Sensitivity of Aeroelastic Properties of an Oscillating ...626791/FULLTEXT01.pdf · Sensitivity of Aeroelastic Properties of an Oscillating LPT ... aeroelastic response, combined

Sensitivity of Aeroelastic Properties of an Oscillating LPT Cascade

Nenad Glodic

Licentiate Thesis 2013

Department of Energy Technology Division of Heat and Power Technology

Royal Institute of Technology 100 44 Stockholm, Sweden

TRITA KRV Report 13/05 ISSN 1100/7990 ISRN KTH/KRV/13/05-SE ISBN 978-91-7501-809-6 © 2013 Nenad Glodic

Licentiate Thesis / Nenad Glodic Page 1

ABSTRACT

Modern turbomachinery design is characterized by a tendency towards thinner, lighter and highly loaded blades, which in turn gives rise to increased sensitivity to flow induced vibration such as flutter. Flutter is a self-excited and self-sustained instability phenomenon that may lead to structural failure due to High Cycle Fatigue (HCF) or material overload. In order to be able to predict potential flutter situations, it is necessary to accurately assess the unsteady aerodynamics during flutter and to understand the physics behind its driving mechanisms. Current numerical tools used for predicting unsteady aerodynamics of vibrating turbomachinery components are capable of modeling the flow field at high level of detail, but may fail in predicting the correct unsteady aerodynamics under certain conditions. Continuous validation of numerical models against experimental data therefore plays significant role in improving the prediction accuracy and reliability of the models. In flutter investigations, it is common to consider aerodynamically symmetric (tuned) setups. Due to manufacturing tolerances, assembly inaccuracies as well as in-service wear, the aerodynamic properties in a blade row may become asymmetric. Such asymmetries can be observed both in terms of steady as well as unsteady aerodynamic properties, and it is of great interest to understand the effects this may have on the aeroelastic stability of the system. Under certain conditions vibratory modes of realistic blade profiles tend to be coupled i.e. the contents of a given mode of vibration include displacements perpendicular and parallel to the chord as well as torsion of the profile. Current design trends for compressor blades that are resulting in low aspect ratio blades potentially reduce the frequency spacing between certain modes (i.e. 2F & 1T). Combined modes are also likely to occur in case of the vibration of a bladed disk with a comparatively soft disk and rigid blades or due to tying blades together in sectors (e.g. in turbines). The present investigation focuses on two areas that are of importance for improving the understanding of aeroelastic behavior of oscillating blade rows. Firstly, aeroelastic properties of combined mode shapes in an oscillating Low Pressure Turbine (LPT) cascade were studied and validity of the mode superposition principle was assessed. Secondly, the effects of aerodynamic mistuning on the aeroelastic properties of the cascade were addressed. The aerodynamic mistuning considered here is caused by blade-to-blade stagger angle variations The work has been carried out as compound experimental and numerical investigation, where numerical results are validated against test data. On the experimental side a test facility comprising an annular sector of seven free-standing LPT blades is used. The aeroelastic response phenomena were studied in the influence coefficient domain where one of the blades is made to oscillate in three-dimensional pure or combined modes, while the unsteady blade surface pressure is acquired on the oscillating blade itself and on the non-oscillating neighbor blades. On the numerical side, a series of numerical simulations were carried out using a commercial CFD code on a full-scale time-marching 3D

Page 2 Licentiate Thesis / Nenad Glodic

viscous model. In accordance with the experimental part the simulations are performed using the influence coefficient approach, with only one blade oscillating. The results of combined modes studies suggest the validity of combining the aeroelastic properties of two modes over the investigated range of operating parameters. Quality parameters, indicating differences in mean absolute and imaginary values of the unsteady response between combined mode data and superposed data, feature values that are well below measurement accuracy of the setup. The findings of aerodynamic mistuning investigations indicate that the effect of de-staggering a single blade on steady aerodynamics in the cascade seem to be predominantly an effect of the change in passage throat. The changes in steady aerodynamics are thereby observed on the unsteady aerodynamics where distinctive effects on flow velocity lead to changes in the local unsteady pressure coefficients. In order to assess the overall aeroelastic stability of a randomly mistuned blade row, a Reduced Order Model (ROM) model is introduced, allowing for probabilistic analyses. From the analyses, an effect of destabilization due to aero-asymmetries was observed. However the observed effect was of moderate magnitude. Keywords: flutter, aeroelastic response, combined modes, bending-torsion flutter, mode superposition, aerodynamic asymmetries, ROM, probabilistic analysis


SAMMANFATTNING

Modern design av turbomaskiner karakteriseras av en tendens till tunnare, lättare och högbelastade skovlar, som i sin tur är känsligare för flödesinducerade vibrationer så som fladder. Fladder är ett självexciterat och självbärande instabilitetsfenomen som kan leda till ett strukturellt haveri p.g.a. utmattning eller överbelastning. För att kunna prediktera potentiella fladdersituationer är det nödvändigt att noggrant studera den instationära aerodynamiken under fladder och att försöka förstå fysiken bakom dess drivande mekanismer. Aktuella numeriska verktyg som används för prediktering av den instationära aerodynamiken orsakad av de vibrerande turbinkomponenterna är kapabla att modellera strömningsfältet på en hög detaljnivå. Kontinuerlig validering av numeriska modeller mot experimentell data spelar en viktig roll för att förbättra predikteringsnoggrannhet av modeller. I fladderundersökningar, är det vanligt att anta ett aerodynamiskt symmetriskt system. På grund av tillverkningstoleranser, onoggrannhet vid montering samt slitage, kan de aerodynamiska egenskaperna i en skovelrad bli asymmetriska. Sådana asymmetrier kan observeras i både stationära samt instationära aerodynamiska egenskaper. Det är av stort intresse att förstå vilka effekter detta kan få på systemets aeroelastiska stabilitet. Vibrationsmoder av realistiska skovelprofiler är nästan alltid kopplade d.v.s. innehållet i en given vibrationsmod inkluderar förskjutningar parallella och vinkelräta mot kordan samt vridning av profilen. Nuvarande designtrender för kompressorskovlar resulterar i låga spann -korda förhållanden som potentiellt minskar frekvensskillnaden mellan vissa moder (t.ex.. 2F & 1T). Kombinerade vibrationsmoder kan sannolikt inträffa vid vibration av en rotorskiva med en jämförelsevis mjuk skiva och styva skovlar eller på grund av i sektorer kopplade skovlar (t.ex. i turbiner). Den här undersökningen fokuserar på två områden som är av betydelse för att förbättra förståelsen av de aeroelastiska egenskaperna i oscillerande skovelgitter. För det första undersöks de aeroelastiska egenskaperna av kombinerade moder i ett oscillerande lågtrycksturbinsgitter och giltighet av superpositionsprincipen av moder bedöms. Vidare undersöks vilka effekter de aerodynamiska asymmetrierna har på de aeroelastiska egenskaperna i gittret. De aerodynamiska asymmetrierna anses här orsakade av profilvinkelvariationer mellan individuella skovlar. Arbetet omfattar både experimentella undersökningar och numeriska simulationer, där de numeriska resultaten har validerats mot provdata. På den experimentella sidan används en provanläggning som innefattar en annulär sektor av sju fristående lågtrycksturbinskovlar. Det aeroelastiska gensvaret i sektorgittret studeras med hjälp av influenskoefficienter, där en av skovlarna oscilleras i tredimensionella, rena eller kombinerade moder medan det instationära trycket på skovelytan uppmäts både på den oscillerande skoveln och på de icke-oscillerande angränsande skovlarna. På den numeriska sidan, genomförs en serie av simuleringar med en kommersiell CFD-kod som använder en fullskalig 3D viskös modell. I enlighet med de experimentella proven görs också simuleringar i influenskoefficientdomänen, där endast en av skovlarna oscilleras.


Resultaten av undersökningarna av kombinerade moder tyder på att för det undersökta området av driftparametrar är det giltigt att kombinera aeroelastiska egenskaper hos två oscillationsmoder. Kvalitetsparametrar som indikerar skillnader i de absoluta och imaginära värdena på den instationära aerodynamiska responsen mellan de uppmätta värdena av kombinerade moder och de superponerade rena moderna har värden som är betydligt lägre än mätnoggrannhet i systemet. Resultaten av aerodynamiska mistuning-undersökningar tyder på att påverkan av ändringar i skovelns profilvinkel på den stationära aerodynamiken i kaskaden verkar främst vara en effekt av areaförändringen i skovelpassagens trängsta sektion. Förändringarna observerades för den instationära aerodynamiken där särskiljande effekter av flödeshastighet leder till förändringar i de lokala instationära tryckkoefficienterna. För att bedöma den totala aeroelastiska stabiliteten hos en slumpmässigt aerodynamiskt mistuned skovelrad, introducerades en modell med reducerade frihetsgrader (Reduced Order Model - ROM), vilket möjliggjort probabilistiska analyser. Från analyserna har en påverkan av destabilisering på grund av aero-asymmetrier observerats men den observerade effekten är av måttlig storlek. Nyckelord: fladder, aeroelastiskt gensvar, kombinerade oscillationsmoder, superposition, aerodynamiska asymmetrier, ROM, probabilistisk analys


PREFACE

The thesis is based on the following papers: 1 Glodic, N., Bartelt, M., Vogt, D. M., Fransson, T.H., 2009 „Aeroelastic Properties of Combined Mode Shapes in an Oscillating LPT

Cascade” Paper presented at ISUAAAT2009 2 Glodic, N., Vogt, D. M., Fransson, T.H., 2011

“Experimental and Numerical Investigation of Mistuned Aerodynamic Influence Coefficients in an Oscillating LPT Cascade” Paper presented at ASME Turbo Expo 2011

The involvement of Dr. Damian Vogt and Prof. Torsten Fransson in the above publications consisted of problem formulation and discussion of results. The contribution of Mr. Michael Bartelt in Paper 1 consisted of supervised numerical computations and experimental testing during his M. Sc. Thesis work. For all publications the underlying material was part of the work elaborated in this thesis.



ACKNOWLEDGMENTS

The present study has been funded by the Swedish Energy Agency, Siemens Industrial Turbomachinery AB, GKN Aerospace (former Volvo Aero Corporation), and the Royal Institute of Technology through the Swedish research program TURBOPOWER, the support of which is gratefully acknowledged. I would like to express my gratitude to Prof. Torsten Fransson at the Chair of Heat and Power Technology at KTH, for giving me opportunity to perform this work at the department and for sharing his enthusiasm regarding aeroelasticity research. Endless thanks go to my supervisor Assoc. Prof. Damian Vogt at KTH, for his enthusiastic and valuable support without which this study would not be possible. To the Turbovib project members from GKN Aerospace (former Volvo Aero), Siemens Industrial Turbomachinery AB, Department of Solid Mechanics and Marcus Wallenberg Laboratory for Sound and Vibration Research at KTH, for their support and productive discussions. It was a pleasure to be a part of such a highly competent team. Special thanks to my colleague Jens Fridh for introducing me to the exciting world of experimental aerodynamics. I would like to thank my colleagues from the Aeromech group and others at the HPT Division for fruitful discussions, mutual motivation and some excellent fishing tours. I would also like to express my gratitude to the technicians in the lab, Stellan Hedberg, Christer Blomqvist, Leif Pettersson, Mikael Schullström and Göran Arntyr, for providing great technical support and for all interesting and stimulating discussions that we had during our coffee brakes. Finally, I would like to express my sincere and deepest gratitude to my wife Vera and to my two beautiful sons, Stefan and Simon, for supporting me and being a warm light through tough times.



TABLE OF CONTENTS

ABSTRACT ........................................................................................................................ 1

SAMMANFATTNING .......................................................................................................... 3

PREFACE ........................................................................................................................... 5

ACKNOWLEDGMENTS ..................................................................................................... 7

TABLE OF CONTENTS ..................................................................................................... 9

LIST OF FIGURES ........................................................................................................... 11

LIST OF TABLES ............................................................................................................. 14

NOMENCLATURE ........................................................................................................... 15

1 BACKGROUND ........................................................................................................ 19

FLUTTER IN TURBOMACHINERY ............................................................................ 19 1.1 MODE SHAPES SENSITIVITY ................................................................................. 23 1.2 EFFECTS OF MISTUNING ...................................................................................... 26 1.3

1.3.1 Aerodynamic Mistuning ................................................................................. 28

2 FUNDAMENTAL CONCEPTS .................................................................................. 31

CLASSICAL APPROACH IN FLUTTER STABILITY PREDICTIONS ................................ 31 2.1 GOVERNING EQUATIONS ..................................................................................... 31 2.2 UNSTEADY AERODYNAMIC FORCES ..................................................................... 32 2.3 STABILITY PARAMETER ........................................................................................ 34 2.4 AERODYNAMIC INFLUENCE COEFFICIENTS ........................................................... 35 2.5 REDUCED ORDER MODEL FOR ADDRESSING AERODYNAMIC MISTUNING ............... 36 2.6

3 OBJECTIVES ........................................................................................................... 39

4 INVESTIGATION METHODOLOGY ......................................................................... 40

5 EXPERIMENTAL INVESTIGATIONS ....................................................................... 41

DESCRIPTION OF TEST SETUP ............................................................................. 41 5.15.1.1 Test Object .................................................................................................... 41 5.1.2 Experimental Facility ..................................................................................... 41 5.1.3 Coordinate Systems ...................................................................................... 44 5.1.4 Convention of Blade Oscillation and Stagger Angles .................................... 45 5.1.5 Measurement Equipment .............................................................................. 47

5.1.5.1 Steady-state measurement setup .......................................................... 47 5.1.5.2 Unsteady measurement setup ............................................................... 48

EXPERIMENTAL TESTING STRATEGY .................................................................... 53 5.2

6 NUMERICAL INVESTIGATIONS ............................................................................. 55

NUMERICAL METHOD .......................................................................................... 55 6.16.1.1 Governing equations ..................................................................................... 55 6.1.2 RANS and Turbulence modelling .................................................................. 57 DESCRIPTION OF NUMERICAL MODEL .................................................................. 60 6.2 SIMULATION APPROACH ....................................................................................... 62 6.3

6.3.1 Aerodynamic mistuning simulations .............................................................. 63

7 RESULTS ................................................................................................................. 65

STEADY-STATE RESULTS .................................................................................... 65 7.17.1.1 Influence of Aerodynamic Asymmetries on Steady Blade Loading ............... 66 UNSTEADY RESPONSE- NUMERICAL PREDICTIONS VS. TEST DATA ....................... 70 7.2

7.2.1 Axial Bending ................................................................................................ 70


7.2.2 Circumferential Bending ................................................................................ 73 7.2.3 Torsion .......................................................................................................... 74 7.2.4 Impact of Flow Velocity ................................................................................. 76 COMBINED MODE SHAPES ................................................................................... 78 7.3

7.3.1 Linear Superposition of Modes ...................................................................... 78 7.3.2 Axial Bending-Torsion ................................................................................... 79

7.3.2.1 Effects of Flow Velocity ......................................................................... 82 7.3.2.2 Effects of Flow Incidence ....................................................................... 85 7.3.2.3 Mode Combination at Different Bending-to-Torsion Amplitude Ratios .. 87 7.3.2.4 Three-Dimensional Effects on Mode Combination ................................ 87

7.3.3 Circumferential Bending-Torsion ................................................................... 88 7.3.3.1 Effects of Flow Velocity ......................................................................... 91

EFFECTS OF AERODYNAMIC MISTUNING ON AEROELASTIC RESPONSE .................. 94 7.47.4.1 Unsteady Response- Mistuned Case ............................................................ 94 7.4.2 Mistuned Aerodynamic Influence Coefficients............................................. 100

7.4.2.1 Impact of Reduced Frequency ............................................................ 102 7.4.3 Mistuned Aeroelastic Model- Probabilistic Analysis .................................... 104

8 SUMMARY .............................................................................................................. 111

CONCLUSIONS .................................................................................................. 111 8.1 RECOMMENDATIONS FOR FUTURE WORK ............................................................ 113 8.2

9 REFERENCES ........................................................................................................ 115


LIST OF FIGURES

Figure 1.1: Collar’s triangle of forces .................................................................... 19 Figure 1.2: Cut-away of a Rolls-Royce Trent 1000 engine (picture courtesy of Rolls-Royce) ......................................................................................................... 20 Figure 1.3: A typical Campbell diagram ................................................................ 20 Figure 1.4: Aerodynamic coupling in a turbomachine blade row (Vogt, 2005) ...... 22 Figure 1.5: Nodal diameter patterns and corresponding instantaneous cascade geometries (adapted from Vogt, 2005) ................................................................. 23 Figure 1.6: Mode shape sensitivity plot at low subsonic Mach number; shaded areas mark stable regions; values display aerodynamic damping parameters (Vogt and Fransson, 2007) ................................................................................... 24 Figure 1.7: The effect of aerodynamic detuning on the critical reduced frequency with the elastic axis and center of gravity located at 40% of the chord (Hoyniak and Fleeter, 1986) ....................................................................................................... 26 Figure 1.8: Effect of mistuning on flutter (Kaza and Kielb, 1983) .......................... 27 Figure 2.1: Investigated orthogonal mode shapes ................................................ 32 Figure 2.2: Response and excitation correlated by the phase angle .................... 34 Figure 2.3: Characteristic stability curve (S-curve) ............................................... 36 Figure 2.4: Schematic illustration of aerodynamic coupling influences in the cascade assuming lumped parameter model ....................................................... 37 Figure 5.1: LPT rotor blade row (including rotor disk) ........................................... 41 Figure 5.2: Schematic drawing of the air supply system ....................................... 42 Figure 5.3: Test facility and sketch of the test module .......................................... 43 Figure 5.4: Inlet and outlet sidewalls; Vogt (2005) ................................................ 43 Figure 5.5: Global test rig coordinate system ....................................................... 44 Figure 5.6: Local on-blade coordinate system ...................................................... 45 Figure 5.7: Orthogonal oscillation modes ............................................................. 46 Figure 5.8: Definition of stagger angle and de-staggering convention .................. 46 Figure 5.9: Distribution of the pressure taps on the blade surface (non-oscillating blade) ................................................................................................................... 47 Figure 5.10: Non-oscillating instrumented blades ................................................. 48 Figure 5.11: Oscillating blade used for the unsteady pressure measurements .... 49 Figure 5.12: Dynamic calibration setup ................................................................ 50 Figure 5.13: Example of a transfer characteristic ................................................. 50 Figure 5.14: Laser signal calibration setup ........................................................... 52 Figure 6.1: Modeled sector cascade ..................................................................... 60 Figure 6.2: Different investigated mesh types; Mårtensson (2005) ...................... 61 Figure 6.3: Time history of unsteady response on the surface of the oscillating blade ..................................................................................................................... 62 Figure 7.1: Steady blade loading at midspan section on blade 0; M2=0.4 ............ 65 Figure 7.2: Impact of blade 0 de-staggering on passage throats .......................... 66 Figure 7.3: Influence of aerodynamic asymmetries on steady blade loading; low subsonic case (M2=0.4); 50% span ...................................................................... 67 Figure 7.4: Influence of aerodynamic asymmetries on steady blade loading; high subsonic velocity (M2=0.8); 50% span .................................................................. 68 Figure 7.5: Experimental data vs. CFD predictions; Steady blade loading at midspan; low subsonic velocity (M2=0.4); ............................................................. 69 Figure 7.6: Unsteady blade surface pressure at midspan; axial bending; k=0.3; M2=0.4 .................................................................................................................. 70


Figure 7.7: Spatially resolved unsteady force component at midspan section on blades -1, 0 and +1 (imaginary and real); IBPA=0deg .......................................... 71 Figure 7.8: Nominal aerodynamic force influence coefficients for blades -1, 0 and +1; axial bending; k=0.3; M2=0.4 .......................................................................... 72 Figure 7.9: Nominal aerodynamic influence coefficients for blades -1, 0 and +1 in complex plane ...................................................................................................... 72 Figure 7.10: Unsteady blade surface pressure at midspan; circumferential bending; k=0.3; M2=0.4 ........................................................................................ 73 Figure 7.11: Nominal integrated aerodynamic influence coefficients for blades -1, 0 and +1; circumferential bending; k=0.3; M2=0.4 ................................................... 74 Figure 7.12: Unsteady blade surface pressure at midspan section; torsion; k=0.3; M2=0.4 .................................................................................................................. 75 Figure 7.13: Nominal aerodynamic influence coefficients for blades -1, 0 and +1; torsion; k=0.3; M2=0.4 .......................................................................................... 76 Figure 7.14: Comparison of measured and predicted unsteady loading on blades -1, 0 and +1 at different velocity levels (M2=0.4 & M2=0.8); axial bending mode; k=0.1 .................................................................................................................... 77 Figure 7.15: Nominal aerodynamic influence coefficients at two different velocity levels; axial bending mode; k=0.1 ........................................................................ 78 Figure 7.16: Unsteady blade surface pressure at midspan, experimental data; combined axial bending-torsion mode at R=1; k=0.1 and M2=0.4; ....................... 79 Figure 7.17: Differences in unsteady pressure data obtained from direct testing of combined mode and result from pure mode superposition ................................... 80 Figure 7.18: Numerical results of combined axial bending-torsion mode at M2=0.4 and k=0.1 .............................................................................................................. 81 Figure 7.19: Variation of quality parameters with reduced frequency; axial bending-torsion; low subsonic velocity M2=0.4; experimental data (left) & numerical results (right); ................................................................................................................... 82 Figure 7.20: Unsteady blade surface pressure at midspan, experimental data; combined axial bending-torsion mode at k=0.1 and M2=0.8 ................................. 83 Figure 7.21: Numerical results of combined axial bending-torsion mode at M2=0.8 and k=0.1 .............................................................................................................. 84 Figure 7.22: Variation of quality parameters with reduced frequency; axial bending-torsion; high subsonic velocity M2=0.8 .................................................................. 84 Figure 7.24: Variation of quality parameters with inflow incidence; blade 0 .......... 86 Figure 7.25: Unsteady response at midspan on blade 0; combined axial bending-torsion mode; at k=0.3 and M2=0.4; amplitude ratio R=0.5 (left) and R=2 (right) . 87 Figure 7.26: Unsteady blade surface pressure at 10% span on blades ±1, experimental data; combined axial bending-torsion mode at k=0.1 and M2=0.4 .. 87 Figure 7.27: Unsteady blade surface pressure at 90% span on blades ±1, experimental data; combined axial bending-torsion mode at k=0.1 and M2=0.4 .. 88 Figure 7.28: Unsteady blade surface pressure at midspan, experimental data; combined circumferential bending-torsion mode at R=1; k=0.1 and M2=0.4; ....... 89 Figure 7.29: Numerical results of combined circumferential bending-torsion mode at M2=0.4 and k=0.1 ............................................................................................. 90 Figure 7.30: Variation of quality parameters with reduced frequency circumferential bending-torsion; low subsonic velocity M2=0.4; .................................................... 90 Figure 7.31: Unsteady blade surface pressure at midspan, experimental data; combined circumferential bending-torsion mode at R=1; k=0.1 and M2=0.8; ....... 91 Figure 7.32: Numerical results of combined circumferential bending-torsion mode at M2=0.8 and k=0.1 ............................................................................................. 92


Figure 7.33: Variation of quality parameters with reduced frequency circumferential bending-torsion; high subsonic velocity M2=0.8; ................................................... 93 Figure 7.34: Unsteady pressure data at midspan on blades ................................ 95 Figure 7.35: Numerical results for unsteady blade surface pressure distribution; axial bending, k=0.3 and M2=0.4; de-stagger angle ∆γ=-2.5°, 0°, 2.5° ................. 96 Figure 7.36: Variation of Cp amplitudes at specific locations on blades -1, 0 and +1; axial bending, k=0.3 and M2=0.4; ................................................................... 96 Figure 7.37: Unsteady pressure response at midspan on blades -, 0 and +1; axial bending; k=0.1 and M2=0.8; de-stagger angle ∆γ=-2.5°, 0°, 2.5° ......................... 97 Figure 7.38: Unsteady pressure response at midspan on blades -, 0 and +1; circumferential bending; k=0.3 and M2=0.4; de-stagger angle ∆γ=-2.5°, 0°, 2.5° . 98 Figure 7.39: Unsteady pressure response at midspan on blades -1, 0 and +1; torsion; k=0.3 and M2=0.4; de-stagger angle ∆γ=-2.5°, 0°, 2.5° ........................... 99 Figure 7.40: Mistuned influence coefficients for blades -1, 0 and +1; axial bending; M2=0.4 & k=0.3; experimental data (black colored) and numerical results (blue colored); ............................................................................................................. 100 Figure 7.41: Mistuned influence coefficients for blades -1, 0 and +1; circumferential bending; M2=0.4 & k=0.3; experimental data (black colored) and numerical results (blue colored); .................................................................................................... 101 Figure 7.42: Mistuned influence coefficients for blades -1, 0 and +1; torsion; M2=0.4 & k=0.3; experimental data (black colored) and numerical results (blue colored); ............................................................................................................. 102 Figure 7.43: Mistuned influence coefficients vs. reduced frequency; axial bending; M2=0.4; experimental data .................................................................................. 102 Figure 7.44: Mistuned influence coefficients vs. reduced frequency; circumferential bending; M2=0.4; experimental data ................................................................... 103 Figure 7.45: Mistuned influence coefficients vs. reduced frequency; torsion; M2=0.4; experimental data .................................................................................. 104 Figure 7.46: Variation of the influence coefficients due to de-stagger on various blades in the cascade; numerical results; axial bending at k=0.3; L1 ................. 105 Figure 7.47: Effect of random mistuning on flutter stability; 20% of blades de-staggered; axial bending at k=0.3 & M2=0.4; numerical results .......................... 106 Figure 7.48: Cumulative probability of least stable mode; axial bending ............ 106 Figure 7.49: Effect of random mistuning on flutter stability; 20% of blades de-staggered; axial bending at M2= 0.8 & k=0.1; numerical results ......................... 107 Figure 7.50: Cumulative probability of least stable mode; axial bending at k=0.1 & M2=0.8 ................................................................................................................ 107 Figure 7.51: Effect of random mistuning on flutter stability; 20% of blades de-staggered; circumferential bending at k=0.3; numerical results .......................... 108 Figure 7.52: Cumulative probability of least stable mode; circumferential bending ........................................................................................................................... 108 Figure 7.53: Effect of random mistuning on flutter stability; 20% of blades de-staggered; torsion at k=0.3; numerical results .................................................... 109 Figure 7.54: Cumulative probability of least stable mode; torsion ...................... 109


LIST OF TABLES

Table 5.1: Key design parameters ........................................................................ 41 Table 5.2: Test conditions .................................................................................... 53 Table 7.1: The impact of aerodynamic asymmetries on minimum stability ......... 110


NOMENCLATURE

Latin Symbols

A blade oscillation amplitude, per-degree basis for 3D consideration, per millimeter (bending) and per radian (torsion), respectively, for 2D consideration (stability parameter)

c blade chord

aekc , aerodynamic stiffness matrix element

aegc , aerodynamic damping matrix element

pc static pressure coefficient refsref

refsp pp

ppc

,,0

,

Apc ,ˆ normalized unsteady pressure coefficient, refdyn

p pA

pc

,

ˆˆ

;

vc specific heat at constant volume

df infinitesimal force component

dm infinitesimal moment component ds infinitesimal arcwise surface component, per unit span

e

direction vector torsion mode (radial)

F complex force vector G damping matrix

h complex mode shape vector

ih oscillation amplitude

i imaginary unit, 1i i internal (thermal) energy

k reduced frequency K stiffness matrix M mass matrix M Mach number m mass mn, blade indices

n

normal vector to surface element N number of blades p pressure

p mean pressure

p~ time-varying perturbation pressure

p unsteady pressure amplitude (complex)

R bending-to-torsion amplitude ratio Q modal displacement vector

r

distance from center of torsion to force realization point t time, time of flight T oscillation period u absolute outflow 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 Greek Symbols

de-stagger angle

torsion orthogonal mode coordinate

circumferential orthogonal mode coordinate

angular coordinate in cylindrical coordinate system eigenvalue axial orthogonal mode coordinate

density

interblade phase angle; phase angle

rotational frequency, f 2

pressure ratio stability parameter

Subscripts

0 total 1 cascade inlet 2 cascade outlet ae aerodynamic

avg average

ax axial damping related to damping disturbance related to disturbance

dyn dynamic

ic influence coefficient outlet cascade outlet phase phase of complex quantity

ref reference

TWM traveling wave mode

torsion direction

circumferential bending direction

axial bending direction


Abbreviations

arc normalized arcwise coordinate; negative branch on suction side, positive on pressure side

avg average CFD computational fluid dynamics COT center of torsion deg degree HCF high cycle fatigue HPT Heat and Power Technology FEM finite element method IBPA interblade phase angle Im imaginary part of complex number INFC influence coefficient KTH Kungliga Tekniska Högskolan (Royal Institute of Technology) LE leading edge LPT low-pressure turbine PS pressure side PSI Pressure System Inc. Re real part of complex number SS suction side TE trailing edge TWM traveling wave mode



1 BACKGROUND

Flutter in Turbomachinery 1.1

Modern aircraft engine design is characterized by a tendency towards thinner, lighter and more highly loaded blades. At the same time, developments in power generation turbines are towards greater power density and larger turbines. These trends give rise to increased sensitivity for flow induced vibrations and result in increasing challenges concerning structural integrity of the engine. Flutter is an aeroelastic instability phenomenon that involves self-excited and self-sustained vibrations of a structure in a flow field. The vibration motion of a blade, initialized by small aerodynamic disturbances in the flow, causes an unsteady pressure field on the blade surface, which results in unsteady aerodynamic forces. These unsteady aerodynamic forces on the blade can either dampen the vibrations or they can feed energy into the structure leading to rapid increase in the vibration amplitude during each cycle. Such a situation will finally lead to extreme blade oscillations and, in a short time period, result in material failure. As such, a correct prediction of the aeroelastic properties represents one of the paramount challenges in the turbomachinery aeromechanical design process. The assessment of flutter involves three different disciplines: fluid mechanics, structural dynamics and solid mechanics. They are embraced by a common discipline, denoted ‘dynamic aeroelasticity’. The need for these different disciplines can be explained by considering the existence of flutter. Referring to Collar’s Triangle (Collar, 1946) in Figure 1.1, flutter arises from an interaction between the unsteady aerodynamic forces resulting from the fluid, the elastic forces of the structure and the inertial forces. During flutter, the balance between these forces is disturbed, resulting in a condition where the fluid feeds energy into the structure leading to larger vibration amplitudes and consequently inducing even larger aerodynamic forcing.

Figure 1.1: Collar’s triangle of forces


The occurrence of flutter in turbomachines is usually limited to slender and long blades that are highly loaded. Typically this situation can be found in fans, low pressure compressors (LPC) and low pressure turbines (LPT). The location of these areas in a typical jet engine is depicted in Figure 1.2.

Figure 1.2: Cut-away of a Rolls-Royce Trent 1000 engine (picture courtesy of Rolls-Royce)

The unsteady pressure fields on blade surfaces may also be induced by external sources. Such blade excitations may result in forced vibrations. The excitation frequency is typically synchronized with the rotational speed of the engine, since the most common forced vibration problems are caused by the relative motion between rotating and non-rotating parts. Resonant vibrations occur if the frequency and shape of the excitation coincides with the natural frequency of a structural mode of the rotor. Such occurrences are traditionally identified by means of a Campbell diagram (an example of which is shown in Figure 1.3) showing the variation of excitation- and natural frequencies with rotational speed.

Figure 1.3: A typical Campbell diagram

Fan LPC LPT


Potential resonant vibration problems can occur at crossings between the engine order lines and the natural frequency lines. Since the occurrence of flutter does not require external excitation, such as engine order, it can appear off the excitation lines. From basic flutter studies in the field of aeronautics, it is known that flutter stability of isolated aircraft wings is mainly influenced by a ratio between the wing mass and the mass of the surrounding air inside a circle with the radius of half chord. This ratio, defined as mass ratio , for wings has typically values of 51 . Mass ratio in turbomachines reaches larger values (typically in a range from 100 to 600) and is therefore not of the same importance as for wing flutter. On the other hand, it was found that the ratio between the blade oscillation frequency, the blade chord and the flow velocity has a great influence on flutter stability in turbomachinery blade rows. This ratio is known as the reduced frequency k and is defined by the time t needed for a fluid particle to travel across the blade chord c divided by the oscillation period .

u

fc

T

tk

2 Eq. 1-1

Flutter occurs below a certain critical reduced frequency; for turbomachinery blades typical values li in the range between 0.1 and 1. Previous studies of fundamental bending and torsion mode shapes conducted by Panovsky and Kielb (2000) indicated that each of these modes becomes unstable for reduced frequency in the range of 0.2 and 0.3. Vogt (2005) has shown that an increase in the reduced frequency tends to lead to an increase in response magnitude and that it has a stabilizing effect on the travelling wave mode stability. From Eq. 1-1, one can notice that for a constant oscillation frequency the value of the critical reduced frequency is approached as flow velocity increases. For small values of reduced frequency the flow has a quasi-steady character and for large values unsteady effects become dominant (Srinivasan, 1997). Controlling the reduced frequency has therefore become an important issue in the aeroelastic design process. The flutter behavior in turbomachines is predominantly affected by the interaction of the whole blade assembly instead of only a single blade. Oscillations of each individual blade in the blade row will induce an aerodynamic response both on the blade itself and on its adjacent blades. An isolated blade might show no tendency to flutter, but when included in a blade row, this assembly might become aeroelastically unstable. The interaction between blades is denoted as aerodynamic coupling and is an important factor when the aerodynamic response of the blades in an oscillating blade row is assessed. Aerodynamic coupling between the blades in a blade row is schematically illustrated in Figure 1.4.


Figure 1.4: Aerodynamic coupling in a turbomachine blade row (Vogt, 2005)

The coupling is such that all blades have an influence on each other, but the coupling influence is strongest between adjacent blades. Flutter studies on turbine blade rows have indicated that the reference blade and its immediate neighbours are dominant contributors to the unsteady behavior of cascades (Panovsky and Kielb, 2000, Vogt, 2005), not taking into account acoustic resonance effects. Acoustic resonance effects are highly significant if there is an empty duct adjacent to the blade row (e.g. fan blades adjacent to the engine intake duct). The oscillation of the blades causes a fluctuating pressure field that propagates up- and downstream of the blade row and can induce acoustic resonance in the adjacent ducts. Acoustic resonance flutter is almost exclusively of relevance in empty ducts adjacent to the considered blade row, as adjacent blade rows tend to suppress resonant behavior (Whitehead, 1973). Wu et al. (2003) highlighted 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, observed as sharp and local drop in the flutter stability margin referred to as “flutter bite”. While investigating mechanisms for wide-chord fan blade flutter, Vahdati et al. (2011) were able to identify two types of flutter: stall flutter (driven by flow separation) and acoustic flutter (driven by intake acoustics). The latter occurs due to wave reflections and was found to be independent of flow separation. It was also shown that acoustic flutter can significantly intensify stall flutter if they occur at the same speed (“flutter bite”). The use of blade frequency (stiffening of the blade) for flutter stability control does not seem to be effective method for suppression of acoustic flutter.


Assuming that all blades on a turbomachinery component (such as rotor) vibrate in the same mode, at the same frequency and at the same amplitude, a so-called travelling wave mode is obtained (Crawley, 1988). From a flutter stability point of view this assumption represents the least stable condition and therefore tends to be over-conservative. A parameter that characterizes the phase lag between two adjacent blades is referred to as interblade phase angle. It translates into the nodal diameter as follows:

Forward travelling wave mode N

ND

2 Eq. 1-2

Backward travelling wave mode N

NDN 2 , Eq. 1-3

where is the interblade phase angle, N is the number of blades and ND is the nodal diameter which gives the order of the travelling wave. Each nodal diameter pattern induces a pair of travelling waves (forward and backward travelling wave). The resulting mode is referred to as travelling wave since it resembles a wave around the circumference of the blade row. An example of nodal diameter patterns and corresponding cascade geometries is shown in Figure 1.5.

Nodal diameter pattern Blade row Nodal diameter pattern Blade row

Nodal diameter ND=1 Nodal diameter ND=3

Figure 1.5: Nodal diameter patterns and corresponding instantaneous cascade

geometries (adapted from Vogt, 2005)

Mode Shapes Sensitivity 1.2 Blades in a bladed disk are structurally coupled through the disk and can even be connected to each other through shrouds i.e. shrouded bladed disks. This coupling will greatly affect the vibration properties of the assembly. Srinisivan (1997) pointed out that bladed disk assemblies will experience system modes influenced by the characteristics of individual blade vibration, the support structure (including part-span and tip shrouds), the operating conditions, damping and the extent of the mistuning among blades. When considering unshrouded bladed disks, one can identify two possible types of modes. The first includes blade dominated modes and corresponds to the situation where long, flexible blades are mounted on a comparatively stiff disk. To assess blade dominated modes one can concentrate on analyzing single blade modes.

−100 0 100

−300

−200

−100

0

100

200

300

400

−100 0 100

−300

−200

−100

0

100

200

300

400

+ - +

+

+ -

-

-


The second type includes disk dominated modes and occurs when small and stiff blades are mounted on a comparatively flexible disk. In the classical approach to flutter stability prediction, the aerodynamic and structural parts are treated separately (Marshall & Imregun, 1996). Since the critical mode shape of the setup is not known a priori, one needs to predict aerodynamic properties for a large number of possible mode shapes. If linearity and non-deforming blade sections are assumed one could span a mode space by investigating three orthogonal modes since linear superposition of modes can be employed when investigating coupled modes. The influence of the mode shape on flutter stability has been investigated by many researchers. The conclusion from studies conducted by Panovsky et al. 1997 and Vogt and Fransson (2004), on a low pressure turbine blade in rigid body oscillations showed that the blade mode shape could have a significant impact on aeroelastic stability. The study has also shown that certain mode shapes are more sensitive to reduced frequency changes than others. The recommendation was to include mode shape sensitivity studies in aeroelastic design phase, which previously was mainly concentrated on studying the reduced frequency of the blade. Panovsky and Kieb (2000) developed a graphical method to assess flutter stability on a two dimensional section of a cascade. The method is based on representing any possible pure rigid body mode as torsion mode around a respective centre of torsion. Modes of bending character (i.e. translational movement of blade) are represented by a center of torsion far away from the reference blade whereas modes of torsion character feature a center of torsion close to the reference blade. An example of a mode shape sensitivity plot (referred to as a stability map or tie-dye plot) is shown in Figure 1.6, where stable and unstable regions related to the torsion axis location are presented.

Figure 1.6: Mode shape sensitivity plot at low subsonic Mach number; shaded areas mark stable regions; values display aerodynamic damping parameters

(Vogt and Fransson, 2007)


Figure 1.6 indicates that mode shape sensitivity is large for torsion type modes as the stability gradients are comparatively large in proximity to the reference blade. Studies conducted by Chernysheva et al. (2001), Kielb et al. (2003) and Vogt (2005) indicate that stability maps have similar appearance for different types of turbine blade geometries. Srinivasan (1997) explains that vibratory modes of realistic blade profiles are almost always coupled i.e. the contents of a given mode of vibration include displacements perpendicular and parallel to the chord as well as torsion of the profile. Current design trends for compressor blades that are resulting in low aspect ratio blades potentially reduce the frequency spacing between certain modes (e.g. 2F & 1T) leading to mode coalescence (Mårtensson et al., 2008). However, coalescence flutter is not a high risk item in the design if a reasonable frequency separation can be established. For the investigated highly loaded transonic fan design a 2% frequency separation allows avoiding aeroelastic mode coupling effects. Another occurrence of combined modes is given by the vibration of bladed disks when having a comparatively soft disk and rigid blades. In such a case, bending and torsion modes are coupled by vibration kinematics and feature a phase lag between the modes of 90°. Coupled bending-torsion modes in fans and compressor rotors were investigated analytically by Bendiksen and Friedman (1980) using a 2D section model .The outcome was that flutter boundaries were considerably affected by introducing coupling between the bending and torsional degrees of freedom. At the same time, there was no appreciable tendency for the bending and torsional frequencies to coalesce as flutter approaches, except at very low reduced frequencies. Coupling between bending and torsion was also found to reduce sensitivity of Single Degree of Freedom (SDOF) torsional flutter boundaries to the elastic axis location. Generally the combination of bending and torsion modes is achieved under a certain phase lag between involved modes. The effect of the phase angle between modes has been addressed by Försching (1991) from a quasi-steady viewpoint. He identified that a phase lag of 90° represents the most unfavorable situation from a stability point of view, as the work produced by the fluid on the structure during the cycle will have positive sign. Försching (1994) pointed out that single mode kinematic coupling in bending and torsion is one of the important structural dynamic features of beam-like turbomachinery blades, which significantly influences their flutter stability. Since the mass ratio of turbomachinery blades is generally much higher than that of an aircraft wing, flutter frequencies and the corresponding mode shapes can be considered to be unaffected by the aerodynamic loading. As such, aeroelastic bending/torsion coalescence flutter coupling between two modes, typical for classical aircraft wing flutter, is less important in turbomachinery blades. He also mentioned that the frequency ratio for the first bending/first torsion mode is typically in the range of 0.3-0.7 and that aerodynamic coupling between these two modes, causing a classic coalescence flutter, is unlikely to occur. Mode coupling


can however occur between 2nd bending and 1st torsion modes since the frequency ratio often falls in the range 0.8-1.2. Hoyniak and Fleeter (1986) showed that, for the investigated unstable supersonic twelve-bladed rotor cascade, bending-torsion coupling effects were found to be significant for a bending-torsion natural frequency ratio between 0.6 and 1.2. Outside this region results are approximately the same as found for the single degree of freedom analysis i.e. similar to results from torsion mode analysis as presented in Figure 1.7. This indicates that for the investigated cascade configurations the bending mode is not coupling with the torsion mode at higher natural frequency ratios.

Figure 1.7: The effect of aerodynamic detuning on the critical reduced frequency with the elastic axis and center of gravity located at 40% of the chord (Hoyniak and

Fleeter, 1986)

Effects of Mistuning 1.3 The aeroelastic stability behavior of turbomachinery blades is sensitive to small variations in natural frequencies and mode shapes between individual blades in a blade row. Due to manufacturing uncertainty and tolerances, turbomachinery blades in bladed-disks (also in blisks) will always differ from one another. Most often when talking about mistuning one is referring to structural mistuning which is defined as blade to blade differences in blade mass, stiffness and damping properties. Mistuning will lead to changes in natural frequencies and mode shapes and can have a major impact on the aeroelastic properties of cascades. Structural mistuning typically has an adverse effect on forced response but a beneficial effect on flutter stability. For forced response, mistuning will cause frequency split and mode localization, which in its turn results in having multiple


peaks in the resonant response curve and a magnification of the response amplitude. Mistuning sensitivity was shown to be strongly depended on structural coupling in bladed disks (Bladh et. al., 1999). In the case of flutter, mistuning breaks the symmetry of a perfect travelling mode and by this tends to suppress flutter. The typical influence of structural mistuning on flutter stability can be observed in root locus plot depicted in Figure 1.8 (Kaza and Kielb, 1983).

Figure 1.8: Effect of mistuning on flutter (Kaza and Kielb, 1983) The complex eigenvalues here present solution to the eigenvalue problem derived from the aeroelastic differential equation of motion. The real part of the eigenvalue in this case represents damping and determines stability. The imaginary part specifies the ratio of the flutter frequency to the reference frequency. One can observe the tendency of moving the least stable modes to the stable region when alternating mistuning is introduced. It can be also noticed that higher level of alternating mistuning cause locus split i.e. the modes separate into high and low frequency groups. The degree of separation increases with the level of mistuning. In analysis of effects of mistuning on flutter stability both structural and aerodynamic coupling mechanisms can play a significant role. Several high fidelity mistuning models have been developed but the emphasis has been put on structural coupling. A few recent publications described high fidelity mistuning models with both structural and aerodynamic coupling (Senturier et al., 2000; Kielb et al., 2004; Kielb, R. et al., 2005; Sladojevic et al., 2006). By comparing the stabilizing effects of mistuning predicted using only aerodynamic coupling and the results from models using aerodynamic and FMM structural coupling, Kielb et al. (2004) showed that the stabilizing effect of mistuning is over predicted in cases where only aerodynamic coupling is included. It is recognized that the beneficial effect of structural mistuning on flutter stability is inhibited by the addition of structural coupling effects. This behavior is caused by the increased spread in tuned frequency due to structural coupling, making the system less sensitive to mistuning.


Previous studies have shown that different mistuning patterns can have very different effect on flutter stability. Crawley and Hall (1985) showed that alternating blade frequencies is a robust optimum arrangement. Finding optimal mistuning patterns for increasing flutter stability was the target of several different studies. Kaza and Kielb (1983) concluded that alternate blade mistuning may be nearly optimal in increasing flutter stability of shroudless fans. A more recent probabilistic flutter study of a mistuned bladed disk, conducted by Kielb et al. (2006), presented a method for identifying beneficial structural mistuning patterns. Försching (1994) indicated that when concerning practical feasibility of optimal blade mistuning as an inverse design procedure, the difficulty is not only in deciding the level of mistuning required, but also in the complexity of the mistuning pattern and the number of different blades which must be manufactured and kept in inventory. 1.3.1 Aerodynamic Mistuning

Although considerable effort has been put into research on structural mistuning effects, there is still a need to address another type of mistuning, known as aerodynamic mistuning or aerodynamic asymmetries. Aerodynamic mistuning refers to aerodynamic non-uniformities due to geometric asymmetries in blade cascades. Although no structural properties of blades are changed, both steady and unsteady loads on blades are affected. Asymmetries may appear as a result of the engine manufacturing process within the frames of manufacturing tolerance or assembly inaccuracies, as well as in-service wear or due to damages. An early study on the effects of aerodynamic mistuning carried out by Hoyniak and Fleeter (1986), looked into how passage-to-passage differences (particularly differences in circumferential blade spacing) in the flow field could affect flutter stability. It was shown that the effect of aerodynamic mistuning is dependent on the location of the elastic axis and the center of gravity. Alternating blade spacing was found to have stabilizing effects on coupled bending-torsion unstalled supersonic flutter. One of the results from this study is presented in Figure 1.7 showing the effect of the aerodynamic detuning on the critical reduced frequency. Sladojevic et al. (2006) investigated the influence of stagger angle variation on aerodynamic damping and frequency shift for a model without structural coupling. The studied blade represented a layer extracted from the fan blade used in civil aero engines and modal properties were artificially created. The investigated mode shape was a mixture of bending and torsion designed with aim to cause instability in lower modes. Modal frequencies were insensitive to stagger angle changes. Stagger angle variations up to 0.5 degrees did not affect the stability of the system. Large alternating stagger angle variations up to 2 degrees were found to have a destabilizing effect. The opposite behavior was found for random patterns i.e. random variations of stagger angle have in this case slightly increased the overall damping of the system. Kielb et al. (2007) addressed the phenomenon of aerodynamic asymmetries on a probabilistic basis. By perturbing the aerodynamic coupling coefficient between individual blades in the blade rows both intentional (i.e. symmetry groups) and


random asymmetries could be studied. It was found that random aerodynamic perturbation could have a destabilizing effect while single blade and alternating perturbations tend to suppress flutter. Stüer et al. (2008) investigated numerically the impact of aerodynamic mistuning (in this case alternating chord length) on the aeroelastic stability of highly staggered low pressure turbine blades with supersonic exit flow. Due to structural coupling in the investigated configuration, there was no net gain in stability when mistuning was introduced. Ekici et al. (2008) investigated the effect of aerodynamic asymmetries caused by alternate blade-to-blade spacing and alternate staggering on the aeroelastic stability of a linear cascade. It was found that alternating spacing improved the stability of the system, which is in line with results presented by Fleeter and Hoyniak (1986). Alternating staggering, on the other hand, was shown to have either stabilizing or destabilizing effect depending on the direction of miss-staggering. Vogt et al. (2009) studied the influence of aerodynamic asymmetries on mode shape sensitivity of an oscillating LPT cascade. The asymmetric aerodynamic perturbation was applied in a generic manner, using perturbation data acquired at negative incidence off-design operation of the setup. It was identified that mode regions that showed greater dependence from asymmetries were torsion-bending types of modes with torsion centre away from the blade pressure side. The scope of previous studies on effects of aerodynamic mistuning has so far been limited to the above described cases. There is a strong need for investigation of additional cases before general conclusions can be made concerning the effect of aerodynamic mistuning on flutter stability in turbomachines. Keeping this in mind, the present study aims to determine the level of change in aeroelastic properties of an oscillating LP turbine blade row when changing the blade-to-blade stagger angle. For the first time ever, as far as the author is aware, the perturbed aerodynamic influence coefficients are obtained directly employing both experimental testing and numerical simulations.



2 FUNDAMENTAL CONCEPTS

Classical Approach in Flutter Stability Predictions 2.1 The classical approach in aeroelastic stability predictions assumes a decoupling of the structure and flow domains (Marshall & Imregun, 1996). Due to the high mass ratios which are characteristic of turbomachines, the structural terms are comparatively large in comparison to the aerodynamic damping term. Structural and aerodynamic parts can be treated as decoupled i.e. on the structural side, free vibration analysis can be used for determination of the mode shapes without introducing aerodynamic loads, while unsteady aerodynamic forces are obtained from pure unsteady aerodynamic analysis with the blade oscillating in the eigen mode shapes obtained from the structural calculation. A similar approach will be employed in the present work, with the exception that the mode shapes of the oscillations are artificial 3D rigid body mode shapes. Flutter stability analysis in the context of the present work is carried out with a goal of determining whether the unsteady flow field has a destabilizing or stabilizing character, rather than the stability analysis for entire flow-structure system.

Governing Equations 2.2 The aeroelastic system presents a system of differential equations describing the interaction between a structure and a fluid. Basic aeroelastic equations of motion state a balance between structural and aerodynamic forces as )(tFXKXGXM ae , Eq. 2-1

where KGM ,, are modal mass, modal damping and modal stiffness matrices,

X is modal coordinate vector and is unsteady modal force vector consisting of

)()()( tFtFtF edisturbancdampingae , Eq. 2-2

where )(tFdamping stands for the aerodynamic damping forces due to the motion of

the blade in the surrounding fluid and )(tF edisturbanc contains forces due to the

upstream and downstream disturbances. In flutter applications the only forces of interest are ones induced by the motion of the blade, thus 0)( tF edisturbanc .

The usual approach to solve the aeroelastic equations is to introduce modal coordinates as following tieQtX )( Eq. 2-3 where is the mode shape, Q modal displacement and is frequency. The aeroelastic equation can now be expressed in frequency domain as

02 QKKGGiM aemaemm Eq. 2-4


where aeG denotes the aerodynamic damping matrix and

aeK the aerodynamic

stiffness matrix respectively The aeroelastic equation can now be recognized as a complex eigenvalue problem where eigenvalues are describing the stability of the system.

Unsteady Aerodynamic Forces 2.3 In the present investigation it has been assumed that the blades are oscillating in three orthogonal mode shapes: axial bending, circumferential bending and torsion. The direction of the bending modes and torsion axis is defined in Figure 2.1.

Figure 2.1: Investigated orthogonal mode shapes

The harmonic motion of the blade can be expressed by complex vector hhhh ˆ,ˆ,ˆˆ describing a 2D representation of an arbitrary mode shape.

Verdon (1987) showed that for small perturbations the unsteady pressure on a blade surface, resulting from an applied harmonic blade motion, can be represented as a harmonic oscillation around a steady mean pressure as

)(ˆ),(),,(~),(),,( hptiepyxptyxpyxptyxp Eq. 2-5 where ),( yxp is the steady mean pressure, ),,(~ tyxp the respective time-varying perturbation part and p is the complex amplitude of pressure perturbation due to harmonic oscillation. A complex unsteady pressure coefficient was introduced by Vogt (2005) and is defined as complex amplitude of unsteady pressure normalized by the reference dynamic head dynrefp , and the oscillation amplitude A :

refdynp pA

pc

,

ˆˆ

Eq. 2-6

The reference dynamic head is taken as the difference between the total and static pressures upstream of the cascade.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

10%50%90%COT


Analogue to that, the unsteady normalized force can be expressed as

refdynpA

Ff

,

ˆˆ

Eq. 2.7

The unsteady normalized force can be obtained by integration of the infinitesimal local unsteady force components around the blade profile. Considering an orthogonal system of three modes the infinitesimal normalized force components per surface element ds are defined as

dsncfd p

ˆˆ Eq. 2-8

dsncfd p

ˆˆ Eq. 2-9

dsecrmd p

)ˆ(ˆ Eq. 2-10

and from these the normalized unsteady force is obtained as

dsfdfˆˆ

Eq. 2-11

For a system of three orthogonal modes assessment of mode shape sensitivity would include integration of the normal forces for all three orthogonal modes and the resulting matrix contains the force component in all three directions, as follows:

fff

fff

fff

F Eq. 2-12

The first index refers to the mode shape causing the force, and the second to the direction in which the force is acting. As already mentioned in the previous section, any arbitrary rigid body mode can be represented using torsion mode representation i.e. by defining the location of a center of torsion close to the reference blade if the oscillation mode has a torsion character or by locating the center of torsion infinitely far away from the blade if motion is of translational character (bending modes). The unsteady aerodynamic force for an arbitrary rigid-body mode is then obtained as

iT

ii hFhF ˆˆˆˆ Eq. 2-13


Stability Parameter 2.4 The unsteady pressure distribution around the blade resulting from an arbitrary motion of the blade will result in unsteady forces on the profile. If the work per oscillation cycle done by the fluid on the structure is positive this means that the fluid exercises work on the structure and thus acts in destabilizing manner. The work per cycle according to Verdon (1987) is defined as

T

ticycle dtevFW

Eq. 2-14

The resulting work depends only on the imaginary part of the force. This means that if the imaginary part of the force is negative the mode is stable. This corresponds to the situation where the response is lagging behind the excitation. This is illustrated in Figure 2.2.

Figure 2.2: Response and excitation correlated by the phase angle

To be able to quantify this behavior, Verdon (1987) introduced a normalized stability parameter (also called the aerodynamic damping coefficient) defined as

ii

icyclei f

h

W Îm,

Eq.2-15

Generally, the stability parameter, considering the three orthogonal modes investigated in the present work can be expressed as

h

W

h

W

h

W cyclecyclecycle Eq. 2-16

This parameter assesses the aeroelastic stability as a sum of the normalized cycle works, resulting from the different motion modes. Furthermore, when two modes oscillate at the same time, each mode gives a contribution to the work on the other mode (i.e. the cross-coupling terms).

unstable region

stable region


Aerodynamic Influence Coefficients 2.5 Flutter analyses of turbomachinery blade cascades are commonly performed under the assumption that all the blades are oscillating at the same amplitude and frequency and in the same mode shape, but at different interblade phase angles relative to each other. This assumption presents the least stable condition and therefore this method is considered as conservative. The presence of small amounts of non-deliberate random mistuning as caused by manufacturing tolerances will only enhance the aeroelastic stability. For small amplitude oscillations the aerodynamic influence superimposes linearly, which allows the overall influence on a blade in a tuned blade row oscillating in travelling wave mode to be described by summation of the individual blade influences, as shown by Crawley (1988).

2

2

,

,

,

,),,(ˆ,,ˆ

Nn

Nn

nimn

ICApm

TWMApezyxczyxc Eq. 2-17

The left-hand side of the equation is referred to as travelling wave mode domain

where ,,

ˆ m

TWMApc is the complex unsteady pressure coefficient at the point ),,( zyx on

a blade m with the cascade oscillating in a travelling wave mode with an interblade phase angle . The right-hand side is referred to as the influence coefficient domain where local aerodynamic influence coefficient is given as the pressure coefficient of the vibrating blade n , acting on the non-vibrating reference blade m at point ),,( zyx .

Aerodynamic influence coefficients are lagged by the interblade phase angle. When superposing the coefficients in Eq. 2-17, the phase lagging is achieved by multiplication with nie where the sign is in agreement with the numbering of the blades in a cascade consisting of a total of N+1 blades. In this case, blade indices are ascending in the direction of the suction side and descending in direction of the pressure side. In practice, the phase lagging means a rotation of the complex influence vector in the complex plane. Analyzing the above explained superposition, it gets apparent that the influence of the oscillating blade (blade index 0) on itself enters with a constant value. The influences from the immediate neighbors will enter with a first harmonic oscillation (due to ie , ie ), the ±2 neighbors with 2nd harmonic, etc. The influence decays rapidly away from the oscillating blade and becomes almost negligible after ±2 blades, as pointed out by Crawley (1988) and Nowinski and Panovsky (2000). The contribution from the pair ±2 is generally of one order of magnitude less than the ones from blades 0 and ±1. This explains why the variation of the stability parameter with respect to the interblade phase angle is single-period sinusoidal, thus commonly has a characteristic S-shape, as shown in Figure 2.3.


Figure 2.3: Characteristic stability curve (S-curve) The S-curve shown in Figure 2.3 indicates that there might be a condition where the influence of the reference blade, which commonly has a stabilizing character, can be overruled by the de-stabilizing coupling influence of its neighbours and the whole blade row attains unstable. In practice, in order to maximize stability, the 0 blade stabilizing contribution needs to be maximized, while the ±1 blade pair influence is minimized, since any stabilizing contribution necessarily results in an equally de-stabilizing contribution at a different interblade phase angle.

Reduced Order Model for Addressing Aerodynamic 2.6Mistuning

The relation between the travelling wave mode domain and the influence coefficient domain, as defined by Crawley (1988) and given in Eq. 2-17, is only valid for a tuned setup i.e. a cyclic symmetric setup. When aerodynamic mistuning is introduced another approach needs to be defined. To account for asymmetries in direct manner it would be necessary to numerically treat the full rotor assembly model, which is very time-consuming. Hence, a model that allows discrete changes in blade-to-blade aeroelastic properties to be accounted for is needed. In the present work a Reduced Order Model (ROM), in which the blades are reduced to single mass points, is used. The model is of cyclic character and has N degrees of freedom, where N corresponds to the number of blades. The model allows for coupled analysis of both structural and aerodynamic part. Taking the aeroelastic equation Eq.2-4 as a starting point, the aerodynamic influence coefficients, which consist of the complex unsteady aerodynamic forces on the blades, are implemented into the system matrices i.e. the aerodynamic damping

aeG and the aerodynamic stiffness matrices aeK .For a given mode, the

aerodynamic stiffness and damping matrices are populated by aerodynamic influence coefficients given by:

−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


0

1,1,

0

2,2,

1

1,2,

1

2,1,

0

1,1,

1

0,1,

1

1,0,

1

1,0,

0

0,0,

.000

.....

0.0

0.

..0

NNg

gg

ggg

Nggg

ae

c

cc

ccc

ccc

G

)Re( ,, ,

mnicaek fc

mn ,

1

)Im( ,, ,

mnicaeg fc

mn Eq. 2-18

Figure 2.4 illustrates schematically the coupling between the blades in the cascade. In the investigated cascade no structural coupling between the blades is assumed and structural damping is neglected. With an appropriate extension of the model, structural damping could be taken into account as well.

Figure 2.4: Schematic illustration of aerodynamic coupling influences in the cascade assuming lumped parameter model

The aerodynamic matrices are of a diagonal character where the diagonal terms are the influence of the blades on themselves and the off-diagonal terms are containing influence of the neighboring blades. The influence coefficient matrices in the present work are of tri-diagonal type, since only the influences from blades -1, 0 and +1 are considered. In cases where all the blades influenced each other, the matrix would be fully populated.

Eq.2-19

When aerodynamic mistuning is applied, the aerodynamic matrices are no longer circulant, as certain elements of the matrix are perturbed. Assuming a case where a single blade n is aerodynamically mistuned, it would imply that 3x3 of the coefficients in the aerodynamic matrix are affected i.e. the matrix row containing blade n, but also the coefficients in rows n −1 and n +1 will be perturbed. The solution of the complex eigenvalue problem given in Eq. 2-4 provides eigenvalues whose imaginary part represents modal frequency and real part represents damping. Combining the eigenvalue solution given by

2

2

4

4

2 M

MKGi

M

G Eq.2-20


with the expression for the aerodynamic damping influence coefficient expressed in Eq.2-18, a relation between the stability parameter and the eigenvalue solution can be derived as:

)()Re(2)Îm( Absmfi Eq.2-21


3 OBJECTIVES

The objective of the present work is to investigate the aeroelastic properties of combined modes and to study the influence of aerodynamic mistuning on the aeroelastic properties of an oscillating low pressure turbine (LPT) cascade. This main objective is broken down into the following sub-objectives:

To broaden knowledge about mode shape sensitivity of an oscillating LPT cascade, with the main focus placed on investigations of combined bending-torsion mode shapes. The investigation involves both numerical simulations and experimental testing, where validation of the numerical model is carried out. Parametric studies involving different reduced frequencies, velocity levels, incidence angles and bending-to-torsion amplitude ratios are conducted. This part can be considered as a continuation of the studies conducted by Vogt (2005) where sensitivity of pure mode shapes has been addressed.

To validate the principle of linear mode superposition (bending-torsion)

within a certain range of blade vibration amplitudes as well as for a variety of combined mode shapes.

To understand the effects of aerodynamic mistuning on aeroelastic

behavior of turbomachinery blade rows. In particular, determining the level of change in aeroelastic properties of an oscillating LP turbine blade cascade upon change in the blade stagger angle.

To quantify the accuracy of the numerical model in predictions of the aeroelastic response in both symmetric and asymmetric cascades. Mistuned aerodynamic influence coefficients obtained from the numerical simulations are validated against the results from experimental testing of the mistuned cascade.

To develop and apply a Reduced Order Model (ROM) for assessing the

flutter stability of aerodynamically mistuned blade rows.

To quantify the sensitivity of flutter stability with respect to random or intentional aerodynamic perturbations of blade-to-blade aeroelastic properties.

Licentiat Thesis / Nenad Glodic Page 40

4 INVESTIGATION METHODOLOGY

The current work has been formulated in order to experimentally and numerically investigate mechanisms related to aeroelastic response sensitivity towards combined mode shapes and aerodynamic mistuning. On the experimental side, influence coefficient test data are acquired in a controlled-oscillation test facility (Vogt, 2005). The facility comprises an annular sector of low-pressure turbine blades, one of which can be made to oscillate in different rigid-body modes while the response data is acquired on several blades in the cascade. The resulting mode shapes are three orthogonal modes (two bending and one torsion mode), as well as combined bending-torsion modes. On the numerical side, a commercial CFD code (ANSYS CFX v11) is employed using a full-scale time-marching 3D viscous model (Vogt et al., 2007). Simulations are performed using the influence coefficient approach with only one blade oscillating. Numerical results are validated against the experimental data. The work put forward in the first part of the study aims to validate the assumption of linear combination of modes based on an experimental approach. The experiments are supported numerically and the simulation results are processed so as to assess the validity of the assumption on a numerical basis. The strategy was to directly acquire the aeroelastic response of combined modes and to compare these with the results of superposition of the properties of the corresponding pure mode components. To evaluate the validity of linear combination certain quality parameters are defined, giving a measure of the accuracy of mode combination. In order to investigate the validity in a parametric manner, the investigation is extended over a range of reduced frequencies, two different velocity levels and different bending-to-torsion amplitude ratios. In addition, the impact of negative inflow incidence angles is addressed so as to confirm the validity of the linear superposition in regions with separated flows.

Studies of the effects of aerodynamic mistuning consisted of experimental and numerical investigations of aerodynamically mistuned influence coefficients. Aerodynamic mistuning is introduced as a variation of the blade-to-blade stagger angle. The perturbed influence coefficients are obtained directly by employing both experimental testing and numerical simulations. Numerically obtained aeroelastic responses are validated through comparison with the experimental data. In order to study the impact of de-staggering in a comprehensive manner investigations are conducted for different mode shapes introducing different blade-to-blade stagger angles and at various reduced frequencies. To assess the aeroelastic stability of aerodynamically mistuned blade rows, a Reduced Order Model (ROM), in which the blades are reduced to single mass points, is implemented and employed. The model allows the introduction of mistuned force influence coefficients into aerodynamic matrices in the aeroelastic equation, and opens up the possibility of investigations of different mistuned setups in a probabilistic manner.


5 EXPERIMENTAL INVESTIGATIONS

Description of Test Setup 5.1 5.1.1 Test Object

The test object is a high subsonic LPT rotor blade row featuring LP blades with a fully 3D geometry. The profiles were designed at Royal Institute of Technology (KTH) and reflect typical aerodynamic properties of aero engine low pressure turbines. The blade row is depicted in Figure 5.1 and the design parameters are shown in Table 5.1.

To simplify the experimental setup, the cascade does not rotate. In nominal operation, loading on the blades is uniformly high, containing radial variation due to the annular geometry. The setup consists of mechanically decoupled (i.e. unshrouded blades) featuring a nominal tip clearance of 1% blade height. Hub and tip sections of the blade have constant radii to allow arbitrary oscillation of the blade. In order to adapt the profile to the test rig and to improve instrumentation possibilities, the blades have been scaled by a factor of 1.5 comparing to the real engine. 5.1.2 Experimental Facility

The test facility employed in present work was developed within the framework of the study conducted by Vogt (2005) and is located at KTH in Stockholm. The airflow is supplied by a 1 MW screw compressor delivering about 5 kg/s of air at 4bar maximum pressure and 303K. A system of valves allows control of the inlet mass flow and the targeted operating point to be set. Air temperature is adjusted

Figure 5.1: LPT rotor blade row (including rotor disk)

Table 5.1: Key design parameters

Parameter Unit ValueReal chord (midspan)

mm 50

Axial chord (midspan)

mm 45

Span mm 97 Pitch (midspan)

deg 4.5

Solidity (midspan)

- 0.68

Aspect ratio - 1.94 Radius ratio - 1.25 Hub radius mm 383 Shroud radius

mm 480

Tip clearance

mm 1


by a system of air coolers controlled by a PID controller which maintains constant temperature within ±0.1K. The facility is operated at atmospheric outlet conditions, discharging air to the ambient, as indicated in Figure 5.2.

Figure 5.2: Schematic drawing of the air supply system

The pre-conditioned and pressurized air supply first enters a circular inlet plenum that serves as a flow conditioner, containing a flow straightener and a setup of different turbulence grids. A uniform turbulence intensity of approximately Tu=2% and a length scale l=2mm is achieved. Afterwards the flow is directed through a bell mouth and a variable annular sector channel, which allows variation of the inflow angles into the cascade, as shown in Figure 5.3. By means of this variable inlet sector duct, different inflow directions can be set continuously in a range between α=-30° and +30°, with the nominal inflow given at α=-26°. The test section comprises seven free standing blades forming 8 full passages (cascade sidewalls are shaped as pressure and suction sides of the blade profile respectively). One of the blades in the cascade can be made to oscillate in controlled 3D rigid body modes. The oscillation of the blade in different mode shapes is achieved by employing a mechanical blade actuation mechanism. A detailed description of the mechanism is included in Vogt (2005). The actuator located below the hub is connected to the oscillating blade by means of a hexagonal joint. The magnitude of the investigated modes is set such that the passage throat is subject to variations in the order of magnitude of 1%. The mechanism contains two co-rotating circular eccentric cams, which induce a sinusoidal oscillatory movement of the blade. By controlling the phase lag between the two cams, the achieved mode can be varied continuously from pure bending through combined motions to pure torsion. Blade-to-blade stagger angle variations, investigated in this study, are obtained by turning the actuator connected to the blade.


Figure 5.3: Test facility and sketch of the test module

After passing the cascade, the flow is discharged through an adjustable annular sector duct, leading into a fully circular outlet plenum. Adaptation of the inclination the sidewalls of the outlet duct enables high degree of flow periodicity in the cascade to be achieved, as shown by Vogt (2005). Continuous variability of the inlet and outlet duct walls is obtained using a semi flexible structure containing a semi-flexible framework made of steel molded polyurethane. This results in smooth walls which are able to elongate, bow and twist while withstanding normal loads. Figure 5.4 shows the inlet and outlet sidewall structures.

Inlet sidewalls and cascade

Outlet sidewalls

Figure 5.4: Inlet and outlet sidewalls; Vogt (2005)

Inlet settling chamber and flow conditioner

Outlet settling chamber

Adjustable inlet sidewalls for setting inflow direction

Adjustable outlet sidewalls for periodicity control

Test section (Annular sector) One blade oscillated

Flow Flow


The airflow from the outlet plenum of the facility is discharged to the ambient through a double-walled circular silencer. An extensive validation of the test setup was carried out by Vogt (2005). The validation procedure addressed issues such as the three-dimensional flow field, steady and unsteady passage-to-passage periodicity and three-dimensional blade oscillations. It has been verified that the facility is a valid test setup for turbomachinery flutter investigations. 5.1.3 Coordinate Systems

Two coordinate systems are employed in the present work: a global test rig coordinate system and a local on-blade coordinate system. The z-axis of the global coordinate system is aligned with the machine axis and is oriented in main flow direction, as indicated in Figure 5.5.The x-axis of the system leads through the leading edge stagnation point at the hub of the center blade (index 0), while the y-axis is defined according to the right-hand rule and points tangentially in the direction of the negative blade indices. The polar angle of the corresponding cylindrical coordinate system is ascending in the direction of negative blade indices. The radial direction points from the origin outwards.

Figure 5.5: Global test rig coordinate system Blade indices in the cascade are ascending in the direction of the suction side and descending in the direction of the pressure side and range from -3 to +3, as Figure 5.5 indicates. The local on-blade coordinate system is defined by an arcwise and a spanwise coordinate, as shown in Figure 5.6. With the origin at the leading edge stagnation

y-axis

x-axis

z-axis

direction

Origin

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

-1 +1 +3

-2 -3

0 +2


point (at nominal flow) the arcwise direction follows the blade surface up to the trailing edge at constant spanwise height. As such, the arcwise coordinate along the suction side is per definition negative and along the pressure side is positive, see Figure 5.6. The arcwise coordinate is normalized by the local arc length. The spanwise direction has its origin at the hub and follows the blade surface at constant arc. The spanwise coordinate is normalized by the total local channel height and is ascending towards blade tip.

Arcwise coordinates at midspan Plane representation of blade surface

Figure 5.6: Local on-blade coordinate system 5.1.4 Convention of Blade Oscillation and Stagger Angles

The investigated blade oscillation mode shapes are obtained by rigid-body rotation around an axis which can be described as

iwteAt )( Eq. 5-1 The center of the blade oscillation is located at the position of the blade swivel bearing pivot point, which described in global test rig coordinates as being at x=0.375m, y=-0.0038m and z=0.0181m. The axis of rotation for the bending modes lies normal to the radial direction pointing towards the center of oscillation. The axis of rotation for axial bending is parallel to y-axis of the global coordinate system; while the circumferential bending rotation axis is parallel the machine axis. For torsion modes, the axis of rotation collides with the radial direction pointing to the center of blade oscillation. In terms of local blade coordinates the center of rotation is located at 40% axc . The convention of the three investigated orthogonal

modes is depicted in Figure 5.7.

−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, −


Axial bending Circumferential bending

Torsion

Figure 5.7: Orthogonal oscillation modes The stagger angle is per definition the angle between chord line and the turbine axial direction (machine axis). The stagger angle is varied by turning the blade around the axis of rotation of the torsion mode and a positive change in stagger angle corresponds to a positive direction of torsional motion, as Figure 5.8 shows.

Figure 5.8: Definition of stagger angle and de-staggering convention

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

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

Axis of rotation

Direction of positive motion

Axis of rotation


Axis of rotation (out of page)



Torsion axis

Machine axis

Chord line

Stagger angle


5.1.5 Measurement Equipment

The test facility is equipped with both steady-state and unsteady measurement devices. The steady-state instrumentation serves to monitor the flow parameters and determine the mean flow characteristics. The unsteady measurement setup was adapted to determine the response of the aerodynamic system caused by controlled blade oscillation and to monitor blade oscillation parameters during the oscillation. 5.1.5.1 Steady-state measurement setup The steady blade loading in the cascade is mapped by means of a total of 57 arcwise distributed static taps, located at different span positions (10%, 50%, and 90%). The distribution of pressure taps is presented in Figure 5.9. In the present investigations, blade measurements are carried out only for span positions 10%, 50% and 90% of the blades -1, 0 and +1 and data is mainly used for purposes of comparison with numerical results. To assess distribution for one span position, two differently equipped blades are employed – one for the suction and one for the pressure side. The pressure taps on the blade surface are connected by stainless steel tubes of 0.4mm in diameter to the blade root and thereafter by vinyl tubes to the pressure measurement equipment. The steel tubes are embedded into a milled channel in the blade surface that is afterwards covered with resin and polished to achieve smooth surface. On each blade, three additional taps are placed at identical locations to serve as a reference when recombining test data from several runs. Instrumented blades were connected by means of miniature quick disconnect couplers (Scanivalve, 19 channels) under the test section hub so as to allow for time-efficient exchanging of the blades (see Figure 5.10).

Arcwise distribution of pressure taps at 50% of span

Pressure tap distribution on the blade surface

Figure 5.9: Distribution of the pressure taps on the blade surface (non-oscillating

blade) The sampling of blade static pressures is performed by means of a multi-module system containing six PSI9010 pressure scanners units with 16 channels in each. The scanners feature the possibility of using a different reference pressure for each channel. However, for the present investigation atmospheric pressure was used as the reference pressure for each module. The range of the sensors in the system varies between 1 PSI and 30 PSI (210KPa), with accuracy of the system at

−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 0.40

0.2

0.4

0.6

0.8

1

10%

50%

90%


norm

aliz

ed s

panw

ise

coor

dina

te, −


0.05% of the maximum range. The system is connected to the measurement computer via a serial connection. For steady-state measurements each data point represents an average of 20 samples acquired at a rate of 10Hz.

Figure 5.10: Non-oscillating instrumented blades

The measured static pressures on the surface of the blades were reduced to normalized pressure coefficient as

refdyn

refstatp p

pxpxc

,

,)()(

Eq. 5-2

The reference dynamic head refstatrefrefdyn ppp ,,0, was obtained from total

pressure measured in the settling chamber and static pressure measurements at 20% axial chord upstream of the cascade. The global flow parameters, such as atmospheric pressure, mass flow, total inlet pressure, static outlet pressure and total inlet temperature are continuously monitored and logged during both the steady and unsteady measurements. The atmospheric pressure is measured by means of a Solartron barometer with an accuracy of 0.01% (p=±11.5Pa). Mass flow measurements are taken in the air supply system by a standard orifice (accuracy 2%). The inlet and outlet pressures are measured by 15 PSI channels of a 16 channel PSI9016 system (accuracy ±50Pa).The total temperature is measured with a PT100 sensor yielding 0.1K of accuracy. 5.1.5.2 Unsteady measurement setup The unsteady measurement setup contains fast-response instrumentation for measuring the unsteady pressure on the blade surface. The setup also includes devices for characterization and monitoring of the blade oscillations.

SS PS


The unsteady blade surface pressure is measured by means of fast-response pressure transducers mounted in recessed manner i.e. the transducers are placed underneath the hub and are connected via pressure tubes. In this way the transducers are placed away from the deterioration influence of high accelerations within the blade and can be reused for different instrumented blades assemblies in the cascade. This technique allows considerably easier instrumentation of the blades, where often only marginal mounting space exists. The transducers can be connected either to the non-oscillating neighbor blades, described in previous section, in order to measure coupling coefficients or to the instrumented oscillating blade for measuring the direct coefficients (blade 0). The oscillating blade is instrumented with 19 pressure taps (d=0.4mm) at midspan section. From pressure taps miniature spark eroded holes (d=0.9mm) lead to the lower end of the blade and thereafter by means of PVC tubes are connected to the lower part of the blade root. The tubes are molded into a flexible transition part made of cold casting polyurethane. The oscillating blade and arcwise pressure tap distribution on the blade are shown in Figure 5.11.

Instrumented oscillating blade Arcwise distribution of pressure taps

(midspan section)

Figure 5.11: Oscillating blade used for the unsteady pressure measurements Mounting piezoelectric pressure transducers in a recessed manner instead of placing them directly on the location where pressure is to be measured requires that one compensates for possible dissipation in the measured signal due to the distance between the transducer and the pressure tap. Vogt & Fransson (2004) described a technique for dynamic calibration of recessed-mounted pressure transducers. The dynamic calibration setup is depicted in Figure 5.12. Recessing the piezoelectric transducers leads to damping characteristics and lagging of the pressure perturbation, which can be described with dynamic transfer functions containing magnitude ratio and phase lag.


Figure 5.12: Dynamic calibration setup

The calibration setup contains a pulse generator, a reference cavity, and a high-speed data acquisition system. An in-house built pressure pulse generator provides a pressure signal at variable frequency, amplitude, shape and mean level, as described by Vogt (2001). The pressure pulses are delivered through a cavity and applied to a blade pressure tap. At the head of the calibration cavity, a fast-response pressure sensor is mounted for acquiring the reference signal. The pulse signal is applied to a pressure tap which in turn is connected to an identical pressure sensor to the one measuring the reference signal. Post-treating these two signals in the frequency domain yields a complex dynamic transfer function in terms of damping and phase. A typical transfer function is shown in Figure 5.13. Dynamic calibration is performed in the range of 10Hz to 2000Hz, with refined steps within the region of interest (40Hz-300Hz). Having a proper transfer function for each of the blade pressure taps enables reconstruction of adequate unsteady pressure values from the ones acquired by recessed transducers.

Figure 5.13: Example of a transfer characteristic

0 50 100 150 200 250 300 350 4000.2

0.4

0.6

0.8

1

frequency, Hz

dam

ping

, −

0 50 100 150 200 250 300 350 4000

50

100

150

frequency, Hz

phas

e sh

ift, d

eg


During unsteady measurements, the unsteady pressure is measured by means of 20 fast-response pressure transducers of the Kulite type (KULITE XCQ-062 and LQ-080). The signals from the sensors are acquired with a digital high-speed data acquisition system (Kayser Threde KT8000). The system features 32 channels with 14bit A/D conversion for each channel. Maximum sampling rate for all 32 channels simultaneously is 200KHz. The gain and cut-off frequency of the low-pass filter for each channel can be programmed individually. The tests were performed with a gain of 25, no low-pass filtering and at a sampling rate of 20KHz. Taking into account the accuracy of the sensor, resolution of the data acquisition system and transfer function accuracy, the total uncertainty for unsteady pressure measurements was determined to be ±130 Pa. The data reduction procedure was performed using the software package MATLAB, where ensemble-averaging of the unsteady pressure data was done with respect to the oscillation. Data sampling was typically carried out over 200 oscillation periods. The unsteady pressure is normalized by the oscillation amplitude and the reference inlet dynamic head in order to obtain the unsteady pressure coefficient. Signal analysis of unsteady pressure coefficient yields the first harmonic amplitude Apc ,ˆ and phase Acp related to the oscillation as

tieAt )( Eq. 5-3 Apcti

ApAp exctxc

)(ˆ),(~

,, Eq. 5-4

The dynamic characteristics of the blade oscillations are determined by employing different trigger devices located in the actuation mechanism. Two of the devices provide one-per-rev signals from the two actuator cams which can be used to determine relative cam phase lag and to give phase reference for the unsteady pressure measurements. A third trigger provides 100 pulses per blade oscillation period for verifying the constancy of the blade actuation frequency. The motion of the oscillating blade during the measurements is continuously verified by using a point-wise laser vibration measurement system (Optocator, LMI Selcom AB). The working principle of the laser system is based on optical triangulation. The signal from the laser is acquired by the high-speed data acquisition system described in the text above. In order to be able to determine the blade oscillation amplitude from the acquired laser signal, calibration of the signal had to be performed. The laser signal has been calibrated against the actual amplitude measured employing an analogue measurement clock (setup shown in Figure 5.14. The actual oscillation amplitude differs from the theoretical one due to the wear issues in the actuation mechanism. Calibration provides a slope function that correlates the actual oscillation amplitude and the laser signal readings. Validation of the blade oscillation amplitude at variable frequencies showed that the resulting oscillation amplitude increases with increasing frequency. During aerodynamic mistuning tests, the laser system is used to determine the actual de-stagger angle that has been applied on to blade. The signal from the laser is acquired by means of the previously described digital high-speed data acquisition system (Kayser Threde KT8000).


Amplitude measurements [mm] Laser signal [Volt]

Figure 5.14: Laser signal calibration setup

The calibration curves that co-relate the measured laser signal to the motion amplitude are displayed in Figure 5.15. The slope of the calibration curve is used to calculate the actual amplitude of the oscillation from the laser signal acquired during the flutter measurements. Accuracy of the oscillation amplitude measurements was determined to be ±0.05mm. The laser signal is also used to determine the phase of the oscillation i.e. to give phase reference for the unsteady pressure measurements. Accuracy of the oscillation phase measurements was determined to ±2 degrees. The total uncertainty for the unsteady pressure phase measurements was determined to ±20 degrees. This average uncertainty was estimated through taking into account uncertainty of the pressure phase determination during the ensemble-averaging of the pressure signals and is stated at 95% confidence level assuming normal probability distribution of the error. It also includes the uncertainty of the oscillation phase measurements from the laser signal.

Figure 5.15: Calibration curves of the laser signal for the investigated orthogonal

modes

−1 −0.5 0 0.5 11

1.5

2

2.5

3

measured amplitude, deg

lase

r si

gnal

, V

slopetorsion

=0.22

slopeaxial

=−1.08

slopecirc

=0.57


Experimental Testing Strategy 5.2 The experimental investigations have been divided in two main testing campaigns: 1) combined mode testing and 2) aerodynamic mistuning testing. Combined mode testing consisted of measuring the unsteady response data for the blade oscillating in two different combined mode shapes: axial bending-torsion and circumferential bending-torsion mode. The combined modes featured 90° phase lag between bending and torsion. Two velocity levels have been investigated: low subsonic velocity (M2=0.4) and high subsonic velocity (M2=0.8). The velocity levels are determined based on values measured at 20% axial chord downstream of the cascade. In order to investigate the validity of the principle of linear superposition of mode shapes in a parametric manner, a series of experimental tests was performed for the combined modes at different reduced frequencies and different bending–torsion amplitude ratios. Tests were conducted at reduced frequencies in the range of k=0.1 to 0.4 for low subsonic velocity level and up to k=0.2 for the high subsonic velocity level. The investigated bending-torsion amplitude ratios at low subsonic velocity levels were R=1, R=0.5 and R=2. In addition, the inflow incidence was varied from nominal (α=-26deg, ”nom”), to negative incidence (α=0deg, ”off1”) and high negative incidence (α=14deg, ”off2”). Unsteady blade surface pressure data were acquired at midspan on the oscillating blade (blade 0) as well as the two neighboring blades (blades -1 and +1). An overview of the investigated operating points is included in Table 5.2. Table 5.2: Test conditions

Parameter Unit M04 M08

nom off1 off2 nom off1 off2 m kg/s 2.4 2.4 2.4 4.95 4.95 4.95

T01 K 303 303 303 304 304 304 p01 kPa 111.7 111.9 112 153.5 155.9 156 p2 kPa 102 102 102 106.4 106.4 106.4

201 - 1.095 1.097 1.098 1.442 1.465 1.466 kmax - 0.4 0.4 0.4 0.2 0.2 0.2

The aerodynamic mistuning tests included investigations of the three pure orthogonal mode shapes (axial bending, circumferential bending and torsion). During the tests, the stagger angle of the oscillating blade was gradually varied within the range of =-2.5deg to +2.5deg. The velocity levels and investigated reduced frequencies remained the same as for the combined mode tests, while incidence angle was kept nominal. Aerodynamically mistuned influence coefficients were determined from the unsteady surface pressures acquired at the midspan section of blades -1, 0 and +1.



6 NUMERICAL INVESTIGATIONS

Numerical Method 6.1 Numerical simulations are carried out using a commercial CFD code (ANSYS CFX v11). The solver is using a full-scale time-marching 3D viscous model. The underlying equations, i.e. the three dimensional Navier-Stokes equations in their conservation form, are solved using a Finite Volume method, where the equations are integrated over finite control volumes. As such, the solution domain is subdivided into a finite number of control volumes employing a suitable grid, which defines the control boundaries around a computational node located at the center of each control volume. 6.1.1 Governing equations

In fluid dynamics, the fluid flow is governed by the conservation laws for mass, momentum and energy. The basic conservation laws are formulated by using Leibniz-Reynolds transport theorem, which is an integral relation stating that changes in some intensive property defined over a control volume must be equal to what is lost (or gained) through the boundaries of the volume plus what is created/consumed by sources and sinks inside the control volume. The three-dimensional conservation equations (mass conservation, x-,y- and z-momentum equations and energy equation) for a compressible fluid can be given in differential form (Versteeg & Malalasekera, 1995) as follows

0

ut

Eq. 6-1

Mx

zxyxxx Szyx

p

Dt

Du

Eq. 6-2

My

zyyyxy Szy

p

xDt

Dv

Eq. 6-3

Mz

zzyzxzS

z

p

zxDt

Dw

Eq. 6-4

E

zzyzxz

zyyyxy

zxyxxx

STk

z

w

y

w

x

w

z

v

y

v

x

v

z

u

y

u

x

u

uDt

DE

Eq. 6-5

where the source terms MzMyMx SSS ,, include contributions due to body forces only.


The specific energy of the fluid E in Eq.6-5 is defined as the sum of internal

(thermal) energy i and kinetic energy 222

2

1wvu , while the effects of

gravitational potential energy changes are included as energy source term (gravitational force is regarded as a body force). It is common practice to extract the changes of the kinetic energy in order to obtain an equation for internal energyi . This is done by taking the part of energy attributable to the kinetic energy, which can be obtained by multiplying the momentum equations with corresponding velocity components and adding them together, and subtracting it from the above presented energy equation (Eq.6-5)

i

zzyzxzzy

yyxyzxyxxx

STk

z

w

y

w

x

w

z

v

y

v

x

v

z

u

y

u

x

u

upDt

Di

Eq.6-6

Among the unknowns in the above presented conservation equations are four thermodynamic variables: ip,, and T . The relationship between these thermodynamic variables can be obtained through the assumption of thermodynamic equilibrium, where the state of substance in the equilibrium can be described by means of just two state variables. This yields, for a perfect gas, the well-known equations of state:

RTp and Tci v Eq. 6-7

In the flow of compressible fluids the equations of state provide the linkage between the energy equation and the mass conservation and momentum equations. The governing equations also contain the viscous stress components ij , and in a

Newtonian fluid the viscous stresses are proportional to the rate of deformation or the strain rate. For compressible flow, Newton’s law of viscosity involves two constants of proportionality: dynamic viscosity , to relate stresses to deformation, and second viscosity , to relate stresses to volumetric deformation. The second viscosity is very difficult to determine and is often neglected. The shear stress components can thereby be expressed as

ux

uxx

2 uy

vyy

2 uz

wxx

2

x

v

y

uyxxy

x

w

z

uzxxz

y

w

z

vzyyz Eq. 6-8

The introduction of the above stress components into the momentum equations (Eq. 6-2 - Eq. 6-4) yields the so called Navier-Stokes equations, which after some re-writing can be given as

MxSux

p

Dt

Du

Eq. 6-9


MySvy

p

Dt

Dv

Eq. 6-10

MzSwz

p

Dt

Dw

Eq. 6-11

Introducing a Newtonian model for viscous stresses in the internal energy equation (Eq. 6-6) gives after some rearrangement

iSTkupDt

Di

, Eq.6-11

where all the effects due to viscous stresses are described by the dissipation function . Having established a system of seven equations (five partial differential flow equations and two algebraic state equations) with seven unknowns, the system is mathematically closed and can properly model the dynamics of the time-dependent three-dimensional flow of a compressible Newtonian fluid. The introduction of a general variable into the conservative form of the fluid flow equations discussed above, yields a so-called general transport equation describing the transport of a fluid property through a continuum

Sut

)(

Eq. 6-12

The left hand side of the equation contains the rate of change term and the convective term, while on the right hand side are the diffusive term (where is diffusion coefficient) and the source term. Equation 6-12 is used as a starting point for computational procedures in the finite volume method, where the transport equation is integrated over a three-dimensional control volume. 6.1.2 RANS and Turbulence modelling

The majority of flows of engineering significance are turbulent. Turbulence consists of fluctuations in the flow field in time and space. It is a complex, three- dimensional, unsteady process consisting of a wide range of length and time scales. Turbulence occurs when the inertial forces in the fluid become significant compared to the viscous forces, and is characterized by a high Reynolds Number. In principle, the Navier-Stokes equations describe both laminar and turbulent flows without the need for additional information. However, turbulent flows at realistic Reynolds numbers span a large range of turbulent length and time scales, and would generally involve length scales much smaller than the smallest finite volume mesh, which can be practically used in a numerical analysis. Although being possible, the direct numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult and expensive in computational efforts. To enable the effects of turbulence to be predicted, time-averaged equations such as the Reynolds-averaged Navier-Stokes equations (RANS), supplemented with turbulence models (such as the k - or k -model), are used. Due to the averaging procedure, information from the full Navier-Stokes equations is lost and it is supplied back into the code by the turbulence model. The idea behind the


equations is Reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities. The Reynolds decomposition defines flow property as the sum of a steady mean (time

average) component and a time varying fluctuating component t with zero mean value. Hence

)()( tt Eq. 6-13 The mean or averaged component is given by:

tt

t

dttt

)(1 Eq. 6-14

where t is a time scale that is large relative to the turbulent fluctuations, but small relative to the time scale to which the equations are solved. For compressible flows, the averaging is actually weighted by density (Favre-averaging). For simplicity, it is assumed that density fluctuations are negligible, but mean density variations are not. Substituting the averaged quantities into the instantaneous transport equations results in the Reynolds averaged flow equations for turbulent compressible flows: Continuity:

0~

Ut

Eq. 6-15

Momentum:

MxSz

wu

y

vu

x

uU

x

PUU

t

U

2~~~

~ Eq. 6-16

MySz

wv

y

v

x

vuV

y

PUV

t

V

2~~~

~ Eq. 6-17

MzSz

w

y

wv

x

wuW

z

PUW

t

W

2~~~

~ Eq.6-18

Scalar transport equation:

Sz

w

y

v

x

uU

t

~~~~

Eq. 6-19

In the presented equations the “overbar” indicates a time-averaged variable and the “tilde” indicates a density-weighted or Favre-averaged variable (Versteeg & Malalasekera, 1995) Extra terms appear in the Reynolds averaged flow equations due to interactions between the various turbulent fluctuations. These additional terms ( jivv ) are

the so-called Reynolds stresses, arising from the non-linear convective term in the un-averaged equations. On the energy equation an equivalent term is found, the Reynolds flux.


In order to be able to compute turbulent flows with RANS equations, it is necessary to develop turbulence models to approximate the Reynolds stresses and thus close the RANS equation system. The RANS turbulence models are classified on the basis of the additional transport equations that need to be solved together with the RANS flow equations. Most widely used are the two-equation models (such as k or k model), consisting of two additional partial differential equations for the different turbulence quantities. The model used in the present work is the standard k model, where the gradient diffusion hypothesis is used to relate the Reynolds stresses to the mean velocity gradients and the turbulent viscosity. The model delivers an additional equation for turbulent kinetic energy k and one for the dissipation rate . The equations are coupled with the momentum equations via the eddy viscosity. The model is based on the eddy viscosity concept stating that the effective viscosity is achieved as the sum of dynamic and turbulent viscosity, which describes the increase of the viscosity due to turbulent fluctuations. The turbulent viscosity is modeled as the product of a turbulent velocity and turbulent length scale. The turbulence velocity scale and length scale l are defined using the turbulent kinetic energy k and the eddy dissipation as follows:

k ,

3

2

kl Eq. 6-20

Applying dimensional analysis, the eddy viscosity t can be specified as

2kClCt Eq. 6-21

where C is a dimensionless constant.

The standard k model (Launder and Spalding, 1974) uses the following transport equations for kinetic energy and eddy dissipation:

ijijtk

t SSkUkt

k.2

Eq. 6-22

k

CSSk

CUt ijij

t2

21 .

Eq. 6-23

where ijS

is the mean component of the rate of deformation, while

,,,, 21 kCCC are adjustable constants, usually having values that are arrived

at by a comprehensive fitting for a wide range of turbulent flows. At high Reynolds numbers the viscous sublayer of the boundary layer becomes very thin, so that an increasing mesh density near the wall is necessary. In situations like this the standard k model avoids the need to integrate the model equations all the way to the wall surface by making use of a universal behavior of the flow in this region. In order to model the near-wall flow properly without resolving the turbulence equations in the boundary layer, wall functions are used and thus a lot of computational effort is saved. These empirical functions relate the local wall shear stress to the mean velocity relying on the existence of a logarithmic region in the velocity profile. Wall functions need the first calculation node to be placed on the log-law region. At low Reynolds numbers the log-law is not valid so modifications to the k model equations (Eq. 6-21 through 6-23) need to be performed (reviewed in Patel et al.,1985).


Description of Numerical Model 6.2 The simulated domain contains a sector of seven freestanding blades where for the unsteady simulations a harmonic sinusoidal motion of the center blade in various rigid-body modes has been employed. The motion of the oscillating blade is described by a set of equations and imposed as a moving mesh boundary condition. The geometry of the modeled sector is depicted in Figure 6.1.

Figure 6.1: Modeled sector cascade The effects of having a finite cascade of blades on the implementation of the influence coefficient technique has been previously addressed by Vogt (2005) where a comparison between results obtained from simulations in the influence coefficient domain and results from travelling wave mode simulations were compared. The outcome of that investigation showed that the employed method of simulating in the influence coefficient domain in a circumferentially limited sector cascade is a valid representation as the observed differences were negligible. In the present investigations the boundary conditions on the sidewalls of the modeled sector were pre-set to be rotationally periodic, in other words, the lateral side walls are not modeled. Based on the findings presented by Vogt (2005), where a highly accurate third order Euler approximation is compared with a lower first order discretization scheme, it was decided that a second order backward Euler approximation should be used in the present investigations. Although it was demonstrated that low order numerical schemes can generally be used for these kinds of flow situations, the choice was made to use higher order scheme due to the fact that the first order scheme may lead to minor differences in response magnitude due to an increased numerical damping behavior. For the discretization of the convective terms, the High Resolution advection scheme is selected. The temperature throughout the flow is predicted with the Total Energy heat transfer model. It models the transport of enthalpy and includes kinetic effects and is especially suitable and recommended for compressible flows. All simulations are conducted using a standard k-ε turbulence model with wall functions. In order to be able to relate the current numerical results with the ones obtained by Vogt (2005), the same grid was also employed in the present work. The mesh was created in VOLMOP grid generator, a part of an in-house code from Volvo Aero.


The mesh topology consists of an H-O multiblock structure with an O-grid around the blades and H-grid in the blade-to-blade passages. Mårtensson and Vogt (2005) carried out an extensive mesh sensitivity study where several different mesh topologies and resolutions were tested (Figure 6.2). The unsteady simulations were performed using an in-house CFD solver employing a linearized Euler method for prediction of the aeroelastic properties. The simulations were conducted in the travelling wave mode with phase-lagged periodic boundary conditions. The conclusion was that unsteady results depend both on proper mesh distribution and mesh resolution i.e. having a too coarse mesh and disadvantageous mesh distribution might seriously affect the correctness of the unsteady simulation results. The differences in unsteady results between medium and fine meshes were however found to be small. Hence, the medium mesh was chosen as a good compromise between accuracy and computational effort. Although these meshes were constructed for an inviscid solver, they are still considered as valid for the present investigations since the viscous effects in the investigated subsonic cascade have low significance. The current simulation domain consists of 507276 hexahedral volume elements with 540498 nodes.

Coarse Medium Fine

Figure 6.2: Different investigated mesh types; Mårtensson (2005) In order to optimize the simulation effort, tip clearance has not been modelled. Results from previous investigations (Vogt et al., 2007) have shown that models without tip clearance feature a more favorable convergence while the prediction accuracy at midspan is not affected considerably. The investigation carried out by Glodic et al. (2012) confirmed these findings and showed that models without tip clearance were equally able to predict the unsteady response at midspan as models with tip clearance included. At the inlet of the domain a total pressure profile boundary condition has been applied. To account for the viscous effects from the hub and shroud, the total pressure distribution at the inlet includes a turbulent boundary layer and is generated by specifying the values of total pressure, static pressure and the boundary layer thickness expressed as a percentage of the channel height. 5% was determined to be suitable considering the characteristics of the test case. At the outlet the average static pressure value was specified. Turbulence intensity was pre-set to a medium level (5%). Both the Tu-level as well as the BL thickness has been measured. A BL thickness of 5% has been verified, while the Tu-level was measured to be 2%. However, the simulation results obtained with 5% Tu-level did not change when the level was set to 2%.


Hub, shroud and blades are defined as smooth walls with no-slip condition. The side walls are rotationally periodic interfaces. The flow regime in the domain is pre-defined as subsonic.

Simulation approach 6.3 The simulations using ANSYS-CFX were performed as follows: first a steady state solution for a specific operating point was obtained. This solution is then used as initial value for the transient simulation. In the unsteady simulations, the motion of the center blade was prescribed by a set of equations for the requested mode. The correctness of the imposed motion is confirmed by an analytical model. A time-marching solution was acquired spanning typically 3 oscillation periods. The flow could be regarded as time periodic already after the second oscillation cycle (criterion used: >0.5% phase-locked difference). A time trace of unsteady pressure coefficient at a chosen point on the blade surface is presented in Figure 6.3. One oscillation period was resolved by 20 time steps and with three iteration loops per time step.

Figure 6.3: Time history of unsteady response on the surface of the oscillating blade

ANSYS CFX offers a possibility of applying mesh deformation on particular mesh regions if the motion of geometry domain in this region is known. In this case the motion of the blade is determined in terms of node displacements relative to the initial mesh. The nodes of the mesh are displaced in accordance with specified values defined by expressions written in CFX Expression Language (CEL) code. Mesh deformation of the other sub-domains or boundaries are either unspecified (as in case of mesh on the hub and shroud) or constrained (all remaining sub-domains or boundaries). The oscillation of the blade 0 is assumed to be a 3D rigid body movement where the blade can oscillate in three orthogonal modes (axial bending, circumferential bending and torsion) or in combined bending-torsion modes. Rigid body movement is an idealization where deformations are ignored. Accordingly, the blade movements are described as ordinary rigid body motion via vector analysis by rotating the blade around the corresponding axes by the angle φ. The time-dependent angle is defined as

0 0.2 0.4 0.6 0.8 1−3.52

−3.5

−3.48

−3.46

−3.44

−3.42

−3.4

normalized time, t/T

pres

sure

coe

ffici

ent C

p, −

Time history at point span 0.5 arc −0.2 on blade 0

osc.period 1osc.period 2osc. period 3osc.period 4


T

tt

2cosmax (6.24)

This angle of rotation has in current simulations never exceeded value of 0.25 degrees i.e. maximum displacement during the blade oscillations was considerably less than 1 % of the blade chord, which ensures that non-linear effects in the unsteady aerodynamics are avoided. Simulation data post-processing includes a reduction of time dependent data to complex unsteady pressures and calculation of stability data. Time-resolved blade surface pressure data is exported form ANSYS CFX into Matlab, where transformation into harmonic complex pressure data is performed. 6.3.1 Aerodynamic mistuning simulations

The variation of blade-to-blade stagger angle is introduced in the same manner as the blade oscillation. The stagger angle of an individual blade is changed by rotating the blade around the torsion axis with a targeted de-stagger angle. The rotation is applied in a first time step of transient simulation. The blade oscillation is thereafter introduced first after the flow field around the de-staggered blade has stabilized and a steady state is reached (typically after a time period corresponding to two oscillation periods). The blade is thereafter oscillated for three oscillation periods as the solution is converged already after the second period of oscillation. The applied de-stagger angles were in range of -2.5° to +2.5° degrees. In order to assess the perturbed influence coefficients matrices described in section 2.6, in addition to the nominal case simulation, at least three more simulations per each investigated stagger have been performed: one simulation where de-stagger is applied on the oscillating blade and two additional simulations where each of the neighboring blades was de-staggered. Together with the nominal geometry simulation, it is in overall 4 simulations needed if the aerodynamic influence coefficients on blades -1, 0 and +1 are considered. The resulting perturbed matrix is of tri-diagonal character where diagonal terms are the influence of the blades on themselves and the off-diagonal terms are containing the influence of the neighboring blades. Aeroelastic stability analyses of a randomly mistuned blade rows are in this case constrained by a requirement to have sufficient spacing between the de-staggered blades to assume that the influence of the two de-staggered blades is not affecting each other. Consequently, between two de-staggered blades there must be at least four nominal blades to avoid interference between perturbation influences.



7 RESULTS

The results obtained from test data and numerical simulations are included in the present section. First, steady blade loading results are presented for tuned and mistuned cascades. Thereafter flutter data, in form of unsteady response on the surface of blades, is presented and discussed. The focus here is put on the midspan section of the blades, where both test data as well as numerical results are available. On the neighboring non-oscillating blades, beside the midspan section, two additional span sections are considered (10% and 90% span). The unsteady results are covering two areas: response properties of combined mode shapes (i.e. validation of mode linear superposition principle) and influence of aerodynamic mistuning on the cascade aerodynamic influence coefficients. The results from the numerical simulations are correlated to the test data in order to validate the numerical model. Deeper aeroelastic stability analyses are carried out for both nominal and mistuned blade rows.

Steady-State Results 7.1 The steady blade loading data is presented in form of pressure coefficient distribution along the normalized arcwise coordinate where the leading edge is located at the origin, with negative values spanning the suction side and positive values spanning the pressure side. Blade indices are included in the upper right corner of the respective graph. Figure 7.1 shows the pressure coefficient distribution measured at three different incidence angles and low subsonic velocity (M2=0.4). For all three distributions a suction peak is observed on the fore suction side around arc=-0.11, which is followed by a deceleration towards the trailing edge. From flow visualization it is identified that this suction peak coincides with a laminar-turbulent transition of the boundary layer (Vogt, 2005).

Figure 7.1: Steady blade loading at midspan section on blade 0; M2=0.4 With increasing negative incidence, the leading edge stagnation point is moved onto the fore suction side. Simultaneously the suction peak is being moved

−0.4 −0.2 0 0.2 0.4−5

−4

−3

−2

−1

0

1


Cp,

−

nomoff1off2

off1

nom

off2


downstream and is decreasing in value. For the nominal incidence angle the static pressure coefficient on the pressure side decreases monotonically from leading edge to trailing edge. The effects of the flow separation on fore pressure side, due to the negative incidence, are observed through a local decrease in pressure coefficient. It is apparent that the flow separation region increases with increasing negative incidence. It will be shown later that these effects are also recognizable when unsteady response at negative incidence angles is discussed. 7.1.1 Influence of Aerodynamic Asymmetries on Steady Blade Loading

The variation in blade-to-blade stagger angle affects passage shape with largest relative change at passage throat. The affected passages are indicated in Figure 7.2. It is to be noted that a reduction in throat size leads to increased blockage and thereby increased pressure in the passage. On the other hand an increase in throat size leads to opening up the passage and consequently a decrease in pressure.

Figure 7.2: Impact of blade 0 de-staggering on passage throats The influence of the stagger angle variation on the steady blade loading is assessed in Figure 7.3 by means of loading distribution at midspan section on the de-staggered blade (denoted with blade index 0) and its neighboring blades ±1. The observed suction peak on blades -1 and 0 is the most susceptible point when changing the blockage and the largest pressure change is observed at this location. The increased blockage in a passage weakens the pressure gradients around the peak and the peak becomes less pronounced, while for an increase in passage throat, stronger gradients are present. As a result of increase in suction side passage throat due to negative de-staggering, lower pressure levels are observed on the suction side of the blade 0. A deceleration from the suction peak towards the trailing edge is growing in strength with increasing negative de-stagger angles. The region close to the trailing edge on the suction side seems not to be considerably affected by change in stagger angle. On the pressure side, higher pressure levels could be observed for negative de-stagger angles as the throat area towards blade +1 decreases leading to increased blockage. The opposite is observed for positive de-stagger angles. On the fore pressure side a distinct local drop in pressure is observed for blade de-staggered by a positive angle. Beside the change in Cp level, the shape of the curves on the aft pressure side is unchanged with different stagger angles.

throat+1 throat-1


Figure 7.3: Influence of aerodynamic asymmetries on steady blade loading; low

subsonic case (M2=0.4); 50% span The effects of the change in stagger angle on blade 0 are propagating on the neighbour blades, mainly on the primary surfaces that enclose passage together with the de-staggered blade 0. It is obvious that the strongest impact is on the suction side of blade -1, where pressure levels are changing significantly due to de-staggering. Negative de-staggering of blade 0 is causing a decrease in throat area between blade 0 and blade -1. Thus the pressure level on the suction side of blade -1 has also increased. At the same time deceleration from the suction peak towards the trailing edge is less pronounced such that for de-stagger angle of ∆γ=-2° the pressure coefficient is fairly constant from arc=-0.2 towards the trailing edge. For the positive de-stagger angle the suction peak value on blade -1 becomes considerably lower. A very strong pressure gradient causes that pressure coefficient value recovers rapidly after the suction peak and close to the trailing edge it reaches the same level as for the nominal and positive stagger angles. The pressure side of the blade -1 is not affected by change in stagger angle on the blade 0. On blade +1 the effects are noticeable on both pressure and suction side. The differences are most visible on the aft part of the pressure side, where the passage throat is located. The suction side of blade +1 which is secondary surface i.e. faces away towards blade +2, seems to be affected by blade 0 de-staggering as well. Possible explanation for this is that the variation of stagger angle will affect the downstream flow pattern of the cascade. The steady state analysis conducted by Vogt and Fransson (2000), where the possibility of improving flow periodicity in the cascade by means of controlling the downstream flow direction was addressed, indicate that modification of downstream flow direction is leading to locally changed pressure. The suction sides of the blades tend to be more

−0.4 −0.2 0 0.2 0.4

−5

−4

−3

−2

−1

0

1


Cp,

−

00000

Δγ=−2°Δγ=−1°Δγ=0°Δγ=1°Δγ=2°

−0.4 −0.2 0 0.2 0.4

−5

−4

−3

−2

−1

0

1


Cp,

−

+1+1+1+1+1

Δγ=−2°Δγ=−1°Δγ=0°Δγ=1°Δγ=2°

−0.4 −0.2 0 0.2 0.4

−5

−4

−3

−2

−1

0

1


Cp,

−

−1−1−1−1−1

Δγ=−2°Δγ=−1°Δγ=0°Δγ=1°Δγ=2°

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2

+2

+1

0

−1

−2

+2

+1

0

−1

−2


susceptible to changes in downstream pressure, which explains the pressure variations observed on the suction side of blade +1, even though this is a secondary surface. The effects of the change in stagger angle on the steady flow field are expected to propagate on the aeroelastic response when the blade oscillation is introduced. Steady blade loading distributions acquired at high subsonic velocity are depicted in Figure 7.4. Similar to the observations made on distributions at low subsonic velocity, steady blade loading at high velocity is largely dependent on the throat variations induced by blade-to-blade stagger angle variations.

Figure 7.4: Influence of aerodynamic asymmetries on steady blade loading; high

subsonic velocity (M2=0.8); 50% span The largest change in pressure is observed at the suction side of blade -1 as well as on either surfaces of the de-staggered blade. In addition to that it can be observed that the negative de-staggering of blade 0 seems to have larger impact on the loading than the positive one. For negative de-stagger angles distinct suction peak on blade -1 cannot be recognized anymore and pressure decreases almost uniformly from about arc=-0.11 towards the trailing edge. Pressures variation is still consistent with the change of the passage throat i.e. negative de-staggering implies reduced throat in passage -1 which results in increased pressure on the surfaces enclosing this passage. The pressure side of blade 0 seems to be less affected than the suction side. On blade +1 smaller change is observed not only at the pressure side which faces towards the de-staggered blade, but also on the secondary surface of the blade. This pressure variation seems to increase when going towards the trailing edge. This, as already discussed for low subsonic levels, might be caused by a local change in flow

−0.4 −0.2 0 0.2 0.4

−5

−4

−3

−2

−1

0

1


Cp,

−

00000

Δγ=−2°Δγ=−1°Δγ=0°Δγ=1°Δγ=2°

−0.4 −0.2 0 0.2 0.4

−5

−4

−3

−2

−1

0

1


Cp,

−

+1+1+1+1+1

Δγ=−2°Δγ=−1°Δγ=0°Δγ=1°Δγ=2°

−0.4 −0.2 0 0.2 0.4

−5

−4

−3

−2

−1

0

1


Cp,

−

−1−1−1−1−1

Δγ=−2°Δγ=−1°Δγ=0°Δγ=1°Δγ=2°

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2

+2

+1

0

−1

−2

+2

+1

0

−1

−2


direction downstream of the cascade resulting in local pressure variations on the positive indexed blades. The behavior observed in measured data is well captured even in numerically predicted results. Figure 7.5 shows predicted steady loading on blades -1, 0 and +1 together with corresponding test data. It is observed that there is a good agreement between the CFD results and experimental data. There are some differences that could be pointed out: pressure variation on the suction side of blade +1 is not so obvious in the numerical results comparing to the experiments. A possible reason could be in different boundary condition on the outlet sidewalls (in computations the boundary was defined as rotational periodic) that might interact differently with changed downstream flow pattern due to de-staggering of the blade 0. On the suction side of blade -1 and for the positive de-stagger angle, the numerical results indicate lower values of pressure coefficient and slightly stronger gradients than ones obtained from experiments. Despite these discrepancies, overall agreement is on a satisfying level.

Figure 7.5: Experimental data vs. CFD predictions; Steady blade loading at

midspan; low subsonic velocity (M2=0.4);

−0.4 −0.2 0 0.2 0.4−6

−5

−4

−3

−2

−1

0

1

0


Cp,

−

Δγ=−2° CFDΔγ=−2° dataΔγ=0° CFDΔγ=0° dataΔγ=2° CFDΔγ=2° data

−0.4 −0.2 0 0.2 0.4−6

−5

−4

−3

−2

−1

0

1

+1


Cp,

−


−0.4 −0.2 0 0.2 0.4−6

−5

−4

−3

−2

−1

0

1

−1


Cp,

−


0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2

+2

+1

0

−1

−2

+2

+1

0

−1

−2


Unsteady Response- Numerical Predictions vs. Test Data 7.2 The present section contains unsteady aeroelastic response data obtained for various blades in the cascade during the controlled blade oscillation. Data is acquired at midspan section of blades -1, 0 and +1 for three different orthogonal oscillation mode shapes, various reduced frequencies and at two velocity levels. Blade-to-blade stagger angles were for this part of investigation kept identical i.e. the cascade was geometrically symmetric. The experiments have been supported numerically at specific operating points and simulation results are validated against the test data. Unsteady response data is presented in form of normalized unsteady pressure coefficients along the blade profile. Spatially resolved aerodynamic forces are calculated and nominal aerodynamic influence coefficients in form of integrated complex force components at midspan section are presented. 7.2.1 Axial Bending

Unsteady response data is presented in the form of normalized unsteady pressure coefficients along the blade profile. The response distributions for blades -1 through +1 at axial bending mode are shown in Figure 7.6. The figures contain unsteady pressure amplitude plotted in the respective top window and the response phase in the bottom window. Presented results are obtained at low subsonic velocity level (L1) and reduced frequency of k=0.3.

Figure 7.6: Unsteady blade surface pressure at midspan; axial bending; k=0.3;

M2=0.4

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

00

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

+1+1

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−1−1

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2

+2

+1

0

−1

−2


It is observed that on the suction side of blades 0 and -1 the response amplitude peaks around arc=-0.11, which was previously identified as the location of the suction peak in the steady blade loading. On blade +1 the major part of the response is observed on the pressure side facing the oscillating blade. The pressure phase suggests that the unsteady response primarily involves a flow passage and its respective surfaces. The unsteady pressure on the suction side of blade 0 and the pressure side of blade +1 is mainly in phase with the blade motion, while the pressure side on blade 0 and the suction side of blade -1 indicate opposite phase behavior i.e. flow is 180deg out-of-phase. Figure 7.6 indicates also that both amplitude and phase of the response are well predicted by the numerical model. The response magnitude in the region of the suction peak on blades -1 and 0 is slightly over predicted by the numerical model. Discrepancies in phase are observed on the aft suction side of blade 0 and aft suction side on blade +1; however the response amplitudes are very low in these regions. In general, the numerical model seems to capture the overall behavior of the unsteady flow well. Spatially resolved complex force coefficients calculated from the above presented pressure distributions are contained in Figure 7.7. The force component considered here is the unsteady force acting in the direction of the oscillation i.e. in this case the axial component. The imaginary and real parts of the force components are plotted against normalized arcwise coordinate. Presented curves are calculated at IBPA=0 deg. From the stability point of view, one should note that when concerning distributed imaginary force contributions, positive values of distributed force components have destabilizing effect while negative values indicate stabilizing effect. The numerical results generally correlate well to test data. Local differences are observed, especially in the imaginary force distribution on blade 0 and on the pressure side of blade +1. The real part of the force shows an overall good agreement between data and numerical results.

Figure 7.7: Spatially resolved unsteady force component at midspan section on

blades -1, 0 and +1 (imaginary and real); IBPA=0deg Integration of the distributed force contributions along the profile yields aerodynamic force influence coefficients presented in Figure 7.8. From an overall perspective there is a fair agreement between predicted and measured values. It is however apparent that relatively small differences in unsteady pressure

−0.6 −0.4 −0.2 0 0.2 0.4−0.05

0

0.05

Re(

dfξ),

− bl−1

expnum

−0.6 −0.4 −0.2 0 0.2 0.4−0.05

0

0.05

Re(

dfξ),

− bl0

−0.6 −0.4 −0.2 0 0.2 0.4−0.05

0

0.05

Re(

dfξ),

− bl+1


−0.6 −0.4 −0.2 0 0.2 0.4−0.02

0

0.02

Im(d

f ξ),− destabilizing

stabilizing

bl−1

expnum

−0.6 −0.4 −0.2 0 0.2 0.4−0.02

0

0.02

Im(d


stabilizing

bl0

−0.6 −0.4 −0.2 0 0.2 0.4−0.02

0

0.02

Im(d


stabilizing

bl+1



magnitude and phase can translate into considerable differences in value of integrated force coefficients. The largest differences are observed for blade 0, where experimental data indicate a more stabilizing behavior than predicted.

Figure 7.8: Nominal aerodynamic force influence coefficients for blades -1, 0 and +1; axial bending; k=0.3; M2=0.4

To take into consideration the measurement uncertainties Figure 7.9 shows the aerodynamic influence coefficients plotted in a complex plane, where the uncertainty of the measurement is presented with error bars on the measured INFCs. Plotted uncertainty is obtained assuming normal probabilistic distribution of error around the measured value within 90% confidence interval. It is clear that relatively small error in the unsteady pressure amplitude and phase measurements can propagate onto integrated force influence coefficients in such a way that the uncertainty of the calculated coefficients is highly significant.

Figure 7.9: Nominal aerodynamic influence coefficients for blades -1, 0 and +1 in complex plane

−1 0 1−5

0

5

Im(F

ξ)

expnum

−1 0 1−5

0

5

Re(

Fξ)

Blade indices

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

Re(Fξ)

Im(F

ξ)

blade −1 expblade −1 numblade 0 expblade 0 numblade 1 expblade 1 num


7.2.2 Circumferential Bending

Figure 7.10 shows unsteady blade loading for blades -1 through +1 at circumferential bending mode. The presented data is acquired at low subsonic velocity (L1) and reduced frequency of k=0.3. The observed response is qualitatively similar to the one at axial bending mode, though featuring lower response amplitude. Suction peaks on blade 0 and -1 are also present for the circumferential bending mode, yet slightly less pronounced than for the case of axial bending. Numerical predictions seem to over-predict response amplitude in these suction peak regions, while for the rest of surfaces the amplitude is very well captured by the numerical tool. A moderate peak in amplitude can also be observed on the pressure side of blade +1, around arc=0.3.

Figure 7.10: Unsteady blade surface pressure at midspan; circumferential

bending; k=0.3; M2=0.4 Concerning the phase of the response, there is an overall good agreement between the numerical results and the test data. However, some minor discrepancies in phase can be observed on the pressure side of blade 0 as well as a slight phase shift along the suction side of blade -1. Discrepancies observed on the secondary surfaces can be neglected since the response amplitudes are very low in these regions. The response along the suction side of blade 0 is in phase with motion, while on the pressure side it starts with out-of-phase, but it approaches in-phase behavior when going towards the aft part. The phases on the primary surfaces i.e. on suction side of blade -1 and pressure side of blade -1 do not agree with behavior observed on their respective adjacent surfaces on blade 0.

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

00

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

+1+1

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−1−1

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2

+2

+1

0

−1

−2


This indicates that for circumferential mode the involvement of the passage flow is not present in the same extent as it was observed for the axial bending mode. The integrated aerodynamic force influence coefficients for the circumferential bending are shown in Figure 7.11. In this case the forces acting in circumferential direction are considered. The force coefficients are calculated at the midspan section of the blades. The largest differences between predicted coefficients and experimentally obtained ones are observed for blade 0, where in this case experimental data indicate a less stabilizing behavior than predicted. Differences are also present for blade +1 where predicted value is higher. Regarding the real part, largest difference is observed on blade -1. However, from an overall perspective there is a fair agreement between predicted and measured values.

Figure 7.11: Nominal integrated aerodynamic influence coefficients for blades -1,

0 and +1; circumferential bending; k=0.3; M2=0.4 7.2.3 Torsion

Response data for torsion mode is depicted in Figure 7.12. Similar to bending modes, significant response magnitude can be observed on the oscillating blade and adjacent surfaces facing to it (pressure side of blade +1 and suction side of blade -1). Looking at the response amplitude on blade 0, it is noted that the suction peak on the fore suction side is less pronounced than in case of axial bending mode. Numerical results display over-predicted response magnitudes in this region. Similar is observed for the suction side of blade -1 where predicted suction peak is considerably higher than the measured one. Response on the pressure side of blade 0 seems to be slightly over-predicted. Secondary surfaces feature low response. The phase on the suction side of the oscillating blade is in-phase with motion, in contrast to the pressure side where out-of-phase behavior is present. The phases on the primary surfaces of the adjacent blades show similar behavior to their counterpart surfaces on the oscillating blade i.e. pressure side of blade +1 features in-phase behavior and suction side of blade -1 indicates out of phase behavior relative to the blade motion. Hence, the involvement of the passage flow seems to be dominant also for the torsion mode. The response phase on the suction side of blade +1, faced away from the oscillating blade, is equal to the phase on the pressure side, apart from the small region on the fore

−1 0 1−5

0

5

Im(F

η)

expnum

−1 0 1−5

0

5

Re(

Fη)

Blade indices


suction side where a phase jump is observed. Numerically predicted response phase on all three considered blades are in very good agreement with measured ones.

Figure 7.12: Unsteady blade surface pressure at midspan section; torsion; k=0.3;

M2=0.4 The observed differences between predicted and measured response amplitudes for torsion mode are propagating in calculations of the integrated aerodynamic force coefficients. Hence, there are differences in the integrated values, mostly pronounced in real part of the influence coefficients on blades 0 and +1, as shown in Figure 7.13. The integrated force coefficient values on blade -1 are rather small and it seems that the large differences between predicted and measured response amplitudes observed on this blade are not obvious when comparing integrated values.

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

00

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

+1+1

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−1−1

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2

+2

+1

0

−1

−2


Figure 7.13: Nominal aerodynamic influence coefficients for blades -1, 0 and +1;

torsion; k=0.3; M2=0.4 7.2.4 Impact of Flow Velocity

In order to assess the impact of flow velocity on prediction accuracy of the used numerical model a comparison between predicted and measured unsteady response at low and high subsonic velocity levels is shown in Figure 7.14. The figure contains the unsteady blade loading obtained at midspan sections of blades -1, 0 and +1. The analyzed mode is axial bending mode at reduced frequency of k=0.1. It is observed that the characteristics of the response distributions are not changing considerably between the two operating points. The magnitude of the measured response around suction peak on blade -1 and 0 is slightly lower at high velocities, while response level is slightly higher on the pressure side of blades 0 and -1. The phase of the response has not changed, and remains similar when velocity is increased. When comparing prediction accuracy, it is noted that for low subsonic velocity both response magnitude and phase are predicted with very satisfying precision. Meanwhile, at high subsonic velocity level, the numerical model tends to over-predict response magnitudes when comparing to the measured values. The phase of response is however predicted quite accurately, with some minor discrepancies observed on the suction side of blades 0 and -1. The overall agreement between simulated results and measured data is considered to be on a satisfying level.

−1 0 1−5

0

5

Im(F

ζ)

expnum

−1 0 1

−5

0

5R

e(F

ζ)

Blade indices


Low subsonic level High subsonic level

Figure 7.14: Comparison of measured and predicted unsteady loading on blades -

1, 0 and +1 at different velocity levels (M2=0.4 & M2=0.8); axial bending mode; k=0.1

The aerodynamic force influence coefficients calculated for these two velocity levels are presented in Figure 7.15. From an overall perspective there is a fair agreement between predicted and measured values, both at low and high subsonic velocity levels. It is however apparent that relatively small differences in unsteady pressure magnitude and phase can translate into considerable differences in value of integrated force coefficients.

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

00

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

00

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−1−1

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

−1−1

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

+1+1

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

+1+1

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


expnum


Low subsonic level High subsonic level

Figure 7.15: Nominal aerodynamic influence coefficients at two different velocity

levels; axial bending mode; k=0.1

Combined Mode Shapes 7.3 The present section focuses on aeroelastic properties of combined mode shapes in the oscillating cascade. The investigated combined modes are a combination of bending and torsion modes at 90 degrees out-of-phase. The obtained modes are axial bending-torsion and circumferential bending-torsion. The validity of the principle of linear superposition of pure modes is analyzed by comparing directly measured combined mode data with the results of the mode superposition. The experimental data is supported by numerical results. 7.3.1 Linear Superposition of Modes

According to the principle of superposition, under the assumption of linearity, the aeroelastic properties of combined modes are obtained by linearly combining the properties of the individual mode components at a given phase lag. Mode combination is realized at different bending-to-torsion ratios and phase angles. The linear superposition of the modes is mathematically described as

itorsionpbendingp

SAp ec

Rcc ,,, ˆ

1ˆˆ Eq. 7.1

where R stands for bending-to-torsion amplitude ratio and φ is the phase lag between the modes. For evaluating the validity of linear combination two parameters are defined giving a measure of the accuracy of mode combination. The first parameter addresses differences in unsteady response magnitude and is defined as

erpospcombpabsp ccc ,sup,, ˆˆ Eq. 7.2

−1 0 1−5

0

5Im

(Fξ)

expnum

−1 0 1−5

0

5

Re(

Fξ)

Blade indices

−1 0 1−5

0

5

Im(F

ξ)

expnum

−1 0 1−5

0

5

Re(

Fξ)

Blade indices


The second parameter addresses differences in imaginary part of the pressure coefficient and is defined as

erpospcombpp ccc sup,,Im, ÎmÎm Eq. 7.3

In order to investigate in a parametric manner the validity of the principle of linear superposition, a series of experimental testing were performed for combined axial bending-torsion and circumferential bending-torsion mode at different reduced frequencies. The investigated amplitude ratios in case of axial bending-torsion mode were R=1, R=0.5 and R=2. In addition to that the inflow incidence has been varied from nominal (α=-26deg,”nom”), to negative incidence (α=0deg,”off1”) and high negative incidence (α=14deg,”off2”). Unsteady blade surface pressure data were acquired at midspan on the oscillating blades as well as on the two neighbour blades. 7.3.2 Axial Bending-Torsion

Unsteady response data acquired at midspan section on blades -1 through +1 for the combined axial bending-torsion mode at low subsonic velocity level (L1) is presented in Figure 7.16. Data is obtained at reduced frequency of k=0.1 and bending-to-torsion amplitude ratio R=1.

Figure 7.16: Unsteady blade surface pressure at midspan, experimental data;

combined axial bending-torsion mode at R=1; k=0.1 and M2=0.4;

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−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,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


+1

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


−1

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2


The figure includes unsteady response data from combined modes (“comb”) as directly tested as well as superposition of pure axial bending and torsion mode (“superpos”). Similar to the unsteady response for pure modes discussed in previous sections, the combined mode response features largest amplitudes at the suction side of the oscillating blade and neighboring blade -1, with peak around arc=-0.11. Secondary surfaces feature a rather low response. The phase on the suction side of the oscillating blade starts of as in-phase with motion at the fore part and approaches to off-phase behavior as going towards the trailing edge. On the pressure side it starts as off-phase and decreases in value as the trailing edge is reached. Similar to what was observed for its pure mode components, involvement of the passage flow seems to be dominant even for the axial bending-torsion mode. Thus, the phases on the primary surfaces of the adjacent blades show similar behavior to their counterpart surfaces on the oscillating blade The comparisons between combined and superposed data suggest a good agreement, featuring errors generally below the measurement accuracy. Smaller differences are only present locally, mostly distinct on the oscillating blade. In order to address eventual differences on a more intimate basis the aforementioned quality parameters are determined. Figure 7.17 shows the resolved differences in absolute and imaginary value of the unsteady pressure coefficient for blades -1, 0 and +1. The imaginary value is here taken as quality indicator as it is the part that finally yields the aerodynamic damping contribution.

Figure 7.17: Differences in unsteady pressure data obtained from direct testing of

combined mode and result from pure mode superposition Both parameters show moderate values over the blade arc lying largely within ±0.05 normalized Cp. Largest differences are observed around arc=-0.2 at the

−0.4 −0.2 0 0.2 0.4−0.1

−0.05

0

0.05

0.1

Δ C

p am

p, − 0

measurement accuracy

−0.4 −0.2 0 0.2 0.4−0.1

−0.05

0

0.05

0.1

Δ C

p Im



−0.4 −0.2 0 0.2 0.4−0.1

−0.05

0

0.05

0.1

Δ C

p am

p, − +1


−0.4 −0.2 0 0.2 0.4−0.1

−0.05

0

0.05

0.1

Δ C

p Im



−0.4 −0.2 0 0.2 0.4−0.1

−0.05

0

0.05

0.1

Δ C

p am

p, − −1


−0.4 −0.2 0 0.2 0.4−0.1

−0.05

0

0.05

0.1

Δ C

p Im




suction side of the oscillating blade. It is to be noted that the aforementioned measurement accuracy translates to ±0.065 in normalized Cp value; hence the observed differences are below measurement accuracy. A series of computations was carried out in parallel with experimental investigations to investigate in parametric manner aeroelastic properties of combined mode shapes and to validate mode superposition. Comparable to the results shown in Figure 7.17, the results from simulations for the same case are plotted in Figure 7.18. The numerical model seems to capture successfully the response behavior at all investigated blades. The correlation between directly simulated combined modes and superposed modes at M2=0.4 indicates very good agreement, where differences in pressure distributions are barely visible.

Figure 7.18: Numerical results of combined axial bending-torsion mode at M2=0.4

and k=0.1 The variations of the quality parameters with reduced frequency are included in Figure 7.19 for all measured blades. Quality parameters shown in the left figure are based on experimental data, while the numerical simulation results are shown to the right. Here the average values are shown as described in Eq. 7.2 and 7.3. Two observations are made: first it is apparent that the differences lie fairly constant and well below measurement accuracy for both the differences in absolute as well as imaginary value. Second it is noted that the differences on blade 0 (i.e. the oscillating blade) tend to show larger values than on the adjacent blades. It is believed that this is partially associated with the increased complexity when measuring the unsteady pressure on an oscillating blade instead of a non-oscillating blade. From an overall perspective it is believed that the quality in

−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


0Linear SuperpositionCombined Motion

−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


+1Linear SuperpositionCombined Motion

−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


−1Linear SuperpositionCombined Motion

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2


agreement of measured combined and superposed modes is a result from measurement accuracy as well as limitations in setting the exact mode shape. The latter aspect is inherently present due to using gear drives in the oscillation actuator featuring a necessary minimum backlash. The quality parameters calculated for numerical simulation results feature low values. For the mean differences in Cp amplitude the values are around one order of magnitude less than experimentally observed. It is also interesting to note that the difference in imaginary value show a similar trend as the experimental data in that the correlations on the oscillating blade tend to show larger differences.

Test data Numerical results

Figure 7.19: Variation of quality parameters with reduced frequency; axial

bending-torsion; low subsonic velocity M2=0.4; experimental data (left) & numerical results (right);

7.3.2.1 Effects of Flow Velocity Next, attention is drawn towards the combined modes at high subsonic operating point (M2=0.8). Unsteady response data measured at midspan section on blades -1 through +1 for the combined axial bending-torsion mode at reduced frequency of k=0.1 is presented in Figure 7.20 The pressure response features similar behavior to what was observed at low subsonic velocity. The largest response amplitudes are observed on the primary surfaces (oscillating blade and surfaces faced to it). The secondary surface on blade +1 i.e. its suction side features rather considerable response magnitudes at the aft part, increasing towards the trailing edge. This behavior was also observed for the torsion mode flutter (Vogt 2005) and it is believed to be due to variations in passage outflow direction that influences the downstream flow pattern of the cascade. Vogt and Fransson (2000) highlighted that modifications in flow direction downstream of the cascade lead to local pressure variations and in case of torsion mode this seems to be more pronounced on the positive indexed neighboring blades. As the high dynamic pressures are present on suction sides of blades, these tend to be more susceptible to changes in downstream pressure. The behavior observed at the combined axial bending-torsion mode where response amplitude increases towards the aft part of the suction side of blade +1 indicates clearly the effects of the torsion component in the combined mode.

0.1 0.2 0.3 0.4 0.50

0.05

0.1


mea

n Δ

Cp

amp,

−

blade −1blade 0blade +1

0.1 0.2 0.3 0.4 0.50

0.05

0.1

k, reduced frequency

mea

n Δ

Cp

Im,−


0.1 0.2 0.3 0.4 0.50

0.05

0.1

mea

n Δ

Cp

amp,

−


0.1 0.2 0.3 0.4 0.50

0.05

0.1


mea

n Δ

Cp

Im,−


The phase on the suction side of the oscillating blade starts of as in-phase with motion at the fore part and approaches to -90deg phase as going towards the trailing edge. On the pressure side, out-of-phase behavior is observed. The involvement of the passage flow seems to be dominant even in this case. Thus, the phases on the primary surfaces of the adjacent blades show similar behavior to their counterpart surfaces on the oscillating blade.


combined axial bending-torsion mode at k=0.1 and M2=0.8 Differences between combined mode data and superposed data at high subsonic velocity are present locally and are fairly small. Discrepancies in amplitude are observed on the aft suction side on the oscillating blade as well as on blade -1 downstream of the suction peak. Small differences in magnitude are even observed along the pressure side on blade +1. The response phases obtained from superposition correlate very well to the phases measured for the combined mode. The results from numerical simulations for combined axial-bending torsion mode at high subsonic velocity are shown in Figure 7.21. It is observed that noticeable although very small differences in response amplitude are present around suction peaks on blades 0 and -1. These differences are more pronounced at higher reduced frequency as shown by Glodic et al. (2009). Looking at the quality parameter values for numerical results included in Figure 7.22, noticeably greater differences than in case of low subsonic flow are observed. The differences at reduced frequency of k=0.5 are almost one order of magnitude larger than the

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−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,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


+1

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


−1

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2


ones at low subsonic velocity (see Figure 7.19) and are comparable to the ones experimentally determined. The quality parameters calculated for experimental data show moderate values over the investigated frequency range indicating that the observed differences are below measurement accuracy (previously determined to ±0.065 in normalized Cp value).

Figure 7.21: Numerical results of combined axial bending-torsion mode at M2=0.8

and k=0.1


Figure 7.22: Variation of quality parameters with reduced frequency; axial

bending-torsion; high subsonic velocity M2=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



−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


−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 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2

0.05 0.1 0.15 0.2 0.250

0.05

0.1


mea

n Δ

Cp

amp,

−


0.1 0.20

0.05

0.1


mea

n Δ

Cp

Im,−


0.1 0.2 0.3 0.4 0.50

0.05

0.1

mea

n Δ

Cp

amp,

−


0.1 0.2 0.3 0.4 0.50

0.05

0.1


mea

n Δ

Cp

Im,−


7.3.2.2 Effects of Flow Incidence The effect of flow incidence on unsteady response for combined axial bending-torsion mode is addressed in this section. The inlet angle of incidence was varied from nominal (α=-26deg) to positive values i.e. negative off-design points as described previously. Figure 7.23 presents unsteady data acquired at low subsonic Mach number on blade -1 through +1 at off-design incidence. Considerable effects on the response amplitude are observed in regions of the fore pressure side on the blades, particularly when going towards the high negative incidence angle.

Off-design “off1” Off-design “off2”

Figure 7.23: Unsteady response on blades 0,-1 & +1; combined axial bending-

torsion mode at k=0.1 and M2=0.4; off-design 1 (left) & off-design 2 (right)

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−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,

−

combsuperpos

−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,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


−1

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


−1

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


+1

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


+1


It is apparent that on the suction side of blade 0 pressure response is largely unaffected by the change in incidence angle, when compared with the nominal incidence distribution shown in Figure 7.16. On the pressure side however the effect of flow separation is visible on the fore part. The effect of separated flow is apparent as locally increased response amplitude with peaks at arc=0.06 at off1, and at arc=0.2 at off2. The phase of the response seems to be unaffected in this region. The suction response peak on blade -1 seems to become sharper and slightly stronger for negative incidence operation. At the same time a small response peak is appearing on the pressure side on blade -1, as an effect of the flow separation caused by negative incidence. The flow separation effects are visible even on blade +1 where a distinct peak is identified at arc=0.06 terminating at arc=0.12 for off1 case, while it extends to arc=0.2 at high negative incidence case. Regardless the region of the blade (and with this the flow being attached or separated) the combined and superposed data lie very well in line with each other.

Figure 7.24: Variation of quality parameters with inflow incidence; blade 0 The variations of quality parameters with inflow incidence determined for blade 0 over the investigated range of reduced frequencies are included in Figure 7.24. Similar to the variation with reduced frequency the parameters hover around a fairly constant value and well below measurement accuracy. This result is an indication that the linear combination of local blade unsteady pressure is also correct for separated flow at the investigated range of reduced frequencies.

0.1 0.2 0.3 0.40

0.05

0.1


mea

n Δ

Cp

amp,

−

nomoff1off2

0.1 0.2 0.3 0.40

0.05

0.1



mea

n Δ

Cp

Im,−


7.3.2.3 Mode Combination at Different Bending-to-Torsion Amplitude Ratios In order to investigate mode combination at various bending-to-torsion amplitude ratios and to assess validity of mode superposition, beside ratio of R=1 two additional ratios were investigated for axial-bending torsion mode at low subsonic Mach number. The effect of amplitude ratio on mode combination is shown in Figure 7.25 by means of data on blade 0. Bending-to-torsion amplitude ratio was changed from R=1 to R=0.5 and R=2. It is apparent that even for these two different ratios the superposed and combined mode data lie very well in line with each other, similar to what was observed in Figure 7.16 the amplitude ratio of R=1.

Figure 7.25: Unsteady response at midspan on blade 0; combined axial bending-

torsion mode; at k=0.3 and M2=0.4; amplitude ratio R=0.5 (left) and R=2 (right) 7.3.2.4 Three-Dimensional Effects on Mode Combination Three-dimensional effects on aeroelastic properties of combined modes are assessed by measuring the unsteady response at 10% and 90% span section on the neighbouring blades ±1.

Figure 7.26: Unsteady blade surface pressure at 10% span on blades ±1,

experimental data; combined axial bending-torsion mode at k=0.1 and M2=0.4

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

R=0.5 combsuperpos

−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,

−

R=2combsuperpos

−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,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


−1

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


+1


Figure 7.27: Unsteady blade surface pressure at 90% span on blades ±1,

experimental data; combined axial bending-torsion mode at k=0.1 and M2=0.4 From the distributions depicted in Figure 7.26 and Figure 7.27 it can be noted that while there is a very good agreement between combined and superposed data on blade -1, some more significant discrepancies in the response magnitudes exist at 90% section on blade +1. These amplitude differences are mainly located on the aft pressure side of the blade. At the same span section, a shift in phase of approximately 50 degrees between combined and superposed data is observed on the suction side of blade +1. 7.3.3 Circumferential Bending-Torsion

Unsteady response data for the combined circumferential bending-torsion mode at low subsonic velocity level (L1) is presented in Figure 7.28. Data is acquired at midspan section on blades -1 through +1, while blade 0 is oscillating at reduced frequency of k=0.1. The response features largest amplitudes at the suction side of blade -1, with a peak around arc=-0.11. The response peak on the suction side of the oscillating blade is not equally strong, but rather smeared out, similar to the response in pure circumferential bending mode. Relatively high response magnitudes are also observed even on the pressure side of blade 0 and blade +1. Secondary surfaces feature low response. The phase along the suction side of the oscillating blade is constant around -90deg, while on the pressure it is slightly shifted from around 70 deg at leading edge towards 40deg at the trailing edge. The response phase on blade +1 is rather constant along the arc, except a small jump around the leading edge (similar to what was observed for pure circumferential mode, but in this case the response phase is out of phase by -90deg). On the suction side of blade -1 phase is constant and values around 90deg, while variations in phase on the pressure side could be neglected since the response magnitude here is low. The correlation between combined and superposed data suggests a very good agreement. Smaller differences in phase are present on the pressure side of blade 0 and blade +1. The observed differences are generally below measurement accuracy.

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6C

p am

p, −

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


−1

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


+1



combined circumferential bending-torsion mode at R=1; k=0.1 and M2=0.4; The observed correlation between combined mode data and superposed data suggests a very good agreement, and confirms validity of superposition. Smaller differences in phase are present on the pressure side of blade 0 and blade +1. The observed differences are generally below measurement accuracy. Numerically obtained pressure response for combined circumferential bending-torsion mode is shown in Figure 7.29. Similar to experimental data, the pressure response is extracted at midspan of blades -1, 0 and +1, at low subsonic velocity and reduced frequency of k=0.1. Comparing to the test data, the response behaviour for combined circumferential bending-torsion mode is in general well predicted by the numerical model. Largest differences between predicted and measured response is observed in the regions of suction peak on blades -1 and 0, where numerical model tend to over-predict the response amplitudes. This was also observed for the predictions of its pure mode components (see Figure 7.10 & Figure 7.12). The predicted phase corresponds well to the measured one. Considering correlation between directly simulated combined mode results and the results of mode superposition perceived in Figure 7.29, it could be concluded that the agreement is very satisfying. From Figure 7.30 it can be seen that the quality parameters determined for numerical results feature low values on all investigated blades. Variations of the parameters with reduced frequency are rather modest, indicating a slight increase in average values with increase in frequency. Figure 7.30 also contains quality parameters based on experimental data, and it is observed that the differences in absolute as well as imaginary values determined

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6C

p am

p, −

combsuperpos

−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,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


+1

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


−1

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2


from test data are considerably higher than the ones calculated for the simulated results. However these differences are still below measurement accuracy for both the differences in absolute as well as imaginary value.

Figure 7.29: Numerical results of combined circumferential bending-torsion mode

at M2=0.4 and k=0.1


Figure 7.30: Variation of quality parameters with reduced frequency

circumferential bending-torsion; low subsonic velocity M2=0.4; experimental data (left) & numerical results (right);

−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



−0.4 −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



−0.4 −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 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2

0.1 0.2 0.3 0.4 0.50

0.05

0.1


mea

n Δ

Cp

amp,

−


0.1 0.2 0.3 0.4 0.50

0.05

0.1


mea

n Δ

Cp

Im,−


0.1 0.2 0.3 0.4 0.50

0.05

0.1

mea

n Δ

Cp

amp,

−


0.1 0.2 0.3 0.4 0.50

0.05

0.1


mea

n Δ

Cp

Im,−


7.3.3.1 Effects of Flow Velocity The impact of flow velocity on of mode superposition in case of circumferential bending-torsion mode at high subsonic operating point (M2=0.8) is addressed in the present section. Unsteady response data measured at midspan section on blades -1 through +1 for the combined circumferential bending-torsion mode at reduced frequency of k=0.1 is shown in Figure 7.31. The pressure response features similar behavior to what was observed at low subsonic velocity (see Figure 7.28). At high subsonic velocity the suction peak on blade -1 is sharper and has moved to arc=-0.15. The response peak on the suction side of the oscillating blade is smeared out and response magnitudes along the arc are in general slightly higher than in case of low Mach number. The pressure response on blade +1 is characterized with increasing amplitude when going from leading edge towards the trailing edge of the blade, which is observed both on suction and pressure side of the blade. The presence of significant response amplitudes on the suction side of blade +1 (secondary surface) indicates that the impact of torsion component is strong in circumferential bending-torsion, since this behavior is characteristic for the torsion at high subsonic Mach number (Vogt, 2005).


combined circumferential bending-torsion mode at R=1; k=0.1 and M2=0.8; A suitable agreement between combined mode and superposed data observed at low subsonic velocity is preserved even for high Mach number. Local differences in magnitude are observed on the fore suction side of blade -1 and on blade 0.

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−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,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


+1

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

−

combsuperpos

−0.4 −0.2 0 0.2 0.4

−200

−100

0

100

200

Cp

phas

e, d

eg


−1

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2


Phase discrepancies are present on the pressure side of blade +1 and fore pressure side of blade 0. The numerically obtained response for combined circumferential bending-torsion at high subsonic Mach number (M2=0.8) is presented in Figure 7.32. The numerical results correspond well to the test data shown above. The amplitude over-prediction on the suction side of blades -1 and 0 is observed even here. The phase of the response is well captured by the numerical model. The agreement between the directly simulated combined mode response and the results of linear mode superposition preservers its good quality.

Figure 7.32: Numerical results of combined circumferential bending-torsion mode at M2=0.8 and k=0.1

Quality parameters calculated for combined circumferential bending-torsion mode at high subsonic velocity are included in Figure 7.33. The variation of the parameters derived from experimental data with change reduced frequency does not indicate any trend, but rather keeps a constant level. Quality parameters based on numerical results show that differences increase with reduced frequency, such that for reduced frequency k=0.5 differences are of the same order of magnitude as the ones calculated for the test data at reduced frequency of k=0.2. Since this was highest reduced frequency tested at high subsonic Mach number, it cannot be estimated if differences between the combined mode data and superposed data would increase for higher reduced frequencies.

−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



−0.4 −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



−0.4 −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 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2



Figure 7.33: Variation of quality parameters with reduced frequency

circumferential bending-torsion; high subsonic velocity M2=0.8; experimental data (left) & numerical results (right);

At this point some main observations based on findings of the combined bending-torsion mode investigations will be brought up:

The numerical model is able of capturing overall behaviour of the aeroelastic response for the two investigated combined bending-torsion modes. Numerical model tend to over-predict magnitude of the response in the region around the suction peak on blades -1 and blade 0.

Quality parameters, indicating differences in mean absolute and imaginary values of the unsteady response between combined mode data and superposed data, feature values that are well below measurements accuracy of the setup.

Numerically obtained quality parameter values are in general much lower than the ones determined for the test data. However, the quality parameters from numerical results at high subsonic Mach number are of same order of magnitude as the experimental ones.

The investigations of the combined modes at off-design operating points i.e. negative incidence angles indicate that regardless the flow being attached or separated the combined and superposed data lie very well in line with each other.

The validity of the mode combination has thereby been demonstrated and verified using conclusive experimental and numerical approach at two different velocity levels.

0.05 0.1 0.15 0.2 0.250

0.05

0.1


mea

n Δ

Cp

amp,

−


0.1 0.20

0.05

0.1


mea

n Δ

Cp

Im,−


0.1 0.2 0.3 0.4 0.50

0.05

0.1

mea

n Δ

Cp

amp,

−


0.1 0.2 0.3 0.4 0.50

0.05

0.1


mea

n Δ

Cp

Im,−


Effects of Aerodynamic Mistuning on Aeroelastic Response 7.4 The present section addresses impact of aerodynamic mistuning on unsteady response in the oscillating LPT cascade. Type of asymmetries that are considered here is variation of blade-to-blade stagger angle. The investigated mode shapes are three orthogonal oscillation modes (axial bending, circumferential bending and torsion). The results are obtained for two different velocity levels (L1, H1) and at various reduced frequencies. The effects of mistuning are discussed both on experimental data and numerical results such as to assess applicability of the numerical model for flutter predictions in the aerodynamically mistuned cascade. Firstly, the impact on the unsteady pressure response is discussed in details. Thereafter level of change of aerodynamic force influence coefficients upon change in blade stagger angle is presented and discussed. Finally, probabilistic stability analyses of randomly mistuned blade rows are carried out and sensitivity of flutter stability towards the investigated type of aerodynamic mistuning is quantified. 7.4.1 Unsteady Response- Mistuned Case

Unsteady response data is presented in the form of normalized unsteady pressure coefficients along the blade profile. The figures contain unsteady pressure amplitude plotted in the respective top window and the response phase in the bottom window. Blade index is included in upper left part of the top window. Figure 7.34 shows the unsteady surface pressure data for the axial bending mode at reduced frequency of k=0.3 and low subsonic outlet Mach number M2=0.4. The unsteady pressure data are acquired at midspan section on blades -1, 0 and +1. Two de-staggered cases are shown, namely +2.5deg and -2.5deg. It is observed that the magnitude of the unsteady pressure coefficients changes moderately for the investigated de-stagger angles. Although distinct trends are measured, it is noticeable that these are in the order of magnitude of the measurement accuracy of the test setup. However, the observed trends are considered statistically significant since the measurements are performed with good repeatability. The effects of aerodynamic mistuning on aeroelastic properties are of less magnitude than the changes observed due to for example mode shape variations, presented in previous sections. The largest change in magnitude is observed around the suction peak (around arc=-0.11) on the blades -1 and 0. The change in magnitude is consistent with the changes in blade loading observed in Figure 7.3 i.e. increased blockage in passage +1 due to positive de-staggering of blade 0 will results in lower response magnitude on the suction side of blade 0, since the velocity in the passage is lower and the gradients around the suction peak are weaker. Lower magnitude is also measured on the pressure side of blade +1. Opposite behavior is noted for the negative de-staggering. The phase of the response seems to be more affected by negative de-staggering which is clearly seen on blade 0 where considerable phase deviation is present on the aft part on the suction side as well as a slight shift in phase on the pressure side. Similar phase deviation is also observed on the suction side on blade -1 and the pressure side on blade +1.The phase variation on the suction side of blade +1 and the pressure side of blade -1 has a smaller significance since the response amplitude on these surfaces is very low.


Figure 7.34: Unsteady pressure data at midspan on blades

-1, 0 and +1; axial bending, k=0.3 and M2=0.4; de-stagger angle ∆γ=-2.5°, 0°, 2.5° Figure 7.35 shows the unsteady response obtained from the numerical simulations of axial bending mode with above mentioned de-stagger angles. In agreement with the experimental data, numerical results show that the most affected regions are located around the suction peak on blade -1 and fore part of blade 0. Predicted phase variations on blade 0 are not as pronounced as it was noted in the measured unsteady response data. It is observed that the change in response magnitude is well captured by the numerical model.

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6C

p am

p, − 0

00

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

− +1

+1+1

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

− −1

−1−1

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2

+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


0Δγ=−2.5Δγ=0Δγ=2.5

−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


+1Δγ=−2.5Δγ=0Δγ=2.5

Destagger on blade 0


Figure 7.35: Numerical results for unsteady blade surface pressure distribution;

axial bending, k=0.3 and M2=0.4; de-stagger angle ∆γ=-2.5°, 0°, 2.5° To be able to look more closely into the trends of response magnitude variation, Figure 7.36 depicts unsteady pressure magnitudes at the locations that are most susceptible to changes. Same trends in magnitude variation can be observed in both numerical results and test data.

Figure 7.36: Variation of Cp amplitudes at specific locations on blades -1, 0 and

+1; axial bending, k=0.3 and M2=0.4; Impact of the blade 0 de-staggering on the unsteady response during axial bending mode has also been investigated at high subsonic velocity level (M2=0.8).

−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


−1Δγ=−2.5Δγ=0Δγ=2.5

0 20 40

−80

−60

−40

−20

0

20

40

60

+2

+1

0

−1

−2

+2

+1

0

−1

−2

−2.5 0 2.50.3

0.35

0.4

0.45

0.5

Cp

amp,

−

Δ γ, deg

bl0, arc= −0.11

expnum

−2.5 0 2.5

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Cp

amp,

−

Δ γ, deg

bl+1, arc= 0.12

expnum

−2.5 0 2.50.2

0.25

0.3

0.35

0.4

0.45

0.5

Cp

amp,

−

Δ γ, deg

bl−1, arc= −0.11

expnum

Destagger on blade 0


The unsteady response for the axial bending mode at reduced frequency of k=0.1 and high subsonic velocity is included in Figure 7.37 . Experimental data and numerical results are plotted next to each other. It is observed that changes in the response amplitudes are of the same character as previously noted for the low subsonic velocity level. However, slightly stronger change in amplitudes is now observed on the fore pressure side on blade zero and around suction peak on blade -1. These variations in magnitude are accompanied with a shift in phase.


Figure 7.37: Unsteady pressure response at midspan on blades -, 0 and +1; axial

bending; k=0.1 and M2=0.8; de-stagger angle ∆γ=-2.5°, 0°, 2.5° Similar to the axial bending mode, in case of the other two investigated orthogonal modes it is observed that the magnitude of the unsteady pressure coefficients changes rather moderately for the investigated de-stagger angles. Figure 7.38

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

− 0

00

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

−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


0Δγ=−2.5Δγ=0Δγ=2.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

− −1

−1−1

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

−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


−1Δγ=−2.5Δγ=0Δγ=2.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

− +1

+1+1

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

−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


+1Δγ=−2.5Δγ=0Δγ=2.5


contains unsteady pressure response for circumferential bending mode at reduced frequency of k=0.3 and low subsonic velocity (M2=0.4).


Figure 7.38: Unsteady pressure response at midspan on blades -, 0 and +1;

circumferential bending; k=0.3 and M2=0.4; de-stagger angle ∆γ=-2.5°, 0°, 2.5° The largest changes in response magnitude are observed on the oscillating blade, where the pressure side and aft part of the suction side (downstream the suction peak) are mostly affected. Relatively strong change is also noted around the suction peak (around arc=-0.11) on blade -1, particularly pronounced for negative de-stagger angle. These changes in magnitudes are captured correct by the numerical model, as well as the amplitude variations on the pressure side of blade +1. Relatively small variations in response phase are measured on all of the

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

− 0

00

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

−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


0Δγ=−2.5Δγ=0Δγ=2.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

− −1

−1−1

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

−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


−1Δγ=−2.5Δγ=0Δγ=2.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

− +1

+1+1

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

−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


+1Δγ=−2.5Δγ=0Δγ=2.5


investigated blades. These variations are somewhat visible in the numerical results as well. The unsteady response for the torsion mode at different stagger angles is shown in Figure 7.39. Again, the largest variations in magnitude are measured around the suction peak on blade -1 and blade 0. Similar to what was observed for the axial bending; it seems that the variations are closely related to the changes in passage throat. The magnitude variations are well captured by the numerical model. The phase of the response does not seem to be affected significantly, beside a phase shift measured along blade 0.


Figure 7.39: Unsteady pressure response at midspan on blades -1, 0 and +1;

torsion; k=0.3 and M2=0.4; de-stagger angle ∆γ=-2.5°, 0°, 2.5°

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

− 0

00

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

−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

200C

p ph

ase,

deg


0Δγ=−2.5Δγ=0Δγ=2.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

− −1

−1−1

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

−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


−1Δγ=−2.5Δγ=0Δγ=2.5

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

Cp

amp,

− +1

+1+1

−0.4 −0.2 0 0.2 0.4−200

−100

0

100

200

Cp

phas

e, d

eg


Δγ=−2.5°Δγ=0°Δγ=2.5°

−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


+1Δγ=−2.5Δγ=0Δγ=2.5


7.4.2 Mistuned Aerodynamic Influence Coefficients

In order to analyze the impact of the blade 0 stagger angle variation on the influence coefficients, the complex force coefficients for axial bending mode are plotted in the complex plane as included in Figure 7.40. The influence coefficients determined for the different stagger angles and blades are marked with markers and enclosed within rectangles highlighting the region of the coefficient variability. It is observed that the influence coefficient obtained from data measured on blade 0 is considerably affected by the change of the stagger angle. The influence coefficients on the neighboring blades seem to be less affected by the change of stagger angle variation. The spread of the numerically predicted coefficients, enclosed by blue dashed line rectangles, is generally less pronounced. The predicted influence variation for blade +1 is considerably smaller than the one measured. It is also noted that no clear trend in variation of the coefficients due to stagger angle change is apparent.

Figure 7.40: Mistuned influence coefficients for blades -1, 0 and +1; axial bending; M2=0.4 & k=0.3; experimental data (black colored) and numerical results (blue

colored); At this stage, a note shall be made on the prediction and measurement accuracy of complex force influence coefficients. Despite the fact that most of the complex pressure data points lie within the measurement accuracy, noticeable differences are apparent in the complex force influence coefficients. The reason for this behavior is to be found in the integration of complex blade surface pressure with respect to a specific orthogonal mode. Thereby, three ingredients play a role: i) the local unsteady pressure magnitude, ii) its phase and iii) the local blade shape. Hence, seemingly small and local deviations in i) and/or ii) might have a major impact on the integrated values depending on their location. This observation applies to both experiments and predictions. As the effects are of systematic nature, the findings of the present study in terms of variability of force influence coefficients are however not affected by this observation.

-6 -4 -2 0 2 4 6

-4

-2

0

2

4

Im(F

)

Re(F)

-2.5-2

-1

0+1

+2

+2.5

-2.5-2-10+1

+2

+2.5

-2.5-2

-1

0 +1

+2

+2.5-2.5

-2

-1 0

+1

+2

+2.5

-2.5-2

-1

0

+1

+2

+2.5

-2.50+2.5

bl0

bl0

bl0 zoom bl+1 zoom

bl-1 zoom


The complex force coefficients obtained for the circumferential bending mode are presented in Figure 7.41. It is interesting to note that for the circumferential bending mode the variability of the influence coefficients measured on the neighboring blades due to blade 0 de-staggering is quite pronounced. The predicted spread of the coefficients is still smaller, in particular for the blade +1, where measured values indicate relatively strong variation in imaginary part while numerical results show more variation of the real part values. A significant shift in real part between measured and predicted coefficients is noted on blade -1. Similar to what was observed for the axial bending mode, no clear trend in variation of the coefficients due to de-staggering is apparent.

Figure 7.41: Mistuned influence coefficients for blades -1, 0 and +1; circumferential bending; M2=0.4 & k=0.3; experimental data (black colored) and

numerical results (blue colored); The complex force coefficients obtained for the torsion mode are presented in Figure 7.42. Some distinct discrepancies between measured and predicted real part values are noted on blades 0 and +1. Secondly, it is observed that variations of the blade 0 coefficients are much more pronounced than variations on the neighboring blades. This applies to both measured and predicted values. The predicted variability of the imaginary part is considerably smaller than the measured ones.

-6 -4 -2 0 2 4 6

-4

-2

0

2

4

Im(F

)

Re(F)

2-2.5

-20+1

+2.5

-2.5-2

0+1

+2.5

bl0 bl0

bl0 zoom

0

-2.5

-20

+1

+2.5

-2.5 -20

+1 +2.5

bl+1 zoom

-2.5

-20

+1 +2.5

-2.5-2

0

+1

+2.5

bl-1 zoom


Figure 7.42: Mistuned influence coefficients for blades -1, 0 and +1; torsion; M2=0.4 & k=0.3; experimental data (black colored) and numerical results (blue

colored);

7.4.2.1 Impact of Reduced Frequency The impact of the reduced frequency is addressed in Figure 7.43. The considered mode is axial bending at low subsonic velocity. The same range of de-stagger angles (within ±2.5deg) was tested at each investigated reduced frequency (k=0.1 to 0.4).

Figure 7.43: Mistuned influence coefficients vs. reduced frequency; axial bending;

M2=0.4; experimental data

-6 -4 -2 0 2 4 6

-4

-2

0

2

4

Im(F

)

Re(F)

−6 −4 −2 0 2 4 6

−4

−2

0

2

4

Re(Fξ)

Im(F

ξ)

k=0.1k=0.2k=0.3k=0.4

bl 0

bl −1 bl +1

-2.5

-2-10

+1

+2+2.5

-2.5-2-1

0+1

+2+2.5

-2.5-2-10 +1

+2+2.5-2.5-2

-10+1

+2+2.5

bl0 bl0

bl+1 zoom 0

-2.5-2-1

0

+1+2+2.5

-2.5 -2-10

+1

+2+2.5

bl-1 zoom


The perturbation of the blade 0 influence coefficients seems to grow with increased frequency. At the same time, the coefficients move to more negative values meaning that blade 0 obtains more stabilizing character with an increase in reduced frequency, which is in line with the finding from previous studies ( Panovsky and Kielb, 2000; Nowinski and Panovsky, 2000; Vogt and Fransson, 2007). Values and size of the rectangles for blade -1 does not change significantly with reduced frequency and influence coefficients remain small. The variability region for blade +1 moves to more positive values with an increase in reduced frequency. Similar behavior can be observed for the circumferential bending mode depicted in Figure 7.44. In circumferential bending mode, the deviation region for blade 0 seem to move to more negative values with increased frequency, while opposite is observed for the neighboring blades. The size of the perturbation regions on neighboring blades does not seem to be affected with change in frequency. The perturbation of the blade 0 coefficients, however, seems to grow with increased frequency.

Figure 7.44: Mistuned influence coefficients vs. reduced frequency; circumferential bending; M2=0.4; experimental data

Deviations of the influence coefficients on the neighboring blades for the torsional mode, shown in Figure 7.45, seem to be much less affected than what is observed for the bending modes. The variability of blade 0 coefficients shows clearly a growing trend with increase in reduced frequency, as well as they move towards more negative values.

−6 −4 −2 0 2 4 6

−4

−2

0

2

4

Re(Fη)

Im(F

η)

k=0.1k=0.2k=0.3k=0.4

bl 0

bl +1

bl −1


Figure 7.45: Mistuned influence coefficients vs. reduced frequency; torsion; M2=0.4; experimental data

A comprehensive study of mistuned influence coefficients at different reduced frequencies, mode shapes and velocity levels led to the conclusion that no clear general trend in change of influence coefficients with stagger variation could be identified. Therefore enclosed deviation regions shown here should be taken rather as a perturbation basis for a probabilistic treatment of mistuned assemblies under assumption that if an applied stagger variation is within boundaries of the here investigated angles, influence coefficients will change inside the marked regions. This opens up for the probabilistic analysis of the flutter stability. 7.4.3 Mistuned Aeroelastic Model- Probabilistic Analysis

On the background of experimental data, a numerical model for assessing the aeroelastic stability of aerodynamically mistuned blade rows is composed. It has been shown above that the numerical model used is capable of predicting the trends and to a large degree also the levels of aerodynamic mistuning based on blade 0 de-stagger data. From this it is concluded that the numerical model is equally applicable for correctly predicting the changes in aeroelastic properties due to blades +1 and -1 respectively being de-staggered. As described earlier, in order to properly assess the perturbed influence coefficients matrices described in section 2.6, in addition to the nominal case simulation, at least three more simulations per each investigated stagger have been performed: one simulation where de-stagger is applied on the oscillating blade and two additional simulations where each of the neighboring non-oscillating blades was de-staggered. Figure 7.46 shows how the influence coefficients on blade -1 through +1 are changing due to de-staggering of each of the considered blades. It should be kept in mind that in all three cases, the oscillating blade is still blade 0.

−6 −4 −2 0 2 4 6

−4

−2

0

2

4

Re(Fζ)

Im(F

ζ)

k=0.1k=0.2k=0.3k=0.4

bl +1bl −1

bl 0


De-stagger on blade 0 De-stagger on blade +1

De-stagger on blade -1 Introduced de-stagger angles [deg]

Figure 7.46: Variation of the influence coefficients due to de-stagger on various

blades in the cascade; numerical results; axial bending at k=0.3; L1 From the figure above it is observed that the influence coefficients can vary considerably even when de-stagger is applied on the stationary neighboring blades (blades -1 and +1). The perturbation is of the same order of magnitude as the one induced by blade 0 de-staggering. Therefore it is important to take these results into account in order to properly assess the aerodynamic INFC matrices for aerodynamically mistuned blade rows. To assess the overall aeroelastic stability of a randomly mistuned blade row, the aforementioned ROM model is built on a probabilistic basis. A fleet of 1000 blade rows is regarded in which every 5th blade (i.e. 20% of all blades) is de-stagger at a random angle between -2.5deg and +2.5deg. In this way spacing between the de-staggered blades is sufficient to assume that the influence of two de-staggered blades is not affecting each other. The results included below are shown in terms of S-curve as well as cumulative probability. Figure 7.47 shows the S-curve of the nominal (i.e. tuned) setup based on linear superposition as well as the discrete ROM model introduced above. The two curves are in line showing the validity of the ROM model. In addition the stability of

−1 0 1

−2

0

2

Im(F

ξ)

−1 0 1

−2

0

2

Re(

Fξ)

Blade indices

−1 0 1

−2

0

2

Im(F

ξ)

−1 0 1

−2

0

2

Re(

Fξ)

Blade indices

−1 0 1

−2

0

2

Im(F

ξ)

−1 0 1

−2

0

2

Re(

Fξ)

Blade indices


the mistuned blade rows is included as a point cloud. It is apparent that the effects on stability are not very drastic amounting to about 15% of the S-curve peak-to-peak amplitude. With respect to the least stable mode it is observed that changes are very small and do not represent a significant danger for having the setup destabilized.

Figure 7.47: Effect of random mistuning on flutter stability; 20% of blades de-staggered; axial bending at k=0.3 & M2=0.4; numerical results

The cumulative probability of the least stable mode is included in Figure 7.48. It is apparent that the majority of the mistuned setups feature higher aeroelastic stability than the tuned setup. About 14% of the regarded fleet however features a minimum stability that is lower than tuned. With respect to the peak-to-peak amplitude of the tuned setup the variability in the least stable mode is around 3%.

Figure 7.48: Cumulative probability of least stable mode; axial bending The overall effects of random mistuning on stability of axial bending mode at high subsonic Mach number are rather similar to what was observed for low velocity level. Maximum perturbation amounts about 10% of peak-to-peak amplitude, while

0.75 0.8 0.85 0.90

0.2

0.4

0.6

0.8

1

Ξ of least stable mode

cum

ulat

ive

prob

abili

ty

ROM, 20% mist.tuned case


the stability parameter values of the least stable mode is affected very little by mistuning.

Figure 7.49: Effect of random mistuning on flutter stability; 20% of blades de-staggered; axial bending at M2= 0.8 & k=0.1; numerical results

The cumulative probability plot, depicted in Figure 7.50, shows that almost all of the mistuned blade rows feature higher featured higher stability than the tuned case. Again, variability of the least stable mode value is very small and amounts less than 2 % with respect to peak-to peak amplitude.

Figure 7.50: Cumulative probability of least stable mode; axial bending at k=0.1 & M2=0.8

The impact of random mistuning on flutter stability for the circumferential bending mode is shown in Figure 7.51. For this bending mode stability parameter is negative in range of interblade phase angles between 0 and 120 degrees, so the mode can be considered as unstable at the investigated operating condition. This is also an indication that the influence from the neighbour blade pair is greater than the influence of the oscillating blade itself since the offset of the S-curve’s

0.04 0.06 0.08 0.1 0.120

0.2

0.4

0.6

0.8

1


cum

ulat

ive

prob

abili

ty



mean value, which corresponds to direct influence of the blade 0 alone, is positive. It is observed that the effects of random de-staggering are quite small; the shape of the curve has remained the same as for the tuned case, with slight increase in peak-to -peak amplitude (amounts to about 3 % with respect to the tuned setup amplitude).

Figure 7.51: Effect of random mistuning on flutter stability; 20% of blades de-staggered; circumferential bending at k=0.3; numerical results

The perturbation of the least stable mode for the considered circumferential bending mode is depicted by the cumulative probability plot included in Figure 7.52. The variation of the least stable mode is such that for all of the 1000 investigated randomly mistuned modes the stability gets less stable in comparison with the tuned case. However, the variability in the least stable mode is very small, corresponding to around 1% of the peak-to-peak amplitude of the tuned setup.

Figure 7.52: Cumulative probability of least stable mode; circumferential bending The impact of random mistuning on flutter stability for the torsion mode is shown in Figure 7.53. Torsion mode seems also to be unstable for the investigated operating condition, again indicating rather strong influence of the blades ±1. The

−2.2 −2.1 −2 −1.9 −1.80

0.2

0.4

0.6

0.8

1


cum

ulat

ive

prob

abili

ty



stability, however, seems to be quite insensitive to the introduced random mistuning. The average change amounts to about 2% of the curves amplitude.

Figure 7.53: Effect of random mistuning on flutter stability; 20% of blades de-staggered; torsion at k=0.3; numerical results

The cumulative probability of the least stable mode in case of torsion mode indicates that all randomly mistuned blade rows feature slightly lower stability than the tuned setup. Nevertheless, the variability of the least stable mode value is very modest and amounts to about 1% of the stability curve amplitude.

Figure 7.54: Cumulative probability of least stable mode; torsion

To summarize the above presented results on influence of blade-to-blade de-staggering on the aeroelastic stability of the setup, Table 7.1 contains variability of the least stable mode values for the three orthogonal modes, based on the 1000 investigated randomly mistuned setups. The variability is also presented as a percentage of the corresponding S-curve peak-to-peak amplitudes.

−0.9 −0.88 −0.86 −0.84 −0.82 −0.8 −0.780

0.2

0.4

0.6

0.8

1


cum

ulat

ive

prob

abili

ty



Table 7.1: The impact of aerodynamic asymmetries on minimum stability

Mode shape

Ξ of least stable mode (tuned setup)

S-curve peak-to-peak amplitude

Variability of least stable mode due to introduced asymmetries (given as % of S-curve peak-to-peak amplitude)

Percentage of setups less stable than tuned setup [%]

Axial bending 0.84 2.53 0.08 (3.1%) 14%

Circ. bending -1.86 9.91 0.1 (1.1%) 100%

Torsion -0.8 6.61 0.09 (1.3%) 100%

From the presented values it is clear that the overall influence of aerodynamic asymmetries on minimum stability values for all investigated modes is rather small, amounting as maximum about 3% of the stability curve peak-to-peak amplitude, observed for the axial bending mode. Despite the fact that in case of circumferential bending and torsion all of 1000 randomly mistuned blade rows exhibited lower minimum stability than the corresponding tuned setup, this could be neglected since the observed variability of the least stable mode was minor.


8 SUMMARY

Conclusions 8.1 The focus of the present work has been on the experimental and numerical investigation of the aeroelastic properties of combined mode shapes and studies of the effects of aerodynamic mistuning on the aeroelastic properties of an oscillating cascade. On the experimental side, a unique test facility that allows controlled flutter investigations was employed. The facility comprises an annular sector of seven free-standing Low Pressure Turbine (LPT) blades. One of the blades was made to oscillate in three-dimensional pure or combined modes while the unsteady blade surface pressure is acquired on the oscillating blade itself and on the non-oscillating neighbor blades. On the numerical side a commercial CFD code (ANSYS CFX v11) has been employed using a full-scale time-marching 3D viscous model. In accordance with the experiments, the simulations have been performed in the influence coefficient domain with only one blade oscillating. The numerical model has been validated through comparison with experimental data. The numerical model is capable of correctly capturing the overall behavior of the unsteady flow in the cascade. In combined mode investigations, the data from pure modes have been combined linearly at specified phase angles so as to obtain the response for a combined mode oscillation. For validation purposes the blade has been made to oscillate in various combined modes, which allowed the combined mode aeroelastic properties to be acquired in a direct manner. In order to draw conclusion on the validity of mode combination over a range of parameters, variations in reduced frequency, flow velocity and inflow incidence have been taken into consideration with the latter leading to local flow separation. The key features of the pure modes as well as the resulting combined modes have been discussed based on local resolved unsteady blade surface pressure data. Quality parameters have been defined that allow the validity of mode combination to be assessed both in terms of absolute as well as the imaginary part of the local unsteady blade surface pressure. From quality parameters, it has been recognized that the observed differences lie well below the measurement accuracy of the present setup, regardless of the reduced frequency, inflow incidence and flow velocity. From simulation results it has been recognized that the validity holds for the employed numerical model. Whereas the agreement is very good at low subsonic flow speeds, moderate differences appear at higher speeds, which are comparable to those observed experimentally. The effect of aerodynamic mistuning on the aeroelastic stability of the oscillating LPT cascade has been investigated experimentally and numerically, employing the above described tools. Aerodynamic mistuning has been introduced as a blade-to-blade stagger angle variation. The study has been performed using the influence coefficient method in which the influence coefficients have been both directly measured and predicted and reassembled so as to yield travelling wave mode results. A set of three pure orthogonal modes has been investigated. The effect of de-staggering a single blade on steady aerodynamics has been explained from test data. The observed effects seem to be predominantly an effect of the change in passage throat. The changes in steady aerodynamics are


observed on the unsteady aerodynamics where distinctive effects on flow velocity lead to changes in local unsteady pressure coefficients. Correlation of the numerical results with the test data with one blade de-staggered has led to the following conclusions: The numerical model used is capable of accurately capturing the differences in

steady aerodynamics induced by the de-staggering The trends in the change of unsteady pressure response during blade

oscillation due to de-staggering are generally well captured. However moderate differences are observed in the absolute values of the response amplitude and phase

There is no clear trend in the variation of complex integrated force influence coefficients, neither in test data nor in numerical results

The numerical model tends to predict a moderately smaller variability of the influence coefficients with de-stagger than measured

Trends with reduced frequency show that the variability due to de-stagger increases for all modes. Blade 0 shows an increasingly stabilizing behavior with an increase in reduced frequency.

A Reduced Order Model (ROM) has been implemented to address the aeroelastic stability of randomly mistuned blade rows for the three investigated mode shapes. The ROM model takes into account mistuned aerodynamic stiffness and damping coefficients. For the present study, only the influence coefficients of blades -1, 0 and +1 have been regarded. These coefficients have been acquired from three individual simulations with blade -1, 0 and +1 de-staggered, respectively. The ROM model has been used to perform a probabilistic analysis of the effect of aerodynamically mistuned influence coefficients involving a fleet of 1000 blade rows leading to the following conclusions: The effect of the present type of aerodynamic mistuning is of a moderate

nature. The largest changes are observed for the axial bending mode and amount to an average of about 15% of the stability curve (peak-to-peak) amplitude. The changes for the remaining two modes (circumferential bending and torsion) are much less pronounced, being less than 3% of the stability curve amplitude.

The overall variation in minimum stability for all modes is rather small, amounting to about 3% of the stability curve (peak-to-peak) amplitude for axial bending modes and about 1% for circumferential bending and torsion mode

While the majority of the randomly mistuned blade rows oscillating in axial bending mode featured higher minimum aeroelastic stability than nominal, the other two modes exhibit the more de-stabilizing effects of the mistuning, where all randomly mistuned blade rows featured lower stability than the tuned case. However this destabilization is of very moderate character.

Based on the above conclusions, it is stated that the investigated type of mistuning can lead to a measurable, though only moderate, change in aeroelastic stability. In the light of industrial practices, under the assumption that the blades are oscillating at one of the modes as tested here and as long as the geometrical blade-to-blade variations in stagger angle are within a range of +-2.5deg, the negative effects of aerodynamic mistuning on aeroelastic stability can be taken care of by applying a certain safety margin. Most probably the effects of structural mistuning would outbalance any destabilizing effect due to aerodynamic mistuning.


However, it should be reminded that the modes investigated here are arbitrary rigid body modes. To be able to establish more general conclusions about the effects of aerodynamic mistuning on aeroelastic stability, the investigations should be extended to embrace parametric studies for real mode shapes.

Recommendations for future work 8.2 Regarding the combined mode investigations and validation of the mode superposition principle, it would be desirable to extent the investigations towards flow regimes where potential “non-linearity” in the flow might be induced. Example of such would be investigations of transonic flows where the presence of choked flows and shock waves could be expected. Another interesting issue would be to assess impact of three-dimensional mechanisms on the validity of the mode superposition, through the investigation of aeroelastic properties of combined modes at different spanwise sections of the oscillating blade. In particular it would be interesting to address the impact of secondary flow structures that are induced by the tip leakage flow due to the existing tip clearance in the cascade. On the numerical side this would then require inclusion of the tip clearance into the numerical model. On the aerodynamic mistuning side, a further development of the ROM model would be required in order to allow assessment of aeroelastic stability of mistuned blade rows where there are no constraints on how large the fraction of the blades that are being mistuned. In the present study, the mistuned blade rows had each 5th blade de-staggered with a random stagger angle. In this way spacing between the de-staggered blades was sufficient to assume that the influence of two de-staggered blades is not affecting each other. A new model should include an approximation scheme in order to overcome this constraint. As the present work focused on the arbitrary rigid body modes, it would be preferable to look into stability maps (tie-dye plots) to see how the effects of aerodynamic asymmetries change when varying the location of the centre of torsion, following a 2D torsion mode representation of a given mode. To be able to generalize the conclusions about the effects of the aerodynamic mistuning, the future investigations should even consider real mode oscillations and should extend to different types of machines.



9 REFERENCES

Bendiksen, O., Friedmann, P.; 1980, "The Effect of Bending-Torsion Coupling on Fan and Compressors Blade Flutter" ASME J. of Engineering for Gas Turbines and Power, Vol. 104, 1982, pp.617-623 Bendiksen O. O.; 1984 "Flutter of Mistuned Turbomachinery Rotors" ASME J. of Engineering for gas turbines and power, Vol. 106, pp. 9, Jan. 1984 Bladh, R., Castanier, R., Pierre, C., 1999; ”Reduced Order Modelling of Vibration Analysis of Mistuned Bladed Disk Assemblies with Shrouds”, J. of Engineering for Gas and Turbine Power, Vol. 121, pp.512-522 Bölcs, A., Fransson, T. H.; 1986 ”Aeroelasticity in Turbomachines –Comparison of Theoretical and Experimental Cascade Results” EPFL Internal Communication No.13 Chernysheva, O. V., Fransson, T. H., Kielb, R. E., Barter, J.; 2001 ”Comparative Analysis of Blade Mode Shape Influence on Flutter of Two- Dimensional Turbine Blades”, Conference proceeding ISABE 2001-1243, Bangalore, India Collar, A. R.; 1946 “The Expanding Domain of Aeroelasticity” Royal Aeronautical Society, London, UK, 1946, pp.613-636 Crawley, E. F.; 1988 “Aeroelastic Formulation for Tuned and Mistuned Rotors” AGARD Manual on Aeroelasticity in Axial-Flow Turbomachines, Vol.2, Structural Dynamics and Aeroelasticity, Chapter 19, AGARD-AG-298 Crawley, E. F.; Hall, K. C.; 1984 "Optimization and Mechanisms of Mistuning in Cascades" ASME Paper 84-GT-196 E. H, Dowell, H. C Curtis, R. H, Scanlan, F, Sisto, 1989; ”A Modern Course in Aeroelasticity” Kluwer Academic Publishers, ISBN 0-7923-0185-4 Ekici, K., Kielb, R. E., Hall, K. C.; 2008 “Aerodynamic Asymmetry Analysis of Unsteady Flows in Turbomachinery” ASME Paper GT2008-51176


Försching, H.; von Diest, K.; 1991 "Flutter Stability of Annular Wings in Incompressible Flow" J. of Fluids and Structures, 5, pp. 47-67. Försching, H.; 1994 "Aeroelastic Stability of Cascades in Turbomachinery” Prog. Aerospace Sci. Vol.30, 1994, pp. 213-266 Glodic N., Bartelt M., Vogt D., Fransson T., 2009 “Aeroelastic Properties of Combined Mode Shapes in an Oscillating LPT Cascade”, Proceedings of the 12th International Symposium on Unsteady Aerodynamics, Aeronautics & Aeroelasticity in Turbomachines, Imperial College London, ISUAAAT 12, I12-S8-2 Glodic N., Vogt D., Fransson T., 2011 “Experimental and Numerical Investigation of Mistuned Aerodynamic Influence Coefficients in an Oscillating LPT Cascade” ASME Paper GT2011-46283 Glodic N., Vogt D., Fransson T., 2012 “Influence of Tip Clearance Modelling in Predictions of Aeroelastic Response in an Oscillating LPT Cascade”, Proceedings of the 13th International Symposium on Unsteady Aerodynamics, Aeronautics & Aeroelasticity in Turbomachines, Tokyo, ISUAAAT 13 Hoyniak, D., Fleeter, S.; 1986 “The Effect of Circumferential Aerodynamic Detuning on Coupled Bending-Torsion Unstalled Supersonic Flutter” J. of Turbomachinery, Vol. 108, 1986, pp. 253-260 Imregun, M.; Ewins, D. J.; 1984 "Aeroelastic Vibration Analysis of Tuned and Mistuned Bladed Systems" Proceedings of the Third International Symposium on Unsteady Aerodynamics of Turbomachines and Propellers, Cambridge, UK, September 1984, pp.149-161. Kaza, K. R. V.; Kielb, R. E.; 1981 "Effects of Mistuning on Bending-Torsion Flutter and Response of a Cascade in Incompressible Flow" NASA TM-81674 Kielb, R. E.; Kaza, K. R. V.; 1983 “Aeroelastic Characteristics of a Cascade of Mistuned Blades in Subsonic and Supersonic Flow” J. of Vibration, Acoustics, Stress, and Reliability in design Vol. 105, 1983, pp. 425-433 Kielb, R. E., Barter, J., Chernysheva, O. V., Fransson, T. H.; 2001 ”Flutter of Low Pressure Turbine Blades with Cyclic Symmetric Modes- A Preliminary Design Method” ASME Paper GT2003-38694


Kielb, R. E., Feiner, D. M., Griffin, J. H., Miyakozawa, T.; 2004 “Flutter of Mistuned Bladed Disks and Blisks with Aerodynamic and FMM Structural Coupling”, ASME Paper GT-2004-54315 Kielb, R. E., Miyakozawa, T., Griffin, J. H., Feiner, D. M., Pai, S.; 2005, “Aerodynamic and Structural Coupling Effects On Mistuned Bladed Disk Response”, 10th HCF Conference, New Orleans, Louisiana, March, 2005 Kielb, R. E., Hall, K. C., 2006 “Probabilistic Flutter Analysis of Mistuned Bladed Disk” ASME Paper GT2006-90847 Kielb, R. E., Hall, K. C., Miyakozawa, T.; 2007 “The effect of unsteady aerodynamic asymmetric perturbations on flutter” ASME Paper GT-2007-27503 Launder, B. E., Spalding, D. B.; 1974 "The numerical computation of turbulent flows", Computer Methods in Applied Mechanics and Engineering, 3(2): 269-289 Mårtensson, H., Gunnsteinsson, S, Vogt, D.; 2008 ”Aeroelastic Properties of Closely Spaced Modes for a Highly Loaded Transonic Fan ", ASME Paper GT2008-51046 Marshall, J.G., Imregun, M.; 1996 “A review of aeroelasticity methods with emphasis on turbomachinery applications”, J. of Fluids and Structures, Vol:10, Pages:237-267, ISSN:0889-9746 Nowinski, M., Panovsky, J. 2000 ”Flutter Mechanisms in Low Pressure Turbine Blades” J. of Engineering for Gas Turbines and Power, Vol. 122, 2000, pp. 82-88 Panovsky, J., Nowinski, M., Bölcs, A.; 1997 ”Flutter of Aircraft Engine Low Pressure Turbine Blades” Presented at the 8th International Symposium of Unsteady Aerodynamics and Aeroelasticity of Turbomachines, Stockholm, Sweden, Sep. 14-18, 1997 Panovsky, J., Kielb, R. E.; 2000 ”A Design Method to Prevent Low Pressure Turbine Blade Flutter” Journal of Engineering for Gas Turbines and Power, Vol. 122, 2000, pp. 89-98 Patel, V. C., Rodi, W. and Scheuerer, G. ; 1985. “Turbulence Models for Near-wall and Low Reynolds Number Flows: A Review”, AIAA J., Vol. 23, No. 9, pp. 1308–1319.


Seinturier, E., Dupont, C., Berthillier, M. and Dumas, M.; 2000, “A New Method to predict flutter in presence of structural mistuning - Application to a wide chord fan stage" Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, Lyon, France, pp. 739-748 Sladojevic, I., Petrov, E. P., Sayma, A. I., Imregun, M., Green, J., S.; 2005, “Investigation of the Influence of Aerodynamic Coupling on Response Levels of Mistuned Bladed Discs with Weak Structural Coupling” ASME Paper GT2005-69050 Sladojevic, I., Sayma, A. I., Imregun, M.; 2007 “Influence of stagger angle variation on aerodynamic damping and frequency shifts” ASME Paper GT2007-28166. Srinivasan, A. V.; 1997 “Flutter and Resonant Vibration Characteristics of Engine Blades” J. of Engineering for Gas Turbines and Power, Vol. 119, 1997, pp. 742-774 Stüer, H. Schmitt, S. Ashcroft, G.; 2008 “Aerodynamic Mistuning of Structurally Coupled Blades” ASME Paper GT2008-50204 Verdon, J. M.; 1987 “Linearized Unsteady Aerodynamic Theory” AGARD Manual on Aeroelasticity in Axial-Flow Turbomachines, Vol. 1, Unsteady Turbomachinery Aerodynamics, Chapter 2, AGARD-AG-298 Versteeg, H.K., Malalasekera, W., 1995 ”An Introduction to Computational Fluid Dynamics: THE FINITE VOLUME METHOD” Pearson Education Limited 1995, 2nd edition 2007; ISBN: 978-0-13-127498-3 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 of Measuring Techniques in Transonic and Supersonic Flows in Cascades and Turbomachines, Florence, Italy, 2000 Vogt, D.M., Fransson, T. H., 2004 “Effect of Blade Mode Shape on the Aeroelastic Stability of a LPT Cascade” Paper presented at the 9th National Turbine Engine HCF Conference, Pinehurst, North Carolina, USA, 2004 Vogt, D. M., 2005 “Experimental Investigations of Three-Dimensional Mechanisms in Low-Pressure Turbine Flutter” Doctoral Thesis, Kungliga Tekniska Högskolan, Department Heat and Energy Technology, Stockholm, 2005hnology, Stockholm, 2005, ISBN 91-7178-034-3


Vogt, D.M., Fransson, T.H., 2007 “Experimental Investigation of Mode Shape Sensitivity of an Oscillating LPT Cascade at Design and Off-Design Conditions”, ASME J. of Engineering for Gas Turbines and Power, 129, 2007, pp.530-541 Vogt, D. M., Mårtensson, H. E., 2007 “Direct Calculation of Aerodynamic Influence Coefficients Using a Commercial CFD Solver” Paper presented at the 18th ISABE Conference in Beijing, September 2-7, 2007 Whitehead, D.S., 1973 “The Effect of Compressibility on Unstalled Torsion Flutter” Report CUED/A-Turbo/TR-51-1973, Cambridge, UK Wu, X., Vahdati, M., Sayma, A.I., Imregun, M., 2003 “A Numerical Investigation of Aeroacoustic Fan Blade Flutter” ASME Paper GT2003-38454

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