1

−2

−3

−0.4 −0.2 0 0.2 0.40

0.02

0.04

∆Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−20

−10

0

10

20


∆Cp

phas

e, −

osc 0osc +1

Figure 5-36. Aerodynamic response on blade +1 measured with oscillating blade at two different indices

meas

osc 0

meas

osc +1

meas

Oscillated blade

Measured blade


The results of the unsteady performance of the facility can be summarized as follows:

• The acoustic noise in the test section could greatly be reduced by discharging the facility to ambient rather than to an exhaust system. It is believed that the exhaust system triggered an acoustic resonance inside the facility.

• Unsteady blade surface pressure data do not show acoustic resonant behavior during blade oscillation.

• Overall non-periodicity amounts to 5% of the local response in amplitude difference and ±10deg in phase difference. Periodicity in response phase gets down to ±5deg in regions of higher response but can also attain locally limited values as large as ±20deg, e.g. at the trailing edge.

Generally it shall at this position be emphasized that the phase distributions might become noisy at low amplitudes due to low signal-to-noise ratios. Given small amplitude values the phase of the unsteady quantity is usually of subordinate importance and can in most cases be neglected. Phase values are nonetheless included in all figures in the present work regardless the amplitude such as to provide consistent appearance.



6 Numerical Prediction of Aerodynamic Influence Coefficients

6.1 Description of Numerical Model

The numerical method used was a small perturbation harmonic approximation of a non-linear Euler method. The method employs a structured multi-block flow solver based on finite volumes and is implemented in the 3D solver VOLSOL as described by Lindström and Mårtensson (2001). While the non-linear Euler method of VOLSOL uses deforming grids at finite amplitude, the linearized method uses an analytically evaluated infinitesimal mesh deformation as a part of the flux calculation to take care of the mesh movement. The underlying non-linear Euler model is a high order finite volume technique allowing for moving meshes by considering a finite volume extending also in time. The inviscid three-dimensional Euler equations on conservative form are considered as follows,

0=∂∂+

∂∂+

∂∂+

∂∂

z

H

y

G

x

F

t

Q, Eq. 6-1

where

( ) ( ) ( )

++

=

+

+=

+

+=

=

wpe

pw

vw

uw

w

H

vpe

vw

pv

uv

v

G

upe

uw

uv

pu

u

F

e

w

v

u

Q2

2

2

,,,

ρρρρ

ρρ

ρρ

ρρ

ρρ

ρρρρ

Eq. 6-2

and ρ is the density. u , v and w are the velocity components in the x , y and z

directions. Furthermore, e is the total energy per unit volume and p is the static

pressure, which is related to the density and total energy through the equation of state for an ideal gas,

( ) ( )

++−−= 222

2

11 wvuep ργ , Eq. 6-3

where γ is the ratio of specific heats.

The non-linear Euler method of VOLSOL as given by Groth et al. (1996) is formulated in the time-space domain. Time is thereby treated as an extra dimension, and not only a parameter, which usually is the case of other methods. Integrating the Euler equations Eq. 6-1 over a control volume with four dimensions (time and space) derives the method. By applying Gauss’ divergence theorem, the following equation is obtained


( ) 0=+++∫Ω∂

zyxt HdSGdSFdSQdS Eq. 6-4

and the corresponding discrete version

( ) 081

=+++∑ =m mzyxt HSGSFSQS Eq. 6-5

follows. In Eq. 6-5,

mtS is the projection of the surface mS in the t direction,

mxS the

projection in the x direction, etc. In the same way, mQ is the value of Q at surface

mS , etc.

At the two faces ( )lkji tzyx ,,, and ( )1,,, +lkji tzyx , Q is known and sought

respectively. To calculate the conserved variables and fluxes at the other surfaces of the time-space control volume, a cell centered upwind biased scheme described by Eriksson (1993) is used. The scheme involves reconstruction of the conserved variables at the faces and calculation of flux vectors based on these reconstructed variables. The reconstruction is done by means of upwind biased characteristic variables. The scheme, which may be viewed as a central scheme with fourth difference smoothing, is third order accurate in space, if effects of grid curvature and stretching are ignored. For the time integration, a three-stage Runge-Kutta scheme is used. Eriksson (1993) has demonstrated that the scheme has very little dispersion. The derivation of the linearized scheme can be started by adding a pseudo-time parameter to the non-linear Euler equations. By finding a steady state solution in pseudo-time τ to the generalized equation

0=∂∂+

∂∂+

∂∂+

∂∂+

∂∂

z

H

y

G

x

F

t

QQ

τ Eq. 6-6

On this form and given that there is a stable iteration technique we eventually arrive

at solutions which satisfy 0=∂∂

τQ

, and the non-linear Euler equations are solved over

one time step. In order to derive the harmonic linear Euler method, a harmonic linearization in the

time direction is constructed around a steady-state solution Q , which satisfies the

non-linear Euler equations on a computational domain with non-moving boundaries. The sought solution is given by

( ) ( ) ( ) ( )( )tizyxQzyxQtzyxQ ⋅⋅+= ωexp,,ˆRe,,,,, Eq. 6-7

In the pseudo-time formulation, if only first order terms in the perturbations are retained

( ) ( ) ( ) ( )( )tizyxQzyxQtzyxQ ⋅⋅+= ωττ exp,,,ˆRe,,,,,,

( ) ( ) ( ) ( )( )tizyxFzyxFtzyxF ⋅⋅+= ωττ exp,,,ˆRe,,,,,, Eq. 6-8

( ) ( )( )tizyxxxx ⋅⋅+= ωexp,,ˆRe

Note that in Eq. 6-8, the functions Q , F and x are complex valued.


By integrating the generalized Eq. 6-8 in the space-time domain in the same manner as Eq. 6-5 and retaining only terms that are leading order in terms of the perturbation, it is found that

( )( )∑

∑

+++−−

++−−=∂∂

mzyxt

mzyx

SHSGSFSQiVQi

SHSGSFVQiVQ

ˆˆˆˆˆ

ˆˆˆˆˆ

ωω

ωτ , Eq. 6-9

where V denotes the space volume of a stationary cell. In the linearized flow solver,

Q is integrated in τ with a Runge-Kutta scheme, until convergence is reached. At

convergence, 0ˆ

=∂∂

τQ

, and the harmonic linearized Euler equations have been

solved. Flutter simulations have been performed in the traveling wave mode domain modeling one passage and applying phase-lagged periodic boundary conditions as sketched in Figure 6-1. The simulations yielded consequently traveling wave mode response data that were decomposed to blade specific data in the influence coefficient domain as given by Eq. 3-15 above.

Figure 6-1. Simulation of flutter in the traveling wave mode domain

inflow

outflow periodic side A

periodic side B

X+IBPA

X

mode


6.2 Validation of Numerical Method

The present numerical method has been validated for the current application with regard to the following aspects:

• Convergence with increased mesh density • Effects of numerical approximation

As additional point the validity of the influence coefficient technique for the present test case in a laterally limited cascade has been addressed.

6.2.1 Mesh Convergence

For the numerical method to be used with confidence it was necessary to show that convergence of both steady and unsteady results with increased mesh density could be achieved. As test object a 3D traveling wave mode model of the present low-pressure turbine profile has been adapted covering one passage and applying phase-lagged periodic boundary conditions. A set of three meshes has been generated differing isotropically by a factor of 1.4 in mesh size. In addition the effect of mesh node distribution in the blade-to-blade plane on prediction accuracy has addressed. The purpose of this was that the mesh convergence campaign was also aimed at delivering an optimum mesh with respect to prediction accuracy and computational effort. The first three meshes were referred to as “coarse”, “medium” and “fine” respectively, while the latter two were referred to as “Mesh 1” and “Mesh 2”. All meshes used compound O-H multiblock structure with an O-domain around the blade and H-domains in the blade-to-blade passage and inlet and outlet section respectively. An overview of mesh parameters is included in Table 6-1.

Mesh name Total nodes Nodes along span

Intent of mesh

Coarse 36700 23 Medium 104976 34 Fine 228044 45

Mesh resolution in steps of 1.4

Mesh 1 59466 31 Mesh 2 59466 31

Smoothness and distribution of mesh

Table 6-1. CFD Mesh parameters Midspan sections of the applied meshes are included in Figure 6-2 and Figure 6-3.


Coarse mesh Medium mesh

Fine mesh

Figure 6-2. Coarse, medium and fine meshes used; midspan shown

Mesh 1 Mesh 2

Figure 6-3. Medium meshes with different node distribution; midspan shown Steady and unsteady results are included in Figure 6-4 close to hub and at midspan. The results are depicted for the high subsonic case and response on blade 0 at axial bending mode. Whereas the steady loading distributions lie well in line and show only minor differences around suction peak (arc=-0.1) major differences can be observed in the predicted unsteady response. First the distributions of the coarse, medium and fine meshes are analyzed. On the suction side the response distributions are similar in magnitude and phase and differ only slightly for the coarse mesh towards trailing edge. On the pressure side good agreement is found in the fore part whereas the coarse mesh predicts higher response towards the trailing edge. These observations are valid close to hub as well as at midspan. Generally the results obtained with


medium and fine mesh correlate well whereas the coarse mesh differs indicating that the employed numerical method converges with increased mesh resolution. As the differences between medium and fine mesh are small in magnitude and phase it has been concluded that the medium mesh was valid for the present simulations.

−0.4 −0.2 0 0.2 0.4−6

−5

−4

−3

−2

−1

0

1

Cp,

−


coarsemediumfineMesh 1Mesh 2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0

Steady loading midspan Unsteady response midspan

−0.4 −0.2 0 0.2 0.4−6

−5

−4

−3

−2

−1

0

1

Cp,

−


coarsemediumfineMesh 1Mesh 2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0

Steady loading close to hub Unsteady response close to hub

Figure 6-4. Steady loading and unsteady response on blade 0 at midspan and close to hub for different meshes

The unsteady results obtained using Mesh 1 and Mesh 2 differ considerably from the other group of meshes. Mesh 1 behaves similar to the coarse mesh on the pressure side. This observation per se is noticeable given the fact that Mesh 1 features the same resolution as the medium mesh. On the suction side at midspan a bump is observed in the Mesh 1 magnitude results between arc=-0.2 and -0.4. The phase on this side is distinctly different from the other results and is at maximum 90deg off. The magnitude distribution obtained from Mesh 2 differs from all other meshes in that there is a considerable increase towards the trailing edge, which occurs on both suction and pressure side at midspan and on pressure side only close to hub. Nevertheless the phase obtained with this mesh is in good agreement with the first group of meshes. The results suggest that the method is converging with increase mesh resolution however also that the distribution of the mesh nodes can play a more dominant role than the resolution itself. Whereas Mesh 1 and Mesh 2 featured approximately the same resolution as the medium mesh, the differences in predicted magnitude and phase are greater than those obtained from changed mesh resolution. This behavior


has been discussed in Mårtensson and Vogt (2005). Although it is not fully understood to date is believed that the combination of too coarse a mesh resolution in combination with poor distribution might lead to the numerical method failing. Finally it is noticeable that although some of the meshes fail or give poor results the prediction of the magnitude of peak response around suction peak is quite coherent for all meshes used.

6.2.2 Effect of Numerical Approximation

The effect of numerical approximation is addressed with simulations on the medium mesh and with two different numerical discretization schemes. Normally a third order approximation scheme is used. Figure 6-5 shows the comparison of a third and a first order scheme for the low subsonic case at axial bending and reduced frequency of k=0.1. The lower order scheme is used to introduce more numerical damping, which might be necessary to allow stable simulations in cases where a numerical unstable situation might establish. Within the framework of this work such situation has been observed when simulating high incidence cases at low subsonic velocity as discussed below.

0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


+13rd order1st order

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−13rd order1st order

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


03rd order1st order

Figure 6-5. Comparison of unsteady response on blades +1 through -1 using different numerical schemes (TWM simulation)

The response distributions tend to show similar behavior for both numerical approximations. On pressure side of blade +1 the response magnitude is slightly over-predicted around arc=0.25. On blade 0 the response lies well in line apart from a region on the fore suction side where the 3rd order scheme leads to over-predicting.


On the suction side of blade -1 the magnitude is generally lower for the lower order discretization. The phase agrees overall well however differs locally on the fore suction side of blade +1. Given the fact that the response is not of major magnitude in this region this is not considered harmful. The results suggest that lower order numerical schemes can be used with confidence however under the awareness that the increased numerical damping might lead to minor differences in response magnitude. This finding has also been confirmed at higher reduced frequencies.

6.2.3 Effect of Finite Cascade on Influence Coefficient Technique

The influence coefficient technique assumes an infinite cascade of blades the responses of which superimpose linearly to yield the traveling wave mode response. In the present test setup the extent of the test section cascade is limited to 8 passages that are limited on either side by lateral sidewalls. In this context the following two points are of interest:

• Does the limited extent of the cascade allow for an accurate representation of the traveling wave mode response?

• Do the lateral sidewalls affect the response in the cascade due to reflections of pressure waves?

To answer these questions a numerical model of the test section comprising 7 blades has been simulated in the influence coefficient domain, i.e. with the middle blade oscillated and the response acquired on several blades in the cascade. The model featured solid sidewalls on either side. These results have been compared to predictions from a tuned cascade fluttering in traveling wave mode, which has been achieved by modeling one passage and applying phase-lagged periodic boundary conditions. The employed meshes are depicted in Figure 6-6. The traveling wave mode results have been decomposed to yield blade specific response distributions and are below compared on blades +2 through -2. The validation results have been calculated at high subsonic conditions and axial bending mode. Below the abbreviations “INFC” are used when referring to the influence coefficient domain and “TWM” to the traveling wave mode domain respectively.

Test section

(influence coefficient domain) Single passage

(traveling wave mode domain)

Figure 6-6. Test section and single passage traveling wave mode model


Figure 6-7 shows the results at midspan. Firstly it is observed that the unsteady response decays rapidly away from the oscillating blade and that the response on blades +2 and -2 is already of very little magnitude. The dominant contributions are present on blades 0 and -1 while blade +1 features moderate response. From a general perspective INFC and TWM results agree very well. The positive index blades +1 and +2 tend to show minor differences in response magnitude on the suction side as well to some degree in response phase. A possible explanation for this behavior might be the vicinity and direct exposure of these sides to the downstream outer sidewall, which might induce pressure reflections. The differences are however small compared to the overall magnitude and can with good confidence be neglected.

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


+2TWMINFC

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


+1TWMINFC

0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0TWMINFC

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−2TWMINFC

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1TWMINFC

Figure 6-7. Comparison of unsteady response on blades +2 through -2 from TWM and INFC simulation


Comparisons between INFC and TWM models suggest that the employed method of testing in the influence coefficient domain in a circumferentially limited sector cascade is numerically correct and that the observed differences are negligible. Furthermore is shown that the influence decays rapidly for blade pairs further away than +1 and -1 and that the dominant influence arises from the oscillating blade and its direct neighbors.


7 Investigation Strategy

7.1 Flutter Testing

To meet the objectives of this work of describing three-dimensional effects during flutter aerodynamic damping data were acquired at different spanwise positions on the blades. Firstly the flutter phenomenon was investigated in detail at midspan with regard to variations in engine design parameters of interest such as to establish a background for identifying three-dimensional effects. These parameters were blade mode shape, reduced frequency, velocity level and inflow incidence. Three different velocity levels have been tested as were low subsonic (M2,avg=0.37), medium subsonic (M2,avg=0.62) and high subsonic (M2,avg=0.71) referred to as M=0.4, M=0.6 and M=0.8 respectively. The inlet incidence has been varied from nominal (incidence=0deg) to two off-design cases, one at moderate (-20deg) and one at high incidence (-37deg) such as to force the flow to separate. Each operating point has been assessed by means of blade loading measurements and inlet and outlet flow field traverses. At each point a set of three orthogonal modes (axial bending, circumferential bending, torsion) has been tested such as to describe a mode space as outlined above. Each mode testing covered a range of reduced frequencies. For safety reasons during the present tests the maximum achievable blade oscillation frequency has been limited leading to k=0.5 for the low subsonic, k=0.4 for the medium subsonic and k=0.3 for the high subsonic operating point respectively. It shall be noted that these limitations can be overcome in future test series. An overview of the test conditions is included in Table 7-1. The listed flow parameters represent mass-averaged values. Denotation of operating point Par Unit L1 L2 L3 M1 M2 M3 H1 H2 H3 m& 1 kg/s 2.36 2.36 2.36 3.64 3.64 3.64 4.89 4.89 4.89 T01 K 303 303 303 303 303 303 303 303 303 p01 kPa 112.3 112.6 112.8 128.4 129.5 130.0 160.6 163.7 165.0 p1 kPa 109.2 109.4 109.2 123.6 123.6 121.7 151.1 154.1 155.3 p2 kPa 102.9 102.9 102.9 105.8 105.8 105.8 107.5 107.5 107.5 Π01 2* - 1.096 1.096 1.099 1.238 1.24 1.237 1.53 1.58 1.58 M1 - 0.21 0.20 0.20 0.28 0.27 0.28 0.30 0.29 0.30 M2 - 0.37 0.37 0.37 0.62 0.62 0.62 0.71 0.71 0.71 α1 deg -23.9 -3.4 14 -21.1 -0.5 15.4 -22.8 -2.1 15.1

α2 deg 56.8 56.8 56.8 57.4 57.4 57.4 57.1 57.1 57.1 kmax - 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3

Table 7-1. Overview of measured test conditions (passage-averaged values) For investigation of the flutter phenomenon at midspan influence coefficient data has been acquired on the oscillating blade, the direct neighbors (±1) and one higher

1 The indicated values refer to the mass flow measured at the orifice and is slightly larger than the actual mass flow through the test section due to leakage around the test section


blade pair (±2). The data were superimposed to yield response data in the traveling wave mode domain and combined to aeroelastic stability plots. For assessment of three-dimensional effects the focus has been put on the direct neighbor blades (indices ±1) only as no adequate instrumentation was available on the oscillating blade. Consequently only the coupling contribution to the total stability could be addressed in the traveling wave mode domain.

7.2 Unsteady CFD Simulations

Unsteady CFD simulations were performed using a linearized inviscid model for the prediction of aeroelastic properties. The simulations have been performed in the traveling wave mode domain with one passage modeled at the same modes as employed during testing (axial bending, circumferential bending and torsion). The traveling wave mode results were decomposed to influence coefficient data for correlation to test data. In order to assess the three-dimensional effects during flutter a strategy has been chosen that was based on models with different extension. A plane 2D model was used and compared to a more advanced 3D full scale model. Whereas the blade oscillation modes in the 3D model were specified as rigid body rotations as present in the experiment the modes were transposed to their respective plane counterpart (i.1. translation) for the 2D model. In a further step the 3D model has been refined by including tip clearance as was the case in the test setup. The respective meshes are included in Figure 7-1. Despite the fact that viscous phenomena play an important role in tip clearance flow this case was intentionally solved with the same inviscid model. The justification for this lies therein that although the effect of tip leakage flow leads to viscous interaction when deploying into the main passage flow field its cause is rooted in the pressure difference over the blade tip, which is well captured when using the inviscid model. The numerical study was aimed at quantifying prediction accuracy when using different models. Additionally the results from the different models were correlated to test data such as to assess the physical mechanisms that are underlying flutter.

Without tip clearance With tip clearance

Figure 7-1. Computational meshes used for 3D simulations


8 Results The present section contains test data and CFD simulation results using the aforementioned methods. First steady-state and flutter test data are presented and discussed in detail. The addressed three-dimensional mechanisms in flutter are thereby introduced on the background of the effects of mode shape, reduced frequency, velocity level and flow incidence studied at midspan. Finally CFD results are correlated to test data.

8.1 Steady-State Test Data

Steady blade loading data at the three velocity levels are shown in Figure 8-1 including distributions at three spanwise sections (10%, 50% and 90% span) as well as contour plots of the blade surface such as to give a global picture. All distributions show high decrease in static pressure coefficient on the fore suction side around the leading edge terminated by a distinct suction peak around arc=-0.11, which is followed by a region of fairly constant loading. This location corresponds well to the position of the blade where the highly bowed fore part of the blade transitions to the fairly straight aft part. The level of the plateau in static pressure coefficient downstream of the suction peak increases gradually from hub to tip revealing the radial pressure gradient that is due to the annular shape of the test section. Furthermore it is noticeable that the location of peak suction moves from arc=-0.15 close to hub over arc=-0.11 at midspan to arc=-0.08 close to tip, which is due to the three-dimensional twisted shape of the blade profile. Close to the tip the static pressure coefficient dips slightly downstream of arc=-0.22 and recovers around arc=-0.35. It is believed that this is due to presence of the tip leakage flow, which leads to an unloading of the profile in the region between peak suction at arc=-0.08 and arc=-0.22, thereafter to an over-loading and finally to a recovery towards the trailing edge. This observation is valid for all three velocity levels. Close to the hub on the suction side the pressure coefficients is fairly constant right downstream of peak suction and decreases thereafter distinctly around arc=-0.35. From a geometrical analysis on the blade row geometry it is observed that the passage throat is located around this arcwise distance. The observed decrease in static pressure coefficient might be due to the blockage effectuated by the pressure side branch of the horseshoe vortex emerging from the positive neighbor blade. This finding is also evidenced from flow visualization pictures that are included below. On the pressure side the static pressure coefficient decreases monotonically from leading to trailing edge. As was the case on the suction side the pressure coefficient increases gradually from close to hub to close to tip indicating the presence of a radial pressure gradient. At all three velocity levels it is observed that the distribution close to hub dips locally around arc=0.02, which is interpreted as small local separation bubble at this section. Note that in the present test setup profile close to hub is operated at negative incidence due to the fact that the inflow to the blade row is constant over span rather than skewed as in a real engine. The observed separation is believed being an effect of this local off-design operation.


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Figure 8-1. Effect of flow velocity level on steady blade loading; low subsonic to high subsonic; operating point 1 (zero incidence)


The effect of inflow incidence on steady loading is assessed in Figure 8-3 for the low subsonic case by means of spanwise pressure coefficient distributions and contour plots. On the suction side effects are obvious in the magnitude of peak suction that decreases from nominal operation (L1) over off-design 1 (L2) to off-design 2 (L3) incidence case. This effect is most pronounced close to blade tip and is believed being due to the unloading of the blade with increased negative off-design operation. Close to hub the previously observed local decrease in pressure coefficient around arc=-0.35, which was believed being due blockage of the endwall vortex in the throat, gets gradually less distinct at increased negative off-design operation. This fact consolidates the interpretation of secondary flow structures acting in this region; by increasingly unloading the profile due to negative incidence the secondary flow towards the endwalls get weakened leading to less interaction on the aft suction side. On the pressure side major differences are observed on the fore part. As incidence increases from zero to high negative incidence a region of decreasing pressure coefficient is establishing centered around arc=0.05 and span=0.3. This region emerges from the local separation observed at nominal operation close to hub and extends to 90% span up to arc=0.02 at off-design 1 incidence and up to arc=0.12 at off-design 2 incidence respectively. The following trend is thereby visible: as incidence increases towards negative the fore part of the pressure side starts to separate starting close to hub and extending over the entire measured blade span at mean negative incidence. The separation is visible by distinct local reduction of the static pressure coefficient and is believed being ended at the location where the off-design pressure distributions coincide with the nominal ones. The corresponding values have been extracted from the pressure distributions and plotted on the blade surface in Figure 8-2 such as to highlight the extent of the separation bubble at various incidence. Note that for the large negative incidence case the separation close to hub extends as far as to 66% of the pressure side.

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L1L2L3

Figure 8-2. Termination lines of pressure side separation bubble from steady loading data; low subsonic velocity level


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Figure 8-3. Effect of off-design operation on steady blade loading; low subsonic; operating points L1 through L3


To provide qualitative affirmation of the aforementioned findings surface flow visualization has been performed at the various inflow incidence cases. For that purpose regions of the blades have been painted with a mixture of titanium oxide and silicone oil (Rhodosil 47V1000) that have been pigmented and applied as follows:

• The hub upstream of the blade has been painted yellow between cax=-0.05 and cax =-0.15 and one pitch to either side of the visualized blade

• The leading edge has been painted blue between arc=-0.06 and arc=0.06 over the entire blade span

• The blade tip has been painted green between cax =0 and cax =0.4 The mixture and painting was made such that it was at the verge to flow due to gravity. The blade was painted when mounted in the test section and rapidly exposed to flow such as to avoid mixing of the colors. Exposure time to flow has been found from trials being around five minutes. Thereafter the applied color has spread over the blade surface and most of the silicone oil has evaporated leaving back surface flow streaks. The blades have thereafter been demounted and photographed.

View on suction side View on pressure side

Figure 8-4. Blade surface flow visualization results and corresponding steady-state blade loading data; operating point L1

Post-processing of the pictures included projection of the pressure taps onto the photographed blade surface. For this purpose an algorithm has been developed using the software package MATLAB that allowed basing the projection of the tap grid on four distinct points (trailing edge corner at hub and tip, leading edge corner at hub and tip) by graphical input. Consequently contours of static pressure coefficient


could be overlaid to the flow visualization picture allowing for qualitative assessment of observed features. In addition a number of observed flow features were highlighted by graphically included splines as were

• Hub corner vortex separation line on suction side • Tip corner vortex separation line on suction side • Laminar-turbulent transition line on suction side • End of separation bubble (reattachment line) on pressure side

The flow visualization pictures for the nominal operating are included in Figure 8-4. The transition line is visible as distinct change in pigment density in mean flow direction ranging from hub to tip and is located shortly downstream of arc=-0.11. From loading data peak suction has been tracked to this point suggesting that transition is induced by the local change in blade loading. Upstream of the transition line on the fore suction side high density of pressure coefficient isolines are visible marking the rapid increase in flow velocity in this region. Downstream of the transition line the isolines become increasingly oblique with proximity to blade trailing edge indicating the establishment of the radial pressure gradient. Close to hub and tip the separation lines of the corner vortices are clearly pronounced; close to hub the upwash from the hub endwall is visible by yellow pigment that has been transported up to 10% span onto the suction side. The deployment of the tip leakage vortex is identified from reduced blue pigment density towards the blade tip extending down to below 80% span. It is noticeable that none of the green pigment that originally has been painted on the blade tip has been transported onto the suction side. This indicates that the tip leakage flow has the character of a jet that deploys into the main passage flow. Due to this interaction a corner vortex is generated with counter-clockwise sense of rotation when seen from upstream leading to fainting of blue pigment in this region. On the pressure side the end of the separation bubble is visible from opposite pigment streak directions on the fore part extending up to 30% span. The position of the separation bubble agrees well with the observations made in blade loading and lies between arc=0.02 and 0.06. This means that the location identified in the blade loading where the off-design loading comes in line with the nominal one lies downstream of the separation bubble reattachment line. Furthermore gradual decrease in static pressure coefficient towards blade trailing edge is visible with increased obliqueness indicating the presence of radial pressure gradient. The flow visualization pictures for the off-design 1 case are included in Figure 8-5. It features similar phenomena as observed for the nominal case. The transition line still collides with the taps arc=-0.11, which agrees with the pressure coefficient distributions in that the location of peak suction is unchanged at this operating point. The extents of the up- and downwash regions towards hub and tip are slightly smaller and terminate below 10% and above 80% span respectively. On the pressure side is distinctly visible that the separation bubble has grown and is now extending over the entire blade span. The locations for the separation bubble end are between arc=0.06 and 0.12 at 10% span and between arc=0.02 and 0.06 at 90% span and thereby corresponds well to the values identified from the blade loading. Downstream of the separation bubble little difference is found between the pressure coefficients contours and surface flow structure between nominal and off-design 1 operation.




Finally the flow visualization pictures for the off-design 2 case are included in Figure 8-6. As for the two other cases the location of the transition line is largely unchanged. On the fore suction side between transition line and leading edge it is now observed that the isolines are packed much closer, which is due to the stagnation line having moved onto the suction side. The obliqueness of the isolines in this region is due to the twisted shape of the blade profile. The regions of hub and tip corner vortices are qualitatively similar to the two other cases; whereas the separation line at the hub largely coincides with the one at off-design 1, it moved to higher span at the tip. It is believed that this is a direct indication for the reduction in blade loading as off-design operation moves to negative incidence angles. Consequently the driving force behind the tip leakage flow, i.e. the pressure difference across the tip, decreases. The contour distribution on the aft suction side largely agrees with the two other operating points. On the pressure side the growth of the separation bubble is clearly visible; at 10% span the reattachment line is moved to between arc=0.2 and 0.3 whereas at 90% it is located at arc=0.12. Once more these values stand in good agreement with the ones found from loading measurements. At this operating point a contour of negative pressure coefficient is visible at the leading edge indicating the acceleration of the flow in that region.




The steady-state test data can be summarized as follows:

• The present profile features uniformly high loading featuring a distinct suction peak at arc=-0.11. From surface flow visualization it has been recognized that the suction peak triggers flow transition from laminar to turbulent

• The fore suction side between leading edge and suction peak is characterized by aggressive acceleration of the flow with slightly oblique isolines due to the twisted shape of the profile

• On the aft suction side the loading is increasingly dominated by a radial pressure gradient that establishes as circumferential flow direction increases

• A hub corner vortex is present that leads to an upwash of hub boundary layer fluid onto the suction side. The extent of the upwash region thereby is at maximum at nominal operation and decreases with negative incidence, i.e. unloading of the profile

• A tip corner vortex is present, which is driven by the jet-like leakage flow over blade tip. The vortex region gets smaller with negative incidence, i.e. unloading of the profile

• A separation bubble is present on the fore suction side starting at the leading edge. The presence of the bubble has been identified from both loading data and surface streak lines. At nominal operation the extent of the bubble is limited to a small region close to the hub and extents towards the tip and trailing edge with negative incidence.


8.2 Flutter Test Data

The current section presents unsteady aeroelastic response data that has been acquired on various blades in the cascade during controlled flutter testing. Introductory the response phenomenon is discussed based on data at midspan where a comprehensive data set spanning five blades (+2 through -2) at various modes, reduced frequencies, velocity level and inflow incidence was acquired. Each of these parameters and its respective effect on the unsteady response is addressed in detail. In the light of these findings three-dimensional effects in the aeroelastic response are addressed. At note shall be made on the presentation of data; the unsteady response, which is per definition complex with respect to the blade motion, is here presented as absolute magnitude (i.e. Cp amplitude) and phase. In order to avoid phase values larger than 360deg the phase has been wrapped into a range of ±180deg. The practical implication of this is that the phase jumps 360deg when lying outside this region. The reader is reminded of that a phase value of +180deg is equal to -180deg.

8.2.1 Aeroelastic Response at Different Modes

Firstly flutter test data are discussed for the three tested orthogonal modes. For this purpose the nominal subsonic case at reduced frequency of k=0.1 is adapted. Figure 8-7 depicts the unsteady response on blades +2 through -2 for the axial bending mode. Significant response magnitude can be observed on the oscillating blade (index 0) and its directly adjacent surfaces, i.e. pressure side of blade +1 and suction side of blade -1, while the response shows substantially minor magnitude on the surfaces of the direct neighbor blades facing away from the oscillating blade and on blades +2 and -2. On blade 0 a local response peak can be noted on the fore suction side around arc=-0.11 whereupon it is decreasing to either side. On the aft part of the suction side towards the trailing edge the response magnitude is increasing again. On the pressure side of blade +1 the response is moderate and of almost constant magnitude with a minor peak around arc=0.3 while the response is low on the suction side. On blade -1 a pronounced peak is observed on the suction side at arc=-0.11 with monotonic decrease towards the trailing edge. On the entire pressure side of this blade the response is of insignificant magnitude. Blade +2 and -2 feature insignificant response magnitude, which lies well below measurement accuracy. The response phase on blade 0 starts off in-phase on the fore suction side and rotates downstream of arc=-0.22 to out-of-phase at arc=-0.41. Thereafter it drops back to in-phase at the trailing edge. On the pressure side the phase tends to lie out-of-phase with respect to the blade motion suggesting that the pressure decreases upon positive motion of the blade. The surfaces that are directly exposed to the oscillating blade behave similar in phase as their index 0 counterparts; whereas the pressure side of blade +1 lies in phase, the suction side of blade -1 shows out-of-phase behavior. The sides facing away from the oscillating blade, i.e. suction side on blade +1 and pressure side on blade -1, start off at 180deg out-of-phase relative to their respective companion and decrease towards trailing edge. On blades +2 and -2 respectively the phase lies well in line with their +1 and -1 counterparts.


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Figure 8-7. Unsteady response on blades +2 through -2 at midspan; operating point L1; axial bending, k=0.1

These observations suggest the following:

• The major response is detected on the primary surfaces, i.e. the surfaces enclosing the oscillating blade

• Surfaces that face away from the oscillating blade show negligible magnitude. These surfaces are below referred to as secondary surfaces

• A response peak is present at arc=-0.11, which corresponds to peak suction and laminar-turbulent transition

• Phases on adjacent primary surfaces tend to agree and thereby indicate that the unsteady response involves the passage flow


Response data for the circumferential bending mode are included in Figure 8-8. The observed response magnitudes tend to be of much lower magnitude than in case of the axial bending mode. Again, on blade 0 a distinct response peak is observed at arc=-0.11. The response decreases thereafter to either side and stays low although a slight increase is observed on the aft pressure side. On the pressure side of blade +1 the response is generally low and culminates in a moderate peak at arc=0.3. On the suction side of blade -1 the response equals the one observed at axial bending mode however with a much less pronounced response peak. The secondary surfaces as well as blades +2 and -2 show negligible responses. On the fore suction side of blade 0 the phase starts again off as in-phase and stays largely constant towards the trailing edge. On the fore pressure side the aforementioned jump of 180deg out-of-phase is observed however here followed by an equally large decrease. Other than in case of the axial bending mode the primary surfaces of blade +1 and -1 do not necessarily show in-phase behavior with their oscillating blade counterparts; whereas the pressure side of blade +1 lies out-of-phase with suction side of blade 0, the suction side of blade -1 lies in-phase with the fore part of the pressure side of blade 0. Blades +2 and -2 again show similar behavior in phase as their respective +1 and -1 neighbors. These observations suggest the following:

• The response at circumferential bending mode equals qualitatively the response at axial bending mode but features much lower magnitude

• The phases on adjacent primary surfaces do not necessarily agree and thus indicate that the involvement of the passage is not obvious for this mode

• Despite the low response magnitude the phases indicate that the response phenomenon extends further than one passage away to either side of the oscillating blade

Finally response data for the torsion mode shall be analyzed in Figure 8-9. The response magnitude on blade 0 shows similar behavior as for the two other modes with a response peak being present at arc=-0.11. This peak is however less distinct as for the two bending modes and appears to be smeared out. The pressure side of blade 0 shows constant low magnitude. On blade +1 the magnitude starts off moderately around blade leading edge ad increases gradually to either side. The increase is thereby somewhat larger on the pressure side, which is a primary surface to the oscillating blade. On the suction side of blade -1 the response shows qualitatively and quantitatively similar behavior to the response at axial bending mode with a very distinct response peak at arc=-0.11. The pressure side of this blade shows negligible amplitude. Other than for the two bending modes there is a very small response on the suction side of blade +2. Blade -2 features continuous low response.


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Figure 8-8. Unsteady response on blades +2 through -2 at midspan; operating point L1; circumferential bending, k=0.1

The phase on the oscillating blade shows similar behavior to what has been observed for the axial bending mode; on the suction side it starts off in-phase with the motion while it is out-of-phase on the pressure side. For the present mode the phase is largely constant on wither side towards the trailing edge. The primary surfaces of blades +1 and -1 indicate largely in-phase behavior with their oscillating blade counterparts. Apart from a region around the leading edge the phase is constant all around blade +1. On the pressure side of blade -1 the phase decreases gradually towards the trailing edge. Blades +2 and -2 again show again similar behavior in phase as their respective +1 and -1 neighbors.



• The response magnitude for the torsion mode differs qualitatively on blades 0 and +1 from the one measured at the two bending modes

• On blade -1 the response agrees qualitatively and quantitatively well to the response at axial bending mode

• A measurable response is present on the suction sides of blades +1 and +2, i.e. two secondary surfaces

• As has been observed for the two bending modes the response phenomenon seems to extend further than one passage away to either side of the oscillating blade

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−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1

Figure 8-9. Unsteady response on blades +2 through -2 at midspan; operating point L1; torsion, k=0.1


At this position a preliminary analysis of the response phenomenon shall be made. Despite the differences in the investigated modes test data indicate that the response phenomenon shows similar features across all modes. One of these is a response peak at arc=-0.11, which has been pointed out as position of peak suction above. For all modes this response peak is in to a certain extent present on the suction sides of blades 0 and -1. The observation of that the axial bending resembles more the torsion mode than the circumferential bending mode shall be explained from a geometrical consideration of the instantaneous blade row shape. Phase data suggest that the response phenomenon primarily involves a flow passage and its respective surfaces. During blade oscillation the instantaneous passage shape changes due to the movement of the oscillating blade featuring the largest relative change at passage throat; for the torsion mode the changes in the two adjacent passages are thereby not symmetric and are larger in the pressure side passage. In terms of quasi-steady flow, i.e. infinitely slow blade displacement, reduced throat size leads to increased blockage and therefore to increase in pressure in the passage. This explains the phase on the suction side of blade 0 being in phase with the motion, i.e. the pressure in the suction side passage increases upon movement of the blade in direction of the suction side, which is equal to reducing the throat size in this passage. The respective difference in throat size of the two adjacent passages is listed in Table 8-1. It is evident that the axial bending mode and torsion mode feature the largest impact on throat size.

Mode Amplitude ∆throat+1, % ∆throat-1, % Axial 1mm 9.86 9.81 Circ 1mm 7.46 7.48 Torsion 1deg 3.01 8.80

Table 8-1. Impact of mode shape on relative change in throat size (amplitudes indicated at midspan)

By changing the blockage in a passage the most susceptible point will be the suction peak due presence of inherently high dynamic pressure and large changes in loading gradient. A pseudo unsteady pressure coefficient is therefore formulated that yields from the product of static pressure coefficient and its second derivative as given by

2

2

darc

CpdCpCp pseudo ⋅= Eq. 8-1

throat+1 throat-1


A comparison between this pseudo Cp and the unsteady Cp measured on blade -1 at axial bending mode is included in Figure 8-10. Note the striking qualitative overall consistency between the pseudo value and the real response distribution.

−0.4 −0.2 0 0.2 0.40

1

2

3

4


Cp ps

eudo

, −

Pseudo unsteady Cp

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

Unsteady Cp on blade -1, axial bending mode

Figure 8-10. Comparison of product of blade loading and its second derivative to aeroelastic response data

Next the aerodynamic coupling phenomenon for the secondary surfaces is discussed. Point of origin shall be the change in throat. There are two mechanisms that are supposedly acting; firstly as the blockage in the passages adjacent to the oscillating blade changes a new balance tends to establish instantaneously in the rest of the blade row. This balance might partially be affected by local deviations in downstream pressure, partially by locally affected inflow to passages as a passage with little blocking will give passage to a larger amount of flow leading to a local sucking effect. These interpretations have been reported previously by Vogt and Fransson (2004) from up- and downstream probe traverses with the blade deflected statically. The second mechanism in mind is effectuated by variations in passage outflow direction as imposed during torsion mode. These variations will affect the downstream flow pattern of the cascade. A parallel can be drawn to a steady-state analysis performed on the test section by Vogt and Fransson (2000) where the potential in cascade periodicity control by means of control of downstream flow direction has been assessed. It has been recognized that modifications in flow direction downstream of the cascade are leading to locally changed pressure. In the case of torsion flutter this effect seems to be more pronounced on the positive indexed neighbors of the oscillating blade. As the suction sides of the blades tend to be more susceptible to changes in downstream pressure given the high dynamic pressure present an explanation for the response magnitude on the secondary surfaces of blades +1 and +2 is given.


8.2.2 Effect of Reduced Frequency on Aeroelastic Response

The effect of reduced frequency on the unsteady response is presented below on data acquired at low reduced subsonic Mach number. Given the fact that the response on blades +2 and -2 generally is of negligible magnitude the focus is here only put on blades -1 through +1, i.e. the oscillating blade and its direct neighbors. Figure 8-11 depicts the unsteady loading distributions at midspan on blades +1, 0 and -1 for different reduced frequencies at axial bending mode. The response magnitude on the suction side of blade +1 is constantly small and within measurement accuracy apart from a limited region on the fore part. The magnitude on the pressure side is generally higher and lies roughly in phase with the blade motion. An increase in magnitude with increasing reduced frequency is observed with a local maximum at arc=0.3 becoming increasingly pronounced. The response on blade 0 peaks around arc=-0.11 for all measured reduced frequencies and generally shows increase in response level with increasing reduced frequencies. It is noticeable that the response distribution changes considerably above k=0.3; on the suction side the response magnitude increases heavily on the aft part whereas the level of peak response around peak suction only increases moderately. On the pressure side the increase is more gradual. At all reduced frequencies the phase on the suction side starts off in-phase with the blade motion and increases towards the trailing edge. Whereas the response up to k=0.3 features falling tendency at the trailing edge it continuous increasing at reduced frequencies above. On the pressure side the response lies fairly out-of-phase regardless the reduced frequency.

0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


+1k 0.1k 0.2k 0.3k 0.4k 0.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1k 0.1k 0.2k 0.3k 0.4k 0.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0k 0.1k 0.2k 0.3k 0.4k 0.5

Figure 8-11. Effect of reduced frequency on unsteady response on blades +1, 0, -1; operating point L1; axial bending


On the suction side of blade -1 the response is similar for all reduced frequencies featuring a pronounced peak at arc=-0.11 with falling tendency to either side. The response varies little with increase in reduced frequency and lies out-of-phase with the blade motion featuring a decrease in phase towards the trailing edge. On the pressure side of this blade the response magnitude is well below measurement accuracy. At all reduced frequencies the phase changes thereby gradually from leading to trailing edge.

0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

egSS normalized arcwise coordinate, − PS

+1k 0.1k 0.2k 0.3k 0.4k 0.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1k 0.1k 0.2k 0.3k 0.4k 0.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0k 0.1k 0.2k 0.3k 0.4k 0.5

Figure 8-12. Effect of reduced frequency on unsteady response on blades +1, 0, -1; operating point L1; circumferential bending

Response data at different reduced frequencies for the circumferential bending mode are included in Figure 8-12. On blade +1 the response magnitude shows little variation with reduces frequency however considerable differences are present in phase on the fore pressure side; whereas the response tends to lie out-of-phase to the blade motion at low reduced frequencies it changes gradually to in-phase with increase in reduced frequency. On the suction side of this blade the phase varies considerably with increase in reduced frequency but given the low response magnitude this is not considered of interest. On blade 0 it is observed that the response magnitude increases gradually with increase in reduced frequency while maintaining its character of moderate but fairly constant level featuring a broad peak around arc=-0.11. The phase shows little variation on the fore suction side but larger differences downstream of arc=-0.29, i.e. downstream of the throat. On blade -1 similar behavior is observed on the suction side with gradually larger response magnitude as reduced frequency increases. Slight change in phase form out-of-


phase towards in-phase is thereby observed. On the pressure side, which is a secondary surface, the magnitude is low whereas the phase varies considerably. Response data for the torsion mode are lastly included in Figure 8-13. The response on the pressure side of blade +1 shows increasing tendency in magnitude from leading to trailing edge while the phase varies only little, especially on the fore pressure side. On this suction side of this blade the response seems show little variation with increasing reduced frequency. On blade 0 the response features similar behavior to what has been observed for the other modes namely increasing magnitude with increasing reduced frequency. The phase is thereby fairly unaffected on the fore part of the blade and lags increasingly on the aft part. On the suction side of blade -1 the overall response characteristics are preserved with increase in magnitude as reduced frequency increases.

0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


+1k 0.1k 0.2k 0.3k 0.4k 0.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1k 0.1k 0.2k 0.3k 0.4k 0.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0k 0.1k 0.2k 0.3k 0.4k 0.5

Figure 8-13. Effect of reduced frequency on unsteady response on blades +1, 0, -1; operating point L1; torsion


• An increase in reduced frequency tends to lead to an increase in response magnitude, which is mainly seen on the primary surfaces

• On the oscillating blade the response shows little change in phase on the fore part with increased variations on the aft part

• The primary sides of the adjacent blades show different behavior in phase; whereas the positive neighbor is mainly affected on the fore part, the negative neighbor shows largest variations on the aft part


• The secondary faces seem to be largely unaffected in response magnitude by increasing reduced frequency

The analysis of the flutter phenomenon initiated above shall at this position be extended towards frequency aspects. From the quasi-steady analysis it has been recognized that the change of throat and by this the change in passage flow is assumed having a major influence on the unsteady response. This is certainly valid at increased reduced frequencies but here the time aspect of the propagation of a disturbance comes into play. A change in flow property at a certain position propagates at speed of sound relative to the flow, i.e. at an absolute Mach number of

aMM prop −= upstream and aMM prop += downstream from the source of

disturbance. This source is here primarily imagined to be located at the passage throat that upon closing “sends” a signal of increased blockage upstream and upon opening reduced blocking respectively. Expressed in other words the aeroelastic response phenomenon seems to some degree be dominated by pulsating passage flow at increased time delay with increased reduced frequency. For low subsonic Mach numbers the propagation speed of the induced disturbance is less affected by reduced frequency but is believed experiencing greater variations as flow velocity level approaches sonic conditions. This consideration assesses the frequency dependency of the aeroelastic response only to a certain degree. An explanation of the increased response magnitude is however still outstanding. For this purpose the physical mechanism underlying aeroelastic response shall be composed by the following contributions:

• The harmonic unsteady pressure at the regarded location, therein including the contribution that is induced due to the motion of the blade, i.e. the dynamic realization in pressure of blade oscillation

• The harmonic pressure at the regarded location that is due to the motion through a spatially varying pressure field

This is expressed by

( )phDt

Dp ∇⋅+−=

rφρ~ Eq. 8-2

The motion induced term thereby enters the unsteady response at increased magnitude as reduced frequency increases. As direct consequence this effect is especially pronounced on the oscillating blade itself as has been observed above, but to some degree also applies to primary surfaces of the adjacent blades. Here the suction side of the negative neighbor seems to be more affected by this effect.


8.2.3 Effect of Flow Velocity Level on Aeroelastic Response

The effect of velocity level on the unsteady response is presented below on data acquired at nominal inflow incidence. As in the presentation of the effect reduced frequency the response phenomenon is discussed only on blades -1 through +1, i.e. the oscillating blade and its direct neighbors. Note that the three velocity levels are referred to by M=0.4 for the low, M=0.6 for the medium and M=0.8 for the high subsonic respectively despite the exact passage-averaged Mach number value. Figure 8-14 includes data for the axial bending mode. On blade +1 the response shows very little change with increase in flow velocity. The reader might at this position be reminded of that the unsteady response magnitude is normalized by the inlet dynamic head, i.e. the equality of the values suggests that the absolute response increases with velocity level. It shall also be mentioned that for the two higher velocity levels the data point at arc=0.02 has been omitted due to transducer overload. On blade 0 the character of the response at low subsonic Mach number is preserved with increased flow velocity. The magnitude of the response peak on the suction side is almost unchanged while the response increases moderately at the trailing edge with increased velocity level. The phase is thereby almost unchanged on the fore suction and pressure side but differs slightly on the aft blade, predominantly downstream of arc=-0.29, i.e. the passage throat. On blade -1 the response changes very little with increased velocity apart from the phase on the suction side.

0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


+1M 0.4M 0.6M 0.8

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1M 0.4M 0.6M 0.8

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0M 0.4M 0.6M 0.8

Figure 8-14. Effect of flow velocity level on unsteady response on blades +1, 0, -1; operating points L1, M1 and H1; axial bending


Data for the circumferential bending mode are included in Figure 8-15. Generally the differences with increased flow velocity are very small. The only noticeable differences are observed on the respective response peaks on the suction sides of blades 0 and -1 although it has to be mentioned that these differences are moderate. The most significant differences in response phase are observed at arc=-0.4 on blade 0, i.e. right downstream of the passage throat.

0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

egSS normalized arcwise coordinate, − PS

+1M 0.4M 0.6M 0.8

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1M 0.4M 0.6M 0.8

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0M 0.4M 0.6M 0.8

Figure 8-15. Effect of flow velocity level on unsteady response on blades +1, 0, -1; operating points L1, M1 and H1; circumferential bending

Data for the torsion mode included in Figure 8-16 indicate that the effects of flow velocity level on response magnitude are more pronounced than for the two bending modes. On blade +1 gradual increase in response phase is observed all around the blade with increase in velocity level. The phase on the pressure side, which his a primary surface, is thereby fairly unchanged. On blade 0 and -1 the response magnitude is mainly affected on the suction side while the response on the pressure side shows little to no influence. The phase on the fore suction side of these blades shows that at the two higher velocity levels moderate local lagging of the response is occurring in the region between leading edge and arc=-0.22. This lagging is approximately equal for the medium and the high subsonic case.


0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


+1M 0.4M 0.6M 0.8

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1M 0.4M 0.6M 0.8

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0M 0.4M 0.6M 0.8

Figure 8-16. Effect of flow velocity level on unsteady response on blades +1, 0, -1; operating points L1, M1 and H1; torsion


• Very moderate increase in (normalized) response magnitude with increased velocity level is observed where the response phenomenon mainly seems to be dominated by the unsteady passage flow (i.e. throat-dominated). This indicates that the absolute response magnitude increases with increased flow velocity

• Comparatively larger increase in (normalized) response magnitude with increased velocity level is observed where the response phenomenon is believed being dominated by the instantaneous blade row outflow (i.e. outflow-dominated). This is the case for the suction sides on the adjacent blades in case of torsion mode.

• Considerable changes in response phase downstream of the passage throat on the suction side of the oscillating blade indicate that the passage flow behaves differently up- and downstream of the throat. The upstream part of the passage seems thereby to be much less affected by changes in flow velocity. The changes in the downstream part are however not obvious as no major differences are found in static blade loading of outlet flow field. It is therefore believed that the differences arise from changed dynamic behavior of secondary passage flow structures, e.g. tip vortex


8.2.4 Effect of Flow Incidence on Aeroelastic Response

The effect of flow incidence on the unsteady response is presented below on data acquired at low subsonic Mach number. As for the discussion above the results presented here cover blades -1 through +1, i.e. the oscillating blade and its direct neighbors. Figure 8-17 includes data for the axial bending mode. Considerable differences are observed on the fore pressure side of blade +1; whereas the response is fairly constant for the nominal case it features a distinct peak at off-design 1 case terminating at arc=0.12 that broadens to arc=0.2 at the high incidence case. Note that these locations coincide with the end of the respective separation bubble as discussed above from steady data. The response shows thereby little change in phase throughout the separated region. On blade 0 the response changes rather dramatically its character; whereas the response featured a broad peak around arc=-0.11 at nominal incidence this peak still exists for the two off-design cases but gets much sharper. In addition a second response peak is appearing at arc=0.12 that considerably increases in magnitude for the high incidence case. This increase is attributed to the separated flow in this region. The phase shows only little difference in this region. On blade -1 negative incidence operation leads to slight increase in peak response magnitude at arc=-0.11. Also on this blade a second peak in response magnitude is appearing at arc=0.12 for the high incidence case that different from the two other blades is accompanied by a drastic change in phase.

0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


+1nomoff1off2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1nomoff1off2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0nomoff1off2

Figure 8-17. Effect of flow incidence on unsteady response on blades +1, 0, -1; operating points L1, L2 and L3; axial bending


Response data for the circumferential bending mode are included in Figure 8-18. The response magnitude on blade +1 is largely unchanged with increased negative incidence. The phase is changing drastically in the region of separated flow as well as also on the fore suction side for the high incidence case but given the small response magnitude in these regions, this is considered negligible. On blade 0 the above made observation of that the response magnitude decreases on the suction side right at the leading edge and thereby pronounces the response peak is also valid here. On the fore pressure side moderate effects in response magnitude are visible while the phase changes drastically in the separated region. The response magnitude on blade -1 is similar to blade +1 largely unaffected by changed inflow incidence. It is however noticeable that the phase varies considerably in the separated region as well as on the fore suction side.

0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200C

p ph

ase,

deg


+1nomoff1off2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1nomoff1off2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0nomoff1off2

Figure 8-18. Effect of flow incidence on unsteady response on blades +1, 0, -1; operating points L1, L2 and L3; circumferential bending

Response data for the torsion mode are at last included in Figure 8-19. On blade +1 and 0 considerable effects of negative incidence are visible in the separated region on the fore pressure side by a remarkable increase in response magnitude. Similar to the axial bending mode the phase seems thereby largely unaffected. The largest effects are however observed on blade -1 on the suctions side where the response peak at arc=-0.11 increases substantially with increased negative incidence. Similar to the axial bending mode a secondary peak is appearing on the fore pressure side at high incidence that comes together with a considerable change in phase.


0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


+1nomoff1off2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1nomoff1off2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0nomoff1off2

Figure 8-19. Effect of flow incidence on unsteady response on blades +1, 0, -1; operating points L1, L2 and L3; torsion


• Separated flow as it is present on the fore pressure side at high negative incidence primarily leads to a considerable increase in response phase. Downstream of the reattachment point the response magnitude features attached flow behavior

• Depending on the mode the response phase in the separated flow region can show substantially different behavior; whereas the response phase is largely unaffected at axial bending and torsion mode it shows large differences at circumferential bending

• Operation at high incidence might also lead to considerable increase in response magnitude around peak suction although the flow in this region is fully attached. It is believed that this is due to change in blade loading induced by migration of the stagnation point onto the suction side as well as due to modified secondary flow structure upon unloading of the profile


8.3 Three-Dimensional Effects of Aeroelastic Response

In this section the three-dimensional effects of the unsteady aeroelastic response are presented on the direct neighbor blades +1 and -1. Note that the instrumentation of the oscillating blade (index 0) included only taps at midspan and therefore data on this blade are not included here. Figure 8-20 shows response data on blade +1 at low subsonic Mach number for axial bending, circumferential bending and torsion mode respectively. The data are presented as arcwise distributions at three spanwise positions (10%, 50% and 90% span) and as 2D contour plots of the entire measured blade surface such as to give a global picture. At axial bending mode two narrow regions of high response magnitude are visible at the leading edge. One region is extending on the pressure side from close to hub to 70% span, the other one starts between 50% and 70% span on the suction side and extends to close to tip. It is believed that the locations of these high response regions are due to the twisted shape of the profile inducing locally incidence, which tends to be negative towards the hub and positive towards the tip. A second region of higher magnitude is visible on the pressure side between arc=0.2 and arc=0.3. This region appears around 50% span and extends to close to tip. On the pressure side the response magnitude is constantly low apart from a tiny region close to tip at arc=-0.22. It is possible that this is due to the tip vortex that discharges into main flow field downstream of the suction peak, see also discussion of steady-state data above. Whereas the response phase is fairly constant on the pressure side it increases gradually from hub to tip on the suction side. The response data at circumferential bending shows consistent low magnitude with a limited region of higher magnitude on the pressure side centered around arc=0.3. The phase is largely constant out-of-phase to the blade motion apart from a region around the leading edge. The torsion mode response features a local response minimum at the leading edge and a broad maximum close to the trailing edge downstream of arc=0.3. This high response region is dominated by a local peak around midspan and narrows from hub to tip, which is believed being due to radial variations in steady static pressure coefficient due to the radial pressure gradient. Response data on blade -1 are included in Figure 8-21 in the same manner as for blade +1. The response at axial bending mode is dominated by a pronounced peak around arc=-0.11, which has been identified as location of peak suction above. This peak gains gradually in strength from hub to tip. The phase is largely constant across blade span apart from the pressure side where very low response magnitude is observed. A distinct increase in phase close to hub downstream of arc=-0.29 is noticeable and it might be possible that this is caused by the secondary flow in this region as identified from steady-state flow visualization. At circumferential bending mode the response shows qualitatively similar behavior to the axial bending mode with a local response maximum being observed at peak suction growing from hub to tip however at overall smaller magnitude. The phase is largely constant over span inside this region but differs considerably downstream of it


on the aft suction side. Whereas little differences are visible between 10% and 50% span the response at 90% span rotates by more than 90deg. A possible reason for this might be found in the effect of tip leakage flow. The torsion mode response features a distinct peak around arc=-0.11 that ranges almost equally from hub to tip. On the fore suction side the observed obliqueness of the contours is believed being due to the twisted shape of the profile. The reduction in response magnitude towards blade tip is explained by the unloading of the profile in this region. The response phase is almost perfectly constant all over the blade surface.

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Torsion

Figure 8-20. Spanwise variation of aeroelastic response on blade +1; operating point L1


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0.4

Torsion

Figure 8-21. Spanwise variation of aeroelastic response on blade -1; operating point L1



• There are clearly identifiable three-dimensional effects in the aeroelastic response on the adjacent blades. The effects are mainly observed on the primary surfaces but appear also on the secondary surfaces, especially on the fore part

• There are two types of three-dimensional effects, one being characterized by increasing amplitude from hub to tip, the other one featuring local variations in response magnitude at different positions on the blade

• The cause for the first type of three-dimensionality is believed being found in the mode shape of the blade as this type is exclusively observed for the two bending modes, which feature linearly increasing amplitude from hub to tip

• The second type of three-dimensionality is on the other hand believed being due to three-dimensional effects of the mean flow field. The most prominent of these effects are seen in radial variation in steady loading (due to the annular shape of the cascade as well as the twisted shape of the profile), local effects close to the hub endwall due to the hub corner vortex and local effects close to the tip endwall due to tip leakage. Among those the hub corner vortex shows least effect while the two other effects are similar order of magnitude

• Whereas radial variations in steady loading tend to lead to variations in response magnitude, e.g. around the leading edge, tip leakage flow affects the aeroelastic response in that the response magnitudes are slightly lowered

• Depending on the mode the tip leakage flow seems to affect the response phase considerably. This suggest that the flow structures induced by the tip leakage flow (it is here especially thought of the tip vortex) may have own unsteady properties, which in turn can affect the unsteady loading on the blade. Most effect of this phenomenon is observed at circumferential bending mode


These observations shall at this position be correlated to flow visualization results from above. The axial bending and the torsion mode are regarded here as they feature most of the observed three-dimensional phenomena and show similar level in response magnitude. Response magnitude data overlaid to flow visualization results on the primary surfaces for the axial bending mode are included in Figure 8-22. On the suction side it is observed that the response peak is located right upstream of the transition line with increasing magnitude from hub to tip. This increase can be seen from a characteristic V-shape of the contour lines in this region. At the transition line the contours suggest a slight positive radial gradient in response magnitude from hub to tip due to moderate backwards lean of the contour lines. On the aft suction side downstream of the transition line the contours align largely with blade span and show minor deflections close to hub and close to tip. On the pressure side most activity is observed close to the leading edge especially close to hub where a separation bubble has been identified. Right at the leading edge the response seems to decrease over blade span, which is seen from the upwards-down V-shape of the contours. While a limited region of higher response magnitude is observed close to the hub at around mid pressure side the response equalizes over span as trailing edge is approached.

View on suction side, blade -1 View on pressure side, blade +1

Figure 8-22. Blade surface flow visualization results and corresponding unsteady response amplitudes on primary surfaces; operating point L1; axial bending

Correlations for the torsion mode are included in Figure 8-23. On the suction side it is again observed that the response peak is located right upstream of the transition line. Other than at the axial bending mode the response magnitude does not a positive


gradient from hub to tip at this mode. At the transition line the contour lines are very well aligned with the transition line itself, which underlines a dominant effect of the three-dimensionality of the mean flow field. Towards the tip the contour lines are slightly bended towards the leading edge, which indicates a moderate decrease in response magnitude close to the tip. It is believed that this decrease is due to the tip leakage flow. On the pressure side high activity is again observed close to the leading edge however at lower magnitude than at axial bending mode. A local maximum on the aft pressure side suggests that effects close to hub and casing endwalls are leading to local reduction in response magnitude. These effects are however not identified in the present visualization.

View on suction side, blade -1 View on pressure side, blade +1

Figure 8-23. Blade surface flow visualization results and corresponding unsteady response amplitudes on primary surfaces; operating point L1; torsion


Next the behavior of the observed three-dimensional character of the aeroelastic response at off-design conditions is addressed. To conclude from the discussion of effects of flow incidence that is included above the main focus has been put on blade +1 where clearly identifiable phenomena have been observed. Figure 8-24 includes response data on blade +1 for the two off-design points at low subsonic Mach number. It is observed that a region of high response magnitude exists on the fore pressure side starting at the leading edge and that this region grows in extent with increasingly negative incidence. The high response region varies in spanwise direction and features maximum response around midspan. At off-design 1 point distinct reduction in extent of the high response region is observed in the upper half of the blade. At off-design 2 point however the response decreases both towards hub and tip suggesting that endwall effects come into play with greater importance. The rest of the blade shows little three-dimensional behavior and features largely the phenomena observed at nominal conditions.

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Figure 8-24. Spanwise variation of aeroelastic response on blade +1; operating points L2 and L3, axial bending

To attribute the observed response pattern to flow phenomena more in detail correlation to flow visualization results are included in Figure 8-25. The correlation is here made on pressure side of blade +1 only for the two off-design operating points at axial bending mode. At off-design 1 operating point the region of high response lies well in line with the reattachment line; while it decays rapidly towards the leading edge, the decay towards the trailing edge is stretched out over a larger arcwise distance, which is believed being due to the local structure of the reattaching flow in


that region. The observations are to a large degree also valid for the high incidence case (off-design 2) in that the response peak largely collides with the reattachment line and that the decay of the peak is somewhat stretched towards the trailing edge. Due to the deviations of the contours towards the hub endwall away from the reattachment line at this operating point it gets obvious that the magnitude of the aeroelastic response is exclusively determined by the effects that could be tracked by surface flow visualization. It is supposed that the response is affected by a phenomenon, which is not visible in the blade surface flow pattern. As the separation bubble itself is seen as a rather unstable flow feature due to the fact that it contains low momentum fluid it might be possible that the pressure side branch of the hub horseshoe vortex leads to local variations in response magnitude in this region. These observations add the following suggestions:

• Reattachment of separated flow leads to local high aeroelastic response featuring a larger gradient upstream of the reattachment point and a comparatively smaller gradient downstream of it

• Flow regions that contain low momentum fluid might be influenced by secondary flow structures such that the aeroelastic response varies locally

Off-design 1 (L2) Off-design 2 (L3)

Figure 8-25. Blade surface flow visualization results and corresponding unsteady response amplitudes; operating points L2 and L3; axial bending


8.4 Correlation of CFD Results to Test Data

This section contains correlations of steady and unsteady CFD results to test data and addresses aspects relating to prediction accuracy of the used models. Comparisons of predicted and measured steady blade loading at low and high subsonic operating points are shown in Figure 8-26. The predictions have been made using the 3D model without tip clearance (mesh medium). The predicted loading lies well in line with test data and indicates that the major phenomena have been captured. On the suction side radial variations in the loading are captured in terms of arcwise varying location of the suction peak with increasing span and decreasing Cp. Whereas the magnitude of the suction peak is well predicted close to hub and at midspan CFD overpredicts close to tip. It is believed that the reason for this effect is the absence of tip clearance in the model, which is leading to unloading of the profile in the tip region in the experiment. On the aft suction side close to tip it can also be observed that the measured Cp lies slightly below the predicted, which is believed being due to interaction effects of the tip leakage jet with the main passage flow. On the pressure side the Cp level as well as radial variation are well captured. On the fore suction side close to tip CFD results lie above test data indicating that the measured pressure in this region is lower. This observation stands in good agreement with the aforementioned unloading of the profile close to tip. The predicted loading distributions feature local peaks around the leading edge (at arc=0.02 close to hub, at arc=-0.01 at midspan and at arc=-0.02 close to tip), which are probably due to defective geometrical representation of the profile due to discontinuities in the surface geometry. Given the local character of this effect no measures have been undertaken to eliminate of this problem.

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data 10%CFD 10%data 50%CFD 50%data 90%CFD 90%

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Low Subsonic (L1) High subsonic (H1)

Figure 8-26. Comparison of predicted and measured steady loading; nominal low subsonic (L1) and high subsonic (H1)


Comparisons of the predicted and measured unsteady response at midspan for the three investigated modes are included in Figure 8-27 through Figure 8-29 for the low subsonic operating point. CFD results are included from the 2D and the 3D model without tip clearance. At this point the focus is put on the prediction accuracy of unsteady loading during flutter depending on the dimension of the model.

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Figure 8-27. Comparison of measured and predicted unsteady response on blades +2 through -2 at midspan; operating point L1; axial bending, k=0.1

Correlations for the axial bending mode are included in Figure 8-27. CFD results capture the overall aeroelastic properties well in that the unsteady response decays rapidly away from the oscillating blade. Whereas minor differences are obvious on the further neighbor blades +2 and -2 as well as on the positive neighbor blade (index +1) considerable differences are observed on blade -1 and the oscillating blade itself (index 0). Response magnitude and phase are very well predicted on blade +1 by the


3D model. 2D CFD results agree largely in order of magnitude however predict qualitatively different response distribution on suction and pressure side than test data. On the oscillating blade the 3D model overpredicts the response peak on the fore suction side heavily however agrees qualitatively well with test data. The 2D results suggest relatively much greater response magnitude on the pressure side and a distinct increase on the aft pressure side, which is not confirmed by test data. The phase is well captured on the fore suction and entire pressure side but neither of the models is able to capture the measured increase in phase downstream of the response peak towards the trailing edge. The response on blade -1 is somewhat overpredicted for both models. From a qualitative point of view the 3D model captures the character of the response more accurately while the 2D model suggests a hump in the response magnitude distribution around arc=-0.3, which is approximately at the passage throat. Correlations for the circumferential bending mode are included in Figure 8-28. As observed at the axial bending mode both CFD models capture the qualitative character of the aeroelastic response well with rapid decay of the unsteady response away from the oscillating blade. Whereas the magnitude on the further neighbors is captured well within measurement accuracy the phase differs considerably. Given the small response magnitude this difference is however not considered harmful for the prediction of aeroelastic stability. On blades +1 through -1 it is evident that the 3D model performs much better in that it captures the unsteady response qualitatively and also quantitatively correct from an overall perspective. The 3D model performs considerably better on the oscillating blade in prediction of response magnitude and phase, especially on the pressure side. Again 2D results suggest a strong increase in response magnitude towards the trailing edge, which is not observed from test data. Both models however tend to overpredict the response on blade -1 while the qualitative character of the response magnitude is captured well by the 3D model. As at the axial bending mode the 2D model suggests again a hump in response magnitude distribution at around arc=-0.3.


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Figure 8-28. Comparison of measured and predicted unsteady response on blades +2 through -2 at midspan; operating point L1; circumferential bending, k=0.1

At last correlations for the torsion mode are included in Figure 8-29. Similar to the two bending modes it is observed that both models are capable of predicting the overall characteristic of aeroelastic behavior with the major part of the response present on the three middle blades and comparatively much lower response on the blades further away. On blade +1 the response is well captured by the 3D model in magnitude as well as in phase; the 2D model differs locally on the fore part of the blade and suggests qualitatively more balanced response distribution around the blade. The response on the oscillating blade is heavily overpredicted by the 3D model but agrees better with test data in a qualitative manner. Again, the 2D model suggests a more balanced response magnitude distribution on pressure and suction


side as well as a strong increase in response magnitude towards the trailing edge. Nevertheless both models predict the phase correctly. On blade -1 both models overpredict the unsteady response; whereas the 2D model gives slightly better agreement in magnitude of the response peak the qualitative distribution is captured more accurately with the 3D model. The phase lies well in line on the suction side but differs on the pressure side due to low response magnitude.

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−1data3D2D

Figure 8-29. Comparison of measured and predicted unsteady response on blades +2 through -2 at midspan; operating point L1; torsion, k=0.1



• The unsteady linearized Euler method is capable of qualitatively predicting the characteristics of the aeroelastic response during flutter for the three orthogonal modes in 2D as well as 3D resolution

• The 3D model generally correlates more accurately to test data from a qualitative point of view

• The 2D model tends to fail locally in that it predicts a different response character, e.g. locally increased response magnitude towards trailing edge

• Both models tend to overpredict aeroelastic response especially on the oscillating blade (index 0) and its pressure side neighbor (index -1). The response on the suction side neighbor (index +1) is generally well predicted

Next the prediction accuracy at different reduced frequencies is addressed. For this purpose the response on blade -1 at axial bending and reduced frequencies k=0.3 and k=0.5 mode is regarded in Figure 8-30. It is observed that both models tend to overpredict the unsteady response magnitude at reduced frequency k=0.3 whereas better agreement is found at k=0.5. This is due to an opposite trend in test data and CFD predictions; while test data suggest an increase in aeroelastic response with increasing reduced frequency both models predict a slight decrease in response magnitude. The qualitative character of the predicted distributions is preserved at higher reduced frequencies with the 2D results featuring a hump downstream of the response peak at around arc=-0.3. The phase lies well in line at both reduced frequencies. These observations add the following suggestion:

• Whereas CFD tends to predict slightly reduced response magnitude with increasing reduced frequency test data suggest a moderate increase. The overall character of the response is preserved at higher reduced frequencies

−0.4 −0.2 0 0.2 0.40

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k=0.3 k=0.5

Figure 8-30. Comparison of measured and predicted unsteady response on blade -1 at different reduced frequencies; operating point L1; axial bending


In a next step the effect of model detailing on prediction accuracy of the aeroelastic response is addressed by correlating 3D CFD results from models with and without tip clearance to test data. Prior to discussing response properties the attention shall be drawn to the difference in steady-state flow features from the two different models. Figure 8-31 includes visualization of total pressure on axial planes downstream of the cascade. The model without tip clearance predicts a uniform total pressure field that features distinct wake regions ranging from hub to tip. By including tip clearance a region of low total pressure is observed close to tip. Note that this region is not due to boundary layer effects as these are not present in the inviscid model. It is believed that the deficit in total pressure arises from interaction of the tip leakage jet with the main flow field. It is also interesting to note that the shape of the wake slightly differs from the model without tip clearance, which is probably due to presence of the tip vortex. In addition minor differences are also visible close to hub suggesting that the including tip clearance might even affect the flow in this region. Although the overall flow structure agrees better to test data when including tip clearance it is repeatedly stated here that it has not been the aim to correctly predict all underlying flow features as such step would imply the use of viscous flow models.

3D inviscid, no tip clearance 3D inviscid, with tip clearance

Figure 8-31. Effect of model detailing on prediction of steady-state flow field; operating point L1

Comparisons of the unsteady aeroelastic response at midspan are included in Figure 8-32 for the axial bending mode. The focus is here being put on the oscillating blade and its direct neighbors. Due to numerical instabilities with the tip clearance model at low subsonic Mach numbers results could only be acquired for the high subsonic case. Comparisons are however still made at low subsonic velocity as a more complete set of test data was available there. The comparisons are still considered valid within the scope of this work as the current profile has shown very little influence of velocity level on aeroelastic response within the addressed Mach number range. The problem of numerical instabilities at certain combinations of model, flow condition and mode has been identified and reported by Mårtensson and Vogt (2005) however its treatment lies outside of the present framework.


Whereas little difference is observed on the blade +1 apart from prediction of the phase on the suction side, it is noticed that the model with tip clearance gives much better prediction in unsteady response on blades 0 and -1. On the oscillating blade (index 0) the predicted response magnitude using the tip clearance model lies almost in line with test data apart from the response peak, which is slightly overpredicted. Furthermore it is noticeable that the agreement of the response phase on the aft suction side gets strikingly better when including tip clearance in the model, which gives rise to the assumption that the tip leakage flow affects the dynamics of the unsteady flow field even at midspan. On blade -1 similar observation is valid in that predicted response magnitude as well as phase agrees much better when including tip clearance.

0 20 40

−80

−60

−40

−20

0

20

40

60

80+2

+1

0

−1

−2

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200C

p ph

ase,

deg


+1data3D3D tc

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1data3D3D tc

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0data3D3D tc

Figure 8-32. Effect of model detailing on prediction of aeroelastic response on blades +1 through -1; operating point L1; axial bending; k=0.1

Figure 8-33 depicts correlations for the torsion mode. On blade +1 the differences between the two models are moderate. From an overall perspective the response magnitude predicted with the tip clearance model agrees better with test data in its respective level on suction and pressure side. Nevertheless considerable differences appear on the fore pressure side when modeling tip clearance. These differences are however not considered harmful given the local level of response magnitude. On the oscillating blade the predicted response magnitude comes closer in line with test data when including tip clearance although there is still a slight overprediction noticed at response peak on the suction side. It is noticeable that the response level decreases almost to half when including tip clearance. Other than for the axial bending mode the phase shows only local and little difference between the two models and agrees


overall well to test data. On blade -1 a considerable reduction in predicted response magnitude is observed for the model with tip clearance leading to much improved agreement to test data. Whereas the overall level of unsteady response is predicted fairly well the sharpness of the response peak is somewhat attenuated. The predicted phase lies very well in line with test data when using the tip clearance model.

0 20 40

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−20

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20

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60

80+2

+1

0

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−2

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0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


+1data3D3D tc

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1data3D3D tc

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6C

p am

p, −

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0data3D3D tc

Figure 8-33. Effect of model detailing on prediction of aeroelastic response on blades +1 through -1; operating point L1; torsion; k=0.1


• By modeling tip clearance the prediction accuracy of the aeroelastic response at midspan increases considerably on the oscillating blade and its direct neighbors

• The tip leakage flow might have considerable effect on the response phase at midspan, especially on the aft suction side of the blade

• Despite using inviscid flow models magnitude and phase of the aeroelastic response at midspan are captured well when modeling tip clearance. This indicates that the aeroelastic response phenomenon is only to a little degree influenced by viscous effects


To further assess prediction accuracy of the different models used correlations are now discussed at different spanwise sections. For this purpose the axial bending and the torsion mode have been selected as different phenomena have been found prevalent for each of these modes; whereas spanwise variations at axial bending mode largely seems being affected by the three-dimensionality of the blade mode shape the three-dimensionality of aeroelastic response at torsion mode is believed being mainly a result from the mean flow field. Correlations for the axial bending mode are included in Figure 8-34 for blade -1. From correlations at midspan it has been recognized that the tip clearance model features better accuracy in predicting the aeroelastic response. Similar conclusions can be drawn from correlations close to hub and close to tip; using the tip clearance model greatly reduces the magnitude of predicted response and thereby agrees much better to test data. Whereas the phase seems to be little affected close to hub by the type of model used a rising tendency is observed on the aft suction side for the model with tip clearance. The rising tendency in phase close to hub on the aft suction side is interpreted as effect from the hub endwall vortex that is transported onto the blade in this region as described from steady flow visualization results above. This phenomenon is not captured by either of the models due to the absence of viscosity. The situation at the tip suggests that the discharge of the tip leakage jet into the main flow field is leading to a modified flow dynamics, which is manifested by a change in phase on the aft suction side. What concerns the response magnitude close to tip it is observed that the tip clearance model predicts a rather wild distribution, which indicates the difficulties of the flow model used when dealing with shear flows. Generally poor agreement is found for the phase on the pressure side but given the small amplitude levels this is not considered harmful.

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1data3D3D tc

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1data3D3D tc

10% span 90% span

Figure 8-34. Comparison of measured and predicted unsteady response on blade -1 close to hub and close to tip; operating point L1; axial bending; k=0.1


Figure 8-35 shows correlations for the torsion mode. Similar observations are made at this mode in that the predicted response magnitude correlates better to test data when modeling tip clearance. It is striking that the effects are that large even close to hub in that the predicted magnitude is approximately reduced to half. Other than at axial bending mode no variations in response phase are visible on the aft suction side from test data as well as predictions suggesting that the supposed effect of hub endwall vortex does not affect the dynamics of the flow field at this mode. As at axial bending mode the predicted response magnitude close to tip shows wild behavior when using the tip clearance model. Again it is believed that this gives an indication for failure of the model in this part of the flow where interaction effects play a major role. The differences are largest downstream of the response peak with a peak around arc=-0.3, which is located approximately at the throat. However no effect on response phase is observed.

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1data3D3D tc

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1data3D3D tc

10% span 90% span

Figure 8-35. Comparison of measured and predicted unsteady response on blade -1 close to hub and close to tip; operating point L1; torsion; k=0.1


• By modeling tip clearance prediction accuracy of the aeroelastic response is not only improved at midspan but also close to hub and close to tip. The improvement consists largely in reduction in predicted response magnitude

• Wiggly distributions of predicted response magnitude close to tip indicate minor local failure of the inviscid flow model when simulating tip clearance flow. This is not surprising when considering the highly viscosity-dominated flow phenomena in this region


As final aspect in the correlation of CFD results to test data the capabilities for predicting aeroelastic properties at large off-design operation shall be addressed. It was thereby aimed at forcing the flow model to fail at high negative incidence and by this get a measure for prediction accuracy at nominal conditions. Incipiently the prediction of steady loading is analyzed and compared to nominal conditions in Figure 8-36. Whereas fair agreement is found at nominal conditions the predictions at off-design conditions differ substantially on the fore pressure side and aft suction side. On the fore pressure side a separation bubble has been identified at this operating point from steady loading data and flow visualization results stretching from the leading edge to arc=0.3 close to hub, arc=0.2 at midspan and arc=0.12 close to tip. Whereas the separation bubble can be identified from test data as local deficit in static pressure coefficient, the predicted distributions do not suggest the presence of a separation. Nonetheless the transition of the stagnation point onto the fore suction side is captured correctly as is also the local acceleration of the flow around the leading edge onto the pressure side at high negative incidence. The differences on the aft suction side suggest an overprediction of loading in this region compared to test data.

−0.4 −0.2 0 0.2 0.4−5

−4

−3

−2

−1

0

1

Cp,

−



−0.4 −0.2 0 0.2 0.4

−5

−4

−3

−2

−1

0

1C

p, −



Nominal (L1) High negative incidence (L3)

Figure 8-36. Comparison of predicted and measured steady loading at various incidence; low subsonic

Comparisons of predicted and measured unsteady response at axial bending mode are included in Figure 8-37 for blades -1 through +1. In the light of the aforementioned observations at high incidence it is revised here that the largest effect has been noticed on the fore parts of the pressure sides of blades +1 and 0 whereas blade -1 has shown comparatively minor variations. On blade +1 it is observed that the measured local maximum in response magnitude in the region of the separation bubble is not predicted correctly. With respect to the overall characteristics of the unsteady response the agreement can be judged as fair. On blade 0 the agreement becomes astonishingly good in that the response peak on the fore suction side as well as the second peak on the fore pressure side is captured correctly. It is also noticeable that the model is even capable of correctly capturing the phase despite the separated nature of the flow. On blade -1 the response magnitude is considerably underpredicted although the qualitative character agrees well, which differs from the previous observations made. Additionally the predicted phase veers largely downstream of the response peak.


0 20 40

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40

60

80+2

+1

0

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−2

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0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


+1data3D

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


−1data3D

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0data3D

Figure 8-37. Comparison of measured and predicted unsteady response on blades +1 through -1 at high negative incidence; operating point L3; axial bending

In order to further assess prediction accuracy at high negative incidence correlations on the oscillating blade (index 0) at the two other modes are included in Figure 8-38. Other than at axial bending mode the predictions deteriorate in both response magnitude and phase. At circumferential bending a very flat response distribution is measured however two distinct peaks are predicted that differ considerably in their respective relative response level. The prediction of the phase starts off correct at the leading edge but fails as trailing edge is approached. At torsion mode the prediction of response magnitude agrees slightly better, especially the response peak on the fore pressure side. The model however tends to fail largely in predicting the response phase. It is remarkable that the distribution of the predicted response on blade 0 looks almost identical for all three investigated modes although considerable differences are obvious from test data. The model thereby seems to loose capability of predicting mode shape effects at this operating point. Nevertheless it is at the same time surprising that the character of the aeroelastic response at high negative incidence with a secondary response peak on the fore pressure side is captured fairly well although the inviscid model failed in predicting the separated steady flow field.


−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0data3D

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


0data3D

Circumferential bending Torsion

Figure 8-38. Comparison of measured and predicted unsteady response on blade 0 at high negative incidence; operating point L3; circumferential bending and torsion


• The inviscid model fails in predicting steady loading at high negative incidence, mainly in the region of separated flow. This finding is not surprising when regarding the fact that this region is heavily dominated by viscous effects

• Despite this deficiency in predicting steady loading the characteristic of the aeroelastic response is captured fairly well on the oscillating blade at axial bending mode with a secondary response peak being present on the fore pressure side

• The prediction of aeroelastic response at high negative incidence deteriorates considerably from an overall perspective compared to the situation at nominal conditions

• The fact that the aeroelastic response is very similar at all three investigated modes gives rise to the speculation that the unsteady flow field is dominated by a “super-mode” that is excited equally regardless of the blade oscillation mode. Indeed the unsteady simulations at this high incidence case featured unstable behavior that was described as spurious resonances in Mårtensson and Vogt (2005). These resonances established at all modes and emerged from a region close to the leading edge. It is believed that the instability is due to an underlying physical instability of the steady solution that gets unleashed when solving unsteady. As described in the aforementioned publication the solution instability could be remedied by using a lower order numerical approximation scheme.

At this position it shall be noted that the observed aeroelastic phenomena at high negative incidence might be classified as non-synchronous vibrations rather than classical flutter. Hall et al. (2004) refer to non-synchronous vibrations as being caused by a well-ordered flow instability, similar to e.g. rotating stall, having inherently distinct dynamic character. It might indeed be possible that the observed instability needed to be classified as non-synchronous vibration; however this conclusion can currently not be drawn from the performed linearized harmonic analyses.



9 Discussion Up to this point the aeroelastic response phenomenon in three-dimensional environment has been presented and its dependency on various design parameters has been discussed. Furthermore the three-dimensional effects during flutter have been identified and correlations to prediction results from different models have been assessed. Now the discussion shall be directed to the following two issues:

• Quantification of the three-dimensionality of the aeroelastic response and description of the effect on aeroelastic stability

• Prediction accuracy of the different models with regard to flutter stability Rather than based on unsteady response data the present discussion is carried out on the basis of resolved infinitesimal normal force components on a specific section. The theoretical background thereof has been elucidated above in section 3.2. The advantage of such consideration is that the contribution of the unsteady response to aeroelastic stability rather than the response itself is evidenced at each of the three orthogonal components. To assess the three-dimensionality of the aeroelastic response more intimately the infinitesimal force components are integrated within different arcwise fragments of the blade profile. The locations of these fragments yielded from the analysis of steady and unsteady blade loading properties and are to understand from a phenomenological point of view as follows

• Fragment “fore SS”: suction side from leading edge to downstream of peak suction (arc=-0.2)

• Fragment “aft SS”: rest of suction side • Fragment “fore PS”: pressure side from leading edge to averaged end of

separation bubble (arc=0.15) • Fragment “aft PS”; rest of pressure side

The fragmenting of the blade profile is sketched in Figure 9-1.

−0.01 0 0.01 0.02 0.03 0.04 0.05−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

−0.2

0

0.15

0.46

Figure 9-1. Fragmenting of profile into arcwise regions To address mode shape sensitivity of the cascade the normal force components are integrated for all three orthogonal modes to yield blade influence coefficients as

aft SS

fore SS

fore PS

aft PS SS suction side PS pressure side


outlined above and recombined to stability maps reflecting aeroelastic stability for a torsion mode representation of any 2D rigid-body mode at a given spanwise section. Such representation means that torsion, bending and intermediate modes are represented by a fictional center of torsion that is yielding from the (rigid-body) blade mode shape; due to the translational character of bending modes the respective center of torsion is located infinitely far away from the blade. The principle of torsion mode representation of various rigid-body modes is sketched in Figure 9-2.

Figure 9-2. Torsion mode representation of rigid-body modes The respective force for an arbitrary rigid-body mode is obtained from

[ ] hFhFT

iˆˆ= Eq. 9-1

Where [ ]F denotes the matrix containing the force components for all orthogonal

modes and all directions and ζηξ ahhh ˆ,ˆ,ˆˆ = a vector describing a 2D representation

of an arbitrary blade mode shape. The work matrix comprises the following elements

[ ]

=

ζζζηζξ

ηζηηηξ

ξζξηξξ

fff

fff

fff

F Eq. 9-2

where the first index refers to the mode causing the force and the second to the direction in which the force is acting. The diagonal terms thus represent forces done by the unsteady response on the mode causing the response. The off-diagonal terms reflect forces done by the response on the other modes. To determine aeroelastic stability the work per cycle is regarded, which is obtained from the product of force and motion as outlined above.

Fictional center of torsion mode A

Mode A

Fictional center of torsion mode B

Mode B


9.1 Quantification of Three-Dimensional Mechanisms during Flutter

The three-dimensional effects during flutter are quantified by means of partial force components that are obtained from integration over the aforementioned arcwise fragments. The resulting values thus include both effects of spanwise variations of the phase-related aeroelastic response as well as their local realization on the blade profile. To accentuate the three-dimensionality of the aeroelastic phenomenon the forces have been normalized by their respective local amplitude. The effect of varying amplitudes over span of the bending modes is thereby cancelled out. Furthermore the data are plotted with the spanwise position assigned to the ordinate. In case of complete absence of three-dimensional effects the distributions of force amplitude and phase were thus expected as constant over blade span, i.e. a straight vertical line.

−0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1 destabilizing

Im(dfξ) amp, −

norm

aliz

ed s

panw

ise

coor

dina

te, −

−1 0 1 2 30

0.2

0.4

0.6

0.8

1 destabilizing

Im(dfη) amp, −

norm

aliz

ed s

panw

ise

coor

dina

te, −

Axial bending Circumferential bending

−0.4 −0.2 0 0.2 0.4 0.6

0

0.2

0.4

0.6

0.8

1 stabilizing destabilizing

Im(dmζ) amp, −

norm

aliz

ed s

panw

ise

coor

dina

te, −

aft SSfore SSfore PSaft PStotal

Torsion

Figure 9-3. Spanwise distribution of stability contribution of blade -1; IBPA=0deg; low subsonic L1; k=0.1


Figure 9-3 depicts the contribution to total cascade stability of blade -1 at 0deg interblade phase angle or the three orthogonal modes. Distinct three-dimensional behavior is apparent at all modes in that the local contribution is relatively larger at the hub. At axial bending mode the distribution shows strong variations in the region close to the hub with the main contribution stemming from the aft suction side. While most part of the blade is acting destabilizing the tip section shows moderate stabilizing influence. Similar behavior is observed for the circumferential bending mode but here destabilizing influence is observed for the entire span. Again largest spanwise variations are apparent in the region close to the hub with the rest of the blade varying moderately. At this mode the fore suction side provides the major contribution to stability influence followed by the aft suction side. The entire pressure side features neutral stability behavior. At torsion mode the spanwise variation is characterized by a decreasing magnitude from hub to tip with mostly stabilizing character. At 30% span a pronounced dip is apparent that suggest local destabilizing behavior. This distribution is mainly impressed by the aft suction side. Fore suction and aft pressure side act stabilizing at about similar order of magnitude. Other than the two bending modes this mode features local spanwise increase in stability contribution that is observed to small extent on the fore pressure side and more distinctive on the aft pressure side; while the fore part of the pressure side acts destabilizing the aft part acts stabilizing. The arcwise resolved contributions to stability reveal fundamental differences in the different modes; whereas the major contribution at the axial bending mode originates from the aft suction side it shifts to the fore suction side for the circumferential bending mode. Two issues are believed responsible for this behavior; firstly the stability contribution of the respective modes have to be seen in the light of local blade geometry meaning that the exposure of the blade surface to the respective direction (axially for the axial bending mode, circumferentially for the circumferential mode) plays a major role. In that sense the exposure of the aft suction side is greatest for the axial bending mode while the exposures of fore and aft suction sides get about equally large at the circumferential bending mode. Note also that the twisted shape of the profile affects the exposure over span. Secondly the underlying physical mechanisms that have been discussed above must be taken into consideration. At axial bending mode the importance of the passage throat has been addressed, which affects the unsteady response mainly by pulsating the passage flow. The effect of this has mainly been recognized in level of the response peak on the fore suction side but is certainly also present on the aft suction side. By considering these two issues an explanation is given for the origin of the major contributions to aeroelastic stability. In a next step the variation of stability contribution with interblade phase angle is analyzed for the axial bending mode and blade -1 in Figure 9-4. Three interblade angles are depicted ranging from -90deg to +90deg. Whereas the spanwise total stability contribution features similar negative gradient with increase over span at 180deg as observed at 0deg it is apparent that at -90deg and +90deg the influence towards tip gets relatively larger. The contribution of the pressure side can thereby


largely be neglected. At +90deg interblade phase angle blade -1 acts destabilizing to overall cascade stability over the entire span with the major contribution originating from the aft suction side. The fore pressure side however changes character from stabilizing to destabilizing from hub to tip, which results in an increase in total stability of this blade over span. At -90deg the situation is similar but now with the blade acting stabilizing. Again the contribution on the fore suction side changes character with increase in span leading to increase in stability contribution in combination with the radially varying contribution from the aft suction side.

−2 −1 0 1 20

0.2

0.4

0.6

0.8


Im(dfξ) amp, −

norm

aliz

ed s

panw

ise

coor

dina

te, −

−0.6 −0.4 −0.2 0 0.2

0

0.2

0.4

0.6

0.8

1 stabilizing

Im(dfξ) amp, −

norm

aliz

ed s

panw

ise

coor

dina

te, −

IBPA=90deg IBPA=180deg

−2 −1 0 1 20

0.2

0.4

0.6

0.8


Im(dfξ) amp, −

norm

aliz

ed s

panw

ise

coor

dina

te, −


IBPA=-90deg

Figure 9-4. Variation of stability contribution of blade -1 with interblade phase angle; low subsonic L1; k=0.1; axial bending

The spanwise distributions of stability contribution reveal that distinct three-dimensional effects are present in the realization of aeroelastic response and the respective effect on flutter stability. A general behavior is observed in that the stability contribution is commonly larger at the hub than at the tip. It is believed that this is due to radial balancing of the instantaneous pressure field as well as due to the three-dimensional structure of the mean flow field featuring higher increasing flow velocities


towards the hub. A positive implication of this is that the tip section, which presumably has the largest effect of the incipience of flutter features minor relative contributions to the stability than the other sections of the blade. This would imply that two-dimensional analyses of the tip section alone were over-conservative, which is to judge positive from a design point of view. On the other hand it indicates that the use of three-dimensional methods was essential for refining aeroelastic design methods. However to conclude finally on this it would be necessary to consider spanwise resolved data on the oscillating blade itself such as to include the eigeninfluence.

−4 −3 −2 −1 0 10

0.2

0.4

0.6

0.8

1 stabilizing

Im(dfξ) amp, −

norm

aliz

ed s

panw

ise

coor

dina

te, −

−6 −4 −2 0 20

0.2

0.4

0.6

0.8

1 stabilizing

Im(dfξ) amp, −

norm

aliz

ed s

panw

ise

coor

dina

te, −

Nominal (L1) Off-design 1 (L2)

−6 −4 −2 0 20

0.2

0.4

0.6

0.8

1 stabilizing

Im(dfξ) amp, −

norm

aliz

ed s

panw

ise

coor

dina

te, −


Off-design 2 (L3)

Figure 9-5. Variation of stability contribution of blade +1; IBPA=90deg; low subsonic; k=0.1; axial bending

The influence of flow incidence on stability contribution is addressed next on blade +1 as this blade has shown major influence in steady and unsteady performance with change in incidence. Figure 9-5 depicts the resolved stability contributions at axial bending mode and 90deg interblade phase angle. The blade shows stabilizing character over the entire span with distinct increase towards the hub. It is noticeable that the contribution increases slightly between 70% and 90% span. From the


arcwise resolved distributions to judge it is apparent that the major contribution stems from the aft pressure side while the contribution from the fore pressure side increases with negative incidence. Additionally the contribution from the entire suction side decreases with negative incidence. Despite the relatively large increase of the fore suction side part no change in overall stability behavior is observed with increase in incidence. For the present mode this is largely due to the dominant effect of the aft pressure side that due to its exposure to axial direction overweighs the influences from the other parts of the blade. Finally the three-dimensional effects during flutter shall be quantified relatively to effects of reduced frequency and flow incidence. This is done by three-dimensional contour plots depicting the stability contribution data from blade pair ±1 over blade span, reduced frequency and flow incidence at a certain interblade phase angle as included in Figure 9-6. As the quantification is here made on a graphical basis the relative differences in contour level rather than the absolute values are discussed.

0

0.5

1

0.1

0.3

0.51

2

3

span, −k, −

OP

, −

1 2 30

0.5

1

1.5

Im(d

f ξ) am

p, −

OP, −0 0.5 1

0

0.5

1

1.5Im

(df ξ)

amp,

−

span, −

0.1 0.3 0.50

0.5

1

1.5

Im(d

f ξ) am

p, −

k, −

Figure 9-6. Graphical quantification of three-dimensional effects with respect to effects of reduced frequency and flow incidence

The contour plots contain a large amount of condensed information and are to be read as follows:

• The bottom plane shows the variation in stability contribution over span and reduced frequency at a certain operating point. Each dot represents a data point comprising stability contribution from blade +1 and -1. Note that data points inside the matrix are present but not depicted due to the sake of readability


• The height coordinate denotes various flow incidence operating points. Consequently the left-hand back plane shows the variations in span and incidence while the right-hand back plane shows the variations in reduced frequency and incidence

• Extraction of data along lines parallel to x-, y-, or z-axis would reveal variation of stability in blade span, reduced frequency or operating point respectively. These variations are included in Figure 9-6 crossing at s=0.5, k=0.2 and OP=1, i.e. nominal incidence

0

0.5

1

0.1

0.3

0.51

2

3

span, −

IBPA −90deg

k, −

OP

, −

0

0.5

1

0.1

0.3

0.51

2

3

span, −

IBPA 0deg

k, −O

P, −

0

0.5

1

0.1

0.3

0.51

2

3

span, −

IBPA 180deg

k, −

OP

, −

0

0.5

1

0.1

0.3

0.51

2

3

span, −

IBPA 90deg

k, −

OP

, −

Figure 9-7. Variation of stability contribution in span, reduced frequency and operating point for blade pair ±1; low subsonic; axial bending

Figure 9-7 shows the variation of stability contribution from blade pair ±1 at four interblade phase angles for the axial bending mode. By combining identical dots of the four plots the blade pair ±1 contribution to the S-curve at respective span, reduced frequency and flow incidence would be obtained. The following observations are made:

• The largest variations are apparent at high reduced frequencies and close to tip

• Variations over span and flow incidence are dependent on reduced frequency and are greatest around k=0.3

• Variations over span and thereby three-dimensional effects are of the same order of magnitude as variations in reduced frequency and flow incidence


In the same style the variation in stability contribution at torsion mode are included in Figure 9-8. Other than at axial bending mode the major variations are located around a band at midspan. Whereas moderate variations are noticed in reduced frequency and flow incidence the variations along blade span are notably large. This evidences that three-dimensional effects in aeroelastic stability contribution from the direct neighbors are of considerable magnitude compared to effects of reduced frequency and flow incidence.

0

0.5

1

0.1

0.3

0.51

2

3

span, −

IBPA −90deg

k, −

OP

, −

0

0.5

1

0.1

0.3

0.51

2

3

span, −

IBPA 0deg

k, −

OP

, −

0

0.5

1

0.1

0.3

0.51

2

3

span, −

IBPA 180deg

k, −

OP

, −

0

0.5

1

0.1

0.3

0.51

2

3

span, −

IBPA 90deg

k, −

OP

, −

Figure 9-8. Variation of stability contribution in span, reduced frequency and operating point for blade pair ±1; low subsonic; torsion


9.2 Assessment of CFD Prediction Accuracy of Aeroelastic Stability

The focus shall at this point be directed towards the prediction of the different models used and their applicability when predicting flutter stability. For this purpose the discussion is entirely made at midspan such as to also include the 2D model used. The discussion is firstly carried out on arcwise resolved stability data and then moved over to mode shape sensitivity of measured and predicted aeroelastic stability. A comparison of measured and predicted arcwise resolved stability contribution at midspan of blade -1 at axial bending mode and 90deg interblade phase angle is depicted in Figure 9-9. The figure includes the influences of the axial bending mode on all three orthogonal directions, which correspond to the diagonal component and the two corresponding off-diagonal components.

−0.4 −0.2 0 0.2 0.4

−0.05

0

0.05 destabilizing

stabilizingIm(d

f ξ), −

−0.4 −0.2 0 0.2 0.4

−0.05

0

0.05 destabilizing

stabilizingIm(d

f η), −

−0.4 −0.2 0 0.2 0.4

−5

0

5

x 10−4

destabilizing

stabilizingIm(d

mζ),

−


data3D tc3D2D

Figure 9-9. Comparison of measured and predicted arcwise stability contribution at midspan on blade -1; IBPA=90deg; operating point L1; axial bending

It is apparent that the two 3D models tend to give better agreement than the 2D model. Furthermore clear improvement in prediction accuracy is seen for all components when moving from the 2D model to the 3D without tip clearance and finally to the 3D model with tip clearance. The increased accuracy gets more apparent at the two off-diagonal components, i.e. the influence of the axial bending mode on circumferential and torsion direction respectively. It can however also be noted that the overall character of the stability contribution is predicted fairly well for all models.


Comparisons at the circumferential bending mode included in Figure 9-10 show superior prediction accuracy of the 3D model with tip clearance over the other two models used. Both the 3D model without tip clearance and the 2D model tend to heavily overpredict stability contribution, which is especially apparent in the torsion component leading to overpredicted destabilizing influence.

−0.4 −0.2 0 0.2 0.4−0.04

−0.02

0

0.02

0.04destabilizing

stabilizingIm(d

f ξ), −

−0.4 −0.2 0 0.2 0.4−0.04

−0.02

0

0.02

0.04destabilizing

stabilizingIm(d

f η), −

−0.4 −0.2 0 0.2 0.4−4

−2

0

2

x 10−4

destabilizing

stabilizingIm(d

mζ),

−


data3D tc3D2D

Figure 9-10. Comparison of measured and predicted arcwise stability contribution at midspan on blade -1; IBPA=90deg; operating point L1; circumferential bending

−0.4 −0.2 0 0.2 0.4

−0.05

0

0.05 destabilizing

stabilizingIm(d

f ξ), −

−0.4 −0.2 0 0.2 0.4

−0.05

0

0.05 destabilizing

stabilizingIm(d

f η), −

−0.4 −0.2 0 0.2 0.4

−5

0

5

x 10−4

destabilizing

stabilizingIm(d

mζ),

−


data3D tc3D2D

Figure 9-11. Comparison of measured and predicted arcwise stability contribution at midspan on blade -1; IBPA=90deg; operating point L1; torsion


Finally the comparisons at torsion mode are included in Figure 9-11. Similar behavior is observed for all components in that the 3D model without tip clearance and the 2D model tend to overpredict stability contribution, especially the destabilizing behavior on the suction side of the blade. From an overall perspective it can be concluded that even the simplest of the models, i.e. the 2D model, is capable of capturing the general character of the stability contribution. The magnitude is however only captured correct when employing the 3D model with tip clearance. A more complete picture is obtained from comparisons of mode shape stability in a two-dimensional torsion mode representation of any possible mode. As described above such consideration yields contour plots of the stability behavior at any possible center of torsion and are commonly referred to as stability plots. In the present case only the character of aeroelastic stability contributions is addressed by shading stable regions. Modes that feature the center of torsion outside these regions are consequently unstable. Note that these stability plots only contain information of the stability of the aerodynamic system and do not describe the overall stability of the setup meaning that even if destabilizing influence is present for a certain mode, it does not necessarily imply that the cascade will flutter at this mode.

−3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3

ax* COT, −

pitc

h* CO

T, −

−3 −2 −1 0 1 2 3 4

−3

−2

−1

0

1

2

3

ax* COT, −

pitc

h* CO

T, −

Data CFD 2D

−3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3

ax* COT, −

pitc

h* CO

T, −

−3 −2 −1 0 1 2 3 4

−3

−2

−1

0

1

2

3

ax* COT, −

pitc

h* CO

T, −

CFD 3D CFD 3D with tip clearance

Figure 9-12. Comparison of measured and predicted stability plot at midspan; IBPA=0deg; shaded areas mark stable regions

Maximum stability line

stable

unstable

Coalescence point


A comparison of measured and predicted stability plots is contained in Figure 9-12 for the nominal low subsonic operating point at reduced frequency k=0.1. On a first view the stability maps agree qualitatively well although the stable regions are predicted differently. The stable region is cohesive and broadens away from the reference blade, which gives it a characteristic X-shape with the narrowest point being located outside of the pressure side. This point, which is also referred to as coalescence point, is marking the highest susceptibility of the aeroelastic response to changes in mode shape while a line of maximum stability can be drawn in the center of the stability region as indicated in the figure. Test data suggest that the maximum stability line is roughly aligned perpendicular to the passage throat. This again confirms the interpretation of that the passage throat plays a dominant role in the aeroelastic response; modes that affect the passage throat a lot induce a high aeroelastic response in the cascade, which under respect of force realization and phase relation add in a stabilizing manner at this interblade phase angle. This finding is confirmed by results from Tchernycheva et al. (2001) who found from analyses of different turbine geometries that the maximum stability line is approximately aligned perpendicular to passage throat regardless the blade geometry. The present data thereby go along with the conclusion drawn by Kielb et al. (2003) in that standard maps for turbine mode shape represent a rough but valid preliminary design tool. Differences are however apparent in the orientation of the stability regions from the different models; whereas the 2D model suggests the maximum stability line approximately aligned with the circumferential direction the 3D models predict similar behavior to test data with line being aligned perpendicular to passage throat. It is believed that this discrepancy is mainly due failure of the 2D model in correctly predicting the phase on the oscillating blade at circumferential bending mode, which yields in a more stabilizing contribution. Stability plots at 90deg interblade phase angle are included in Figure 9-13 with test data suggesting almost uniform stable behavior apart from a local unstable region for torsion dominated modes. The predictions differ largely in that both the 2D model and the 3D model without tip clearance suggest unstable regions aligned with the circumferential direction of the cascade. Better agreement to test data is found from the 3D model with tip clearance, which as the only model predicts instability of the torsion dominated modes. However even this model differs from test data in that the unstable region is stretched towards intermediate torsion-bending modes with flex character. By zooming out of the close-field domain it has been recognized that this unstable region terminates similar to test data but after several chords distance.


−3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3

ax* COT, −

pitc

h* CO

T, −

−3 −2 −1 0 1 2 3 4

−3

−2

−1

0

1

2

3

ax* COT, −

pitc

h* CO

T, −

Data CFD 2D

−3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3

ax* COT, −

pitc

h* CO

T, −

−3 −2 −1 0 1 2 3 4

−3

−2

−1

0

1

2

3

ax* COT, −

pitc

h* CO

T, −

CFD 3D CFD 3D with tip clearance

Figure 9-13. Comparison of measured and predicted stability plot at midspan; IBPA=90deg; shaded areas mark stable regions


10 Summary

10.1 Conclusions

Three-dimensional effects in low-pressure turbine flutter have been studied experimentally and numerically in a single blade row setup. For this purpose a new test facility has been build that allowed controlled investigation of aeroelastic properties during flutter in three-dimensional environment. The facility comprised an annular sector cascade of seven free-standing low-pressure turbine blades one of which could be made oscillating in various rigid-body modes while the unsteady aerodynamic response has been measured on various blades at different radial heights yielding spanwise resolved aerodynamic influence coefficients. The flutter phenomenon has been investigated at flow conditions ranging from low subsonic outlet Mach number (M2=0.37) over medium subsonic (M2=0.62) to high subsonic (M2=0.71) and three different inflow incidence angles (0deg, -20deg and -37deg). Three orthogonal modes were tested at variable reduced frequency as were axial bending, circumferential bending and torsion around profile stacking line. The two bending modes featured spanwise variation of local bending amplitude that increased linearly from hub to tip such as to resemble low order modes in real engines. The maximum achieved reduced frequencies varied with flow velocity level and amounted to k=0.5 for the low subsonic, k=0.4 for the medium subsonic and k=0.3 for the high subsonic case respectively. The test setup has been validated at the different flow conditions with regard to relevance for turbomachine flow by means of blade loading and probe traverse measurements as well as surface flow visualization. It has been shown that the flow featured the required three-dimensional characteristics and that satisfying steady and unsteady flow periodicity was achieved. Furthermore the blade oscillation has been characterized and found to be highly sinusoidal. On the numerical side an industrial linearized Euler flutter prediction tool (VOLSOL) has been used. To predict the unsteady aeroelastic response during flutter two different models have been used, one describing the blade row oscillating in traveling wave mode and the other with only one blade oscillating such as to reflect the experimental setup. Back-to-back comparisons of the two models have shown that the differences are negligible leading to the conclusion that the employed methodology was valid from a numerical point of view. Furthermore it has been shown that the numerical method converges with increased mesh density and that the influence of the numerical approximation scheme was of irrelevant magnitude. To recognize the impact of three-dimensional effects on prediction accuracy of aeroelastic stability three different numerical models have been used that differed in degree of complexity; the simplest model comprised a 2D plane representation of the cascade at midspan whereas two more advanced 3D models have been employed one of which modeled tip clearance as present during testing.


The investigation of the flutter phenomenon at midspan leads to the following conclusions:

• The aeroelastic response is largest on the oscillating blade and its immediate neighbors and decays rapidly further away

• The arcwise distribution of response magnitude has similar character for all three investigated modes

• At axial bending and torsion mode dominant involvement of the passage flow embracing the oscillating blade is observed leading to distinct in-phase behavior of opposing surfaces and comparative order of response magnitude between modes. The instantaneous passage throat size has thereby been identified as key driver

• The response magnitude at circumferential bending mode is lowest and little involvement of the passage flow is observed, which is believed being due to least influence on passage throat at this mode

• Increase in reduced frequency mainly leads to increase in response magnitude but also to phase veering especially on the fore part of the blade

• Moderate influence of flow velocity level on unsteady response is observed. Major effects are observed on the aft part of the blade suggesting moderate increase in response magnitude with increasing flow velocity while the fore part of the blade upstream of passage throat seems largely unaffected

• Separated flow due to change in incidence leads to considerable change in response phase compared to attached flow. Negligible influence is observed downstream of the reattachment point

• Change in incidence leads to considerable change in response magnitude in attached flow areas. It is believed that this is a compound effect of change in blade loading as well as change in secondary flow structure upon unloading

In the light thereof the three-dimensional effects during flutter have been assessed on the blades adjacent to the oscillating blade and are summarized as follows:

• The most prominent type of three-dimensional effect features increasing response magnitude from hub to tip and is observed at the two bending modes. It is believed that the spanwise variations are due to the respective blade mode shape

• A second type of three-dimensional effect features locally varying response magnitude and is mainly observed at torsion mode. It is believed that the local variations are caused by three-dimensional effects of the mean flow field. Among those the spanwise variation in blade loading and local effects close to hub and casing endwalls are seen as most prominent

• Whereas the first type leads to considerable increase in force realization towards hub when normalized to its respective plane rigid-body motion the second type leads to distinct variations in local stability behavior

• At axial bending mode the major aeroelastic stability contribution stems from the aft suction side while it is shifted to the fore suction side at circumferential bending mode. This is seen as direct manifestation of the influence of passage throat, which is more dominant at axial bending

• At torsion mode the aft part of the blade dominates the contribution to stability while considerable variations are observed over blade span


Correlations of CFD results to test data lead to the following conclusions:

• The overall character of the aeroelastic response magnitude is captured fairly well regardless the model used while the agreement in phase is not as good

• Whereas the two-dimensional model features distinct local discrepancies considerable improvement is achieved when employing one of the three-dimensional models

• The two-dimensional and the three-dimensional model without tip clearance generally tend to overpredict response magnitude although not uniformly in the cascade. Depending on mode and operating point considerable differences are observed in response phase

• Prediction accuracy is greatly improved when including tip clearance in the three-dimensional model. It is believed that the tip leakage flow globally affects aeroelastic response in that the magnitude generally is lowered and the phase is veered locally. The latter observation suggests that the flow features induced by the tip leakage flow feature separate dynamic behavior than the undisturbed main flow. As the used model was inviscid it can be concluded that this effect is driven by pressure difference rather than viscosity

• In a two-dimensional consideration at midspan aeroelastic stability with regard to mode shape sensitivity can be predicted highly accurate when using the three-dimensional model with tip clearance. The general characteristic of mode shape sensitivity is captured when modeling in 2D however major differences exist. To fully conclude on the ability of the tip clearance model to predict flutter data on blade 0 at different spanwise heights were necessary

• From an application point of view it can be concluded that the two-dimensional model allows for overnight assessment of the aeroelastic stability of the investigated blade row at one operating point. Such assessment was based on a standard set of traveling wave mode simulations comprising three modes, five reduced frequency and 20 interblade phase angles. The obtained accuracy is judged fair enough for a preliminary design keeping in mind that the method is over-conservative in two aspects; firstly the treatment in traveling wave mode assumes a tuned system, which represents the worst situation form a stability point of view. In reality manufacturing tolerances as well as individual wear of components lead to mistuning of the system that generally has a stabilizing effect. This is valid both for two-dimensional and three-dimensional cases. The second aspect is found in the spanwise variation of stability contribution that tends to decrease from hub to tip. From an aeroelastic design point of view the stability analysis would be carried out on the section with the largest amplitude, which for low order modes and unshrouded blades is the tip section. Results from a two-dimensional simulation therefore can be expected to overpredict the response compared to the situation in an annular blade row


10.2 Recommendations and Future Work

Acquisition of spanwise resolved aeroelastic response data on oscillating

To date spanwise resolved aeroelastic response data have only been acquired on the adjacent blades due temporal and budgetary limitations. The acquisition of response data on the oscillating blade itself at different spanwise positions would however be necessary to conclude on the effect of three-dimensional mechanisms during flutter on overall local stability. This step would add on the existing set of data and is therefore recommended as logical continuation of the present work. Extension of test matrix towards transonic flow conditions

Present results suggest little influence of flow velocity on aeroelastic response in the investigated range that ranged from low to high subsonic and it would therefore be of greatest interest to determine up to which velocity level this behavior is preserved. At transonic flow conditions the cascade would be operated chocked at passage throat, which then had a major effect on the so-called passage-dominated modes (axial, torsion). With respect to the maximum allowable mass flow rate extension towards higher flow velocities might practically be possible by reducing the circumferential extent of the cascade. Variation in tip clearance

Based on the comparisons from 3D CFD results using models with and without tip clearance it has been recognized that the modeling of tip clearance is essential for correctly capturing the aeroelastic response during flutter. It is believed that the tip leakage flow induces secondary flow structures that feature different dynamics than the undisturbed main flow and thus are capable of affecting the aeroelastic response considerably. It is therefore proposed to systematically address the influence of varying tip clearance in the present three-dimensional setup. On one hand tests should be run with sealed tip (a flexible rubber sealing vulcanized onto blade tip is suggested) to provide correlation data for the 3D model without tip clearance. The amount of tip leakage flow should then be varied stepwise around the present nominal tip clearance of 1% within the kinematical possible range (i.e. guarantee touch-free oscillation of blade).


Testing of real deforming modes

The present modes were run with the blade oscillating as rigid body while in real engines the blades are deforming elastically. Considering the basic low order blade modes (flex, edgewise, torsion) spanwise amplitude variation is in reality present for all three modes but could so far only be achieved for the two bending modes. Real deforming modes could be achieved by choosing an appropriate material such as to allow excitation of low order modes at the present frequencies. The necessary material properties were thereby to be determined from the proportionality of

ρEf prop. . Practically it is suggested to use blades made of polyurethane as the

elasticity of this material can be varied within a large range. To achieve specific modes the deforming blades could be equipped with integrated structures of greater stiffness such as steel springs as depicted in Figure 10-1. Testing of locally deforming modes

In real engines locally deforming modes exist such as local corner modes or stripe modes that lead to a deformation of profile shape. Such modes feature distinct three-dimensionality that might interact with the three-dimensional features of the mean field in a non-obvious way. The proposed objective would be to investigate the three-dimensional effects at locally deforming modes such as to get a more complete picture of the mechanisms that are underlying the global aeroelastic response and to address the prediction accuracy of CFD various models. Similar to the proposed real deforming modes the blades could be manufactured from flexible material with integrated stiff structures that are engineered to achieve the required modes. A proposition for the test object is included in Figure 10-1.

Real deforming low order modes Locally deforming modes

Figure 10-1. Proposition for real and locally deforming test objects

Stiff core including entities for connecting blade to actuator

Integrated springs to engineer stiffness


Numerical simulations using viscous models

The present simulations were performed using inviscid models but it is known that the flow in turbomachine blade rows is largely affected by viscous effects especially towards the endwalls. Comparisons of measured and predicted mean flow field using inviscid models have shown that the main secondary flow features (mostly vortices) are captured although their exact strength and extent does not agree to test data due to the fact that the deployment is largely affected by viscous effects. The influence of the physical mechanisms modeled is apparent from inviscid and viscous steady-state predictions included in Figure 10-2. The recommended investigation is proposed as logical continuation of the available results such as to quantify viscous effects during flutter in the light of the already addressed effects of model dimension, resolution and detailing.

3D inviscid with tip clearance 3D viscous with tip clearance

Figure 10-2. Predicted steady-state total pressure distribution from inviscid and viscous models

Investigation of observed numerical instabilities

Especially at low flow velocities numerical instabilities have been observed in the unsteady simulations that lead to some sort of resonance behavior. The instabilities were mainly observed at off-design conditions as well as when using the model with tip clearance which gives rise to the assumption that flow structures that are naturally unstable but are converged to a pseudo-stable steady solution are the underlying cause. The problem has been reported in Mårtensson and Vogt (2005) and the use of lower order numerical approximation schemes has been demonstrated as partial remedy for theses instabilities. As the problem is not solved completely it is recommended to systematically investigate the cause for these instabilities. The instabilities that are believed being due to the Kármán vortex street downstream of the profile might be avoided by sharpening the trailing edge of the profile. To suppress other instabilities that presumably are due to vortex structures (tip clearance model, off-design operation) the only possibility might be to use viscous models.


11 References Bellenot, C., Lalive d’Epinay, J., 1950 “Selbsterregte Schaufelschwingungen” Brown Boveri Mitteilungen, October 1950, pp.368-376 Bendiksen, O.O., Friedmann, P.P., 1982 "The Effect of Bending-Torsion Coupling on Fan and Compressor Blade Flutter" ASME J. of Engineering for Gas Turbines and Power, Vol. 104, 1982, pp.617-623 Bölcs, A., Fransson, T.H., 1986 “Aeroelasticity in Turbomachines – Comparison of Theoretical and Experimental Results” Communication du Laboratoire de Thermique Appliqué et de Turbomachines, No. 13, EPFL, Lausanne, Switzerland, 1986 Bell, D.L., He, L., 2000 “Three-Dimensional Unsteady Flow for an Oscillating Turbine Blade and the Influence of Tip Leakage” ASME J. of Turbomachinery, Vol. 122, 2000, pp.93-101 Buffum, D.H., Fleeter, S., 1990 “Oscillating Cascade Aerodynamics by an Experimental Influence Coefficient Technique” AIAA J. of Propulsion, Vol. 6, 1990, pp.612-620 Buffum, D.H., Fleeter, S., 1991 “Wind Tunnel Wall Effects in a Linear Oscillating Cascade” ASME J. of Turbomachinery, Vol. 115, 1991, pp.147-156 Buffum, D.H., Fleeter, S., 1994 “Effect of Wind Tunnel Acoustic Modes on Linear Oscillating Cascade Aerodynamics” ASME J. of Turbomachinery, Vol. 116, 1994, pp.513-524 Buffum, D.H., Capece, V.R., King, A.J., El-Aini, Y.M., 1998 “Oscillating Cascade Aerodynamics at Large Mean Incidence” ASME J. of Turbomachinery, Vol. 120, 1998, pp.122-130 Carta, F.O., St.Hilaire, A.O., 1980 “Effect of Interblade Phase Angle and Incidence Angle on Cascade Pitching Stability” ASME J. of Engineering for Gas Turbines and Power, Vol. 102, 1980, pp.391-396 Carta, F.O., 1983 “Unsteady Gapwise Periodicity of Oscillating Cascaded Airfoils” ASME J. of Engineering for Gas Turbines and Power, Vol. 105, 1983, pp.565-574


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Ewins, D.J., 1988 “Structural Dynamic Characteristics of Bladed Assemblies” AGARD Manual on Aeroelasticity in Axial-Flow Turbomachines, Vol. 2, Structural Dynamics and Aeroelasticity, Chapter 15, AGARD-AG-298 Försching, 1991 "Flutter Stability of Annular Wings in Incompressible Flow" J. of Fluids and Structures, Vol. 5, 1991, pp. 47-67 Fransson, T.H., Pandolfi, M., 1986 ”Numerical Investigation of Unsteady Subsonic Compressible Flow through an Oscillating Cascade” ASME Paper 86-GT-304 Fransson, T.H., 1999 “Aeroelasticity in Turbomachines” VKI lecture series, May 1999, Rhode-Saint-Genèse, Belgium Fransson, T.H., Vogt, D.M., 2003 “A New Facility for Investigating Flutter in Axial Flow Turbomachines” Paper presented at the 8th National Turbine Engine High Cycle Fatigue (HCF) Conference, Monterey, California, USA, 2003 Frey, K.K., Fleeter, S., 1999 “Combined-Simultaneous Gust and Oscillating Compressor Blade Aerodynamics” ASME Paper 99-GT-414 Giles, M.B., 1988 “Calculation of Unsteady Wake/Rotor Interaction” AIAA J. of Propulsion, Vol. 4, 1988, pp.356-362 Greitzer, E.M., Tan, C.S., Wisler, D.C., Adamczyk, J.J., Strazisar, A.J., 1994 “Unsteady Flow in Turbomachines: Where’s the Beef?” Paper presented at the 1994 ASME Winter Meeting, Chicago, Illinois, USA Groth, P., Mårtensson, H., Eriksson, L.E., 1996 “Validation of a 4D Finite Volume Method for Blade Flutter” ASME Paper 96-GT-429 Hall, K.C., Crawley, E.F., 1989 “Calculation of Unsteady Flows in Turbomachinery Using the Linearized Euler Equations” AIAA J. of Propulsion, Vol. 27, 1989, pp.777-787 Hall, K.C., Ekici, K., Voytovych, D.M., 2003 “Multistage Coupling for Unsteady Flows in Turbomachinery” Proceedings 10th International Symposium on Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines (ISUAAAT), Durham, North Carolina, USA, 2003


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Appendix I: Blade Profile Description

Profile Denotations

Leading edge

Trailing edge

Flow direction

Suction side

Pressure side

Arcwise direction (pressure side)

Arcwise direction (suction side)

Spanwise direction Blade tip

Hub


Profile Data

Hub Section span arc x, mm y, mm z, mm 0.000 -0.540 382.223 25.576 46.324 0.000 -0.530 382.300 24.391 46.501 0.000 -0.473 382.662 17.804 42.973 0.000 -0.388 382.983 8.427 37.263 0.000 -0.321 383.075 1.273 32.009 0.000 -0.264 383.055 -4.103 27.031 0.000 -0.203 382.973 -8.922 20.585 0.000 -0.141 382.913 -11.210 12.909 0.000 -0.084 382.963 -9.325 5.872 0.000 -0.038 383.041 -5.208 1.574 0.000 -0.009 383.073 -1.764 -0.182 0.000 0.005 383.077 0.236 0.231 0.000 0.020 383.076 0.549 1.997 0.000 0.054 383.077 0.081 6.082 0.000 0.105 383.076 0.366 12.394 0.000 0.167 383.070 2.320 19.720 0.000 0.231 383.031 5.954 26.743 0.000 0.290 382.937 10.372 32.544 0.000 0.337 382.809 14.354 36.658 0.000 0.378 382.647 18.146 39.981 0.000 0.421 382.423 22.388 43.216 0.000 0.450 382.237 25.363 45.256 0.000 0.459 382.218 25.653 46.173 Midspan span arc x, mm y, mm z, mm 0.500 -0.540 430.563 28.514 44.075 0.500 -0.533 430.606 27.859 44.479 0.500 -0.489 430.904 22.957 41.913 0.500 -0.409 431.257 14.438 36.913 0.500 -0.337 431.436 6.892 31.885 0.500 -0.281 431.483 1.409 27.633 0.500 -0.219 431.467 -4.166 22.287 0.500 -0.155 431.417 -8.637 15.814 0.500 -0.096 431.412 -9.996 8.678 0.500 -0.047 431.479 -7.552 3.172 0.500 -0.014 431.485 -4.505 0.407 0.500 0.000 431.485 -2.857 -0.072 0.500 0.012 431.489 -1.824 1.206 0.500 0.038 431.531 -1.902 4.462 0.500 0.086 431.532 -1.130 10.277 0.500 0.146 431.524 1.466 17.210 0.500 0.211 431.482 5.766 23.959 0.500 0.273 431.379 10.830 29.693 0.500 0.324 431.233 15.499 33.987 0.500 0.367 431.065 19.560 37.239 0.500 0.413 430.824 24.237 40.589 0.500 0.448 430.604 27.853 42.963 0.500 0.459 430.560 28.558 43.936


Tip Section span arc x, mm y, mm z, mm 1.000 -0.540 478.898 31.354 41.916 1.000 -0.534 478.938 30.729 42.196 1.000 -0.492 479.201 26.317 39.810 1.000 -0.416 479.565 18.534 35.246 1.000 -0.345 479.788 11.396 30.670 1.000 -0.288 479.887 5.879 26.733 1.000 -0.224 479.923 0.016 21.899 1.000 -0.159 479.894 -5.307 16.252 1.000 -0.098 479.844 -8.751 9.964 1.000 -0.049 479.845 -8.691 4.139 1.000 -0.016 479.877 -6.698 0.765 1.000 -0.002 479.893 -5.347 -0.004 1.000 0.008 479.904 -4.269 0.936 1.000 0.034 479.907 -3.952 4.118 1.000 0.080 479.913 -2.387 9.667 1.000 0.139 479.909 1.191 16.128 1.000 0.203 479.862 6.328 22.341 1.000 0.266 479.753 12.005 27.666 1.000 0.318 479.598 17.125 31.746 1.000 0.362 479.421 21.638 34.974 1.000 0.412 479.168 26.811 38.382 1.000 0.449 478.940 30.716 40.841 1.000 0.459 478.895 31.405 41.795

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