Advanced Numerical Simulation of Corner Separation in a Linear Compressor Cascade

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Feng Gao

To cite this version:

Feng Gao. Advanced Numerical Simulation of Corner Separation in a Linear Compressor Cascade. Fluid mechanics [physics.class-ph]. Ecole Centrale de Lyon, 2014. English. �tel-01068581�

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Num´ero d’ordre : 2014-08 Ann´ee 2014

TH `

ESE

pr´esent´ee pour obtenir le titre de

DOCTEUR DE L’ ´

ECOLE CENTRALE DE LYON

´

Ecole Doctorale M´ecanique, ´

Energ´etique, G´enie Civil et Acoustique

sp´ecialit´e

M ´

ECANIQUE

Advanced Numerical Simulation of

Corner Separation

in a Linear Compressor Cascade

Soutenue le 10 avril 2014 `a l’ ´

Ecole Centrale de Lyon

par

Feng GAO

Devant le jury compos´e de :

P. SAGAUT

Universit´e Pierre et Marie Curie

Rapporteur, Pr´esident du jury

J. CHEW

Universit´e du Surrey, Royaume-Uni Rapporteur

R. DENOS

Commission europ´eenne

Examinateur

G. DUFOUR ISAE

Examinateur

J. BOUDET

LMFA, ´

Ecole Centrale de Lyon

Co-directeur

X. OTTAVY

LMFA, CNRS

Co-directeur

L. LU

Universit´e Beihang, Chine

Co-directeur international

L. SHAO

LMFA, CNRS

Directeur

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I dedicate this thesis to Prof. Francis Leboeuf,

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Acknowledgements

This thesis is supported by the China Scholarship Council (CSC), the French Na-tional Research Agency (ANR), the NaNa-tional Natural Science Foundation of China (NSFC) and the Grand Équipement National de Calcul Intensif(GENCI). It is su-pervised by Dr. Jérôme Boudet, Dr. Xavier Ottavy, Dr. Liang Shao and Prof. Fran-cis Leboeuf from LMFA at École Centrale de Lyon in France, and Prof. Lipeng Lu from the School of Energy and Power Engineering at Beihang University in China. I deeply appreciate all the tutors for giving me the opportunity to do research in this strong international group, and providing me freedom, equality and fraternity during the thesis.

My first gratitude goes to Jérôme, who is not only a co-supervisor of this thesis, but also a great friend. I appreciate his constant encouragement, guidance, trust, and in particular his tolerance. Je remercie également Cécile, Gabriel et Paul pour leurs

soutiens.

My sincere thanks goes to Xavier, who is a great friend rather than a co-supervisor, for his great support, help and guidance. His optimism makes me more confident in facing all the problems. Je remercie aussi Zabou, Naomi, Mac´eo et Ruben, qui

me font sentir la chaleur malgr´e loin de Chine.

I would like to express my sincere gratitude to my supervises Prof. Lipeng Lu and Dr. Liang Shao for their guidance not only on turbulence but also on people skills. I have greatly benefited from the discussions about transcultura with Liang, which allow me to try thinking about the cultural differences between China and France, and also those between turbomachine and turbulence.

I would like to express the deepest appreciation to Prof. Francis Leboeuf, super-visor of my thesis, who passed away ten months before my thesis defense. I will never forget his brown hat, his elegance, his kindness, his enthusiasm, his conta-gious optimism... I will never forget our discussion in the most difficult period of my thesis – “Je fais confiance en ce mod`ele...” – which gives me confidence to

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accomplish this work. I hope Francis could be happy to see this work.

I want to thank Prof. Pierre Sagaut and Prof. John Chew for accepting to review this manuscript. Their precious comments helped to improve the final manuscript. I would also like to thank Dr. R´emy D´enos and Dr. Guillaume Dufour for accepting to be members of the jury.

This thesis would not have been possible without the constant and great help of Jo¨elle Caro, who is always available for replying all my questions and for resolving all the technical problems that I encountered. I want to thank Micka¨el Philit for the enormous help, without whom I would have already given up. I also thank Adrien Cahuzac for his help on Turb’Flow.

The experimental work is done by Wei Ma and Gherardo Zambonini, it is impos-sible to accomplish this thesis without the full access of the experimental database provided by them. I appreciate them for their high quality work. I am in debt to Wei Ma for all the help concerning and not concerning the research. I would like to thank “les sous-soliens”, S´ebastien, Pierre, Benoˆıt, Gilbert for their generous support to ensure the experimental work.

The “Turb’Flow” team must not be forgotten, this work would not have been possi-ble without their contributions. I would like to express my gratitude to Pascal Fer-rand, St´ephane Aubert, Yannick Rozenberg, Lionel Gamet, Laurent Soulat, Laurent Smati... (Sorry, I do not recognize all the people, but I owe you my best thanks.) I owe a special gratitude to Faouzi Laadhari, an “anonymous co-supervisor”, for the discussions about the turbulent boundary layer. I thank Antoine Godard for the discussion about the interpolation method, which allows me to do the analyses on the huge LES database within the last two months of this thesis. I appreciate Jian Fang, Jian Ye and Le Fang for the help and discussions through e-mails. I would like to thank Philippe Eyraud for the help in the library. I want to thank Dan-Gabriel Calugaru and Bernard Barbier for ensuring the operations of the supercomputers and the network, from which I have greatly benefited during these years.

I received generous support and encouragement from colleges and friends, I en-joy the best time with them. I want to thank Zhe Li, Guanyu Zhou, Gang Huang and Changwei Zhou, Lu Zhang (Annabelle) and Benoit Hennebelle, Xu Chen, Guillaume Despres, Pierre Gougeon, Zebin Zhang, Emmanuel Benichou, Joannes Schreiber, Alexis Giauque, Tianli Huang, Kai Zhang, Yu Fan, Xingrong Huang, Shanghao Su, Bo Li, Bo Qu, Ding Tong, Hong Jiang, Yu Bai... I would like to address a special gratitude to Annabelle, “la directrice des Diaos”, for directing “le

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v

I would like to express my sincere gratitude to Prof. Michel Lance, director of the Laboratoire de M´ecanique des Fluides et d’Acoustique for welcoming me in LMFA. I would also like to thank Prof. Isabelle Tr´ebinjac, directrice of the tur-bomachinery team. I want to thank Christine Lance, Fatima El Boukhrissi and Marie-Gabrielle Perriaux for the help concerning the administrative procedures. I want to thank Dominique Berthet for providing a comfortable environment in the office.

Je tiens à remercier “mes profs. de français”, Jean-Marie Duchemin, Ghislaine Ngo Boum, Jean-Louis Marie et Arlène Taulet, pour m’avoir aidé à améliorer mon français.

I want to express my appreciation to the friends who have shared with me many impressive moments, who have shared with me happiness and sadness during the

preparation of PhD thesis: Wei Cui(_{队长), Tianxiao Yang(大饼), Weiguang Zhang(光),}

Wei Hou(猴子), Fang Zhu(芳芳姐) and Jun Xu(师傅), Peiyi Wang, Guotun Hu...

Because of you, the PhD life became so wonderful.

感谢我的父母一直以来对我不懈的支持和鼓励。感谢你们在远方日日夜夜的

思_{念，感谢你们对我没能常回家看看的体谅，感谢你们每个周末守在电话机}

前的耐心等待_{。我想对你}们_{说：“谢谢你们，你们辛苦了！”}

La dernière mais non des moindres, je voudrais remercier du fond du cœur la per-sonne qui m’a accompagnée tout au long de cette thèse, et avec qui je partage ma vie, Liuqing, pour son soutien de chaque instant et son amour de tous les jours. Je t’aime!

Je voudrais dire désolé à mon futur enfant, qui n’est pas encore planifié à venir dans ce monde, à cause du fait que ton papa a fait beaucoup de calculs. Si un jour, tu trouves cette thèse et que tu peux comprendre le français, j’espère que tu ne m’en voudras pas. Mais je te promets que tu bénéficeras des années que ton papa a passé avec toutes les personnes mentionnées ci-dessus!

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Abstract

The increasing demand to reduce the mass of aircraft jet engines and emissions of aircraft propulsion requires to make the compression system of engines more com-pact, since this component accounts for about 40%-50% of the total mass. How-ever, at a given overall pressure ratio, decreasing the number of stages will raise the compressor blade loading per stage. The blade loading is extremely restricted by different three-dimensional flow loss mechanisms. One of them is the corner separation that forms between the blade suction side and the hub or shroud.

Although some works previously investigated the mechanisms and the parameters of corner separation, it is still difficult to propose an effective control method of the corner separation. That is mainly due to two reasons: (i) the lack of knowledge of the physical mechanisms, (ii) the nowadays classical RANS1turbulence models are not capable to accurately predict the corner separation, since they cannot correctly describe the turbulent transport mechanisms.

RANS1 and LES2 simulations are here presented on a compressor cascade

con-figuration, in comparison with experimental data obtained at LMFA (from sepa-rate works). The RANS approach globally over-estimates the corner separation, whereas a significant improvement is achieved with the LES, especially for the blade surface static pressure coefficient and the total pressure losses. The corner separation region, which is the main source of the total pressure losses, is shown to generate large-scale energy-containing eddies. The bimodal histograms of the streamwise velocity that were observed experimentally seem to be confirmed by the LES results. Concerning the streamwise velocity fluctuations (RMS), both the experiment and the LES show some profiles with two peaks. Finally, thanks to the LES approach, the turbulent kinetic energy budget, which represents the bal-ance between the production, dissipation and transport terms, are computed and analyzed. This may help the improvement of RANS turbulence modeling.

1 _{RANS: Reynolds-averaged Navier-Stokes}

2 _{LES: large-eddy simulation}

Keywords: corner separation, compressor cascade, large-eddy simulation, Reynolds-vii

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R´esum´e

La demande croissante pour alléger les moteurs d’avions et diminuer les émissions polluantes de la propulsion aéronautique réclame à rendre plus compact le système

de compression des moteurs, qui repr´esente environ 40%-50% de la masse

to-tale. Or, à taux de compression global égal, la réduction du nombre d’étage ex-ige d’un point de vue aérodynamique une augmentation de la charge des aubes de compresseur par étage. La charge d’aube est aujourd’hui limitée car elle induit différents mécanismes de pertes tridimensionnelles très pénalisant. L’un des plus importants est le décollement de coin qui se forme à la jonction entre l’extrados de l’aube et le moyeu ou le carter.

Bien que des travaux existent sur les mécanismes et paramètres intervenant dans le décollement de coin, il est encore difficile de proposer une méthode de contrôle ef-ficace. Cela est principalement dû à deux raisons : (i) le manque de compréhension fine des mécanismes physiques, (ii) l’utilisation pour la conception de modèles de turbulence classiques de type RANS1qui ne sont pas capables de prédire précisément le décollement de coin, car ils ne peuvent pas décrire correctement les mécanismes de transport turbulent.

Des simulations de type RANS1 et LES2 sont présentées dans cette thèse sur une configuration de grille d’aubes de compresseur, et comparées avec les données expérimentales obtenues au LMFA (issues de travaux séparés). L’approche RANS surestime globalement le décollement de coin. Une amélioration significative est obtenue par la méthode LES, en particulier pour le coefficient de pression statique sur l’aube et les pertes de pression totale. Ces résultats montrent que la zone de décollement de coin, qui est la source principale des pertes, génère des tourbillons de grande échelle associés à de forts niveaux d’énergie. Les histogrammes bi-modaux de la vitesse tangentielle qui ont été observés expérimentalement semblent confirmés par les résultats LES. En ce qui concerne les amplitudes des fluctua-tions de vitesse tangentielle, les résultats expérimentaux et ceux de la LES met-tent en évidence deux pics sur certains profils perpendiculaires aux parois. Enfin, grâce à l’approche LES, les bilans de l’énergie cinétique turbulente sont calculés et analysés. Ils décrivent l’équilibre entre les termes de production, de dissipation

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1_{RANS: Reynolds-averaged Navier-Stokes}

2_{LES: large-eddy simulation}_{= simulation des grandes ´echelles}

Mots-clés : décollement de coin, grille d’aubes de compresseur, simulation des grandes échelles, Reynolds-averaged Navier-Stokes, bilan d’énergie cinétique tur-bulente, couche limite turbulente.

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2.3.2.2 Reynolds-Averaged Navier-Stokes . . . 41 2.3.2.3 Large-eddy simulation . . . 42 2.3.3 RANS method . . . 42 2.3.3.1 Averaged equations . . . 42 2.3.3.2 Turbulence model . . . 44 2.3.4 LES method . . . 45 2.3.4.1 Filtered equations . . . 45 2.3.4.2 SGS model . . . 48 2.3.4.3 Mean-flow extraction . . . 49

2.3.5 Reynolds stress budget . . . 50

2.3.6 Triggering turbulence . . . 51

2.4 Implementation of the simulations . . . 53

2.4.1 RANS/URANS simulation . . . 53

2.4.1.1 Mesh . . . 53

2.4.1.2 Boundary conditions . . . 55

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Contents xiii

2.4.2 LES simulation . . . 57

2.4.2.1 Mesh . . . 57

2.4.2.2 Boundary conditions . . . 58

2.4.2.3 Initial condition . . . 60

2.4.2.4 Spatial and temporal discretization . . . 60

2.5 List of the calculations . . . 60

3 Mean aerodynamics of the compressor cascade 63 3.1 Inflow conditions . . . 63

3.1.1 Inflow boundary layer evolution . . . 64

3.1.2 Mean velocity profile . . . 65

3.1.3 Turbulent fluctuations . . . 66

3.1.4 Spectrum . . . 68

3.2 Statistical convergence . . . 69

3.3 Flow within the cascade passage: reference configuration . . . 71

3.3.1 Classification of the corner separation . . . 71

3.3.2 Global performance . . . 72

3.3.3 Topology . . . 74

3.3.4 Streamlines . . . 75

3.3.5 Blade surface static pressure coefficient . . . 76

3.3.6 End-wall static pressure coefficient . . . 79

3.3.7 Total pressure losses . . . 81

3.3.8 Downstream flow evolution . . . 82

3.3.9 Extent of the corner separation . . . 85

3.4 Conclusion . . . 86

4 Corner separation parameter study 89 4.1 Numerical parameters . . . 89

4.1.1 Spatial interpolation scheme . . . 89

4.1.2 Artificial viscosity of the centered spatial scheme . . . 90

4.1.3 Outlet boundary condition . . . 91

4.1.4 Turbulence model . . . 92

4.2 Physical impact parameters . . . 96

4.2.1 Incidence angle . . . 97

4.2.2 Inlet boundary layer . . . 101

4.2.2.1 Inflow boundary layer thickness . . . 101

4.2.2.2 Inflow TKE level . . . 103

4.2.2.3 Inflow angle fluctuation . . . 104

4.2.3 Tripping bands on the blade . . . 104

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5 Turbulent characteristics of the corner separation 109

5.1 Subgrid activity . . . 109

5.2 Subgrid-scale viscosity ratio . . . 110

5.3 Unsteadiness of the corner separation . . . 111

5.3.1 One-point spectral analysis of velocity . . . 111

5.3.2 Energy integral length scale . . . 113

5.3.3 Visualization of the turbulent coherent structures . . . 116

5.3.4 Bimodal histogram . . . 118

5.4 Turbulent boundary layers . . . 120

5.4.1 Boundary layer on the suction surface close to the mid-span . . . 121

5.4.1.1 Mean flow . . . 121

5.4.1.2 Second-order statistics . . . 124

5.4.1.3 Turbulent anisotropy . . . 127

5.4.1.4 Turbulent kinetic energy budget . . . 131

5.4.2 Boundary layer on the suction surface close to the end-wall . . . 134

5.4.2.1 Mean flow . . . 135

5.4.3 End-wall boundary layer within the passage . . . 144

5.4.3.1 Mean flow . . . 145

5.5 Conclusion . . . 150

6 Conclusions and Perspectives 153 6.1 Conclusions . . . 153

6.2 Perspectives . . . 154

A Reynolds stress budget equation 155

B Test case: Turbulent boundary layer at moderate Reynolds number 161

C Turb’Split 167

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List of Figures

1 Gas-turbine engine pressure ratio trends. (data partially from [Gunston,1998]) 2

2 High-loss regions in a multistage compressor [Wisler,1985]. . . 3

3 Nature of the flow in an axial flow compressor rotor passage [Lakshminarayana, 1996, p. 11]. . . 3

1.1 Blade suction surface flow visualization [Dring et al.,1982]. . . 8

1.2 Flow visualization in an axial flow water pump [Zierke & Straka,1996]. . . 8

1.3 Second stage stator vane flow visualization on suction side [Joslyn & Dring, 1985]. . . 9

1.4 Stator flow visualization [Dong et al.,1987]. . . 10

1.5 Total pressure coefficient contours, downstream of the test stator [Barankiewicz & Hathaway,1998]. . . 11

1.6 Oil visualization of stator hub and suction surface [Friedrichs et al.,2001]. . . . 11

1.7 Blade surface flow visualization: top, Experiment; bottom, CFD. [Gbadebo et al.,2005] . . . 13

1.8 Corner separated flow topology by CFD [Lewin et al.,2010]. . . 14

1.9 Classification of critical points [Dallmann,1983]. . . 15

1.10 Topology of corner separation [Schulz et al.,1990]. . . 17

1.11 Topology of corner separation [Hah & Loellbach,1999]. . . 17

1.12 Topology of corner separation [Lewin et al.,2010]. . . 18

1.13 Lei’s criterion for massive corner separation [Lei et al.,2008]. . . 26

2.1 General view of the wind tunnel in LMFA of Ecole Centrale de Lyon. . . 31

2.2 Sketch of the wind tunnel: side and vertical views. . . 31

2.3 Sketch of the blade: (a) blade in original thickness distribution; (b) mean cam-ber line; (c) blade based on the mean camcam-ber line. . . 32

2.4 Notation of the cascade. . . 33

2.5 Sketch of the tripping bands. . . 34

2.6 Available experimental results. . . 35

2.7 PIV measurement zones on a plane. . . 38

2.8 LDA measurement stations, squares indicate the beginning points of the mea-surement stations. . . 39

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2.9 Effect of tripping band on turbulent transition. Extracted from [Cousteix,1989]. 52

2.10 View of the computational domain. . . 54

2.11 Views of the mesh. . . 54

2.12 View of parallelization. . . 55

2.13 y+_{on the blade and end-wall. . . .} ₅₅

2.14 y+_{on the blade and end-wall in LES.} _{. . . .} ₅₇

2.15 Sketch of inflow condition. . . 59

3.1 Evolution of the boundary layer thicknesses. . . 65

3.2 Evolution of the shape factor and the friction coefficient. . . 65

3.3 Mean velocity profiles. . . 66

3.4 Reynolds stress and TKE. . . 67

3.5 TKE balance. . . 67

3.6 Inflow boundary layer velocity spectrum aroundy+ _{= 85. . . .} ₆₉

3.7 Statistical convergence at mid-span. Black line: instantaneous velocity signal. Red line: cumulative first-order moment convergence. Green line: cumulative second-order moment convergence. Blue line: cumulative third-order moment convergence. . . 70

3.8 Statistical convergence in the corner separation region. Same notations as Fig.3.7. 70 3.9 Statistical convergence in the high loss center. Same notations as Fig.3.7. . . . 71

3.10 Classification of the corner separation. . . 72

3.11 Oil visualization of the ECL experiment . . . 75

3.12 Topology of hub-corner separation . . . 76

3.13 Streamlines around the corner separation: the walls are colored by static pres-sure coefficient from_{−0.25 to 0.25, the streamlines are colored by helicity. . .} 77

3.14 Mean static pressure coefficient on the blade suction surface. . . 78

3.15 Mean static pressure coefficient on the blade pressure surface. . . 78

3.16 Static pressure coefficient around the blade, at various spanwise positions. . . . 80

3.17 Mean static pressure coefficient on end-wall. . . 81

3.18 Mean total pressure loss coefficient at outlet section 1. . . 82

3.19 C∗ pt at outlet section 1. . . 83

3.20 Downstream flow evolution. . . 84

3.21 Pitchwise-averaged quantities evolution. . . 85

3.22 Cpt,globalevolution. . . 85

3.23 Sketch of the boundary layer stations. . . 86

3.24 Relative displacement thicknessδ1,r(s∗, z/h). . . 87

4.1 Influence of the numerical spatial scheme: Cptcontours at the outlet section 1. . 90

4.2 Influence of the numerical spatial scheme: velocity profiles at the outlet section 1,z/h = 0.11. . . 90

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List of Figures xvii

4.4 Impact of the artificial viscosity: velocity profiles at the outlet section 1,z/h =

0.11. . . 92

4.5 Impact of the outlet boundary condition: velocity profiles at outlet section 1, z/h = 0.11. . . 92

4.6 Impact of the turbulence model: static pressure coefficient on the blade suction surface. . . 93

4.7 Impact of the turbulence model: static pressure coefficient on the blade pressure surface. . . 94

4.8 Impact of the turbulence model: static pressure coefficient around the blade. . . 94

4.9 Impact of the turbulence model: static pressure coefficient on the end-wall. . . 95

4.10 Impact of the turbulence model: total pressure loss coefficient at the outlet sec-tion 1. . . 95

4.11 Impact of the turbulence model: Cpt,globalaxial evolution. . . 96

4.12 Impact of the turbulence model: evolution of the tangential velocity us/U∞ close to the mid-span. . . 97

4.13 Cp distributions on the blade suction side. Top: experiment; Bottom: RANS. . . 97

4.14 Cp distributions on the blade pressure side. Top: experiment; bottom: RANS. . 98

4.15 Cp distributions on the end-wall: top, experiment; bottom, RANS. . . 99

4.16 Static pressure coefficient around the blade. . . 100

4.17 CLof the blade. . . 100

4.18 Investigated inflowδ1,∞and the corresponding velocity profiles. . . 101

4.19 Cp distributions on the blade suction side, with different inflowδ1. . . 102

4.20 Static pressure coefficient around the blade. . . 102

4.21 Cptcontours at the outlet section 1, for different inflowδ1. . . 103

4.22 Global performances, for different inflowδ1. . . 103

4.23 Inflow TKE profiles. . . 104

4.24 CLandCpt,global. . . 104

4.25 Impact of the inflow angle fluctuation: CLandCpt,global. . . 105

4.26 Sketches of the blade with and without tripping bands. . . 105

4.27 Influence of the tripping bands on the laminar-turbulent transition ati = 4◦_{, oil} visualization extracted from [Ma,2012]. . . 106

4.28 Influence of the tripping bands on theCp distribution around the blade close to the mid-span ati = 0◦_{, extracted from [}_Ma_,₂₀₁₂_{]. . . .} ₁₀₆

4.29 LDA measurement lines and the corresponding velocity vector profiles close to the mid-span. . . 107

4.30 Evolution of the tangential velocityus/U∞at the mid-span. . . 108

5.1 Subgrid-activity parameter. . . 110

5.2 Subgrid-scale viscosity and RANS eddy viscosity. (a) and (b):z/h = 180/370 = 48.6%; (c) and (d): z/h = 5/370 = 1.4%. . . 111

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5.3 Numerical probe locations with instantaneous streamlines, close to the end-wall

(z/h = 1.4%). . . 112

5.4 Numerical probe locations with time-averaged streamlines, close to the end-wall (z/h = 1.4%). . . 113

5.5 One-dimensional velocity spectra. From bottom to top: P1 to P8. . . 114

5.6 One-dimensional velocity spectra. From bottom to top: P9 to P16. . . 114

5.7 Energy integral length scales at the probes located in Fig.5.4. . . 115

5.8 3-D view (from the mid-span to the end-wall) of the turbulent coherent struc-tures: iso-surfaces ofQ criterion, colored by the velocity magnitude. . . 117

5.9 View of the turbulent coherent structures in the passage: iso-surfaces ofQ cri-terion, colored by the velocity magnitude. . . 118

5.10 Inflow velocity PDF (at hot-wire measurement station 2 in Fig.2.6,y+_{= 85). .} ₁₁₉

5.11 Velocity histograms on the bimodal point P3. . . 120

5.12 Sketch of the boundary layer stations at the mid-span. . . 121

5.13 Velocity decomposition in the blade coordinate system. . . 122

5.14 Tangential velocity_husi/U∞close to the mid-span: Comparison between LES, LDA and PIV results. (vertical bars indicate the positional uncertainties of the LDA measurement points) . . . 123

5.15 Wall normal velocity _hu_n_i/U_∞ close to the mid-span: Comparison between LES, LDA and PIV results. (vertical bars indicate the positional uncertainties of the LDA measurement points) . . . 124

5.16 The RMS of the streamwise fluctuating velocityu′ s,rms/U∞ close to the end-wall: Comparison between LES, LDA and PIV results. (vertical bars indicate the positional uncertainties of the LDA measurement points) . . . 126

5.17 The RMS of the wall-normal fluctuating velocityu′ n,rms/U∞ close to the end-wall: Comparison between LES, LDA and PIV results. (vertical bars indicate the positional uncertainties of the LDA measurement points) . . . 127

5.18 Reynolds shear stress_−hu′_su′ ni/U∞2 close to the end-wall: Comparison between LES, LDA and PIV results. (vertical bars indicate the positional uncertainties of the LDA measurement points) . . . 128

5.19 Sketch of the anisotropy-invariant map, fromLumley & Newman[1977]. . . . 129

5.20 Color scheme used to denote distance in anisotropy-invariant map. . . 130

5.21 Development of the turbulent anisotropy along the blade suction side close to the mid-span. . . 131

5.22 Turbulent anisotropy map at the first stations∗ _{= 0.21, for three different} span-wise positions . . . 132

5.23 TKE budget close to the mid-span. . . 133

5.24 Sketch of the boundary layer stations, on the suction side close to the end-wall. 134 5.25 Tangential velocity_hu_s_i/U_∞ on the suction surface close to the end-wall: (a) linear scale; (b) log scale. . . 135

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List of Figures xix

5.26 Wall-normal velocity_huni/U∞on the suction surface close to the end-wall: (a) linear scale; (b) log scale. . . 137 5.27 Spanwise velocity component_huzi/U∞ on the blade suction side close to the

end-wall, in semi-log scale. . . 137 5.28 Streamwise Reynolds normal stressu′

s,rms/U∞: (a) linear scale; (b) log scale. . 138 5.29 The RMS of the wall-normal fluctuating velocity u′

n,rms/U∞: (a) linear scale; (b) log scale. . . 140 5.30 Reynolds shear stress_−hu′_su′

ni/U∞2: (a) linear scale; (b) log scale. . . 141 5.31 Development of the turbulent-anisotropy map along the blade suction side close

to the end-wall. . . 142 5.32 TKE budget close to the end-wall. . . 144 5.33 Sketch of the boundary layer stations. . . 145 5.34 Tangential, wall-normal and spanwise velocity components along the cascade

passage: (a) linear scale; (b) semi-log scale. . . 146 5.35 The RMS of the fluctuating velocities along the cascade passage: (a) linear

scale; (b) semi-log scale. . . 147 5.36 Development of the turbulent-anisotropy map along the compressor cascade

passage. . . 148

5.37 TKE budget along the compressor cascade passage: (a) LES; (b) RANS. . . 149

B.1 Turbulent boundary layer evolution. . . 163 B.2 Turbulent boundary layer profile. . . 164 B.3 Turbulent kinetic energy budget. . . 165 C.1 Process-based parallelization. . . 167 C.2 Block-based parallelization. . . 168 C.3 LES load balancing. . . 168

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Nomenclature

Roman Symbols

A [m2] Flow area

AR [-] Blade aspect ratio

B [-] Blockage coefficient

c [m] Blade chord

ca [m] Blade axial chord

CL [-] Blade lift coefficient

Cp [-] Static pressure coefficient,Cp = (Ps− Ps,∞)/(1₂ρU∞2 ) Cpt [-] Total pressure loss coefficient,Cpt = (Pt,∞− Pt)/(1₂ρU∞2)

Cs [-] Standard Smagorinsky constant:Cs= 0.18

D [Rad] Lei’s diffusion parameter

DF [-] Lieblein diffusion factor

DH [-] De Haller number f [s−1] Frequency fc [s−1] Cutoff frequency h [m] Blade span H12 [-] Shape factor i [◦] Incidence angle

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i∗ _[◦_] _{Optimum incidence angle}

J [-] Jacobian matrix

k [m2/s2] Turbulent kinetic energy

l [-] Normalized energy integral length scale

N [-] Node number

n [m] Perpendicular distance to blade suction surface

p [m/s] Trace of Jacobian matrix

Ps [Pa] Static pressure

Pt [Pa] Total pressure

Q [s−2] Q-criterion

q [m2/s2] Determinant of Jacobian matrix

r [m] Radius

Re [-] Reynolds number

S [-] Lei’s stall indicator

S [-] Saddle point number

s [m] Pitch of the cascade

Sij [s−1] Rate-of-strain tensor

s, n, z [m] Curvilinear abscissas of cascade

t [s] Time

U [-] Uncertainty

U [m/s] Velocity magnitude

un [m/s] Velocity component perpendicular to blade suction surface

us [m/s] Tangential velocity relative to blade suction surface

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List of Figures xxiii

u, v, w [m/s] Instantaneous velocity components inx, y, z directions, respectively

uz [m/s] Spanwise velocity component

x, y, z [m] Cartesian coordinates of the cascade

Greek Symbols

αi [◦] Attack angle

β1 [◦] Actual inflow angle

β′

1 [◦] Design inflow angle

β2 [◦] Actual outflow angle

β′

2 [◦] Design outflow angle

C∗

pt [-] Pitchwise-mass-averaged total pressure loss coefficient

∆ [m] Grid spacing

δ, δ99 [m] 99% Boundary layer thickness (u(δ) = 99%u∞)

δ0 [◦] Flow deviation angle

δ1, δ∗ [m] Displacement thickness of boundary layer

δ2, θ [m] Momentum thickness of boundary layer

∆η [◦] Additional turning angle due to skewed incoming end-wall boundary layer

δ1,r [-] Relative displacement thickness

∆Uθ [m/s] Change of circumferential velocity

γ [◦] Stagger angle

µ [kg/m/s] Dynamic molecular viscosity

µsgs [kg/m/s] SGS eddy viscosity

µt [kg/m/s] Eddy viscosity

ν [m2/s] Kinematic viscosity:ν = µ/ρ

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Ωij [s−1] Rate-of-rotation tensor

ω+ _[-] _{Normalized frequency:}_ω+ ₌ 2πνf

u2 τ

φ [-] Flow coefficient

Φuu [-] Normalized velocity spectrum: Φuu(ω+) = φuu_2πν(f )

φuu [m2/s] Velocity spectrum

Πij [kg/m/s2]Subgrid-scale tensor

ρ [kg/m3] Density

σ [-] Solidity

τij [kg/m/s2]Viscous stress tensor τtij [kg/m/s

2_{]Reynolds stress:} _τ

tij = −hρu

′′ iu′′ji

θ [◦] Flow turning angle

ϕ [◦] Camber angle

Superscripts

q′′ _{Fluctuating quantity after Favre average:}_q′′_{= q − [q], [q}′′_{] = 0, hq}′′_{i 6= 0}

q Filtered quantity

q′ _{Fluctuating quantity after time average:} _q′ _{= q − hqi, hq}′_{i = 0}

q+ _{Quantity in wall unit}

− →_q _{Vectorial quantity} e q Favre-filtered quantity:q = ρq/ρ_e Subscripts e Effective

global Global quantity

∞ Reference quantity

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List of Figures xxv

ps Pressure side

rms Root mean square

ss Suction side

Other Symbols

[q] Favre-averaged quantity:_{[q] = hρqi/hρi}

hqi Time-averaged quantity

Acronyms

ANR L’Agence National de la Recherche (The French National Research Agency) AXIOOM Advanced eXperiments and sImulations of cOmplex flOws in turboMachines CDA Controlling Diffusion Airfoil

CFD Computational Fluid Dynamics

CSC China Scholarship Council

CS Corner separation

DDES Delayed Detached-Eddy Simulation DNS Direct Numerical Simulation

LDA Laser Doppler Anemometry LDV Laser Doppler velocimetry

LES Large-Eddy Simulation

LMFA Laboratoire de M´ecanique des Fluides et d’Acoustique NSFC National Natural Science Foundation of China

PDF Probability Density Function

PIV Particle Image Velocimetry

PVD Prescribed Velocity Distribution: A blade profile of the Whittle lab, University of Cam-bridge.

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RANS Reynolds-Averaged Navier-Stokes

SAS Scale-adaptive simulation method

SGS Subgrid scale

TKE Turbulent Kinetic Energy

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Introduction

Background

Engineers continuously strive to reduce the costs of aircraft propulsion. This trend is en-forced in the recent years for two main reasons. One is the impact of the economic crisis, the other is the reduction of the nonrenewable fossil fuels.

There are two possible ways to achieve this purpose. A first solution is to increase the efficiency of aircraft engines, particularly in increasing the overall pressure ratio of compressors [Wang, 2009b]. As shown in Fig. 1, since the 1960s, the overall pressure ratio of aircraft engines is constantly increasing from 13 to 42. A second solution is to decrease the weight of aircraft engines, especially in reducing the number of compressor stages. Actually, compressor represents about 40-50% of engine weight [Steffens & Sch¨affler,2000]. The trend of increasing the overall pressure ratio and reducing the stage number can be observed in the motorization evolution of Boeing 777 [Godard,2010]. From PW 4084 to GE 90-115B (shown in Tab.1), the overall pressure ratio increased, the stage number of compressor and turbine decreased and the thrust specific fuel consumption decreased. This means that, the loading of each stage (blade loading) increased.

However, the blade loading is limited by many three-dimensional flow losses in

compres-sors. These high-loss regions are located and summarized by Wisler [1985] and

Lakshmi-narayana [1996], shown in Fig. 2 and Fig. 3 respectively. These high-loss regions originates from the three dimensional end-wall boundary layers, flow separations, leakages, secondary flows and shocks.

During the last decades, the impact of hub-corner separation in reducing the blade loading has been emphasized by many researchers. Among these areDong et al.[1987],Schulz et al.

[1990],Yocum & O’Brien[1993],Zierke & Straka[1996],Hah & Loellbach[1999],Gbadebo

et al.[2005],Lei[2006], Choi et al.[2008],Lewin et al.[2010], Ma et al.[2013a]. The main causes of hub-corner separation are: (1) strong adverse pressure gradients; (2) secondary flows in the passage; (3) the end-wall and the blade suction surface boundary layer mixing, (4) the

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1965 1975 1985 1995 2005 Year of Certification 0 10 20 30 40 O ve ra ll Pr essu re R at io , S ea L ev el JT3D Conway 508

Spey 555 Tay 611 Tay 651

Conway 550 Spey 505 Spey 512 Spey 512-14 JT8D-1 _JT8D-17 JT8D-219 JT9D-3A JT9D-70JT9D-7R4G PW4052 PW4165 PW4084 TF39-1 CF6-6 CF6-50A CF6-50E CF6-80C2A8 CF6-80E1A2 CF6-80E1A4 CFM56-2 CFM56-3C CFM56-5B CFM56-5C4 RB211-22 RB211-524D4 Trent 775 Trent 890 GE90-115B

Figure 1: Gas-turbine engine pressure ratio trends. (data partially from [Gunston,1998])

Table 1: Motorization evolution of Boeing 777.

PW 4084 GE 90-115B

Overall pressure ratio 40 42

Stage number of compressor 18 14

1F + 6LPC + 11HPC 1F + 4LPC + 9HPC

Stage number of turbine 9 8

2HPT + 7LPT 2HPT + 6LPT

Thrust specific fuel consumption 0.55 0.52

First certification April 1994 July 2003

(FAA) (FAA)

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Introduction 3

Figure 2: High-loss regions in a multistage compressor [Wisler,1985].

Figure 3: Nature of the flow in an axial flow compressor rotor passage [Lakshminarayana,1996, p. 11].

horseshoe vortex (when the blade leading edge is thick) [Ma, 2012]. As hub-corner separation can reduce the compressor efficiency, and lead to negative consequences, it is requested to control or reduce the occurrence of hub-corner separation.

Some flow control strategies have been investigated on specific blades by researchers, such as Mu & Lu [2007], Gbadebo et al.[2008], Godard [2010], Sachdeva [2010], Marsan et al.

[2012]. But, it is still very difficult to attain an effective strategy for controlling hub-corner separation. There are probably two main reasons:

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the parameters controlling the onset and the size of separation [Gbadebo et al.,2005] [Ma,

2012].

2. In parallel with the development of the computer resources, the computational fluid dy-namics (CFD) codes based on Reynolds-averaged Navier-Stokes (RANS) equations have integrated the turbomachinery design process. But the existing RANS turbulent models are very limited in predicting compressor hub-corner separation. Therefore, the control strategies which are developed from these turbulent models are not always reliable [Ning & Xu,2001] [Wang,2009a] [Liu et al.,2011].

Fortunately, with the rapid development of computer technology, it becomes possible to use large-eddy simulation (LES) for investigating the flow mechanisms in complex geometries [Boudet et al.,2007] [Ye,2009] [Gand,2011]. Large-eddy simulation has proved to be capable to predict the turbomachinery flows [Cahuzac, 2012].

Research objectives

This thesis is supported by CSC1_{and the ANR}2_-NSFC3_{project AXIOOM}4_{, an international}

collaborative project between the ´Ecole Centrale de Lyonin France and the Beihang University in China.

Some previous experimental work has been done byMa[2012] who established an original and accurate database.

The objectives of the work in this thesis are:

1. to carry out simulations with both (U)RANS and LES approaches, and to evaluate the numerical results in comparison with the experimental database fromMa[2012],

2. to study parameters controlling the hub-corner separation with the (U)RANS results, 3. to reinforce the practical use of LES in turbomachines,

4. to explore the turbulent energy transport nature in the corner separation region.

1

China Scholarship Council

2

Agence National de la Recherche (French National Research Agency)

3

National Natural Science Foundation of China

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Introduction 5

Thesis outline

Chapter1reviews previous works on corner separation in axial turbomachines, with partic-ular attention to criteria, parameters and topologies of the corner separation.

In Chapter2, the experimental configuration, the blade geometry and the experimental ap-proaches used in the thesis ofMa[2012] are recalled. Then the numerical methods employed in this thesis are introduced, including the (U)RANS and LES approaches, the turbulence/subgrid-scale modelings, the budget analysis, etc.

Chapter 3 concerns the mean aerodynamics of corner separation. The RANS and

time-averaged LES results are analyzed in comparison with the experiment results.

The parameters controlling the onset and the size of the corner separation are studied in Chapter4using the RANS results.

The turbulent characteristics of the corner separation are analyzed in Chapter 5using the LES results. The budget of turbulent kinetic energy (TKE) and Reynolds stresses are investi-gated, which lightens a new avenue for turbulence model improvement oriented towards corner separation.

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Chapter 1 Corner separation in axial turbomachines

Corner separation is a common feature in axial turbomachines. It is a major source of losses, at the junction between the hub (end-wall) and the blade suction surface. Corner sepa-ration, interacting with the tip leakage flow and the main flow, is one of the precursors of the rotating stall [Choi et al.,2008]. Many research works have been conducted to investigate axial compressor corner separation. In this chapter, firstly, previous works on corner separation of rotor, stator, and linear compressor cascade will be reviewed. Secondly, criteria for corner sep-aration will be introduced. Thirdly, the main parameters known to influence corner sepsep-aration will be summarized. And finally, some works about the turbulent modeling in corner separation will be summed up.

1.1 In compressors/annular cascades

Rotors

In some investigations, corner separation was observed in rotors of compressors, even in axial flow water pumps.

Dring et al.[1982] investigated an isolated compressor rotor facility with high blade load-ing and low blade aspect ratio, to investigate the three-dimensional flows in compressors. Near the hub, a high-loss region associated with corner separation was clearly observed. When in-creasing the blade loading, the corner separation at the junction between the hub and the blade suction surface extended to full blade height separation (shown in Fig.1.1).

Wisler[1985] constructed a multi-stage, low-speed, large-scale research compressor rig. A large region of corner separated flow was found on the Rotor I blade suction surface near the

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hub, at peak efficiency operating point. As sketched in Fig.2, corner separation was considered to be a main source of losses in the axial compressor.

Corner separation occurs not only in compressors, but also in water pumps.Zierke & Straka

[1996] investigated experimentally the three-dimensional flow in an axial flow water pump by oil visualization. A three-dimensional corner separation including a spiral node was observed on the rotor suction surface near the hub, while a two-dimensional-like separation extended over most of the span.

(a) Lower blade loading (b) Higher blade loading

Figure 1.1: Blade suction surface flow visualization [Dring et al.,1982].

(a) Sketch of the suction surface flow pattern (b) Oil visualization near the hub

Figure 1.2: Flow visualization in an axial flow water pump [Zierke & Straka,1996].

Stators

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1.1. In compressors/annular cascades 9

The rotating rig of Dring et al.[1982] was upgraded by Joslyn & Dring[1985] to a two-stage compressor test model. The second two-stage stator aerodynamics was carefully studied at three operating points: a high flow coefficient operating point (φ = 0.55), the design operating point (φ = 0.51) and a near-stall operating point (φ = 0.45). As shown in Fig. 1.3, corner separations were observed on the blade suction side near the hub at all these three flow coeffi-cients. At the high flow coefficient operating point (φ = 0.55), the stator blade corner separation occurred at about 60% of the axial chord, and extended up to 36% of the span at the trailing edge. Decreasing the flow coefficient toφ = 0.45, the corner separation grew significantly, the separating point went upwards to26% of the axial chord, and extended up to 75% of the span at the trailing edge. It was suggested that the corner separation was the major feature in the stator flow, and the main source of losses and blockage. It was also found that, at near-stall operating point, the rotor corner separation resulted in an important increase of the incidence angle of the stator, thus causing a significant growth of the stator corner separation.

Flow

(a)φ = 0.55 (b)φ = 0.51 (c)φ = 0.45

Figure 1.3: Second stage stator vane flow visualization on suction side [Joslyn & Dring,1985].

In the investigation ofWisler[1985], the hub of Rotor A was designed to avoid the corner separation observed in Rotor I. However, the corner separation was then found on the stator.

Dong et al.[1987] studied the three-dimensional flow in a single-stage, low-speed, high-reaction axial compressor. Corner separations were found at both hub and shroud and of the stator, with a larger extent near the hub, as shown in Fig.1.4. However, no evidence of corner separation was observed on the rotor hub.

Schulz & Gallus[1988],Schulz et al.[1990] carried out an experimental investigation of the three-dimensional flow in a compressor rig. The experiments were performed at five different stator incidence angles. The oil flow visualization revealed that corner separation occurred at all these five incidence angles, with or without the upstream rotor. The corner separation was also evidenced, by the blade surface static pressure coefficient, with the presence of a constant pressure region on the suction side near the blade trailing edge. A region of total pressure losses was also observed downstream the corner region.

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Shroud Hub

(a) Stator suction surface (b) Stator hub

Figure 1.4: Stator flow visualization [Dong et al.,1987].

On the same configuration asSchulz & Gallus[1988],Hah & Loellbach[1999] performed some numerical investigations to study the evolution of the corner separation. They found that, a single vortex with two legs was formed at the corner separation region, its legs jointed on the hub with two counter-rotating vortices (see Fig.1.11(b)). They suggested that, the formation of this three-dimensional single vortex was the major mechanism of the corner separation.

Li et al.[1992] investigated experimentally an annular compressor cascade, at low Reynolds number and low Mach number. The flow was considered at four incidence angles. Corner sep-aration was located by oil visualization, passage velocity measurements and downstream total pressure losses. The author observed that the separated flow is highly turbulent. They also claimed that the severe corner separation may result in much higher losses than wake and sec-ondary flows. The highest momentum losses of the separated flow are found in axial direction. In the study ofBarankiewicz & Hathaway[1998], the General Electric’s Energy Efficient Engine blade was modeled and installed on the NASA Lewis research center’s low-speed axial compressor, to investigate the impact of the stator hub leakage. Three flow coefficients were chosen for the investigation. According to the downstream total pressure coefficient contours shown in Fig. 1.5, corner separation was identified at all the three flow coefficients (design point, peak pressure, near stall).

Friedrichs et al.[2001] did experimental investigations with two different stators in a single-stage, high loading, low-speed axial compressor. A significant corner separation was found on baseline Stator A at both design operating point and part load operating point, as shown in Fig.1.6.

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1.1. In compressors/annular cascades 11

(a) Design operating point (φ = 0.395)

(b) Peak pressure operating point (φ = 0.35)

(c) Near-Stall operating point

(φ = 0.31)

Figure 1.5: Total pressure coefficient contours, downstream of the test stator [Barankiewicz & Hathaway,1998].

(a) Design operating point (φ = 0.45)

(b) Part load operating point (φ = 0.37)

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1.2 In linear cascades

During the last two decades, in order to investigate the fundamental mechanisms of cor-ner separation, some people used linear compressor cascades, which eliminates the difficulties linked to the rotation and some of the technological effects, such as leakage, non uniform in-flow, etc. This can also allow experiments with higher dimension and improve significantly the spatial resolution of the measurements. Compared to annular cascades, the geometry of lin-ear cascades are simpler, and instrumentations and its displacements are often easier. Without changing the solidity, linear cascades permit to use blade with large span height so as to uncor-relate the physics at mid-span from the one close to the end-walls. However the periodicity of the flow is more difficult to obtain with linear cascade.

Yocum & O’Brien[1993] used a low-speed linear compressor cascade with 18 blades of

simple geometry and a solidity of unity. The aspect ratio was set to be 4.7 to preclude the interaction between the corner separations from both the two end-walls. The measurements were carried out for three stagger angles and a large range of attack angles. Corner separation was captured by surface and smoke visualizations, mean velocity measurements and blade static pressure distributions. Blade stagger angle was shown to be a key element in determining the characteristics of corner separation.

A series of more detailed investigations was carried out by Gbadebo et al. ([Gbadebo,

2003], [Gbadebo et al., 2005]), based on the previous work of Bolger[1999]. Two cascades were experimentally and numerically studied. Cascade 1 consisted of five NACA65 blades, while cascade 2 consisted of five modern PVD (prescribed velocity distribution) blades. The results were presented at incidence angle0◦ _{for cascade 1, and at incidence angles}₋₇◦ _and₀◦ for cascade 2. As shown in Fig.1.7, both the tuft visualization on cascade 1 and oil visualization on cascade 2 indicated a corner separation on these two cascades. It appears that corner sepa-ration is highly three-dimensional, and that the formation of corner sepasepa-ration is closely related with the leading-edge horseshoe vortex and the associated end-wall dividing streamlines that emanate from the leading-edge saddle point.

Lewin et al.[2010] performed some numerical investigations on a large-scale low-speed

linear compressor cascade, following the experimental work ofSellschopp[1995]. The cascade of NACA65 blade profiles was studied with two different stagger angles, and increasing load-ings. Corner separation was identified by oil visualization and downstream total pressure loss coefficient. Three vortices are observed in a developed corner separation. The end-wall focus (F2), which can be seen in Fig.1.8, was thought to dominate the characteristics of a developed corner stall. The fluid leaves off the end-wall from this point.

A high-loading linear compressor cascade with low-aspect ratio controlling diffusion airfoil (CDA) blading was experimentally investigated byZander et al.[2009]. And later,Steger et al.

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1.2. In linear cascades 13

(a) Cascade 1, incidence angle:0◦

(b) Cascade 2, incidence angle:0◦

Figure 1.7: Blade surface flow visualization: top, Experiment; bottom, CFD. [Gbadebo et al.,

2005]

[2010] studied numerically this cascade by using delayed detached-eddy simulation (DDES) proposed by Spalart et al. [2006]. Suction surface laminar-turbulent transition bubble, corner separation and mid-span 2D separation were observed, both experimentally and numerically. The DDES method was found to be superior to the RANS method in predicting the flow in this cascade. However, some discrepancies on the blade surface static pressure coefficient could be observed between DDES and experiment near the end-wall, demonstrating the insufficiency of DDES in the corner separation region.

Recently, an experimental investigation of intermittent corner separation in a linear com-pressor cascade was carried out byMa et al.[2013a], at Laboratoire de M´ecanique des Fluides

et d’Acoustique (LMFA), Ecole Centrale de Lyon. A very detailed characterization of corner

separation at different incidence angles was achieved, using different measurement methods, including oil and tuft flow visualization, hot-wire measurements, stationary wall pressure mea-surements, five-hole probe meamea-surements, 2D-PIV and 2D-LDA. Particularly, a bimodal phe-nomenon was found in the corner separation region, which is not yet clearly understood.

Wang & Yuan [2013] simulated the configuration of Ma et al. [2013a] using a scale-adaptive simulation method (SAS) proposed by Menter & Egorov [2010]. A good descrip-tion of the blade static pressure coefficient was achieved, although some discrepancies appear on the downstream total pressure loss coefficient. They suggested that the unsteadiness of the

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(a) Inlet angle of50◦

(b) Inlet angle of56◦

Figure 1.8: Corner separated flow topology by CFD [Lewin et al.,2010].

corner separation is dominated by the “backbone” and the “precursor” vortices, as well as the “induced” vortex that is generated by the interaction between these two vortices.

1.3 Topology

Topological analysis is a useful tool for studying the flow properties on the surface of a body. It can help to obtain a direct understanding of the separation.

The critical point theory and the corner separation topology will be reviewed in this section.

1.3.1 Critical point theory

Defining a wall friction vector −→τ = τx−→i + τy−→j , the critical points are the points where the wall friction are zero, i.e. bothτx andτy equal to zero.

The critical point theory on three dimensional separation was reviewed in detail byGbadebo

[2003] andSachdeva [2010]. Basically, the critical point is classified by the trace (p) and the determinant (_{q) of the 2 × 2 Jacobian matrix in Eq. (}1.1).

J =     ∂u ∂x ∂u ∂y ∂v ∂x ∂v ∂y     (1.1)

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1.3. Topology 15

The trace (p) and the determinant (q) are given in Eq. (1.2) and Eq. (1.3), respectively,

p = ∂u ∂x + ∂v ∂y (1.2) q = ∂u ∂x ∂v ∂y − ∂u ∂y ∂v ∂x (1.3)

The classification of critical points suggested by Dallmann [1983], is drawn in Fig. 1.9.

Tobak & Peake [1982] and Perry & Chong [1987] proved the critical point is topologically unstable when occurs on the parabola or on the axes of Fig. 1.9. A topology with streamlines that connect two saddle points is proven to be topological unsteady, unless it occurs at a sharp corner between two surfaces. The stable critical points are summarized in Tab.1.1.

Figure 1.9: Classification of critical points [Dallmann,1983].

Table 1.1: Stable critical points.

Critical point Relation betweenp and q

Saddle point q < 0

Regular point q > 0, p2 _{< 4q} Spiral node/focus q > 0, p2 _{> 4q}

The most popular rule for critical points is the “index rule” ofFlegg [1974]. In Fig. 1.9, each of regular nodes, foci, node-foci, star-nodes and centers is noted as a node with an index of_{1, and a saddle point is assigned an index of −1, deriving}

X

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whereN is the number of nodes, and S the number of saddles.

It was shown byGbadebo[2003] that the index rule follows Eq. (1.5) for a single passage without blade tip clearance, while a passage with blade tip clearance satisfies Eq. (1.6).

X

N −XS = 0 (without tip clearance) (1.5)

X

N −XS = 2 (with tip clearance) (1.6)

1.3.2 Topology of corner separation

As suggested byD´elery[2001], the phenomenon of 3-D separation, which is different from that of 2-D separation, is nearly independent of Reynolds number.

The first topology of compressor blade corner separation is proposed bySchulz & Gallus

[1988] and Schulz et al. [1990], as shown in Fig. 1.10. This topology was obtained from a compressor cascade at a stator inlet angle of49.2◦_{. In Fig.}_1.10(a)_{, big areas of corner flow} separation are clearly seen on both the hub and the blade suction surface. The onset of the corner separation occurs at the junction between these two surfaces, close to the blade leading edge. Near the separation point, a vortex (c) is formed, since the main flow is suddenly obstructed, and the backflow inside the corner separation region migrates upstream and coils up with another vortex (d). A vortex on the hub (marked as a) and a vortex on the blade suction surface (marked as b) can be observed with the help of surface limiting streamlines. The fluid flows along the limiting streamlines starting at the separation point, and rolls away from the hub and blade suction surface. Points (a) and (b) seem like foci (“saddle” points, according toSchulz & Gallus

[1988] andSchulz et al.[1990]). The corner separation is closed off by the limiting streamlines on the hub and the blade suction surface, and a ring vortex is supposed to join the two vortices (a) and (b) somewhere near the trailing edge. This topology of corner separation is sketched in Fig.1.10(b).

By numerical investigation of the compressor cascade of Schulz & Gallus [1988], with RANS method,Hah & Loellbach[1999] proposed another type of corner separation topology (see Fig.1.11). Two distinct vortices on the hub are suggested to be the dominant features of the corner separation. One vortex lies near the blade suction side at about80% axial chord, and the other is located close to the blade trailing edge. These two counter-rotating vortices roll outward the hub, and finally connect outside the hub boundary layer. Therefore, as sketched in Fig.1.11(b), the two hub vortices are two legs of one single vortex. Due to the strong fluid motion around this vortex, a reverse flow region and a limiting streamline are formed on the blade suction surface. In the same paper,Hah & Loellbach[1999] also investigated numerically the corner flow in Rotor 37 (“blind test case” in 1994, [Strazisar & Denton, 1995], [Suder &

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1.3. Topology 17

Celestina,1996], [Denton, 1997]). A weak vortex structure is identified, the vortex tube forms on the hub and finally diffuses in the main flow.

Flow

(a)

Flow

(b)

Figure 1.10: Topology of corner separation [Schulz et al.,1990].

(a)

Flow

(b)

Figure 1.11: Topology of corner separation [Hah & Loellbach,1999].

Recently,Lewin et al.[2010] provided a more detailed topology for corner separated flows (see Fig.1.8 and Fig.1.12). In Fig.1.12(b), three foci and their associated vortices were iden-tified (see the ribbons), interacting with each others. One is on the suction surface near the trailing edge, the others are located on the hub. According to the authors, the cores of the hub vortices do not connect together as described byHah & Loellbach[1999]. Furthermore, they indicate that the focus F2 (shown in Fig.1.12(b)) is the dominant characteristic of a developed corner separation. This focus, missing in undeveloped corner separation cases, increases with increasing loading. At inlet angle of50◦_{, the characteristic line I (see Fig.}_1.8(a)_{) is downstream} of line II. Only a small fraction of the inlet boundary layer flow lifts off the end-wall at focus

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Fl ow (a)

F2

Flow

(b)

Figure 1.12: Topology of corner separation [Lewin et al.,2010].

F2. When increasing the inlet angle from50◦_to₅₆◦ _{(see Fig.}_1.8(b)_{), the critical lines I and II} change their mutual position, and the corner separation develops.

1.4 Extent of three-dimensional corner separation

The region of three-dimensional corner separation is bounded on the end-wall and blade suction side by the skin-friction lines which emanate from the node-saddle point on the juncture between the blade suction side and the end-wall (e.g. the NS point in Fig.1.8), so it is relatively easy to quantify the extent of the corner separation on the blade suction surface and on the end-wall. The issue is then to determine the boundary layer thickness on the suction surface.

Gbadebo [2003] suggested a relative displacement thickness to determine the effect of

three-dimensional separation on blockage near the blade trailing edge. Its definition is the displacement thickness at any radius minus the displacement thickness at midspan. Since the suction surface boundary layer at midspan usually does not separate, this relative displacement thickness could evaluate the relative contribution from the corner separated boundary layer.

When the blade is uniform spanwise, this relative displacement thickness can be calculated for each chordwise position. The displacement thickness at curvilinear abscissa (s) and radius (r) is defined as: δ1(s, r) = Z δ 0 1 − ρu_ρ s(s, r, n) ∞us,∞(s) dn (1.7)

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1.5. Exit blockage coefficient 19

indicates the perpendicular distance to the suction surface, and the subscript_{∞ means the} ref-erence quantity.Gbadebo et al.[2005] used the local mid-pitch quantity as the reference value, while in this thesis the reference quantities are taken on the last point of the measuring-lines (the length of the measuring-lines are identical).

Therefore, the relative displacement thicknessδ1,r(s, r) normalized by the blade chord c is defined as:

δ1,r(s, r) =

δ1(s, r) − δ1(s, rm)

c (1.8)

where thermis the mid-span radius.

1.5 Exit blockage coefficient

The blockage coefficient defined byCumpsty[1989] is expressed as

B = 1 − A_Ae (1.9)

where Ae is effective flow area, and A is geometric flow area. The definition of the effective flow area will be introduced later in the same section.

Inspired by the approach ofKhalid et al.[1999], Gbadebo[2003] proposed an estimation of the exit blockage coefficient to measure the effect of corner separation.

Supposing a one-dimensional compression process where the total pressure is conservative, from the Bernoulli’s equation, the inlet and outlet dynamics pressure are given by:

1 2ρU 2 1 = Pt− Ps,1 (1.10) 1 2ρU 2 2 = Pt− Ps,2 (1.11)

where the subscripts 1 and 2 designate inlet and outlet respectively.

By subtracting Eq. (1.11) from Eq. (1.10), the static pressure rise coefficient is given by:

Cp = Ps,2− Ps,1 1 2ρU 2 1 = 1 − U2 U1 2 (1.12)

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Under the mass flow conservative principle, the effective outflow area is the area on which the inlet mass flow goes across with the outlet flow velocity, as expressed below:

A2,e= A1 U1 U2

(1.13) whereA1, andA2,eis the inlet and effective outlet flow areas respectively.

WithA2 being the geometric outlet flow area, the blockage coefficient can be derived by combining Eqs. (1.12) and (1.13) with Eq. (1.9):

B = 1 − A_A2,e 2 = 1 − A1 A2(1 − Cp )−0.5 _{= 1 −} cos β1 cos β2(1 − Cp )−0.5 (1.14)

whereβ1 andβ2are the actual inlet and outlet flow angles (mid-span value for a real compres-sor).

1.6 The parameters of corner separation

1.6.1 Loading

Increasing compressor loading generally increases the spread and the intensity of corner separation, as revealed by many researchers. In the experiment of Dring et al. [1982] and

Joslyn & Dring[1985], while increasing the blade loading, a corner separation developed into a full-span separation on the rotor. In the second-stage stator, increasing the blade loading resulted in a dramatic growth of the stator corner separation, and the blockage due to the corner separation reached nearly40% with an extension of nearly 70% of the span. The same trends were observed in the investigations ofSchulz et al. [1990], Li et al. [1992], Barankiewicz & Hathaway[1998] andGbadebo et al.[2005].

1.6.2 Inflow boundary layer

Gbadebo[2003] examined experimentally and numerically the effect of the inlet boundary

layer thickness. Imposing a thickened boundary layer (of 2.6 times as the reference thickness), increased corner separation and losses were identified. Consistently, through the numerical investigations, Lei [2006] suggested that the size of the corner separation increases with the thickness of the incoming boundary layer.

Gbadebo [2003] conducted a RANS simulation with the mixing length model to further

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1.6. The parameters of corner separation 21

To do this, they increased the turbulence level within the thickened inflow boundary layer by modifying the mixing length of the mixing length turbulence model. Based on these numerical results, they presumed that the high turbulence level within the thickened inlet boundary layer brought high momentum fluid from the free-stream into the boundary layer, thus suppresses the further growth of separation, and the extra losses that were observed were generated by the turbulent mixing within the boundary layer.

Demargne & Longley [2000] used a hub cavity leakage to model the incoming boundary

layer skewness. They observed that the size of the high-loss region associated with the corner separation is related to the change in tangential momentum thickness of the end-wall boundary layer. The size of the corner separation decreases when increasing the ratio of leakage velocity to the free stream tangential velocity (associated with the boundary layer skewness.)

1.6.3 Free-stream turbulence intensity

When studying the rotor-stator interaction, Schulz et al.[1990] pointed out that, the up-stream rotor wakes (where the turbulence intensity was 13%) yielded a mean upstream turbu-lence intensity of7% (much higher than in the case without upstream rotor where T u = 1.2%). This high turbulence intensity suppressed the laminar-turbulent transition bubble on the blade suction side. They concluded that the massive corner separation and the losses near the hub was significantly decreased, mostly owing to the wakes-induced transition at the blade leading edge which suppressed the transition bubble. The same observation was achieved by Ottavy et al.

[2002] through the investigation over a highly-loaded compressor-like flat plate.

As reported by Schreiber et al.[2002], atRec ≈ 2 × 106, increasing turbulence intensity from0.7% to 4-5%, the laminar-turbulent transition bubble was removed, and the bypass tran-sition became dominant, at the same time the trantran-sition location moved upstream from30%c to 10%c. Schreiber et al. [2002] did not mention the corner separation nor the losses, but it can be observed in the figures that the corner separation is suppressed when the transition moves forward. So the author of the present thesis considers that the forward movement of the laminar-turbulent transition can reduce the corner separation and the losses.

1.6.4 Clearance flow

In the study of Dong et al.[1987], the stator corner separation was significantly reduced by a hub clearance (the hub is not rotating), because the high momentum leakage flow through the gap from the pressure side to the suction side re-energized the low-momentum flow on the suction side and thus decreased corner separation.