CFD Simulation of Complex Flows in Turbomachinery and Robust Optimization of Blade Design

FACULTY OF ENGINEERING

Department of Mechanical Engineering

CFD Simulation of Complex

Flows in Turbomachinery and

Robust Optimization of Blade

Design

Submitted to the Department of Mechanical Engineering Doctor of Philosophy

at the Vrije Universiteit Brussel July 2010¡À (Xiaodong Wang)

Advisors: Prof. Chris Lacor

Prof. Charles Hirsch

Prof. Shun Kang

CFD Simulation of Complex Flows in Turbomachineryand Robust Optimization of Blade Design

by

Xiaodong Wang

Submitted to the Department of Mechanical Engineering

Doctor of Philosophy

at the Vrije Universiteit Brussel

July 2010

Advisors: Prof. Chris Lacor

Prof. Charles Hirsch

Prof. Shun Kang

Print: Silhouet, Maldegem

c©2010 Xiaodong Wang

2010 Uitgeverij VUBPRESS Brussels University Press

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Abstract

Turbomachinery is a kind of widely used equipment in industrial engineer-

ing, which usually has complex configurations. The performance of turbo-

machinery is related closely to the complicated internal flow. A good under-

standing of the internal flow is the keystone of the high performance turbo-

machinery design. Moreover, the increasing rigorous demands of modern

aerodynamic designs require advanced optimal design methodologies. The

main research work of the present thesis is contributed to these two points.

The first part of this thesis focuses on the numerical analysis of the

film cooling flow and 3D unsteady flow within turbomachinery. A relatively

simple model of a flat plate with a single square cooling hole is adopted to

investigate the effect of the blowing ratio on the local flow structure nearby

a cooling hole. Steady and unsteady RANS simulations and DES simula-

tions are performed. The simulated results show a stable symmetric vortex

structure when the blowing ratio is low (< 0.8); with the increase of blow-

ing ratio, the flow structure becomes unstable and evolves gradually into a

new stable asymmetric flow structure. While the cooling flow closing to the

flat plane are warped into the dominated CVP, which decrease the cooling

effect greatly. Therefore, the blowing ratio should be lower than a criti-

cal blowing ratio. Further steady simulations are performed on the AGTB

planar cascades with film cooling at the leading edge. Three different cool-

ing holes configurations of slots, radial inclined holes and straightforward

holes are simulated respectively under different blowing ratios. Based on

the validation to experiment data, the analysis mainly focuses on the im-

pacts of the cooling flow on the main flows and the related additional losses.

The simulation results show a large separation flow exists downstream the

slots which leads to the deterioration of cooling effect and large additional

losses; The cooling flows issue from the inclined holes are distributed more

uniformly than that from the straightforward holes. However, the addi-

tional losses of the former are a little bit larger than that of the latter.

For 3D unsteady flows in blade passages, a low speed axial turbine

i

from AIST (Japan) is adopted as the simulated model. Unsteady simu-

lations using nonlinear harmonic (NLH) method are performed. The dis-

cussions mainly focus on the unsteady properties of flow within blade pas-

sages based on the validation to experiment data. The comparison shows

that the compressibility of the fluid used in simulations has great influence

on the simulated results for the low speed flow simulation. The simulated

result using incompressible fluid has a good agreement with the experi-

ment data, while the performance of the simulation using compressible

fluid is not so good. However, the simulated results also show the effec-

tiveness of the NLH method on treatment of unsteady R/S interactions.

Increasing the number of harmonics can improve the simulated results. In

general, if only one R/S interface exists, using 2 harmonics can provide a

satisfied result since the third order harmonic is quite small.

The second part of the present thesis, an aerodynamic optimization

framework based on CFD simulations, multi-objective optimization algo-

rithm and artificial neural network (ANN) is developed. In this frame-

work, the most widely used multi-objective genetic algorithm NSGA-II is

employed. Firstly the coupled method of NSGA-II and ANN is tested us-

ing a set of classical mathematical testing problems, ZDT problems. It is

shown that the optimized results of NSGA-II are stuck easily into a local

optimum. An improved crowding distance is proposed to adjust the select-

ing strategy to enhance the capability of jumping out the local optimum

due to the predication error of ANN. Meanwhile, a novel Coarse-to-Fine

iteration strategy is employed to improve the prediction accuracy of ar-

tificial neural network to approach the real Pareto front gradually. The

improved results of testing problems illustrated the effects of improving

strategies. A practical industrial application is performed on the multi

working points optimization on the NASA Rotor37. A trade-off solution

is chosen using multi-criteria decision method from the converged Pareto

front, which shows better performance than that of the single objective

optimization. A robust optimization is also performed on rotor37 under

stochastic outlet static pressures. A non-intrusive probabilistic collocation

method is employed to quantify the uncertainty. The overall performance

of the optimized results is better than that of the origin design, and the

sensitiveness to fluctuations of the outlet static pressure is decreased si-

multaneously.

The final chapter of this thesis summarizes the conclusions of the

present PhD research and discusses future challenges for high credible

CFD analysis and robust optimization of turbomachinery in general.

ii

ÁÁÁß²Å´ó+¥A^Ǒ2 OǑǑE,aÅ§Ù5UÙSÜE,6Ä'"Ïd§©ÛÚ)ÙSÜE,6Ä´JpÙ5UÄ:Ú'"Ó§duDÚ“¢-?”O®ØU÷vFJpO§up&Ýê[Úk?`zOǑ´ß²Å+ïÄ:"Ø©(Ü±þü¡SNmïÄ"©1Ü©ÌÏLê[ïÄß²Å¥íe%6ÄÚn½~6Ä"éuíe%6Ä§Äk±²Ý/e%Ǒ.§ÏL½~RANS[Ú½~URANSÚDES[§ïÄNº'ée%NC6ÄKǑ§«e%eiµX(9Ùu"[(JL²3$Nº'e£u0.8¤§e%ei6Ä¥½é¡(¶XNº'O\§6Ä§,ÅìLÞ½é¡("d§²L¡e%í6ò\é=µé§Ø|ue%"ÏdNº'Ø´L"3dÄ:þ§±AGTB²¡»Ǒ~?1¡ Æe%½~ê["ïÄØÓNº'e§e%©OǑø¿!»Ú¹eíe%6Ä"3¢?1é'yÄ:þ§©Ûe%6éÌ6KǑÚÚÚå"'(Jw«§ø¿e%´3ø¿eiÚå©l6§e%z§O¶e%í6©Ùe%í6þ!§Úå'e%"éu¡ÏSn½~6Ä§±AIST£F¤,$¶6µÓǑ.§æ^5Å?1½~ê[§¼¡ÏS6Ä½~A5§¿¢?1é'y"é'(Jw«§éu$ß²Å[§óØ 5éu[(JKǑé"æ^ØØó[(J¢ÎÜé¶ ^Ø ó[(J¢k "Ó[(JǑL²§5Å±k/[Ä·Z¶O\Åê§±Jp[°Ý"éu3=·¡ß²Å§æ^Å®±¼÷¿°Ý"©1Ü©uAûß²ÅíÄ`zO§ÄuCFD[!õ8I`zÚ<ó ²ä(ÜÍÜ`z"T`zæ^ A^Ǒ2NSGA-IIõ8I¢D?1`z"Äk^;.êÆÿÁ~—ZDTÿÁ¯K8éÍÜ?1ÿÁ§y¢du ²äzØ3§NSGA-II`z(JúÛÜ`"©JÑU?Pål§iii

±kUõNSGA-IIaÑÛÜÙå"Óæ^«Coarse-to-

FineSüÑ§ÏLÅÚ%Cý¢` ÷§JpÔöþ§? Jp<ó ²äýÿ°Ý"êÆÿÁ~(JyU?üÑk5"¢SóA^~f±NASA=f37Ǒ.§?1õó¹:íÄ`z§¼`)8"²Lõá5ûüÀJ`z(Juü8I`z(J",æ^Ø(½5CFD§é=f37?13ÑØåǑØ(½5^¹e°`z"Ø(½5æî\ªVÇ:"`z(JOoN5UJp§ éÑØåCz¯a5ü$"©o( ¡ïÄó§¿lJpCFD[&Ý!uäk°5õ8I`z¡é5'uß²ÅíÄOAmïÄó?1&?"

iv

Acknowledgments

When I finished the last word of this thesis, I am so exciting on what I

have done. Four years ago, I had little experience on CFD simulations and

Turbomachinery; I never used the Genetic algorithm and Artificial Neu-

ral Network; I knew nothing about the Polynomial Chaos or Probabilistic

Collocation methods. But, now I finished my PhD thesis using all these

knowledge and methods. I learned a lot in the past four years which will

be the most precious time in my life. There are many people to whom I

wish to express my gratitude. Without their help, I couldn’t finish this

thesis.

The first person whom I would like to gratefully acknowledge is my

promoter in China, Prof. Shun Kang, who gave me the opportunity to do a

PhD under his guidance. I thank him for his valuable suggestions during

my PhD research, as well as for granting me sufficient freedom to pursue

my own ideas.

Secondly, I am greatly indebted to my promoters in Belgium, Prof.

Charles Hirsch and Prof. Chris Lacor. The extensive knowledge and board

vision of Prof.Hirsch in CFD and Turbomachinery fields gave me a deep

impression. The valuable weekly discussions and kind suggestions from

Prof.Lacor make my research work easier. It is a pleasure and a privilege

to work with them.

My gratitude also goes to the IT support of our system administrator

Alain Wery. I greatly appreciate him for his good mood and everlasting

patience through the perpetual stream of requests and computer problems

coming towards him. I am pleased to acknowledge our secretary Jenny

D’haes and my colleagues in STRO of VUB: Ghader Ghorbaniasl, Patryk

Widera, Matteo Parsani, Vivek Agnihotri, Mahdi Zakyani, Willem Decon-

inck, Khairy Elsayed, Floriane Krause, Dean Vucinic. We have nice discus-

sions and good fun at the coffee corner and KK bar. I also wan to thank

the colleagues in NUMECA Inc.: Alban Ligout, Cristian Dinescu, Michel

Pottiez, et al. They gave me lot of technical support on code development;

v

thank Dr. Takayuki Matsunuma from AIST (Japan) for data share and

suggestions. In addition, special thanks also goes to aunt Yan Cao, who

gave me a lot of help when I was in Brussels.

I also want to say thanks to my colleagues in NCEPU: Prof. Xiaodong

Zhang, Liping Dai, Zhonyao Fan, Huirong Wei, Lei Song, Li Ma, Liping

Sun, Junyu Liang, Zuoming Yin, Jingxiong Yin, Wenbo Shao, Jie Li et

al. for the support and help during my research. In addition, the same

thanks also for my classmates in Doctorial Class 0624 and the colleagues

in Numeca-Beijing Ltd.

Last, but certainly not least, I would like to thank my parents, my

sisters, and my girlfriend Jing Li, for the support they have given me

throughout my education and PhD research.

vi

Jury members

President Prof. Johan DECONINCK

Vrije Universiteit Brussel

Vice-president Prof. Rik PINTELON


Secretary Prof. Steve VANLANDUIT


Members Prof. Herman DECONINCK

Von Karman Institute

Prof. Erik DICK

Universiteit Gent

Prof. Charles HIRSCH

NUMECA International

Prof. Xiaodong ZHANG

North China Electric Power University

Promoter Prof. Chris LACOR


Prof. Shun KANG

North China Electric Power University

vii

viii

Contents

1 Introduction 1

1.1 Turbomachinery . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Development of Turbomachinery . . . . . . . . . . . . 2

1.1.2 Research Progress of Internal Flows in Turbomachinery 4

1.2 Computational Fluid Dynamics . . . . . . . . . . . . . . . . . 5

1.2.1 Application of CFD in Turbomachinery . . . . . . . . 5

1.2.2 Verification and Validation of CFD Simulations . . . 7

1.2.3 Non-deterministic CFD Methodologies . . . . . . . . . 8

1.3 Robust optimization of turbomachinery . . . . . . . . . . . . 9

1.3.1 General Aerodynamic Design of Turbomachinery . . 9

1.3.2 Multi-objective Optimization Methods . . . . . . . . . 11

1.3.3 Robust Optimization . . . . . . . . . . . . . . . . . . . 12

2 Complex Flows in Turbomachinery 13

2.1 General Review on Secondary Flows . . . . . . . . . . . . . . 13

2.1.1 Classical Secondary Flow Models . . . . . . . . . . . . 14

2.1.2 Modern Secondary Flow Models . . . . . . . . . . . . . 14

2.1.3 Latest Secondary Flow Models . . . . . . . . . . . . . 18

2.2 Unsteady Flow within turbomachinery . . . . . . . . . . . . . 19

2.2.1 Conditional Unsteadiness . . . . . . . . . . . . . . . . 20

2.2.2 Inherent Unsteadiness . . . . . . . . . . . . . . . . . . 21

2.3 Film Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Film Cooling Flow on A Flat Plate . . . . . . . . . . . 26

2.3.2 Film Cooling Flow in Turbine Cascades . . . . . . . . 28

3 Numerical Methods 31

3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 The Mass Conservation Equation . . . . . . . . . . . . 32

3.1.2 The Momentum Conservation Equation . . . . . . . . 32

ix

3.1.3 The Energy Conservation Equation . . . . . . . . . . 34

3.1.4 Rotating Frame of Reference . . . . . . . . . . . . . . 36

3.2 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . 39

3.2.1 General Introduction of Discretization Method . . . . 39

3.2.2 Mesh Generation . . . . . . . . . . . . . . . . . . . . . 39

3.2.3 Spatial Discretization of Convective Term . . . . . . . 41

3.2.4 Spatial Discretization of Diffusive Term . . . . . . . . 42

3.2.5 Explicit Multistage Runge-Kutta Scheme . . . . . . . 42

3.3 Accelerating Convergence Methods . . . . . . . . . . . . . . . 43

3.3.1 Implicit Residual Smoothing . . . . . . . . . . . . . . 43

3.3.2 Multigrid Method . . . . . . . . . . . . . . . . . . . . . 44

3.4 Turbulence Approximation . . . . . . . . . . . . . . . . . . . . 46

3.4.1 Reynolds Averaged Navier-Stokes Equations . . . . . 48

3.4.2 Turbulence model . . . . . . . . . . . . . . . . . . . . . 49

3.4.3 Large Eddy Simulation . . . . . . . . . . . . . . . . . . 55

3.4.4 Detached Eddy Simulation . . . . . . . . . . . . . . . . 57

3.5 Dual-time Stepping Method . . . . . . . . . . . . . . . . . . . 57

3.6 Rotor/Stator Interaction Treatment . . . . . . . . . . . . . . . 58

3.6.1 Steady Simulation Treatment . . . . . . . . . . . . . . 59

3.6.2 Unsteady Simulation Treatment . . . . . . . . . . . . 60

3.6.3 Harmonic Method . . . . . . . . . . . . . . . . . . . . . 60

3.7 Uncertainty Quantification in CFD . . . . . . . . . . . . . . . 63

3.7.1 Intrusive Polynomial Chaos Method . . . . . . . . . . 64

3.7.2 Non-intrusive Polynomial Chaos Method . . . . . . . 67

3.7.3 Non-intrusive Probabilistic Collocation Method . . . 68

4 Numerical Simulations on Film Cooling 71

4.1 Jets in Crossflow on A Flat Plate . . . . . . . . . . . . . . . . 71

4.1.1 Review of Experiment . . . . . . . . . . . . . . . . . . 71

4.1.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . 73

4.1.3 Results of Steady Simulations . . . . . . . . . . . . . . 75

4.1.4 Results of Unsteady Simulations . . . . . . . . . . . . 88

4.2 Cooling Flow in Turbine Cascades . . . . . . . . . . . . . . . 90

4.2.1 Review of the Experiment . . . . . . . . . . . . . . . . 92

4.2.2 Computational Model and Mesh . . . . . . . . . . . . 97

4.2.3 Simulation Results and Analysis . . . . . . . . . . . . 99

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

x

5 Numerical Simulations on Three Dimensional Flows in Axial

Turbine 115

5.1 Review of Experiment . . . . . . . . . . . . . . . . . . . . . . 115

5.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2.1 Geometry Model and Mesh . . . . . . . . . . . . . . . 118

5.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . 118

5.2.3 Numerical Method . . . . . . . . . . . . . . . . . . . . 119

5.3 Simulation Results and Analysis . . . . . . . . . . . . . . . . 120

5.3.1 Time-averaged Simulation Results . . . . . . . . . . . 120

5.3.2 Time-resolved Simulation Results . . . . . . . . . . . 123

5.3.3 Effect of the Harmonic Number . . . . . . . . . . . . . 131

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6 Development of Multi-objective Aerodynamic Optimization

Framework 133

6.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . 133

6.2 General Principles of GA . . . . . . . . . . . . . . . . . . . . . 135

6.2.1 Encoding and Decoding . . . . . . . . . . . . . . . . . . 136

6.2.2 Fitness Function . . . . . . . . . . . . . . . . . . . . . 137

6.2.3 Genetic Operators . . . . . . . . . . . . . . . . . . . . . 137

6.2.4 Control Parameters . . . . . . . . . . . . . . . . . . . . 138

6.3 Multi-objective Genetic Algorithm . . . . . . . . . . . . . . . 140

6.3.1 Pareto Optimal Concept . . . . . . . . . . . . . . . . . 140

6.3.2 NSGA-II . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.3.3 SPEA2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.3.4 Performance Evaluation Criteria . . . . . . . . . . . . 148

6.3.5 Comparison between NSGA-II and SPEA2 . . . . . . 150

6.4 Coupled Method with Approximation Model . . . . . . . . . . 153

6.4.1 Artificial Neural Network . . . . . . . . . . . . . . . . 153

6.4.2 Integration of NSGA-II into Design3D . . . . . . . . . 157

6.4.3 Test Results of the Coupled Method . . . . . . . . . . 158

6.5 Improving Strategies . . . . . . . . . . . . . . . . . . . . . . . 165

6.5.1 Improved Crowding Distance . . . . . . . . . . . . . . 165

6.5.2 Coarse-to-fine Iteration . . . . . . . . . . . . . . . . . . 170

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7 Application of Robust Optimization on An Axial Compressor181

7.1 Design Background of the NASA Rotro37 . . . . . . . . . . . 182

7.1.1 Application Platform . . . . . . . . . . . . . . . . . . . 182

7.1.2 Parametric Model . . . . . . . . . . . . . . . . . . . . . 183

7.1.3 Numerical Model and Mesh . . . . . . . . . . . . . . . 184

xi

7.2 Multiple Working Points Optimization . . . . . . . . . . . . . 185

7.2.1 Setting Statement . . . . . . . . . . . . . . . . . . . . . 185

7.2.2 Optimization Results . . . . . . . . . . . . . . . . . . . 186

7.3 Robust Optimization . . . . . . . . . . . . . . . . . . . . . . . 193

7.3.1 Uncertainty Statement . . . . . . . . . . . . . . . . . . 193

7.3.2 Optimization setting Statement . . . . . . . . . . . . . 194

7.3.3 Optimization Results . . . . . . . . . . . . . . . . . . . 195

7.3.4 Effect of Penalty Setting . . . . . . . . . . . . . . . . . 200

7.3.5 Effect of the Number of Initial Training Samples . . . 201

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

8 Conclusions and Perspectives 205

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

8.1.1 Numerical Simulations of Complex Flows . . . . . . . 205

8.1.2 Multi-objective Optimization of Aerodynamic Design 207

8.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

List of publications 211

List of projects participated in 213

Bibliography 215

xii

List of Figures

1.1 Waterwheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 GP7200 Engine . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 S1 and S2 surfaces . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Impact of CFD on SNECMA fan performance . . . . . . . . . 7

2.1 Classical secondary flow model . . . . . . . . . . . . . . . . . 15

2.2 Modern secondary flow model . . . . . . . . . . . . . . . . . . 16

2.3 Synchronous evolution of horseshoe and passage vortices . . 17

2.4 Structure of inlet horseshoe vortex . . . . . . . . . . . . . . . 17

2.5 Vortex patterns of secondary flows . . . . . . . . . . . . . . . 18

2.6 Endwall vortex pattern . . . . . . . . . . . . . . . . . . . . . . 20

2.7 Periodic wake shedding . . . . . . . . . . . . . . . . . . . . . . 22

2.8 Pressure contour of wake flow . . . . . . . . . . . . . . . . . . 23

2.9 Wake flow in blade passage . . . . . . . . . . . . . . . . . . . 24

2.10 Instantaneous absolute velocity contour pattern at nozzle exit 25

2.11 Blade with cooling holes. . . . . . . . . . . . . . . . . . . . . 26

2.12 Pattern of film cooling. . . . . . . . . . . . . . . . . . . . . . . 26

2.13 vortex systems of JICF . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Rotating frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Control volume approaches. . . . . . . . . . . . . . . . . . . . 40

3.3 Multigrid method . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Full multigrid strategy . . . . . . . . . . . . . . . . . . . . . . 46

4.1 Experiment of JICF . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Computational domain of JICF . . . . . . . . . . . . . . . . . 73

4.3 Mesh for simulations on JICF . . . . . . . . . . . . . . . . . . 74

4.4 Residual convergence curves of steady simulations . . . . . . 75

4.5 Residual convergence curves of DES simulations . . . . . . . 75

4.6 Profile of Vx at the central plane . . . . . . . . . . . . . . . . 77

xiii

4.7 Contour pattern of Vx at the central plane . . . . . . . . . . . 78

4.8 Contour pattern of Vz at the outlet of the cooling hole . . . . 79

4.9 Limiting streamline pattern on the flat plate . . . . . . . . . 81

4.10 Detailed view of Limiting streamline pattern on flat plate . 82

4.11 Spatial streamline pattern of Horseshoe vortex . . . . . . . . 83

4.12 Spatial streamline pattern of vortices system nearby the cool-

ing hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.13 Detailed view of vortices system . . . . . . . . . . . . . . . . . 85

4.14 Top view of streamline pattern of CVP . . . . . . . . . . . . . 86

4.15 Velocity vectors and streamline patterns of the horseshoe

vortex at the central plane . . . . . . . . . . . . . . . . . . . . 86

4.16 Streamline patterns of vortices system . . . . . . . . . . . . . 87

4.17 Origin of the secondary CVP . . . . . . . . . . . . . . . . . . . 88

4.18 Profile of the time averaged Vx at the central plane . . . . . 89

4.19 Time averaged contour pattern of velocity on the flat plate

(100 physical time steps) . . . . . . . . . . . . . . . . . . . . . 91

4.20 Time averaged contour pattern of velocity of the flat plate

(400 physical time steps) . . . . . . . . . . . . . . . . . . . . . 92

4.21 Instantaneous velocity of DES (M=0.5) . . . . . . . . . . . . . 93

4.22 Instantaneous velocity of URANS (M=0.5) . . . . . . . . . . . 93

4.23 Instantaneous velocity of DES (M=1.0) . . . . . . . . . . . . . 94

4.24 Instantaneous velocity of URANS (M=1.0) . . . . . . . . . . . 94

4.25 Instantaneous vorticity isosurface of DES (M=0.5) . . . . . . 95

4.26 Instantaneous vorticity isosurface of URANS (M=0.5) . . . . 95

4.27 Instantaneous vorticity isosurface of DES (M=1.0) . . . . . . 96

4.28 Instantaneous vorticity isosurface of URANS (M=1.0) . . . . 96

4.29 Geometry of AGTB cascade and the cooling configurations . 97

4.30 Butterfly mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.31 Convergence history of steady simulation on AGTB-B1 . . . 99

4.32 Static pressure distributions on the blade surface of different

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.33 Static pressure distributions under the same inlet condition 101

4.34 Static pressure distributions of AGTB-B1 with different blow-

ing ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.35 Static pressure distribution at the leading edge . . . . . . . . 102

4.36 Streamline pattern at midspan of AGTB-S (M=0.7) . . . . . 102

4.37 Streamline patterns of AGTB-S at different spanwise sec-

tions close to the midspan. . . . . . . . . . . . . . . . . . . . . 102

4.38 Streamline patterns of AGTB-B1 at different spanwise sec-


xiv

4.39 Streamline patterns of AGTB-B2 at different spanwise sec-


4.40 Streamline patterns of AGTB-S at midspan . . . . . . . . . . 104

4.41 Limiting streamline patterns on the blade surface of AGTB 105

4.42 Limiting streamline patterns on the blade surface of AGTB-S 106

4.43 Streamline pattern near the cooling slot . . . . . . . . . . . . 107

4.44 Streamline patterns on the blade surface of AGTB-B1 . . . . 108

4.45 Streamline patterns on the blade surface of AGTB-B2 . . . . 109

4.46 Spatial streamline patterns of cooling flow for different models110

4.47 Flow pattern in two cross-sections of AGTB-B1 . . . . . . . . 110

4.48 Losses evolution through the blade passage . . . . . . . . . . 112

4.49 Isolines patterns of total pressure losses of different models 113

5.1 Profiles of blades of the AIST turbine . . . . . . . . . . . . . . 117

5.2 Domain of the AIST turbine model . . . . . . . . . . . . . . . 118

5.3 Mesh for AIST turbine model . . . . . . . . . . . . . . . . . . 118

5.4 Spanwise distribution of velocity at stator inlet . . . . . . . . 119

5.5 Convergence history of simulations on flows in AIST turbine 121

5.6 Steady simulated spanwise distribution of velocity at stator

outlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.7 Comparison of the flow patterns on the stator suction surface. 124

5.8 Velocity contour pattern at the stator exit . . . . . . . . . . . 125

5.9 Spanwise distribution of velocity at stator outlet at different

time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.10 Time-resolved absolute velocity at the stator exit . . . . . . . 127

5.11 Time-resolved entropy increase isolines pattern at midspan 128

5.12 Time-resolved entropy contour patterns at three spanwise

sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.13 Time-resolved relative velocity contour pattern at t = 0 T (ax-

ial position ZRT /Cax,RT = 0.853). The middle figure presents

the steady simulation result using incompressible fluid for

comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.14 Time-resolved relative velocity contour patterns . . . . . . . 130

5.15 Harmonic pressure amplitude on the blade surface of rotor

at midspan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.16 Isolines of Entropy increase at midspan. . . . . . . . . . . . . 132

6.1 Pareto front with two objectives. . . . . . . . . . . . . . . . . 140

6.2 Procedures of NSGA-II and SPEA2. . . . . . . . . . . . . . . 149

6.3 The real Pareto fronts of ZDTs problems. . . . . . . . . . . . 152

6.4 Example of the configuration of BPNN. . . . . . . . . . . . . 154

xv

6.5 Transfer functions of ANN. . . . . . . . . . . . . . . . . . . . . 155

6.6 Pareto front of ZDT1 problem using coupled optimization

method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.7 Pareto front of ZDT2 problem using ANN. . . . . . . . . . . . 160

6.8 Convergence and spacing metrics. . . . . . . . . . . . . . . . 160

6.9 Pareto front of ZDT∗ problems. . . . . . . . . . . . . . . . . . 162

6.10 Pareto front of ZDT1∗. . . . . . . . . . . . . . . . . . . . . . . 162

6.11 Pareto front of ZDT2∗. . . . . . . . . . . . . . . . . . . . . . . 163

6.12 Effect of DoE on optimized results of ZDT1∗. . . . . . . . . . 164

6.13 Effect of DoE on optimized results of ZDT2∗. . . . . . . . . . 164

6.14 Effect of DoE on convergence and spacing metrics of ZDT∗. . 165

6.15 Example of ith generation of population . . . . . . . . . . . . 166

6.16 Selection process using the original crowding distance . . . . 167

6.17 Improved selection process using improved crowding distance 168

6.18 Optimized resutls of ZDT1∗ problems using the improved

crowding distance. . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.19 Optimized results of ZDT2∗ problems using the improved

crowding distance. . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.20 Convergence and spacing metrics of ZDT∗ problems. . . . . . 169

6.21 Optimized results of ZDT problems using the improved crowd-

ing distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.22 Example of Fuzzy MCDM. . . . . . . . . . . . . . . . . . . . . 173

6.23 Global flow chart of the optimization framework. . . . . . . . 174

6.24 Improved results of ZDT2 using coarse-to-fine iterations. . . 175

6.25 Convergence history of boundary points . . . . . . . . . . . . 176

6.26 Pareto front of ZDT1 problems using the improved crowding

distance, coarse-to-fine iterations and boundary control . . . 177

6.27 Pareto front of ZDT2 problems using the improved crowding

distance, coarse-to-fine iterations and boundary control . . . 178

6.28 Convergence and spacing metrics of ZDT problems using the

improved coupled method with boundary control . . . . . . . 178

7.1 Parametric model of rotor37 blade. . . . . . . . . . . . . . . . 183

7.2 Swept model and curved model of blades . . . . . . . . . . . . 184

7.3 Convergence and spacing metrics . . . . . . . . . . . . . . . . 186

7.4 Pareto front of multi-objective optimization . . . . . . . . . . 186

7.5 Aerodynamic performance of multi-objective optimization re-

sults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

7.6 Comparison of the aerodynamic performance . . . . . . . . . 189

7.7 Comparison of the blade profiles at different spanwise sections.190

xvi

7.8 Contour patterns of the relative Mach number of the initial

design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7.9 Contour patterns of relative the Mach number of the final

design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7.10 Limiting streamline patterns on the suction surface. . . . . . 192

7.11 Blade geometry of the initial design. . . . . . . . . . . . . . . 196

7.12 Blade geometry of the trade-off design obtained by the multi-

objective optimization. . . . . . . . . . . . . . . . . . . . . . . 196

7.13 Blade geometry of the single objective optimization result us-

ing weighting functions. . . . . . . . . . . . . . . . . . . . . . 197

7.14 Blade geometry of the single objective optimization result us-

ing a combined objective. . . . . . . . . . . . . . . . . . . . . . 197

7.15 Comparison of aerodynamic performances of different opti-

mization designs. . . . . . . . . . . . . . . . . . . . . . . . . . 198

7.16 Convergence and spacing metrics for multi-objective optimiza-

tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7.17 Pareto front of the multi-objective optimization. . . . . . . . 199

7.18 Convergence history using the penalty setting1. . . . . . . . 201



7.21 Convergence history using different number of initial train-

ing samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

xvii

xviii

List of Tables

3.1 Comparison of the approximation method for turbulence . . 47

3.2 Constants of Spalart-Allmaras model . . . . . . . . . . . . . 54

4.1 Boundary conditions and blowing ratios . . . . . . . . . . . . 98

4.2 Boundary conditions for M=0.7 . . . . . . . . . . . . . . . . . 111

4.3 Boundary conditions for M=1.1 . . . . . . . . . . . . . . . . . 111

5.1 Specifications of turbine cascades . . . . . . . . . . . . . . . . 116

6.1 Mathematical test problems for MOGA . . . . . . . . . . . . 151

6.2 Convergence and Spacing metrics . . . . . . . . . . . . . . . . 173

7.1 General configurations of rotor37 . . . . . . . . . . . . . . . . 182

7.2 Discrete levels of design variables . . . . . . . . . . . . . . . 185

7.3 Comparisons of design variables of the multi-objective opti-

mization results . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.4 Comparison of design variables between the final design and

the single objective optimization result . . . . . . . . . . . . . 189

7.5 Distribution of the CPU time in one global iteration. . . . . . 192

7.6 Comparison of the CPU time cost in optimizations with and

without the approximation model . . . . . . . . . . . . . . . . 193

7.7 Collocation points and weights. . . . . . . . . . . . . . . . . . 193

7.8 Settings of the penalty function . . . . . . . . . . . . . . . . . 194

7.9 Values of the design variables . . . . . . . . . . . . . . . . . . 195

7.10 Comparison of the total performances of different optimiza-

tion designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.11 Penalty setting 1. . . . . . . . . . . . . . . . . . . . . . . . . . 200

7.12 Penalty setting 2. . . . . . . . . . . . . . . . . . . . . . . . . . 200

7.13 Penalty setting 3. . . . . . . . . . . . . . . . . . . . . . . . . . 201

xix

7.14 Comparison of the optimization results using different penalty

settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

xx

Nomenclature

1D One dimensional

2D Two dimensional

3D Three dimensional

ANN Artificial Neural Network

BR Blowing ratio

CFD Computational fluid dynamics

CMF Choked mass flow rate

CVP Counter-rotating vortex pair

DES Detached eddy simulation

DNS Direct numerical simulation

FDM Finite difference method

FEM Finite element method

FVM Finite volume method

IPCM Intrusive Polynomial Chaos Method

IRS Implicit Residual Smoothing

JICF Jets in cross-flow

LDV Laser Doppler Velocimetry

LE Leading edge

LES Large eddy simulation

MC Monte Carlo

MCDM Multi-criteria decision making

MOGA Multi-objective genetic algorithm

N-S Navier-Stokes

NIPCM Non-intrusive Polynomial Chaos Method

NIPRCM Non-intrusive Probabilistic Collocation Method

PC Polynomial Chaos

PRC Probabilistic Collocation

R-K Runge-Kutta

RANS Reynolds Averaged Navier-Stokes

xxi

TE Trailing edge

TPC Total pressure coefficient

TWR Thrust-to-weight ratio

URANS Unsteady Reynolds Averaged Navier-Stokes

V&V Verification and Validation

Subscripts

ax Value in axial direction

C Convection term

cf Value associated to the Cross-flow

i Cell or individual or generation index

imp Imposed value

j Cell or individual or generation index

jet Value associated to Jets

l Index of multigrid

max Maximum value

min Minimum value

P Pressure side

p Passage

pc Pressure side corner

r Rotating frame of reference

ref Reference value

RT Value associated to rotor

S Suction side

s sub-grid

sc Suction side corner

sLc Suction side leading edge corner

sPc Pressure side leading edge corner

ST Value associated to stator

t Turbulent

uns Unsteady

V Diffusive term

w Value associated to a wall

wip Wall vortex induced by the passage vortex

Symbols

∆tI Inviscid time step

∆tV Viscous time step

δ2 Second order center difference operator

ǫ Turbulent dissipation rate

κ Thermal conductivity, JK−1m−1s−1, or Karman constant

µ Dynamic viscosity coefficient, kgm−1s−1, or membership func-

xxii

tion

µt dynamic turbulent, or eddy, viscosity

µv Bulk viscosity coefficient, kgm−1s−1

νt kinematic turbulent, or eddy, viscosity

ω Vorticity, s−1

ρ Mass density, kgτxx, ... Viscous stress tensor elements, PaF Body force, NFC Vector format of diffusive term

FV Vector format of convection term

Q Vector format of source term

U Vector format of transition term

u velocity vector

x Vector of coordinate

CP Pressure coefficient, dimensionless

E Specific total energy, Jkg−1

H Specific total enthalpy, Jkg−1

h Specific internal enthalpy, Jkg−1

k Turbulent kinetic energy

l Mixing length

M Blowing ratio

P Penalty function

p Static pressure, PaPt Total pressure, PaPr Prandtl Number

Q Generalized source item

R Residual

r Specific gas constant, about 287Jkg−1K−1 for air

Re Reynolds number, dimensionless

s Entropy, Jkg−1K−1

Sij Strain-rate tensor

t time, sTt Total temperature, Ku x-component of velocity, ms−1

v y-component of velocity, ms−1

v Specific volume, m3kg−1

W Weights

w z-component of velocity, ms−1

Superscripts

′ Fluctuation

i inner layer of B-L model

xxiii

L Value on the finest level of multigrid

l Index of multigrid

n Order of Runge-Kutta method

o outer layer of B-L model

S Value after residual smoothing

s sub-grid

xxiv

Chapter 1

Introduction

As the title suggests, the present thesis is devoted to the investigations on

two parts related to turbomachinery. The first part, namely aerodynamic

analysis for turbomachinery in particular, mainly focuses on the analysis

of the complex flows within the blade passage of turbomachinery, in which

CFD (Computational fluid dynamics) simulation is adopted as analytical

tool. For the second part, a state-of-the-art robust optimization technology

is developed and applied to the optimization of a typical turbomachinery

case—NASA rotor37. In this chapter, a brief introduction of the research

background is presented.

1.1 Turbomachinery

Turbomachinery is widely used equipment in industry, agriculture and

daily life. Such as compressors and turbines in a jet engine; steam turbine

in power plants; propeller for ships, hydraulic turbines for irrigation, wind

turbines for green energy, small fans for CPU cooling, and so on. A common

feature of these devices is that they all work with fluid and have rotating

component. Gorla [1] gives a general definition of turbomachinery which

says “Turbomachinery is a device in which energy transfer occurs between

a flowing fluid and a rotating element due to dynamic action, and results

in a change in pressure and momentum of the fluid”. In general, the ro-

tating element is named rotor which is usually composed of one or several

rows of rotating blades. In general, there also exits a stator which is also

composed of rows of blades, but not rotating. A pair of stator and rotor

constitutes a stage.

1

CHAPTER 1. INTRODUCTION

According to the way of energy transfer, turbomachines are generally

divided into two main categories. The first category is used primarily to

generate power which is called Turbine, including steam turbines, gas tur-

bines and hydraulic turbines. The main function of the second category

is to increase the total pressure of the working fluid by consuming power

which includes compressors, pumps and fans. According to inlet and outlet

flow directions, turbomachines can be classified into two types: axial tur-

bomachinery and radial turbomachinery. However, this thesis only focuses

on the axial turbomachinery. More detail classification and description

about the configurations can be found in [2].

1.1.1 Development of Turbomachinery

The usage of turbomachinery has a long history. It is recorded that the

waterwheel, a kind of primitive turbomachinery, was invented and used for

power generation more than hundred years ago. As illustrated in Figure

1.1, although the configuration is simple, it does follow the same basic

principle with other complicated modern turbomachineries, for instance

the compressor and the gas turbine in a jet engine.

Figure 1.1: Waterwheel (Tiangong Kaiwu, by Yingxing Song, 1637).

A jet engine, for example the latest GP7200 engine (Pratt & Whiteney)

as shown in Figure 1.2, is generally composed of compressor, combustion

chamber and gas turbine. The air is compressed in compressor before en-

2

1.1. TURBOMACHINERY

tering the combustion chamber where it is mixed with fuel and combus-

tion occurs. Then the gas with high pressure and high temperature flows

through gas turbines and leaves the engine through a nozzle. While ex-

panding through the turbine blades, power is released from the gas and

drives the turbine rotating.

Figure 1.2: GP7200 Jet fan engine (Pratt & Whiteney).

The jet engine was first invented in the 1930s∼1940s, which gave the

opportunity of rapid development to turbomachinery. From the initial tur-

bojet engine to the modern turbofan engine with large bypass ratio, the evo-

lution of jet engine requires more advanced compressors and turbines with

higher stage pressure ratio and higher efficiency. Since 1988, the military

of USA launched a series of research projects to develop advanced turbines,

such as “IHPTET” (Integrated High Performance Turbine Engine Technol-

ogy), “VAATE”(Versatile Affordable Advanced Turbine Engines ) etc. The

primary goal is to double the thrust-to-weight ratio (TWR) of engine which

will reach to 15∼20, decrease the fuel consumption ratio by 15%∼30%.

Compressor and turbine are two core components of jet engine. The per-

formance of a jet engine strongly depends on the design level of compressor

and turbine. Therefore, significant researching efforts have been spent on

improving the performance of turbomachinery. Today, the modern com-

pressor stage has an efficiency of about 90% and the modern turbine stage

has an efficiency of up to 95% [3]. Further improvements become more

and more difficult and require much deeper understanding of the flow field

inside of the turbomachinery.

Meanwhile, in industrial field, steam turbine and gas turbine are the

main instruments of power generation. Due to the energy crisis, design

of advanced turbine with higher efficiency is much more crucial than ever

before. Therefore, similar strong demands of improving the performance

3


of turbomachinery are also brought forward.

1.1.2 Research Progress of Internal Flows in Turbo-

machinery

The design of turbomachinery is a complex task due to the complicated

flow phenomena and interaction of multi-disciplines which involves aero-

dynamics, heat transfer, structural dynamic, control theory, materials and

manufacture engineering etc. Among these design processes, aerodynamic

analysis is the keystone of the design, which decides the performance of

turbomachinery directly. While, without numerical technologies (CFD sim-

ulation and numerical optimization), it is impossible to meet the increasing

rigorous requirements of design. Hence, the research on numerical aerody-

namic analysis and numerical design of turbomachinery are outstandingly

important, which gives the motivation of the present Ph.D research.

The aerodynamic performance of turbomachinery mainly depends on

the complex internal flows which usually are strongly three dimensional,

viscous and unsteady. The flows in blade passages may be laminar, tur-

bulent and transitional, and may include wake flow and secondary flows

etc. There also may exist other complicated flow phenomena, such as tran-

sition, boundary layer separation, shock and shock-boundary layer inter-

action, the unsteady interaction between the blade rows, the interactions

between the blade row and endwall, etc. In 1999, a NASA report of “Nu-

merical Simulation of Complex Turbomachinery Flows” [3] stated four typ-

ical complex flows in turbomachinery which have been investigated exten-

sively and may remain being the key research problems of turbomachinery

in next few decades. These flows are:

1 Unsteady flow

2 Transition to turbulence

3 Film cooling

4 Three dimensional flow in turbine including tip leakage effect

The investigation in the present thesis will cover the film cooling flows and

3D unsteady flows in turbines. A detailed literature review and discussion

of existing flow models are included in Chapter 2, followed by simulations

of film cooling on a flat plate and planar cascades in Chapter 4. The inves-

tigation of three dimensional unsteady flow in turbines with tip leakage is

presented in Chapter 5.

4

1.2. COMPUTATIONAL FLUID DYNAMICS

1.2 Computational Fluid Dynamics

Due to the complexity of the internal flow in turbomachinery, numerical

simulations are extensively used for the investigation as an important

auxiliary tool to experiments which usually are expensive and time con-

suming, even difficult to perform at some specific conditions, for instance

the High-altitude experiments.

“Computational Fluid Dynamics, known today as CFD, is defined as

the set of methodologies that enable the computer to provide us with a

numerical simulation of fluid flows”, a definition of CFD is summarized

by Hirsch in his book [4]. A fluid field can be described by a set of Par-

tial Differential Equations (PDEs) which is known as Navier-Stokes (N-S)

equations. However, this set of PDEs is highly nonlinear so that analytical

solutions are available only for a limited number of simple cases. For most

cases, even it is unknown if the analytical solution exists. CFD solves the

PDEs using numerical methodologies, which gives a practical alternative

solution for fluid mechanic problems. Usually three basic methodologies

are necessary for a whole CFD simulation, including spatial discretiza-

tion methods (preprocess module), time integration method and turbulent

flow approximation method (solver), flow visualization method (postpro-

cess module). While, for certain complicated cases, some additional meth-

ods are needed, for instant the treatment of R/S interaction of turboma-

chinery. The numerical methods adopted in the presented thesis are intro-

duced in Chapter 3 in detail.

1.2.1 Application of CFD in Turbomachinery

The aerodynamic performance of turbomachinery is related to the physi-

cal properties of the internal flow which is full three dimensional, viscous

and unsteady. However, it is still difficult to simulate the internal flow in

an exact way. Some approximation models of the internal flow were pro-

posed and used in practical designs, which usually are called aerodynamic

models. With the development of computation method, the aerodynamic

models go through four levels: 1D and 2D model, quasi 3D model, full 3D

model and unsteady full 3D model.

In the 1940s and the early 1950s, due to the weak computational

power, great simplifications had to be introduced into the N-S equations.

The one dimensional mass flow rate conservation model or simple radial

equilibrium equation model were widely used [5]. While some additional

empirical models have to be used to revise the results which usually takes

lots of experiments and the accuracy is far away from being satisfactory.

5


In 1952, the proposal of quasi-three-dimensional flow theory by Wu [6]

became a milestone of the numerical simulation of flow in turbomachin-

ery. The quasi-three-dimensional flow theory, also known as two relative

stream surface method, is based on two families of stream surface, named

S1/S2 stream surface. The S1 stream surfaces are blade-to-blade surfaces

as shown in Figure 1.3, which are usually assumed as surfaces of revolu-

tion for simplification. The S1 calculation is the basis for defining the de-

tailed blade shape. The S2 stream surfaces are hub-to-tip surfaces, which

represent the spanwise profiles of the flow. Therefore, the calculation of

S2 stream surfaces is usually called “throughflow calculation”. The quasi-

three-dimensional calculation is performed with the iterations between S1

calculation and S2 calculation. This method is so successful in the pe-

riod with weak computational power that it becomes the dominated de-

sign method till 1980s. As Denton commented “Wu’s S1/S2 approach was

far ahead of his time” [7]. Afterward, several numerical schemes based on

through-flow method are proposed in 1960s, such as the streamline curva-

ture method [8] and the matrix method [9] (also called the stream function

method [7]). Due to the simplicity and ability to cope with mixed subsonic-

supersonic flow, the streamline curvature method became the most widely

used through-flow method [10]. Even today, some improved streamline

curvature methods are still used in some investigations [11].

Figure 1.3: Illustration of S1 and S2 surfaces (Novack, 1967 [8]).

With the development of computer technology, the Reynolds Averaged

Navier-Stokes (RANS) simulations are developed rapidly since 1980s. In

the same time, a couple of turbulence models are proposed successively to

complete RANS model. In most design processes, the steady RANS simu-

lations give satisfied prediction of overall performance. While in elaborate

design processes, unsteady RANS (URANS) simulations are needed since

6

1.2. COMPUTATIONAL FLUID DYNAMICS

the flows in turbomachinery are highly unsteady. Respecting to the ap-

proximation level of geometry, CFD simulation of turbomachinery devel-

oped from 2D to 3D, from planar cascade to annular cascade, from single

blade passage to whole ring, from single stage to multi stages. The in-

crease of model accuracy to the real geometry has significant effects on

turbomachinery design. Figure 1.4 shows the impact of CFD on the perfor-

mance improvement of aircraft engine in SNECMA (France) over a period

of almost 30 years. The evolution, from the initial use of simple 2D po-

tential flow models in the early 1970s to the current applications of full 3D

Navier-Stokes code, has led to a overall gain in efficiency close to 10 points.

Figure 1.4: Impact of CFD on SNECMA fan performance, over a period of 30 years

(Escuret, 1998 [12]).

In recent years, Large Eddy Simulation (LES) and Direct Numerical

Simulation (DNS) are getting more popular, which can predict turbulent

effects with more accuracy, at the cost however of significantly increased

CPU time. For simulations of turbomachinery, the current computation

power is far away from affording such high costly computations. The un-

steady RANS simulations of whole stage or multi stages will be the focus

of the current research.

1.2.2 Verification and Validation of CFD Simulations

With the fast development of CFD application in industrial engineering,

the quality and credibility of industrial application of CFD becomes a criti-

7


cal question faced by engineers, especially for the designer of turbomachin-

ery which needs sophisticated design. And it has also become an important

research topic of CFD [13, 14] in recent years. In 1996, AIAA [15] proposed

a guide for verification and validation of CFD simulations, in which two

components are suggested. Firstly, the accuracy of the solution to a com-

putational model should be assessed primarily by comparison with known

solutions, called Verification, to ensure the accuracy of numerical method-

ology; secondly, the simulated results should be compared with experimen-

tal data, called Validation, to assess approximation of the numerical model

used in CFD simulation to real physical model. The sources of numerical

error, such as grid density, computational domain scale, which exist mostly

in CFD simulation, are listed in references [16, 17].

Since the solver used in the present work, FineTM/Turbo from NU-

MECA Inc., has been verified by a large number of applications, verifica-

tion will not be involved in the present thesis. However, all the simulations

in the present thesis are validated by comparison with experimental data.

1.2.3 Non-deterministic CFD Methodologies

Although the CFD simulations have been performed extensively, so far

most of them are still deterministic simulations which use determinis-

tic models, deterministic boundary conditions and deterministic solvers.

However, for real physical problems, there often exist lots of uncertain

factors caused by the manufacturing process and the variation of work-

ing conditions, which may have significant influence on the performance,

such as the uncertain variation of geometry, uncertainty of boundary con-

ditions, working parameters and flow property. It should be noted that

these factors are different from the numerical error handled in Verification

and Validation (V&V). However, the influences of all these factors have not

been included in the current framework of CFD credibility research. The

aim of the deterministic simulation is searching for the deterministic so-

lution. Obviously, there might be large potential risks if the performance

of a design objective, such as turbine, compressor, aircraft, etc., is sensi-

tive to some uncertain variable. Even a minimal variation of the random

variable could lead then to catastrophic results. Therefore, in recent years,

NASA and European Union launched a series of research projects [18, 19]

for non-deterministic analysis methodology in aero-space research, to ad-

vance the robustness of design and analysis of complex systems. Due to

the high nonlinear property of control equation and turbulence models, it

is more difficult to implement the non-deterministic method in fluid dy-

namics field than in solid mechanics field. Thus, investigations on the

8

1.3. ROBUST OPTIMIZATION OF TURBOMACHINERY

non-deterministic methodology for CFD have been the main focuses and

difficulties in these research projects.

Non-deterministic analysis for CFD normally consists of two intimately

components. The first one is involved in retrieving parametric uncertain

property of random variable itself from the reliable data. The second one

pertains to the influence of uncertain model input or model parameter on

output parameters, namely propagation of uncertainty in flow field. As

can be concluded from experience of many industrial applications that the

uncertain input or model parameters usually are stochastic variables with

some certain probability distribution. However, it still needs a long term

experimental and statical investigation to verify the type of distribution

functions for specific cases. Thus, it is far beyond the scope of present nu-

merical analysis which mainly focuses on the second component. In other

words, the non-deterministic physical problem can be treated as a stochas-

tic process, and the distribution function of random variable is supposed

to be known.

The present thesis will not tend to discuss non-deterministic CFD

methodologies in detail. However, in the second part of the thesis—robust

optimization, the non-deterministic CFD simulations have to be used to

quantify the uncertainty. Therefore, in Chapter 3, the schemes of two

kinds of non-deterministic methods are introduced briefly.

1.3 Robust optimization of turbomachinery

In this section, a general description about the second topic of the present

thesis which is related to the optimal design of turbomachinery is pre-

sented. However, only the aerodynamic design of turbomachinery blades

is considered here since the shape of blade has great impact on the over-

all performance. The optimal design is usually composed of aerodynamic

design method and optimization method. The former is closely related to

CFD simulations, while the latter is mainly involved in numerical opti-

mization algorithms.

1.3.1 General Aerodynamic Design of Turbomachinery

The traditional trial-and-error design method fully depends on the experi-

ence of designers, which can not ensure the design is global optimal. Usu-

ally it is time and labor consuming due to the long design cycle. With

the rapid development of computational technology, the numerical design

method based on CFD and optimization algorithms start to be adopted in

9


aerodynamic design procedures. According to different definitions of op-

timization objectives, the numerical design methods can be classified into

two groups: inverse design and optimal design.

In inverse design, the blade geometry is modified to minimize the dif-

ference of profiles of pressure or velocity between the designed and the

specified. This method is widely used in 1980s and 1990s since it provides

a cheap way for numerical aerodynamic design, such as 2D blade opti-

mization in Meauze[20] and 3D blade optimization in Demeulenaere [21].

Whereas, considerable experience of the designer is still needed to give a

proper profile of pressure or velocity. In optimization design, the geometry

of the blade is sought to maximize the overall performance parameters,

such as efficiency, total pressures ratio, etc. In some sense, it is not so effi-

cient as inverse design which has clear direction at the beginning, while it

makes the design process less dependent on designers’ experience, which

gives the potential possibility of better design than specified by designer.

Nowadays, both of these two methods are used in practical design

and improved by coupling some new numerical technologies. Jameson [22]

used a joint gradient method coupling the inverse design and the gradient

optimization method for 2D airfoil design. Tiow et al [23] expanded this

framework with a global optimization technology, simulated annealing op-

timization algorithm, in the design of transonic axial cascades. Oyama

et al [24] optimized the NASA rotor67 using a parallel genetic algorithm

and full 3D CFD simulations. In their research, thousands of CFD com-

putations are needed to obtain the global optimal solution. Although the

numerical simulation is getting more accurate and fast, it is still time and

computational labor consuming. In that view, these methods are still far

away from industrial application [25]. Hirsch et al. [26, 27] proposed

an efficient integrated optimization platform based on CFD simulation,

which uses genetic algorithm for optimization and Artificial Neural Net-

work (ANN) as an approximation model. A coarse-to-fine approaching

strategy is performed to improve the prediction of approximation model

through the iterations between CFD simulation and optimization. Ahn

and Kim [28] developed a similar optimization technology for optimization

of NASA rotor37. The response surface method and linear programming

were adopted. All these technologies reduced the computational cost and

design time greatly. Many new optimization methods using three dimen-

sional CFD simulations and global optimization algorithms are constantly

being proposed. The research work in the present thesis will follow the

framework of Hirsch et al.

10

1.3. ROBUST OPTIMIZATION OF TURBOMACHINERY

1.3.2 Multi-objective Optimization Methods

In practical optimization design, there often exist multi optimization objec-

tives. The classical optimization method usually converts a multi-objective

optimization problem into a single objective problem using penalty func-

tions or weighting coefficients. However, for most cases, these objectives

often are incompatible. It’s difficult to set the appropriate penalty func-

tions or weighting functions which really depends on the experience and

preferences of the designer. Moreover, only one optimal result is obtained

after optimization. The designers have no alternative options to choose.

In fact, in most cases, there is no “best” solution by nature, but an infinite

number of feasible solutions which represent different levels of trade-off

between the objectives. This set of solutions is called Pareto optimal set

or Pareto Front. A couple of novel Multi-objective optimization algorithms

based on Pareto optimal concept are proposed and applied into optimal de-

sign. They provide a set of non-inferior solutions rather than one “best”

solution, which represents a more reasonable optimal nature.

The practical application of multi-objective optimization algorithms

in engineering design is still far away from success, especially in aerody-

namic design. High computational cost and convergence are two critical

problems needed to be investigated. Poloni et al [29] implemented hy-

brid multi-objective optimization on the fin keel of a sailing yacht based

on CFD simulation. Firstly, they performed a multi-objective genetic al-

gorithm (MOGA) to give an approximation of Pareto-optimal front. Then

a single objective optimization is started from several specific points on

the approximated Pareto-optimal front. In order to reduce the computa-

tional cost, small population size and evolution generations are imposed

in MOGA. However, during the optimization process, the real Pareto front

was not reached. Benini [30] proposed a multi-objective optimization algo-

rithm, named Genetic Diversity Evaluation Method(GeDEM), and applied

it to optimization of NASA rotor37. They used exact evaluation by CFD

simulation rather than approximation model. Although small size popula-

tion of 20 and evolution generations of 100 are used, more than thousands

of CFD computations are still needed to get the converged Pareto front,

which is extremely computational expensive. Moreover, a small size of

population can not provide a smooth Pareto-optimal front which increases

the risk of less convergence.

It therefore can be concluded that a multi-objective optimization is to

be preferred and that, with the present computational power, an approxi-

mation model is necessary for aerodynamic optimization of turbomachin-

ery. A key issue is how to improve the combination effect of approximation

model and multi-objective optimization algorithm. In Chapter 6 of the

11


present thesis, a coupled multi-objective optimization framework based on

the multi-objective genetic algorithm and ANN is developed and tested by

standard mathematical problems. A practical application on multi work-

ing points optimization of NASA roto37 is presented in Chapter 7

1.3.3 Robust Optimization

A further advanced optimization is the robust optimization which involves

the optimization approaches considering the influence of uncertainties,

with the aim of improving the performance of design with less sensitive-

ness to the random parameters. The robust optimization of turbomachin-

ery is a state-of-the-art technology which is reported in literatures only in

recent years. However the differences between robust optimization and

normal optimization only exist in objective definitions and CFD solver. In

robust optimization, the effect of uncertainties usually is one of the objec-

tives to be optimized or one of the constraints, which requests the non-

deterministic CFD solver to handle the uncertainty quantification.

The present robust optimization research is one part of the EU project:

NODESIM-CFD (Non-Deterministic Simulation for CFD Based on Design

Methodologies) [31] which intends to contribute to the development of a

new paradigm for CFD based virtual prototyping by incorporating the rel-

evant operational, geometrical and numerical uncertainties in the simula-

tion process. In Chapter 7, a robust optimization on NASA rotor37 with

uncertainty outlet pressure is presented, where a probabilistic collocation

method is employed to quantify the uncertainties. Different strategies of

objective definition are performed to investigate the effect of objective def-

inition on the optimization results.

12

Chapter 2

Complex Flows in

Turbomachinery

The internal flows within a blade passage of turbomachinery are strongly

three dimensional, viscous flows which may include laminar flow, turbu-

lent flow and transitional flow. Moreover, they are fully unsteady due to the

interactions between blade rows in a stage or multistage machine. There

also exist secondary flows including the flows due to passage vortices in

the endwall range, radial flow near blade surfaces, and tip leakage flow

and leakage vortex, shock and shock boundary layer interaction in high

speed conditions, wakes flows, even some specific flows, for instance film

cooling flows nearby the cooling holes. In the present chapter, a litera-

ture review on investigations of secondary flows, unsteady flows and film

cooling flows is given, followed by discussions on the existing flow models.

2.1 General Review on Secondary Flows

The important 3D viscous flow phenomena within a blade passage of turbo-

machinery are boundary layers and their separations, tip clearance flows

and wakes, which are most responsible of energy losses existing in blade

passage. Hence, the losses in an axial compressor or turbine can be mainly

classified as [32]:

1 Profile losses due to blade boundary layers and their separations and

wake mixing; In high speed condition, shock/boundary layer interac-

13

CHAPTER 2. COMPLEX FLOWS IN TURBOMACHINERY

tion may exist.

2 Endwall boundary layer losses, including secondary flow losses and

tip clearance losses.

3 Mixing losses due to the mixing of various secondary flows, such as

the passage vortex and tip leakage vortex.

Among all these losses, the most complex one is the secondary flow loss.

That is why considerable research on the secondary flow phenomena has

been done in last decades. Secondary flow is defined as the difference be-

tween the real flow and a primary flow, which is related to the development

of boundary layer on endwall and blade surface, the evolution of vortices

in passage, and detached flows. Based on topology analysis and experi-

ments, as well as the numerical simulations in recent decades, a couple of

secondary flow models are proposed which are presented below.

2.1.1 Classical Secondary Flow Models

The so-called classical secondary flow model, as illuminated in Figure 2.1,

is proposed by Hawthorne [33] for the first time according to the theory of

inviscid flow in 1955. This model presents the components of vorticity in

the flow direction when a flow with inlet vorticity is deflected through a

cascade. The main vortex, so-called passage vortex, represents the distri-

bution of secondary circulation, which occurs due to the distortion of the

vortex filaments of the inlet boundary layer passing with the flow through

a curved surface. The vortex sheet at the trailing edge is composed of the

trailing filament vortices and the trailing shed vorticity whose sense of

rotation is opposite to that of the passage vortex.

The classical vortex model attributes the secondary flow losses to the

generation and evolution of passage vortex, which reveals the basic mech-

anism of secondary flow losses. However, this model is relatively simple,

in which the interaction between the inlet boundary layer and blade force

was not considered. Moreover, the vortex system within passage is only

single passage vortex in half of the passage height range with other vor-

tices absence.

2.1.2 Modern Secondary Flow Models

When a shear flow along the solid wall approaches a blade standing on the

wall, the shear flow will be separated from the wall and roll up into a vortex

in front of the blade leading edge. This vortex is called horseshoe vortex due

14

2.1. GENERAL REVIEW ON SECONDARY FLOWS

Figure 2.1: Classical secondary flow model (Hawthorne, 1955 [33]).

to its particular shape. This well known phenomenon is firstly observed in

the flow around cylinders. The oil flow visualizations by Fritsche in 1955

[34] show the evidence of the horseshoe vortex in accelerating cascades. In

1966, Klein presents a finer cascade vortex model with both the passage

and horseshoe vortices as depicted in Figure 2.2(a).

While, the pioneering work for detailed analysis of secondary flow pat-

terns in turbine cascades in general is done in 1977 by Langston et al [35]

who proposed the well known modern vortex model in cascade. Three vor-

tices are presented in this model, as depicted in Figure 2.2(b). Langston

explains the interaction between the horseshoe vortex and the passage

vortex, and the development of the passage vortex. The big differences

between Langston’s model and Klein’s model exist in twofold: [36]

1 Langston clearly postulates that the pressure side leg of the leading

edge horseshoe vortex, which has the same sense of rotation as the

passage vortex, merges with and becomes part of the passage vortex.

2 Langston clarifies that the suction side leg of the leading edge horse-

shoe vortex which rotates in the opposite sense to the passage vortex,

continuing in the suction side endwall corner, while the presentation

of Klein suggests that this vortex is gradually dissipated in contact

with the passage vortex.

The first point from Langston is supported by the light sheet experiment

by Marchal and Sieverding [37] in 1977. While, the results of this ex-

periment also show the counter-rotating vortex, called counter vortex by

15


(a) by Klein, 1966 (b) by Langston, 1977

Figure 2.2: Modern secondary flow model[36].

Langston, in the trailing edge plane on the midspan side of the passage

vortex rather than in the corner, which is not consistent with the second

point from Langston.

An essential piece of information about the evolution of the horseshoe

and passage vortices was contributed by Moore and Smith [38] in 1983.

They found the pressure side leg of horseshoe vortex HP from leading edge

of one blade gathered with the suction side leg of the horseshoe vortex HS

from the adjacent blade leasing edge and the boundary layer on blade. A

more detailed explanation of the synchronous evolution of horseshoe and

passage vortices is given in late 1983 by Sieverding and Van den Bosch

[39], as illustrated in Figure 2.3. The figure shows two stream surfaces,

SS1 and SS2; the former starts upstream inside the endwall boundary

layer, while the latter outside the endwall boundary layer at different axial

positions in the blade passage. Approaching the leading edge the lateral

extremities of stream surface SS1 start to roll up into the two counterrotat-

ing legs of the horseshoe vortex, HP and HS , the main part of the stream

surface being nearly undisturbed. Behind the leading edge plane the whole

stream surface starts slowly to rotate, and gathering with HS and HP to

gradually develop into the passage vortex. The flow visualizations show

that the pressure side leg of the horseshoe vortex, HP , follows basically a

smooth curve through the passage without any noticeable vortical motion,

which would indeed suggest that its core coincides with that of the passage

vortex, while the suction side leg of the horseshoe vortex, HS , wraps itself

around the passage vortex core [36].

Although Sieverding [36] gave an excellent review of the vortex models

up to 1985, where the secondary flow vortex structures were described in

16

2.1. GENERAL REVIEW ON SECONDARY FLOWS

Figure 2.3: Synchronous evolution of horseshoe and passage vortices, (Sieverding

and Van den Bosch, 1983 [39]).

detail, the investigation on secondary flows are more active and new mod-

els continually emerge. In 1986, a more clear flow visualization of horse-

shoe vortex is illustrated by Ishii and Honami [40] using oil film and smoke

wire technique. They present a new flow model of three-dimensional tur-

bulent detached region on the plane of symmetry, as depicted in Figure

2.4. Four types of vortices exist in the detached region which can be di-

vided into two part: region I and II. Region I includes the horseshoe vortex

and corner vortex, while region II contains the detachment point, the sep-

aration and counter vortex.

Figure 2.4: New flow model of three-dimensional turbulent detached region on the

plane of symmetry (Ishii and Honami, 1986 [40]).

17


2.1.3 Latest Secondary Flow Models

In 1987, Sharma and Butler [41] proposed a secondary flow pattern which

is slightly different to that from Langston. This pattern, shown in Fig-

ure 2.5(a), demonstrates that the suction side leg of the horseshoe vortex

wraps itself around the passage vortex instead of adhering to the suction

side. This result is similar to the results of Moore [38] and Sieverding

[39]. However, in 1988, another pattern is given by Goldstein and Spores

[42], shown in Figure 2.5(b), which is different to Sharma’s again. Based

on mass transfer results, they suggested that the suction side leg of the

horseshoe vortex stays above the passage vortex and travels with it. This

flow pattern is similar to that suggested by Jilek [43] in 1986. The ma-

jor difference among these three models is the location of the suction side

leg of the horseshoe vortex. Since it is difficult to be detected due to the

small size, most literatures can not demonstrate the develop of this vortex

clearly.

(a) by Sharma and Butler, 1987 [41] (b) by Goldstein and Spores, 1988 [42]

Figure 2.5: Vortex pattern of secondary flows.

In 1997, a very detailed secondary flow visualization study was per-

formed by Wang [44] et al. They proposed a more comprehensive but more

complicated secondary flow pattern, as illustrated in Figure 2.6, which in-

cludes the passage vortex, the horseshoe vortex, the wall vortex and the

corner vortex. The development of the horseshoe vortex nearby the end-

wall is effected by the boundary layer on endwall and the blade surface.

In modern advanced blade, the leading edge radius of blade is so small

that can be compared with the thick of boundary layer. Hence, the sepa-

ration of boundary layer on endwall generates the multi-vortex structures

18

2.2. UNSTEADY FLOW WITHIN TURBOMACHINERY

at the leading edge of blade, Vph. Due to a strong pressure gradient the

pressure side leg of the horseshoe vortex moves toward the suction side

after it enters the passage. Meanwhile it entrains the main flow and the

inlet boundary layer forming a multi-vortex leg. When this multi-vortex

leg reaches suction surface, at approximately 1/4 of the surface distance

from the leading edge where it encounters the suction side leg Vsh from

adjacent blade, it is gradually squeezed into a single vortex. These two

vortices wrap each other and develop into a strong vortex in blade passage,

the so-called passage vortex, Vp. The passage vortex stays close to the suc-

tion surface, and rotates in a counterclockwise direction when viewing the

flow in the flow direction. With the entraining the main flow and boundary

layer, it is lifted away from the endwall and increases in size as it travels

downstream. Very close to the suction wall a small but very intense vor-

tex is found, which is named Wall Vortex by Wang [44], Vwip. This vortex

originates near the merging point of two legs of the horseshoe vortex and

rotates with reverse direction against the passage vortex. Then, it swept

up on the suction surface by the passage vortex and stays above the latter.

The corner vortices is too small to be caught using smoke flow visualiza-

tion technique, while the surface flow visualization by Jabbari et al. [45]

in 1992 and the local measurement of naphthalene mass transfer by Gold-

stein et al. [46] in 1994 indicate the existence of the corner vortices, which

are illustrated as VsLc, VpLc, Vsc and Vpc in Figure 2.6. Wang et al. suppose

these corner vortices are formed by the separation of the boundary layer

very closed to the leading edge corner.

In 2001, Langston [47] reviewed these new models after the Sieverd-

ing’s review. Laster in the same year, Zhou and Han [48] gave a more

comprehensive review of all these models. They concluded that the good

understanding of the secondary flow in turbomachinery can help greatly

to control the vortices within passage and decrease the losses.

2.2 Unsteady Flow within turbomachinery

During early investigations, the flow in turbomachinery is assumed to be

steady. In blade passage of stator, the absolute velocity of flow is supposed

steady in respect to the absolute reference frame, while in blade passage

of rotor, the relative velocity of flow is supposed to be steady in respect to

the relative reference frame. This hypothesis has been used for decades

in design of turbomachinery. Even today, in the preliminary design of tur-

bomachinery, it is still adopted. However, the flow in turbomachinery is

inherently unsteady, as proven by Dean [49] through an adiabatic friction-

19


Figure 2.6: Endwall vortex pattern (Wang et al., 1997 [44]).

less model of turbomachinery. Greitzer [50] redescribed this model and

gave a more clear explaination. In some sense, the unsteadiness is mainly

due to the relative motions of rotor and stator. Although the steady flow

model using the relative frame of reference in rotor gives an time aver-

aged approximation of the unsteady flow in turbomachinery. However, it

is just suitable for the primary design. In the elaborate design phase, un-

steadiness plays a important role which can increase losses, blade vibra-

tion, noise generation and affect the heat transfer. A good understanding

of the unsteady flow in turbomachinery is necessary for advanced design.

According to Greitzer [51], the unsteady flow in turbomachinery can be

classified into two groups: inherent unsteadiness and conditional unsteadi-

ness.

2.2.1 Conditional Unsteadiness

The conditional unsteadiness is mainly caused by the sudden changes of

the working condition. When turbomachinery is working on the start

stage, acceleration stage or off-design condition, the fluctuation of work-

ing condition might lead to the unsteady rotating stall and surge of com-

20


pressor. Sometimes, the distortion of inlet flow or the asymmetric outlet

condition of vector nozzle also might lead to the unsteadiness.

Rotating Stall

Taking an axial compressor for example, the instantaneous fluctuations of

rotating speed, or inlet pressure or outlet pressure may disturb the flow

in passage. The disturbance may cause the increase of incidence angle of

a part of flow which leads to the separation flow on suction side of blade.

The separation flow will block the mean flow in blade passage and affect

the adjacent range to generate new separated flow. This instability will be

propagated against the rotating direction in circumferential face, leading

to the so-called rotating stall.

Rotating stall is an asymmetric unstable status of the internal flow.

The separation flow rotating with the blade will cause the periodic fluctu-

ation of the pressure on blades surface, and then result in a periodic stress

on blades, which will decrease the life of blades.

Surging

Surging of compressor is marked by a complete breakdown of the continu-

ous steady flow throughout the whole compressor, resulting in large fluctu-

ating of flow with time and also in subsequent mechanical damage to the

compressor. The phenomenon of surging should not be confused with the

stalling of a compressor stage. Usually, the stalling is assumed to be the

cause and the beginning of surging.

2.2.2 Inherent Unsteadiness

The inherent unsteadiness is mainly due to the relative motion and inter-

action between rotor and stator. In general, it involves three parts: the

interaction of potential flows in adjacent blade rows; the interaction be-

tween the wake flow and blade rows downstream; the interaction between

the secondary flows and blade rows.

Interaction between Adjacent Blade Rows

The first part comes from the changing of the relative position of rotor to

stator which results in the periodic fluctuation of the pressure. This fluctu-

ation is propagated both upstream and downstream as disturbance waves.

The researching from Greitzer [51] shows the impact of the disturbance is

21


obvious if the ratio of axial gap between blade rows to the cascade pitch is

lower than 0.5, and which increases with the increase of axial velocity.

Interaction between wake Flow and Blade Rows

The second part, unsteady wake, is a quite common flow phenomenon, not

only in turbomachinery. Due to the thickness of the trailing edge of blade,

the flows after the blade generate a high dissipation region, called wake,

which is similar to the flow passed a circular cylinder where a famous

wake flow Von Karman vortex street can be observed. When a viscous flow

passes a cylinder or an airfoil, a regular vortex shedding can be found

behind the cylinder, which results in a zone with fully turbulent flow and

high dissipation. The pressure on the surface of cylinder will fluctuate

with the vortex shedding. A similar flow phenomenon exists in the bypass

flow after a blade, as shown in Figure 2.7. In this figure, Gehrer et al.

[52] predicts the instantaneous streamline patterns of the periodic vortex

shedding through unsteady simulations with low Re number k − ǫ model.

Figure 2.8 shows the results of unsteady simulation performed by Wang

and He [53], in which the instantaneous pressure contour patterns of wake

for both laminar flow and turbulent flow through unsteady simulations are

presented clearly .

Figure 2.7: Periodic wake shedding (Gehrer et al., 2000 [52]).

The wake flow in multi-stage turbomachinery is more complicated

22


(a) Laminar flow (b) Turbulence flow

Figure 2.8: Pressure contour of wake flow (Wang and He, 2001 [53]).

than vortex shedding after circle cylinder since it will be distorted and

deformed by the blade when flows through the blade row downstream as

shown clearly in Figure 2.9 by Hodson [54]. This unsteady transport pro-

cess could last to the next few blade rows and mix with new wake flows to

forming highly nonuniform unsteady flow in blade passage.

Interaction between Secondary Flows and Blade Rows

The third part is similar to the second one, in which the second flows

are also sheared by the blade rows downstream during the transport pro-

cess. The distortion and mixing of these vortices will enhance the non-

uniformity of the flow. Schlienger et al. [55] investigated the interaction

between secondary flows and blade rows through experiments on a low

speed turbine with two stages. It is found that the characteristic of the

unsteady flow field at the rotor hub exit is primarily a result of the inter-

action between the rotor indigenous passage vortex and the remnants of

the secondary flow structures that are shed from the first stator blade row.

Moreover, there exist interactions among secondary flows, wake and

blade rows, which results in more complicated unsteady flow. Matsunuma

[56] investigated this interaction effect on a low speed turbine of single

stage, with the instantaneous absolute velocity contour pattern at the

nozzle exit shown in Figure 2.10. The experimental results suggest that

the secondary vortices are periodically and three-dimensionally distorted

at the rotor inlet. A curious tangential high turbulence intensity region

23


Figure 2.9: Velocity vector of wake flow perturbation (Stieger, 2005[54]).

spread at the tip side is observed at the front of the rotor, which is be-

cause of the axial stretch of the nozzle wake due to the effects of the nozzle

passage vortex and rotor potential flow field.

2.3 Film Cooling

According to the theory of Carnot cycle, increasing the inlet temperature

of gas turbine is an effective way to increase the efficiency and capacity of

a turbine. In order to enhance the performance of jet engine and gas tur-

bine, the temperature of the gas flowing into a turbine blade passage has

been raised continually in recent years, which might result in the dam-

age of blades, especially the leading edge (LE) which is exposed to the hot

gas directly. Although new high temperature materials have been investi-

gated and used constantly, it is obvious that they couldn’t follow the rising

pace of the inlet temperature. For the aero jet engine with TWR of 10,

the inlet temperature of the turbine has been increased to 1850K∼1950K.

Whereas, the modern materials cannot stand temperature higher than

1200K∼1400K [3]. According to the “IHPTET” and “VAATE” projects, the

TWR of the engine will be increased to 15∼20 in 2020s. At that time, the

24

2.3. FILM COOLING

Figure 2.10: Instantaneous absolute velocity contour pattern at nozzle exit (T. Mat-

sunuma, 2006 [56]).

inlet temperature will be over 2400K which will be far above the temper-

ature can be withstood by modern meterials. So effective cooling methods

for the blades must be used to ensure the turbine works normally and to

extend lifetime of the blades.

Four types of cooling methods, such as Convection cooling, Impinge-

ment cooling, Film cooling and Effusion cooling [57], and their hybrid

methods are used in practical engineering. Usually, the effusion cooling

can provides the best cooling among these four methods, while it is seldom

used because it will weaken the structure strength of blades greatly. Con-

vection cooling and impingement cooling are usually used in conditions

where the temperature is lower than 1600K since they cannot provide pro-

tection to the surface of blades. Film cooling is the only way can be used in

whole range of the temperature is higher than 1600K. Figure 2.11 shows

the configuration of a blade with cooling holes at the leading edge and

trailing edge and the corresponding perspective of intern channels.

The working principle of film cooling can be illustrated as Figure 2.12.

In film cooling, the cold fluid colored in blue jet from the cooling holes or

slots distributed on blade surface. It is curved downstream under the press

and friction of hot main flow colored in pink in the figure, then forms a thin

cooling film on the surface which separates the blade surface from hot gas.

25


(a) Cooling holes at the leading edge

and trailing edge

(b) Perspective of cooling holes

Figure 2.11: Blade with cooling holes.

Meanwhile, it takes the sporadic flames and radiant heat to downstream.

Hence, it can protect the blade surface effectively.

Figure 2.12: Pattern of film cooling.

2.3.1 Film Cooling Flow on A Flat Plate

Many investigations focus on a flat plate model since it is relatively sim-

ple and could provide the basic research of film cooling in a cascade. The

flow structure nearby the cooling hole has great impact on the cooling ef-

fect that indicates a good understanding of the flow structure nearby the

cooling hole is the keystone of improving the cooling effect. Film cooling

26

2.3. FILM COOLING

flow mixed with the main flow downstream of the holes forms a typical

jets-in-crossflow (JICF) problem which involves a system of vortices flow.

Margason provided a very comprehensive review of investigations on JICF

[58] during the past fifty years till 1990s, in which several distinguishable

features of JICF were noted :

1 When the jets flow out from the exit of cooling hole, it is defected

by main crossflow to follow a curved path. At the same time, due

to shearing of the cross flow along the lateral edges, the jets form a

counter-rotating vortex pair (CVP), also called kidney vortex, which

dominates the flow structure nearby the cooling hole;

2 Similar to the flow around cylinder, the cross flow near to the exit of

cooling hole forms a stagnation point and a horseshoe vortex. How-

ever, it is much weaker and smaller than CVP.

3 The main flow, which bypassed the jets flow, forms a wake region

downstream which is the weakest. One analogy relates it to the

Karman-Benard vortex street. Another concept describes it as an

unsteady flow separation region.

This three-vortices model is also illustrated in Figure 2.13(a). The CVP

is the dominant vortex, while others are all belong to secondary vortices.

Some investigations [59, 60] include the vestiges of the free jet ring vor-

tices as a fourth vortex system, as depicted in Figure 2.13(b) from Fric and

Roshko.

(a) By Margason, 1993 [58] (b) By Fric and Roshko, 1994 [59]

Figure 2.13: Sketch of vortex systems of JICF.

27


Hale et al [61] performed similar investigations on the flow structure

near multiple cooling holes through surface streak experiments and nu-

merical simulations. A secondary counter-rotating vortex pair with the op-

posite rotation sense of the CVP was observed in their experiments. This

secondary CVP was also found by Yuan et al. in [62] where it was named

wall wake vortex. Kang [63] investigated the flow structure nearby a sin-

gle cooling hole in a flat plate using numerical simulations and topological

analysis, and gave a comprehensive explain of the vortex structure of jets

in crossflow.

The flow structure nearby cooling hole is associated with the mixing

process which is influenced by many factors, such as main flow Re num-

ber, hole’s geometry, surface curvature, injection angle, blowing ratio, etc.

However, for the model with certain configurations, the blowing ratio is the

main factor. Hence, large investigated efforts have been contributed to the

effect of blowing ratios. Ajersch et al. [64] investigated the jets in cross-

flow on a flat plate using both experiments and RANS simulations in 1997.

A model of square cooling holes on a flat plate was tested under different

blowing ratios. The investigation results show the CVP clearly for blow-

ing ratios of 1.0 and 1.5, while that is less distinct for blowing ratio of 0.5

where the jet is too weak to penetrate through the inlet turbulent bound-

ary layer. The interactions between jets and the main flow was enhanced

with the increase of the blowing ratio. However, the 3D flow structures

are absent in the discussion. Kang et al. [65] simulated the same model

using RANS simulations and presented the 3D vortex systems and devel-

opment of the vortex rings nearby the cooling hole. Also based on the same

experiment, Tyagi and Acharya [66] and Ma [67], using LES and RANS

simulations respectively, simulated three square cooling holes with differ-

ent aspect ratios under fixed blowing ratio. It is found that the intensity of

CVP will increase with the increase of the aspect ratio.

In the present thesis, DES and RANS simulations are also performed

on the experimental model of Ajersch. The details of experiments and nu-

merical simulations are presented in Chapter 4.

2.3.2 Film Cooling Flow in Turbine Cascades

Although the investigation of jets in cross flow on a flat plate can depict

the flow structure nearby the cooling hole and reveal the interaction be-

havior between the main flow and jets, it is only focus on the local view

and far from the practical application in turbine cascade. However, due

to the impact of difference of geometry, streamwise pressure gradient and

secondary flows in the main flow, the structure of film cooling jets in a

28

2.3. FILM COOLING

turbine cascade may not the same as those on a flat plate.

Schwarz et al. [68, 69] presented their experiment results of film cool-

ing on generic curved surfaces with three different relative strength cur-

vatures. The laterally averaged film cooling effectiveness was observed

to increase with increasing curvature until the cooling film separates on

strongly curved wall. Koc et al. [70] investigated the film cooling effective-

ness of rectangular holes on a curved surface with different curvatures.

They indicated that there exist the optimal curvature and the best blow-

ing ratio for a given curvature.

Meanwhile, the effect of cooling holes’ geometry was also investigated

by many researchers. Hyams and Leylek [71] investigated numerically

the streamwise inclined jets on a flat plate with five distinct film cooling

configurations, including cylindrical, forward-diffused, laterally diffused,

inlet shaped and cusp-shaped holes. They found that the laterally diffused

film hole provides the best cooling effect. Dittmar et al. [72] performed

experiments on the suction side of a turbine guide vane to investigate the

cooling effectiveness of four different cooling hole configurations, such as

cylindrical holes, discrete slots, straight fan-shaped holes and compound

angle fan-shaped holes, with the same blowing ratio. They concluded that

the fan-shaped holes show very good effectiveness values at moderate and

high blowing ratios. However, at low blowing ratios, all configurations

show similar film-cooling effectiveness. Kim et al. [73] also performed ex-

perimental investigations on the influence of injection holes with different

exit shapes at the leading edge of cylindrical body models under various

blowing ratio and found that the conventional cylindrical holes have poor

film cooling performance compared to the shaped holes with diffused exit.

Dieter et al. [74] investigated numerically the mixing phenomena of span-

wise inclined holes. Bunker [75] gave a review on investigations of shaped

hole film cooling over the past 30 years. He summarized the benefits of

shaped film holes using expanded exits of fan and/or laidback geometries

with higher centerline and laterally averaged adiabatic film effectiveness,

and little variation in effectiveness over blowing ratios from 0.5 to 2, low

sensitivity to free-stream turbulence intensity variations.

An excellent series of experimental investigations on film cooling in

turbine cascades were performed by Fottner and his research team (Beeck

et al.[76], 1992, Ardey and Fottner [77] 1997, [78] 1998). They had made a

systemically investigation on the flow field within a high pressure planar

turbine cascade with film cooling injections (holes and slots) at the leading

edge through experiments. The experimental models are designed with

two slot arrays or two cylindrical hole arrays located at the two sides of

blade leading edge. Two different hole geometries, vertical and spanwise

29


inclined respect with to the blade surface, were tested. Detail experiments

data are provided for validation, including the pressure distribution on

blade surface and oil visualization patterns. Theodoridis et al. [79, 80] in-

vestigated the same cascade with slots and vertical holes using 2D and 3D

numerical simulations, respectively. While they only focused on the vali-

dation between simulated results and experiment data, seldom discussion

on the flow field was presented. Wang et al. [81] also simulated the flow in

the same cascade with vertical holes. They found that jets only influence

the pressure distribution in a small local range. A recirculation region is

observed behind the cooling hole, which is related to the Counter rotating

vortex pair generated by interaction between jets and the main flow.

However, the 3D flow structures at different configurations are still

unclear. In addition, the cooling air is usually extracted from the compres-

sor, which will affect the performance of a whole engine. And the interac-

tion between cooling flow and main flow in blade passage may increase the

flow losses greatly. However, the additional losses produced by jets are not

included in the investigations above. Wang and kang [82, 83] simulated

all three models in the same experiment. Based on a good agreement of

the comparison between simulated results and experimental data, the flow

structures for different model and additional losses were discussed. The

details of the simulated results and discussion are presented in Chapter 4.

30

Chapter 3

Governing Equations and

Numerical Methods

In the present chapter, the numerical methods used in this thesis are de-

scribed in detail, including the governing equations of fluid, the classical

finite volume method and turbulence approximation. In particular, the

steady and unsteady treatments to the rotor/stator interaction, including

Domain Scaling Method, Phase Lagged Method and Non-linear Harmonic

Method, are introduced. At the end of this chapter, the mathematical back-

ground of stochastic analysis for non-deterministic CFD simulation is in-

troduced, followed by descriptions of Intrusive Polynomial Chaos Method

and Non-intrusive Probabilistic Collocation Method which are used for un-

certainty quantification.

3.1 Governing Equations

The motion of a fluid can be completed described by the conservation law

for the three basic properties: mass, momentum and energy, which usu-

ally is expressed in a set of equations. When applied to a perfect viscous

fluid, these equations are known as the Navier-Stokes equations, while for

a perfect inviscid fluid, they are known as Euler equations.

31

CHAPTER 3. NUMERICAL METHODS

3.1.1 The Mass Conservation Equation

The law for mass conservation is a general statement of kinematic nature,

which illuminates the empirical fact that the variation of mass flow of a

fluid system equals to the mass flow passed through the boundary of the

system. The mass conservation equation, also called the Continuity Equa-

tion, can be expressed in

∂ρ

∂t+ ∇ · (ρu) = 0, (3.1)

where ρ is the mass density, t is time and u is the velocity vector. ∇ denotes

the divergence operator, then the expansion of the second item in the above

equation is

∇ · (ρu) =∂(ρux)

∂x+

∂(ρuy)

∂y+

∂(ρuz)

∂z, (3.2)

where, ux, uy, uz are the components of u in x, y, z direction, respectively.

For incompressible flow, ρ is constant in whole flow field, then Eq.3.1 can

be reduced to the divergence free condition for the velocity:

∇ ·u = 0. (3.3)

If the flow is steady, then ρ will not change with time, then Eq.3.1 can be

replaced by

∇ · (ρu) = 0. (3.4)

3.1.2 The Momentum Conservation Equation

The momentum conservation law is the expression of the generalized New-

ton’s second law, defining the equation of motion of a fluid which means the

momentum variation of a fluid system equals to the total external forces

imposed on it. The momentum conservation equation involves three com-

ponents:

∂ (ρux)

∂t+ ∇ · (ρuxu) = − ∂p

∂x+

∂τxx

∂x+

∂τyx

∂y+

∂τzx

∂z+ Fx,

∂ (ρuy)

∂t+ ∇ · (ρuyu) = −∂p

∂y+

∂τxy

∂x+

∂τyy

∂y+

∂τzy

∂z+ Fy, (3.5)

∂ (ρuz)

∂t+ ∇ · (ρuzu) = −∂p

∂z+

∂τxz

∂x+

∂τyz

∂y+

∂τzz

∂z+ Fz ,

where p denotes the static pressure, τii and Fi, (i = x, y, z) are the compo-

nents of viscous stress shear tensor τ and body force F. This equation is

32

3.1. GOVERNING EQUATIONS

valid for both Newtonian fluid and non-Newtonian fluid. For Newtonian

fluid, the components of τ are defined by

τxx = 2µ∂ux

∂x+ λ∇ ·u,

τyy = 2µ∂uy

∂y+ λ∇ ·u, (3.6)

τzz = 2µ∂uz

∂z+ λ∇ ·u,

and

τxy = τyx = µ

(∂ux

∂y+

∂uy

∂x

)

,

τxz = τzx = µ

(∂ux

∂z+

∂uz

∂x

)

, (3.7)

τyz = τzy = µ

(∂uy

∂z+

∂uz

∂y

)

,

where µ is the dynamic viscosity coefficient of fluid and λ is the second vis-

cosity coefficient. The latter is related to a viscous stress caused by the

change of the volume, thus it is also called volume viscosity or bulk viscos-

ity. For most fluids, the Stokes’s hypothesis is satisfied [4]:

2µ + 3λ = 0. (3.8)

Another viscosity coefficient ν, called Kinematic Viscosity, is often used

which is defined as ν = µ/ρ. Substituting Eq.3.6 and Eq.3.7 into Eq.3.5

and utilizing Eq.3.8, the N-S equations of motion is obtained:

∂(ρux)

∂t+ ∇ · (ρuxu) = µ∆ux − ∂p

∂x+ Qx,

∂(ρuy)

∂t+ ∇ · (ρuyu) = µ∆uy − ∂p

∂y+ Qy, (3.9)

∂(ρuz)

∂t+ ∇ · (ρuzu) = µ∆uz −

∂p

∂z+ Qz,

33


where ∆ is the Laplace operator. Qi(i = x, y, z) denotes the generalized

source item which can be expanded as

Qx = Fx +∂

∂x

(

µ∂ux

∂x

)

+∂

∂y

(

µ∂uy

∂x

)

+∂

∂z

(

µ∂uz

∂x

)

+∂

∂x(λ∇(u)),

Qy = Fy +∂

∂x

(

µ∂ux

∂y

)

+∂

∂y

(

µ∂uy

∂y

)

+∂

∂z

(

µ∂uz

∂y

)

+∂

∂y(λ∇(u)), (3.10)

Qz = Fz +∂

∂x

(

µ∂ux

∂z

)

+∂

∂y

(

µ∂uy

∂z

)

+∂

∂z

(

µ∂uz

∂z

)

+∂

∂z(λ∇(u)).

According to Graves [84], the shear stress from the volume viscosity can

be neglected for the majority of fluid flows. For incompressible fluids, the

source items can be reduced to the body force F. Therefore, Eq.3.9 can be

rewritten in a vector format:

∂(ρu)

∂t+ u ·∇(ρu) = µ∆(u) − ∇p + F. (3.11)

The second item on the left-hand side of Eq.3.11 is called convective item,

while the first item on the right-hand side is called diffusive item. The ratio

between momentum of the convection item and the diffusion item gives an

important dimensionless number Re, the so-called Reynolds number:

Re =ρ|u|L

µ. (3.12)

Substituting the dynamic viscosity with the kinematic viscosity ν, then

gives

Re =L|u|

ν. (3.13)

3.1.3 The Energy Conservation Equation

The energy conservation law is also referred to as the expression of the

first principle of thermodynamics, which states that the variation of total

energy in a fluid system are the work of the forces acting on the system

34


plus the heat transmitted into the system. Hence, the energy conservation

equation can be expressed by

∂(ρE)

∂t+ ∇ · (ρEu) = ∇ · (k∇T ) + ∇ · (−pu + τ ·u) + WF + qH , (3.14)

where, E denotes the specific total energy, k is the thermal conductivity

coefficient, T is the absolute temperature, WF and qH denote the work of

body forces and the heat source item, respectively. Usually, the specific

total energy E is related to the specific internal energy e and the kinetic

energy by

E = e +|u|22

. (3.15)

Another useful state variable is the total internal enthalpy H which is

defined as

H = E +p

ρ= e +

p

ρ+

|u|22

. (3.16)

The related specific internal enthalpy is

h = e +p

ρ= cp∇T, (3.17)

where, cp is the specific heat, which is can be related to ν and k by a dimen-

sionless number Pr = µcp/k, namely Prandtl Number, which indicates the

the ratio of momentum diffusivity (kinematic viscosity) and thermal diffu-

sivity.

Introducing Eq.3.15 and Eq.3.17 into Eq.3.14 leads to the conservation

equation for enthalpy:

∂(ρh)

∂t+ ∇ · (ρhu) = ∇ · (k∇T ) + ∇ · (τ ·u) + qH . (3.18)

Usually, a additional correlative equation between p and ρ is needed, which

is called state equation. For perfect gas, the state equation is

p = ρRT, (3.19)

where R is the Molar gas constant, which is about 287J/kgK for air. Then, a

closed system of six unknowns (p, T, ρ, ux, uy, uz) with five nonlinear partial

differential equations and a state equation is obtained.

A compact general format of N-S equation can be expressed in:

∂U

∂t+ ∇·FC = ∇·FV + Q. (3.20)

35


The items of equation 3.20 in sequence are transient term, convective term,

diffusive term and source term. The expansion of the transient term U is:

U =

ρρux

ρuy

ρuz

ρE

.

The convection term FC and the diffusive term FV denote the vector as

below, respectively:

FC =

ρu

ρuxu + px

ρuyu + py

ρuzu + pz

(ρE + p)u

, (3.21)

and

FV =

0τx

τy

τz

τu + k∇T

. (3.22)

The last term Q is:

Q =

0Sx

Sy

Sz

WF + qH

.

More details about the governing equations of fluid dynamics and

more comprehensive discussions can be found in Hirsch [4] and White [85].

3.1.4 Rotating Frame of Reference

Accounting for the particular flow situation in turbomachinery, it is neces-

sary to be able to describe the flow behavior relatively to a rotating frame

of reference that is attached to the rotor. Without loss of generality, it is

assumed that the moving part of turbomachinery is rotating steadily with

angular velocity ω around the machine axis along which a coordinate z is

aligned.

36


Define w as a velocity field relative to a rotating system and v = ω× r

as the entrainment velocity. Thus, a observer located outside the rotating

frame observes the velocity

u = w + v = w + ω × r. (3.23)

Introducing Eq.3.23 into Eq.3.1 gives the mass conservation equation in a

rotating frame of reference:

∂ρ

∂t+ ∇ · [ρ(w + v)] = 0. (3.24)

Expanding the above equation gives:

∂ρ

∂t+ ρv + w ·∇ρ + ρ∇ ·w + ρ∇ ·v = 0. (3.25)

The first term in Eq.3.25 indicates the time rate of change of density at a

fixed station in an absolute frame of reference. The second term involves

the spatial change of density registered by a stationary observer. Combin-

ing of these two terms expresses the time rate of change of the velocity

within the rotating frame of reference. Since the last term is zero [2],

Eq.3.23 is reduced to∂rρ

∂t+ ∇ · (ρw) = 0, (3.26)

where, the subscript r refers to the rotating frame of reference. Comparing

Eq.3.26 with Eq.3.1, it is found that the mass conservation equation keeps

the same expression in both stationary and rotating frames of reference.

Without causing confusion, the subscript r can be omitted in general.

Using Eq.3.23, the acceleration is also can be redefined as

du

dt=

∂(w + ω × r)

∂t+ (w + v) ·∇(w + v). (3.27)

Expanding Eq.3.27 and rearranging the items result in

du

dt=

∂w

∂t+

∂v

∂t+ w ·∇(w) + 2w × ω + ω × v. (3.28)

The first item on right-hand side expresses the local acceleration of the

velocity field within the rotating frame of reference. The second term and

third item denote the angular velocity acceleration and the convective term

within the rotating frame of reference,respectively. While, the fourth item

and last item are the Coriolis acceleration and the centrifugal acceleration,

37


Figure 3.1: Coriolis and centripetal forces created by the rotating frame of refer-

ence (Schobeiri, 2005 [2]).

respectively, which are fictitious forces produced as a result of transfor-

mation from stationary frame to rotating frame of reference. Figure 3.1

shows the directions of the velocity and the acceleration, and relationship

between the absolute velocity, relative velocity and rotation. Substituting

the acceleration in Eq.3.11 and Eq.3.14 separately, equations of motion

and energy in rotating frame of reference can be obtained:

∂(ρw)

∂t+

∂(ρv)

∂t+ w ·∇(w) + ω × v + 2ω × w = µ∆(w) − ∇p + F,

(3.29)

d[

ρ(

h + |w|2

2 − |v|2

2

)]

dt=

∂p

∂t+ ∇ · (k∇T ) + ∇ · (τ ·w) + WF + qH .

(3.30)

It should be noted that WF is the work of body forces in rotating frame of

reference, while the subscript r is omitted here. The detailed derivation

process of governing equations in rotating frame of reference can be found

in Schobeiri [2].

38

3.2. FINITE VOLUME METHOD

3.2 Finite Volume Method

3.2.1 General Introduction of Discretization Method

Three families of numerical methods are available for discretization of N-S

equations: Finite Difference Method (FDM), Finite Element Method (FEM)

and Finite Volume Method (FVM). The most traditional method is FDM,

which is based on the properties of Taylor Expansions. In FDM, the deriva-

tive is estimated straightforward by the ratio of two differences according

to the theoretical definition of the derivative. Hence, FDM is probably the

simplest method to apply, but it requires that the mesh must be structured

and the nodes have to be located on the parallel lines of coordinate, which

results in some limitations in application. The second method FEM origi-

nates from the field of structural analysis, which is based on the Weighted

Residual Method. In FEM, a computational domain is divided into many

small sub-domains, called elements, and reassembled after each element

had been analyzed. In fact, it is seldom used in fluid dynamic analysis due

to the large computational cost. By far the most widely applied method in

CFD today is FVM which was developed in 1970s. The strength of FVM

is in its direct connection to the physical flow properties. The basis of this

method relies on the direct discretization of the integral form of the con-

servation law, which distinguishes FVM significantly from FDM.

FVM is employed by the solver used in the present thesis, named EU-

RANUS (EURopean Aerodynamic NUmerical Simulator) from NUMECA

Inc. Therefore a detailed description of FVM is given below. More complete

descriptions about the other two methods can be found in Hirsch [4].

3.2.2 Mesh Generation

In order to solve the N-S equations numerically, the space, also called do-

main, has to be discretized into a set of time-invariant, non-overlapping

volumes (also called cells), which could be polygons, like triangles and

quadrilaterals in 2D, and polyhedra, like tetrahedra, pyramids, prisms

and hexahedra in 3D. The space discretization forming the so-called mesh

or grid. The mesh can be built in a structured way or an unstructured

way. Unstructured mesh is more flexible for complicated geometries, while

the generation process is difficult than that of structured mesh. Moreover,

structured meshes have been highly developed and applied successfully in

simulation of turbomachinery. Hence, structured meshes are adopted in

simulations of this thesis.

In FVM, a control volume has to be set for each cell, where the dif-

39


ferential equations are integrated. There are two kinds control volume

approaches: cell-centered approach or cell-vertex approach. The difference

between these two approaches can be found in Figure 3.2. The left Figure

(a) Cell centered (b) Cell vertex

Figure 3.2: Control volume approaches.

3.2(a) shows the cell centered approach which consists in four cells, P, Q, R

and S. The black dots denote nodes, while the line segment connecting two

nodes is called edge for 2D, or face for 3D. In this approach, the unknowns

are stored at the center of cells, then the control volumes are identical to

the cells exactly. In the cell vertex approach, as depicted in Figure 3.2(b),

the unknowns are stored at the vertices of cells. Thus the control volume

has its corner nodes at the cell centers of the original mesh, and its edges

are line segments directly connecting these cell centers. In the present

work, the cell-centered approach is used.

When a control volume is built, then Eq. 3.20 can be integrated over

the volume:∫

V

∂U

∂tdV +

∫

V

∇·FCdV =

∫

V

∇·FV dV +

∫

V

QdV. (3.31)

Applying Green’s theorem to equation 3.31, one gets∫

V

∂U

∂tdV +

∮

S

FCdS =

∮

S

FV dS +

∫

V

QdV,

where, the subscript V and S stand for the volume and the surface of a

cell. Within this cell, values of the unknowns are constant which leads to

∂U

∂tV +

∑

faces

(FC ·n)∆S =∑

faces

(FV ·n)∆S + QV, (3.32)

40

3.2. FINITE VOLUME METHOD

where n is the unit normal of the cell surface.

3.2.3 Spatial Discretization of Convective Term

There exist several discretization schemes for the convective term, such as

central scheme, first order upwind scheme, QUICK and MUSCL schemes

et al. In present simulations, the second order central scheme is adopted

which is described as below. For a cell i, the fluxes through its edges are

(F ·n)i+1/2 =1

2[(F ·n)i + (F ·n)i+1] − di+1/2, (3.33)

where, di+1/2 stands for the artificial dispassion term. Equation 3.33 is

the standard central scheme. However, as suggested by Hirsch [4], the

averaging of fluxes could be replaced by the averaging of the unknowns

which leads to

(F ·n)i+1/2 = F

(Ui + Ui+1

2

)

·n − di+1/2. (3.34)

In a certain extent, averaging of the unknowns is more robust than aver-

aging of fluxes, especially for high speed flows.

For the dissipation term, the Jameson type dissipation [86] with 2nd

and 4th order derivatives of the unknowns is adopted:

di+1/2 = ε(2)i+1/2(Ui+1 − Ui) − ε

(4)i+1/2(Ui+2 − 3Ui+1 + 3Ui − Ui−1), (3.35)

where, ε(2) and ε(4) denote the 2nd order dissipation and 4th order dissipa-

tion, respectively. They can be calculated as follows:

ε(2)i+1/2 =

1

2κ(2)λ∗

i+1/2max(υi−1, υi, υi+1, υi+2), (3.36)

ε(4)i+1/2 = max

(

0,1

2κ(4)λ∗

i+1/2 − ε(2)i+1/2

)

, (3.37)

In Eq. 3.36, υi stands for the gradient sensor defined by:

υi = max

(∣∣∣∣

pi+1 − 2pi + pi−1

pi+1 + 2pi + pi−1

∣∣∣∣,

∣∣∣∣

Ti+1 − 2Ti + Ti−1

Ti+1 + 2Ti + Ti−1

∣∣∣∣

)

(3.38)

In Eqs. 3.36 and 3.37, λ∗ is the spectral radius multiplied with the cell face

area:

λ∗i+1/2 = |u ·∆S|i+1/2 + c∆Si+1/2, (3.39)

41


where, c is the speed of sound. The coefficient κ(2) and κ(4) are user inputs

and the typical values are:

κ(2) ≈ 1

4and κ(4) ≈ 1

128. (3.40)

From Eq. 3.36 to Eq. 3.40, it is seen that ε(2) is expected to be switched

on in order to suppress the oscillatory behavior in the region of strong

gradients. However, in uniform regions the pressure sensors will be small,

such that the effect of 2nd order dissipation is vanished. The ε(4) provides

some background dissipations which are always active, except in regions

where ε(2) is switched on.

3.2.4 Spatial Discretization of Diffusive Term

For the diffusive term, the discretization is much easier since it is usually

in the purely central way. However, both the unknowns and the gradients

of the unknowns have to be discretized:

Ui+1/2 =1

2(Ui + Ui+1), (3.41)

∆Ui+1/2 =1

2(∆Ui + ∆Ui+1). (3.42)

The gradient ∆U can be obtained from U by using the Green’s theorem:

∆U ≈ 1

Vi

∫

Vi

∆UdV =1

Vi

∮

Si

UdS. (3.43)

3.2.5 Explicit Multistage Runge-Kutta Scheme

A widely used explicit time integration technique, of hight order of accu-

racy, is multi-stage Runge-Kutta (R-K) method, which is described below in

detail. For simplicity, a general ordinary differential equation is employed,

which can be written in:dU

dt= F (U). (3.44)

The basic idea of R-K method is to evaluate the right-hand side of Eq.3.44

at several values of U in the interval, between n∆t:

U1 = Un + α1∆tF (Un),

U2 = Un + α2∆tF (U1),

· · · , (3.45)

U q = Un + ∆tF (U q−1),

Un+1 = U q,

42

3.3. ACCELERATING CONVERGENCE METHODS

The coefficients αi determine the stability area and the order of accuracy of

the R-K scheme. The time step ∆t is obtained by the harmonic averaging

of the inviscid time step ∆tI and the viscous time step ∆tV :

∆t =∆tI∆tV

∆tI + ∆tV. (3.46)

The inviscid and viscous time steps in each cell are scaled with the CFL

number through following equations:

∆tI =CFL ·V

|u∆Si| + |u∆Sj | + |u∆Sk| + c (|∆Si| + |∆Sj | + |∆Sk|),(3.47)

∆tV =CFLρV 2

8µ [|Si|2 + |Sj |2 + |Sz |2 + 2 (|SiSj| + |SiSk| + |SjSk|)]. (3.48)

In simulations of this thesis, an fourth order R-K scheme is adopted with

coefficients being:

α1 = 0.125, α2 = 0.306, α3 = 0.587, α4 = 1.

3.3 Accelerating Convergence Methods

In order to enhance the convergence of simulations, some accelerating con-

vergence methods are employed in EURANUS, such as Implicit Residual

Smoothing (IRS) and Multigrid method.

3.3.1 Implicit Residual Smoothing

Implicit residual smoothing, also called Residual Averaging, is used in

combination with the multi-stage Runge-Kutta method to speed up the

convergence. The smoothed residual is obtained by solving the following

Alternating Direction Implicit (ADI) formulation [87]:

(1 − εiδ2i )(1 − εjδ

2j )(1 − εkδ2

k)RSi,j,k = Ri,j,k, (3.49)

where RSi,j,k is the smoothed residual, Ri,j,k is the original residual. The

δ2i,j,k is the second order center difference operator:

δ2i RS = Ri−1 − 2Ri + Ri+1. (3.50)

43


The smoothed residual is obtained by smoothing successively in the direc-

tions i, j and k. The smoothing parameter ε has the same formulation in i,

j and k direction:

εi >1

2λ∗

i

∆t

V.

3.3.2 Multigrid Method

The implementation of the multigrid method requires a set of meshes from

fine to coarse. The coarser mesh is created by dropping nodes of a finer

mesh in each direction. According to the stability analysis, high frequency

errors are eliminated fast during the iterations on a certain mesh, while

the low frequency errors will be difficult to be eliminated by the same

mesh. A straightforward idea is to solve the equations on a mesh coarser

than the previous one to eliminate the errors with relative high frequency

for the fine mesh. Then the R-K scheme loops are implemented iteratively

between the coarser mesh and finer mesh, as shown in Figure 3.3, which

could accelerate the convergence effectively. According to the numbers of

iteration on different mesh level, there exits V cycle, V-sawtooth cycle, W

cycle, W-sawtooth cycle, F cycle and F-sawtooth cycle. The detailed descrip-

tion of theses strategies can be found in Zhu [88]. The V cycle, as depicted

in Figure 3.3 is used in the present work, which is described below in gen-

eral. Consider a set of meshes denoted with an index l = 1, . . . , L with L

Figure 3.3: Scheme of the V cycle multigrid strategy [89].

being the finest level. Eq. 3.44 on the finest level can be written as:

dUL

dt= FL(UL), (3.51)

44

3.3. ACCELERATING CONVERGENCE METHODS

where, FL is the spatial discretization operator on the finest mesh L. In

like manner, the approximation on a coarser level l is:

dU l

dt= Fl(U

l) + fl, (3.52)

with fl the forcing function which is defined recursively as:

fl = Fl(Ill+1U

l+1) + I ll+1

[fl+1 − Fl+1(U

l+1)], (3.53)

In this equation, I ll+1 and I l

l+1 represent restriction operators of the un-

knowns and the residuals, respectively, which are defined as:

I ll+1R

l+1 =∑

Rl+1, (3.54)

I ll+1U

l+1 =

∑V l+1U l+1

∑V l+1

, (3.55)

where Rl+1 is defined as:

Rl+1 = ff+1 + Fl+1(Ul+1), (3.56)

and V represents the cell volume. The summation is over the 8 fine cells

contained within a coarse cell for a 3D problem. After certain number iter-

ations on mesh l, the same procedures can be implemented till the coarsest

mesh is reached. During the time integration on each level mesh, IRS op-

erator usually is performed before going to the next coarse level. Once

the solutions on the coarsest mesh is smoothed, the coarse-to-fine sweep

of the multigrid cycle is initiated. The current solutions on finer grids are

updated with the solution on the next coarser level:

U l = U l + I ll−1(U

l−1 − I l−1l U l). (3.57)

The operator I ll−1 is a prolongation operator. The rigorous mathematical

treatments about the restriction operator and prolongation operator can

be found in Wesseling [90] and Zhu [88].

Based on the multigrid method, a full multigrid strategy (FMG) can be

performed to provide a good initial solution to accelerate the convergence.

The full multigrid strategy starts the R-K scheme loops with a relative

coarse mesh. The solution on that level will not be interpolated to the next

finer mesh until it converges to a certain accuracy level. Then the solution

on the finer grid is taken as the initial solution for further iteration on that

grid level. The process is recursively used until the finest grid is reached,

which is illustrated clearly in Figure 3.4.

45


Figure 3.4: Scheme of the full multigrid strategy [89].

3.4 Turbulence Approximation

A critical numerical method which has been investigated extensively in

CFD is the approximation of turbulence which is the most important flow

phenomenon in fluid dynamics. Turbulent flow is a fluid region character-

ized by chaotic, stochastic property changes, which includes low momen-

tum diffusion, high momentum convection, rapid variation of pressure and

velocity in space and time. Since Reynolds observed the turbulent flow and

laminar-turbulent transition for the first time in 1883 [91], turbulence has

been investigated for more than one hundred years. However, it is still far

away from the thorough understanding of its inner mechanism. In past

decades, comprehensive efforts are spent on the numerical investigation of

turbulence, and four main numerical methods are developed to deal with

turbulence:Reynolds Averaged Navier-Stokes Simulation (RANS), Large

Eddy Simulation (LES), Direct Numerical Simulation (DNS) and Detached

Eddy Simulation (DES).

Among these four methods, DNS is the most accurate method, while

it is also the most expensive one. Since DNS simulates the whole range

of turbulent statistic fluctuations at all relevant physical scales, the mesh

should be very fine. The size of the smallest eddies is inversely propor-

tional to Re3/4—the well known Kolmogrov scale related to the turbulent

dissipation. Then, the number of mesh points needed for a 3D simulation

should be proportional to Re9/4. As the time step of integration is deter-

mined by the smallest turbulent time scales, which is also proportional to

Re3/4. Therefore, the total computational effort of DNS simulation is pro-

portional to Re3 for homogeneous turbulence [4], which will be extremely

expensive for high Reynolds number flow.

In LES, the smallest turbulence scales are filtered by employing a so-

called Subgrid Scale (SGS) model, while the larger turbulence scales are

46

3.4. TURBULENCE APPROXIMATION

still calculated directly. Therefore a coarser mesh than that of DNS can

be used. LES starts more and more to be applied to industrially relevant

cases in recent years. However, it is still seldom used in simulations of the

complex turbomachinery flow. Moreover, In engineering view, the time ac-

curate details of the turbulent flow have no more practical meaning than

the total effect of turbulence for some industrial designs. In fact, that re-

veals the reason of the prevalence of RANS in engineering applications. In

RANS, the properties of turbulent flow are averaged in time domain which

leads to the mesh size and time integral scale used in RANS are much

larger than in DNS and LES. As a result, RANS simulations run faster

even with smaller computational costs compared to that of DNS and LES.

However, some empirical or semi-empirical models, known as turbulence

model, are required to close the time averaged equations system. Thus,

the accuracy of RANS simulation to a turbulent flow is heavily limited by

the quantity of turbulence model. Although, a coupled of turbulence mod-

els, such as Baldwin-Lomax model, Spalart-Allmaras model, k − ǫ model,

etc, have been developed and applied successfully in plenty of engineering

cases. However, none of them has consistent good performance in all fields

[92]. In 1997, DES is proposed by Spalart [93], which combines the most

favorable aspects of LES and RANS to give a compromised solution. In

DES, the flow close to the wall is simulated using RANS, while in separa-

tion region, the LES simulation is launched. Then the performance and

computational cost of DES are between that of LES and RANS. The com-

parison of the computational requirement of four methods is summarized

by Spalart [94] as listed in Table 3.1.

Table 3.1: Comparison of the approximation methods for turbulence

Method Empiricism Gride size Number of time steps

3D URANS strong 107 103.5

DES strong 108 104

LES weak 1011.5 106.7

DNS none 1016 107.7

With curren computer technology, DNS and LES are excluded for the

simulation of complex flow. DES has shown great potential with some suc-

cessful applications [95]. However, for flows in turbomachinery, RANS is

still the dominant simulation method. Thus, for simulations of film cooling

in Chapter 4, both RANS and DES methods are employed, while for simu-

lations of 3D flow in turbine and optimization of compressor, only RANS is

47


used.

3.4.1 Reynolds Averaged Navier-Stokes Equations

The basic tool required for the derivation of the RANS equations from

the instantaneous Navier-Stokes equations is the Reynolds decomposition.

Reynolds decomposition refers to separation of the flow variable into two

parts, a mean (time-averaged) component and a fluctuating component:

u = u + u′,

p = p + p′,(3.58)

where the overline stands for time averaged value and the superscript “′”stands for fluctuation. Note that, the time averaged value of first order

fluctuation is zero, φ′ = 0, while that of second order is usually not zero.

Introducing Eq.3.58 into Eq.3.3 and Eq.3.11, and averaging them in time

domain result in the Reynolds Averaged Navier-Stokes equations.

∇ ·u = 0, (3.59)

∂(ρu)

∂t+ u ·∇(ρu) = −∇p + µ(u) − ∇ · (u′ ·u′), (3.60)

where, u′ ·u′ represents the correlation between fluctuating velocities, which

is called Reynolds stress tensor. The expanding format of this tensor is

u′ ·u′ =

u′1u

′1 u′

1u′2 u′

1u′3

u′2u

′1 u′

2u′2 u′

2u′3

u′3u

′1 u′

3u′2 u′

3u′3

.

Note that, in Eq.3.60 the body force F is omitted for expressing conve-

nience. For compressible flows, the extra products of density fluctuations

with other fluctuating quantities can be averaged by using a density-weighted

averaging, called Favre-averaging [4], through

φ =ρφ

ρ,

with

φ = φ + φ′′,

ρφ′′ = 0.

48


As mentioned above, the application of RANS requires additional tur-

bulence models to relate the unknown Reynolds stress tensors with mean

quantities. A comprehensive investigations on turbulence models are per-

formed and a wide variety of models are developed. According to the way

to build the relation, there are two main categories, namely the Eddy Vis-

cosity Models (EVM) and Reynolds Stress Models (RSM).

In EVM, a viscosity µt or νt generated by turbulence is introduced

according to the Boussinesq assumption:

u′iu

′j = −νt

(∂ui

∂y+

∂uj

∂x

)

+2

3

(

k + νt∂ui

∂x

)

δij , (3.61)

where, ui, uj are the components of mean velocity. Since the averaged

value of mean value is the same, the overline for mean value is omitted

hereafter. δij is Kronecker delta. k is the turbulent kinetic energy,

k =1

2

(

u′iu

′i

)

(i = 1, 2, 3). (3.62)

According to the number of equations used to determine µt or νt, the EVM

can be classified into Algebraic model, for instance the Baldwin-Lomax

(BL) model; One equation model, for instance the Spalart-Allmaras (SA)

model; Two equation model, for instance the well known k − ǫ model and

the k − ω model. All these models are applied with varying degrees of

success. However, none of the available turbulence models offers a totally

accurate description of turbulent flows [96, 97].

While, in RSM, a set of momentum equations of Reynolds stress and

a length scale determining equation are sought to calculate the Reynolds

stress directly. In some cases, they could provide better approximation of

turbulence than that of EVM. Whereas, the computational efforts need by

RSM is much larger than that of EVM, which results in a less widely use

of RSM compared to EVM.

In the present thesis, SA model is employed due to the robustness

and less computational cost [98], which is introduced below in detail. The

details about other turbulence models can be found in Wilcox [92].

3.4.2 Turbulence model

Baldwin-Lomax model

Baldwin-Lomax (B-L) model is a zero-equation model. The so-called zero

equation model means the turbulent viscosity is related to mean variables

by a algebraic equation, not by a differential equation. B-L model is a

49


double-layer model. The turbulent viscosity of inner layer is determined

by the Prandtl mixing length model; the turbulence viscosity of the outer

layer is determined by the mean flow rate and the length scale.

νt =

νit , n ≤ nc

νot , n ≥ nc

(3.63)

where, n is the normal distance from the calculated point to the wall. nc

is the minimal value of n, when the turbulent viscosity νit equals to the

turbulent viscosity νot .

The inner viscosity νit can be calculated by:

νit = l2|ω|. (3.64)

In the equation above, the mixing length l can be calculated respectively

by:

l = kn(

1 − e−y+/A+)

,

y+ =

(√ρwτw

µw

)n

.

where, k and A+ are constants, which usually can be 0.4 and 26, respec-

tively. ρw and µw are the density and dynamic viscosity of the flow close to

the wall, respectively. τw is the friction force on the wall.

The vorticity ω in Eq.3.64 can be calculated by:

ωi = εijk∂uj

∂xk.

The outer turbulent viscosity νot can be calculated by:

νot = KCcpFwakeFKleb(n). (3.65)

where, K and Ccp are constants, which usually are 0.0168 and 1.6, respec-

tively.

Fwake = min

nmaxFmax,

Cwknmax

[√

(u2 + v2 + w2)max − (u2 + v2 + w2)min

]2

/Fmax.

where, Cwk is constant, usually has a value of 1. The value of nmax and

Fmax are determined by the following function:

F (y) = y|ω||1 − e−n+/A+ |. (3.66)

50


Fmax is the maximum value of F (y) normal to the wall, nmax equals to the

value of y when F (y) is maximum.

In Eq.3.65, Fkleb is the Klebanoff intermittency factor:

FKleb =[1 + 5.5(nCKleb/nmax)6

]−1. (3.67)

where, the constant CKleb usually has a value of 0.3.

Spalart-Allmaras Model

Spalart-Allmaras model, proposed by Spalart and Allmaras [99], is a one-

equation turbulence model which belongs to the eddy viscosity models fam-

ily. Basically, it is a transport equation for the eddy viscosity νt, in which

the Reynolds stresses are given by −u′v′ = 2νtSij . Here, Sij denotes the

strain-rate tensor:

Sij =1

2

(∂u

∂y+

∂v

∂x

)

.

The basic SA model is derived for free shear flows with high Re number

in which the turbulence is found only where vorticity is present. Then the

eddy viscosity νt is given as follows:

dνt

dt= cb1Sνt +

1

σ

[∇· (νt∇νt) + cb2(∇νt)

2]− cw1fw

[νt

d

]2

, (3.68)

where cb1, cb2 and cw1 are all constants with the subscript b stands for “ba-

sic” and w stands for “wall”. The first term on the right-hand side suggests

the effect of the production term, in which S denotes the magnitude of the

vorticity:

S =√

2ΩijΩij ,

Ωij =1

2(∂u

∂y− ∂v

∂x).

The second item indicates the effect of the diffusion term which naturally

focuses on spatial derivatives of νt. The dimensionless number σ is the

turbulent Prandtl number which indicates the ratio between the momen-

tum eddy diffusivity and the heat transfer eddy diffusivity. The third item

reflects the blocking effect of a wall on the eddy viscosity, in which the sub-

script w stands for “wall” and d is the distance to the wall. The construction

of fw is inspired by algebraic models in which the mixing length can be de-

fined as l =√

νt/S. While Spalart and Allmaras suggest to use the square

51


of l/κd for convenience, where κ is the karman Constant. A satisfactory fw

function suggested by them is:

fw = g

(1 + c6

w3

g6 + c6w3

)1/6

,

g = r + cw2(r6 − r), (3.69)

r ≡ ν

Sκ2d2,

However, in the buffer layer and the viscous sublayer, the basic S-A

model needs an additional modification. Spalart and Allmaras introduced

a modified viscosity coefficient ν which equals to νt except in the viscous

region:

νt = νfv1,

where, fv1 is a “switch” function:

fv1 =χ3

χ3 + c3v1

.

Here, χ is the ratio between turbulence viscosity and the molecular viscos-

ity, χ = ν/ν. The subscript v stands for “viscous”.

The production term in Eq.3.68 is improved by replacing S with S:

S ≡ S +ν

κ2d2fv2, (3.70)

fv2 = 1 − χ

1 + χfv1. (3.71)

The function fv2 is constructed, like fv1, to ensure S maintain the log-layer

behavior all the way to the wall. Then the Eq.3.68 has become

dν

dt= cb1Sν +

1

σ

[∇· ((ν + ν)∇ν) + cb2(∇ν)2

]− cw1fw

[ν

d

]2

. (3.72)

By far, the terms provides control over the laminar regions of the shear

layers are absent, which has two aspects: keeping the flow laminar where

desired and obtaining transition where desired. In the laminar region, ν is

usually less than ν, then there will be νt ≤ ν/350 because of the damping

by fv1. In order to make ν is a stable solution, the production term is

multiplied by (1 − ft2) leading to

ft2 = ct3 exp(−ct4χ2).

52


The subscript t stands for “trip” which means that transition in the real

flow is imposed by an actual trip, or that it is natural but its location is

known. In order balance the budget near the wall, an opposite change is

added to the destruction term. In addition, a source term ft1 is added to

initiate transition near the specified trip points in a smoother manner, and

to retain a local formulation. The function ft1 is as follows

ft1 = ct1gt exp

(

−ct2ω2

t

∆U2(d2 + g2

t d2t )

)

.

In above equation, dt is the distance from the computational field point to

the trip, which is on a wall. ωt is the wall vorticity at the trip, and ∆Uis the difference between the velocity at the computational field point and

that at the trip. The odd factor gt is obtained by:

gt = min (0.1, ∆U/(ωt∆xt)) ,

where, ∆x is the grid spacing along the wall at the trip. This factor is

needed only for numerical reasons. Without the grid dependence of gt, the

streamwise influence domain of the trip would scale with the boundary-

layer thickness, which could be very small in the laminar region. As a

result, that domain would easily fall between two streamwise gride points.

The gt factor guarantees that the trip term is nonzero over a few stream-

wise stations. The Gaussian function in ft1 confines the influence domain

of the trip terms as needed which is roughly a semi-ellipse.

Applying these adaptions for laminar region to Eq.3.72, then leads to

a complete SA model:

dν

dt= cb1[1 − ft2]Sν +

1

σ

[∇· ((ν + ν)∇ν) + cb2(∇ν)2

]

−[

cw1fw − cb1

κ2ft2

] [ ν

d

]2

+ ft1∆U2. (3.73)

The constants value in Eq.3.73 suggested by Spalart and Allmaras are

listed in Table 3.2.

k − ǫ model

k − ǫ model adds two equations to the governing equation set, one k equa-

tion and one ǫ equation. The turbulent dissipation rate ǫ is defined as:

ǫ = ν

(∂u′

i

∂xk

)(∂u′

i

∂xk

)

(3.74)

53


Table 3.2: Constants of Spalart-Allmaras model

Constant Value Constant Value

cb1 0.1355 cw3 2.0

cb2 0.622 cv1 7.1

σ 2/3 ct1 1.0

κ 0.41 ct2 2.0

cw1 3.239 ct3 1.2

cw2 0.3 ct4 0.5

The turbulent viscosity νt can be presented in the function of k and ǫ:

νt = Ctk2

ǫ(3.75)

where, Ct is a constant. For the standard k − ǫ model, Ct = 0.09.

Two additional transport equations are needed to calculate k and ǫ.For standard k − ǫ model, these two equations are:

∂k

∂t+

∂kui

∂xi=

∂

∂xj

[(

ν +νt

σk

)∂k

∂xj

]

+ Gk + Gb − ǫ − YM + Sk (3.76)

∂ǫ

∂t+

∂ǫui

∂xi=

∂

∂xj

[(

ν +νt

σǫ

)∂ǫ

∂xj

]

+ G1ǫǫ

k(Gk + G3ǫGb) − G2ǫ

ǫ2

k+ Sǫ

(3.77)

In these two equations, σk and σǫ are the Prandtl number related to k and

ǫ, respectively. Gk is the generated item by k due to the mean velocity

gradient, which can be calculated by:

Gk = νtρ

(∂ui

∂xj+

∂uj

∂xi

)∂ui

∂xj

Gb is the generated item by k due to the flotage, which is zero for incom-

pressible fluid. For compressible fluid, it can be calculated by:

Gb = βgiνtρ

σ

∂T

∂xi

where, σ is the turbulent Prandtl number, which is 0.85. gi denotes the

component of gravitational acceleration in i direction. β denotes the ther-

mal expansion coefficient.

54


YM suggests the effect of fluctuation, for incompressible fluid, which is

zero. For compressible fluid, it can be calculated by:

YM = 2ρǫk

a2

where, a is the sound speed.

The normal value of constants in Eqs.3.76 and 3.77 are as below:

C1ǫ = 1.44, C2ǫ = 1.92, σk = 1.0, σǫ = 1.3

The constant C3ǫ is between [0, 1]. If the direction of mean flow is same to

that of the gravity, G3ǫ = 1; if the direction of mean flow is perpendicular

to the gravity, G3ǫ = 0.

Some modified k − ǫ model were proposed based on the standard one,

such as Realizable k− ǫ model and RNG k− ǫ model. The detail description

about these models can be found in Wilcox [92].

3.4.3 Large Eddy Simulation

As mentioned above, in large eddy simulation, only the large turbulent

eddies are simulated, while the small turbulent scales are filtered using

a so-called subgrid scale (SGS) model. Let φ(x, t) denotes the physical

quantity, while the filtered quantity is denoted by φ(x, t). Then

φ(x, t) =

∫

D

G(x − z)φ(z, t)dz, (3.78)

where, D is the computational domain. G is the filter which satisfies

∫

D

G(x − z)dz = 1. (3.79)

Many filter functions can be considered for LES. For FVM mehtod, the

top-hat filter is often used which is defined as:

G(x − z) = 1, if ‖x − z‖ < ∆2

G(x − z) = 0, otherwise(3.80)

where, ∆ denotes the filter width which is proportional to the size of the

smallest scales not removed by the filter, ∆ = (∆x∆y∆z)1/3. The physical

quantity can be decomposed into a filtered part and a sub-grid part:

φ = φ + φs, (3.81)

55


where the superscript s stands for the sub-grid part. Note that, the over-

line denotes the filter operation in contrary to the time-averaging in case of

RANS. Implementing the filter operation on the incompressible N-S equa-

tions, one obtains:

∇ ·u = 0, (3.82)

∂u

∂t+ u ·∇(u) = µ∆(u) − 1

ρ∇p + ∇ ·τ s. (3.83)

Here, the τ s denotes the sub-grid scale stresses which is given by

τsij = uiuj − ui ·uj, (i, j = x, y, z) (3.84)

Introducing the Eq.3.81 into Eq.3.84 and rearranging the terms, one

obtains

τsij = −us

iusj − (uius

j + ujusi ) + uiuj − uiuj . (3.85)

The first term in Eq.3.85 is similar to the Reynolds stresses. The second

term, namely cross-term or Clark term [100], are dispersive in nature and

largely account for the backscatter effects. The last two terms are called

Loenard stresses together. According to Loenard [101], they are of the same

order of magnitude as the the truncation error. In most applications, the

Clark and Loenard stresses can be neglected and only the Reynolds-like-

stresses remain to be modeled.

The simplest and mostly used SGS model is the Smagorinsky model,

where the subgrid-scale stresses are modeled by employing the Boussinesq

assumption:

τij −1

3τkkδij = −νt∇ ·u. (3.86)

The eddy viscosity νt is modeled by

νt = L2s

√

2SijSij . (3.87)

Here, Ls is the length scale for the subgrid scale, which can be given by

Ls = CsV1/3. (3.88)

where, V is the cell volume and the constant Cs has a value of 0.17. The

detailed description of LES methods can be found in Lesieur [102] and

Jiang [103].

56

3.5. DUAL-TIME STEPPING METHOD

3.4.4 Detached Eddy Simulation

The basic model employed in the majority of DES applications is the S-A

model. As mentioned above, the destruction term in S-A model is propor-

tional to (ν/d)2, where d is the distance to the wall. While, if replacing dwith a length scale

d = min(d, CDES∆), (3.89)

a DES method is obtained. Here, ∆ is the cell size based on the largest

size of cell in x, y and z directions, ∆ = max(∆x, ∆y, ∆z). The empirical

constant CDES has a value of 0.65. Eq.3.89 indicates that near the solid

wall , d ≪ ∆ and the model acts as the standard SA model, while away

from walls where ∆ ≪ d, a one-equation subgrid model is obtained which

is quite similar to the Smagorinsky model.

3.5 Dual-time Stepping Method

For unsteady simulations, a natural way is directly using the Physical

Time Subiteration (PTS) method. However, both the explicit and implicit

PTS methods have some limitations. Due to the stability limitation, the

time step in the explicit method has to be quite small and the common con-

vergence accelerating technologies are difficult to be implemented. That

leads to the computation take unbearable long time. In implicit method,

although without the stability problem, the computational powers needed

are too large for most cases. The Dual-Time Stepping (DTS) method, pro-

posed by Jameson [104] is an great improvement over the classical global

time stepping approach, which has be proved to be an effective approach

through extensive applications. It consists in adding a pseudo-time deriva-

tive terms to the time dependent N-S equations. At each physical time

step, a steady state problem is solved in a pseudo time and all available

acceleration techniques can be applied. The revised N-S equations over a

control volume can be expressed in:

∫

V

∂U

∂tdV +

∫

V

∂U

∂τdV +

∫

V

∇·FCdV =

∫

V

∇·FV dV +

∫

V

SdV, (3.90)

where t is the physical time and τ is pseudo time. The second order upwind

scheme are used for the physical time discretization:

(∂U

∂t

)n+1

=1.5Un+1V n+1 − 2UnV n + 0.5Un−1V n−1

∆t. (3.91)

57


While all other terms in Eq. 3.90 are computed at the same physical time

step n + 1, then Eq. 3.90 can be treated as a steady problem in the pseudo

time τ :

∂U

∂τV n+1 + Runs(U) = 0. (3.92)

Noted that, Runs is the residual of unsteady problem which is calculated

by:

Runs(U) = R(U) +1.5Un+1V n+1 − 2UnV n + 0.5Un−1V n−1

∆t,

where R denotes the residual of steady problem determined by Eq. 3.31.

However, when integrating Eq. 3.92, it should be modified in an implicit

scheme:(

Un+1 − Un

∆τ+

1.5

∆t

)

V n+1 + Runs(U) = 0, (3.93)

where, ∆τ is the pseudo time step for the steady problem. Then a modified

pseudo time step can be obtained:

∆τmod =∆t

∆t + 1.5∆τ∆τ.

The modification is adding a term relate to the physical time step. This

term will become dominant if the physical time step is significantly smaller

than the pseudo time step which has a stabilizing effect there.

3.6 Rotor/Stator Interaction Treatment

In order to use the same solver, the flows in stator and rotor should be

calculated in the stationary frame of reference and the rotating frame

of reference, respectively. However, a critical problem is how to transfer

the information downstream and upstream at the interface of stator and

rotor. The quality of the flow predictions for multistage turbomachinery

strongly depends on the treatment of rotor/stator interaction. Five differ-

ent approaches are available in FineTM/Turbo to simulate the interaction

between rotating and non-rotating blocks. Tow approaches, the Mixing

Plane Method and Frozen Rotor Method are for steady simulations; the

rest three approaches, Domain Scaling method, Phase Lagged Method and

Non-linear Harmonic Method are for unsteady simulations.

58

3.6. ROTOR/STATOR INTERACTION TREATMENT

3.6.1 Steady Simulation Treatment

The simplest treatment of R/S interface is the mixing plane method pro-

posed by Denton [105]. This method assumes the exiting flows of stator

become uniform flows before entering the inlet of domain of rotor. Then,

a pitchwise averaging of the flow solution is needed at the R/S interface

before transferring the information of both sides.

Firstly, the solutions are extrapolated from the inner cells onto the R/S

interface from both sides. For the information transferred from stator to

rotor, the value is interpolated onto the mesh points on the patch belong to

rotor domain, and averaged pitchwisely on these points. The the averaged

value can be interpolated back onto the inner cells of rotor domain. For the

information transferred from rotor to stator, the procedure is same, while

with the contrary interpolation direction. The transferred values can be

the unknowns or the fluxes of the unknowns. The exchange of information

at the interface depends on the direction of the flow which can be local

value which leading to a local conservative coupling method, or can be

the averaged value of per pitchwise row which leading to a conservative

coupling by pitchwise rows method. Note that the meshes on both sides

of the interface should cover the same range in spanwise, the averaging

is performed along the same azimuthal mesh lines. However, a Full Non

Matching Mixing Plane [89] can be used to overcome this limitation.

Better than the isolated simulation on single stator or rotor, the inter-

action of potential flows in considered in this method. However, the impact

of secondary flows and separation flow are erased. This physical approxi-

mation tends to become more acceptable as rotational speed is increased.

The mixing plane method is by far the most often used R/S modeling in

industry design and optimization.

If the exchange of information at the interface is by interpolation di-

rectly without averaging, one has the frozen rotor method. As the name

indicates, the relative position of rotor and stator is fixed. Hence, the result

of the frozen rotor method is equivalent to a certain point of the unsteady

simulation which means the flow solutions will dependent on the relative

position between rotor and stator. Since the information exchange on R/S

interface is through interpolation, the mesh on both sides of the R/S in-

terface should cover the same pitch range. That means the periodic of the

rotor domain and stator domain should be kept the same,

KsPs = KrPr,

where, Ks and Kr are relative prime which stand for the number of pas-

sages in the stator domain and rotor domain, respectively. Ps and Pr de-

note the pitch of stator and rotor separately. An approximation of the blade

59


number can be made if Ks and Kr are large in order to reduce the compu-

tational cost, which is called Domain Scaling. For instance a turbine with

29 blades of stator and 31 blades of rotor can be approximated by a turbine

with 30 blades of both stator and rotor, then only one passage is needed to

mesh for both stator and rotor. However, the simulation results are only

the approximated result to the real model.

The frozen rotor method is used firstly by Brost et al. [106] in simu-

lations of an axial turbine where the simulated results have a good accor-

dance with the transient results of the measurement. While, the flow field

in a passage usually changes a lot during the per period. Therefore, this

method is only used in some specific simulations.

The information exchange processes of mixing plane method and frozen

rotor method depend on the boundary type of the R/S interface. The de-

tail settings for different boundary types and the corresponding exchange

strategies can be found in [89].

3.6.2 Unsteady Simulation Treatment

As mentioned in last subsection, the frozen rotor method just simulates a

specific status of turbomachinery. For unsteady simulation, a natural idea

is to simulate several different transient positions of rotor related to stator

which leading to the traditional unsteady treatment of R/S interface is the

Sliding Mesh Method proposed by Rai [107]. In this method, the computa-

tional domain is divided into two parts: rotor domain and stator domain.

The mesh for rotor domain rotates with rotor. The R/S interface becomes a

sliding face and the exchanges of solution information are through the in-

terpolation to the dummy cells on both side without any averaging. In the

Phase lagged method, only one blade passage for each blade row is needed.

At each time step, the rotor is set at its correct position and equations are

solved for that particular time step for the whole computation domain. The

final solution is therefore a succession of instantaneous solutions for each

increment of the rotor position. These two methods will not be described in

detail here, the main effort is focused on the Non-linear Harmonic Method

introduced below.

3.6.3 Harmonic Method

The sliding mesh method simulates the full unsteady flow, which is still

quite computational expensive for industrial requirements. In 1985, a

novel method is proposed by Adamczyk [108] in which the local effects

60

3.6. ROTOR/STATOR INTERACTION TREATMENT

of the unsteadiness on the time-averaged flow are considered via the so-

called deterministic stress. The conventional nonlinear time-domain in-

tegration, for instance the local time-stepping approach, can be used to

calculate these nonlinear stress terms [109, 110]. However, a multiple-

passage domain, even the whole annulus domain, is needed to be meshed

which leading to expensive computational cost again.

In the past decade, a harmonic frequency-domain methods are de-

veloped, e.g., using potential flow model [111, 112] and Euler equations

[113, 114]. However, all of the previous harmonic methods adopt the lin-

ear assumption, so that the nonlinear interaction between unsteady dis-

turbances and the time-averaged flow is completely neglected. A nonlinear

harmonic method is developed by He [115, 116] following the framework

of Giles[117] which is based on an asymptotic theory. This method solves

the the steady transport equations for the time-averaged flow and the time

harmonics. For turbomachinery, the blade passing frequencies (BPF) are

the fundamentals in time domain of the periodic disturbances from the ad-

jacent bladerows. The solving of the generated perturbation amplitudes in

a row is performed in the frequency domain by a steady transport equa-

tion associated with BPFs and subharmonics. The deterministic stresses

are calculated directly from the in-phase and out-of-phase components of

the solved harmonics. Using this method, only one passage is needed that

saves the computational cost greatly. He et. al. [116, 118], Vilmin et al.

[119] validated this method with simulations on a 3D radial turbine and a

multistage axial compressor. Therefore, this method is adopted in the un-

steady simulation of a low speed axial turbine which is included in Chapter

5. A detail description of this method is given below.

The physical quantity can be decomposed into a time-averaged value

and a sum of perturbations, which in turn can be decomposed into N har-

monics [119]:

U(r, t) = U +∑

U′,

U′ =∑N

k=1

(

UkeIωkt + U−keIω−kt) , (3.94)

where, the harmonic amplitudes Uk and U−k are complex conjugates. In-

troducing Eq. 3.94 into the discretization Eq. 3.32 and implementing a

time-averaging operator, which leading to a Reynolds averaging-like equa-

tion:

∂U

∂tV +

∑

faces

(FC ·n)∆S =∑

faces

(FV ·n)∆S + QV. (3.95)

61


The averaged convection and diffusion terms can be written in

FC =

ρu

ρuxu + px

ρuyu + py

ρuzu + pz

(ρE + p)ρu

+ DetC , (3.96)

and

FV =

0τx

τy

τz

τu + k∇T

+ DetV . (3.97)

While the non-linearity of the perturbation terms introduced two addi-

tional terms DetC and DetV , called deterministic stress, which are rep-

resented in:

DetC =

0

u′x(ρu)′

u′y(ρu)′

u′z(ρu)′

(ρE + p)(ρu)′

and DetV =

0000

τ ′u′

, (3.98)

In order to close the equation system, some additional equations are needed

to calculate the deterministic stresses. For each perturbation, a transport

equation can be obtained by subtracting Eq. 3.95 from Eq. 3.32. If only

retains the first order terms, the transport equation can be expressed in

∂U

∂tV +

∑

faces

(FC ·n)∆S =∑

faces

(FV ·n)∆S + Q′V. (3.99)

The convection term FC and the diffusion term FV are

FC =

ρu

uxρu + ρuux + px

uyρu + ρuuy + py

uzρu + ρuuz + pz

˜(ρE + p)ρu + ρu(ρE + p)

, (3.100)

62

3.7. UNCERTAINTY QUANTIFICATION IN CFD

and

FV =

0τx

τy

τz

τu + ˜k∇T

. (3.101)

Since only the firs order item is involved, Eq. 3.99 is a linearized equation

for all perturbations. For instance, two perturbations f ′ and g′ exist, whose

Fourier expansions are

f ′ =

N∑

k=1

f ′k (3.102)

and

g′ =

N∑

k=1

g′k (3.103)

Due to the orthogonality of Fourier expansion, the deterministic stress

generated by f ′ and g′ can be calculated by

f ′g′ =

N∑

k=1

f ′kg′k

=

N∑

k=1

1

2|f ′||g′| cosφfg, (3.104)

where |f ′| and |g′| are the amplitudes of the perturbations. φfg denotes the

difference of phase angle between f ′ and g′, which is related to the blade

pass frequency. By casting the linearized equation into the frequency do-

main, the harmonic perturbation equation is made space dependent only,

which can be solved using the normal finite volume method.

3.7 Uncertainty Quantification in CFD

In this section, the methodologies used in stochastic flow analysis are dis-

cussed, including the uncertainty quantification methods and the coupling

process with differential equations.

Several methods exist to quantify the uncertainties, which can be clas-

sified into two types. One is statistical method, such as Monte Carlo (MC)

method, sampling method, etc.; the other is non-statistical method, such as

63


perturbation method, Neumann series expansion method, sensitive anal-

ysis, etc. The former is simple and can be easily used for many fields.

But, usually, they are only used as the last resort since they are inefficient

and normally need lots of samples which consume great computational

resources. Perturbation and Neumann series expansion methods are lim-

ited to small perturbations and some simple cases. In recent years, some

efficient analysis methods based on spectral expansion were developed,

such as Polynomial Chaos (PC) method and Probabilistic Collocation (PRC)

method. Both these two methods are investigated in project of NODESIM-

CFD, while only the latter is employed in robust optimization due to the

limitation of time.

3.7.1 Intrusive Polynomial Chaos Method

The basic theory of the Intrusive Polynomial Chaos Method (INPC) was

proposed by Wiener [120] where it is named homogenous chaos. The pio-

neering development work of this method is done by Ghanem and Spanos

[121] in solid mechanics field. In recent years, the applications of this

method in fluid mechanics are expanded. Le Maitre et al. [122] simulated

the 2D incompressible channel flow and convection flow in a cavity with

uniform distributed viscosity coefficient. Xiu and Karniadakis [123] pro-

posed the generalized polynomial chaos (GPC) method for different kinds

of probability distribution function. They found that there exists an opti-

mal polynomial corresponding to the specific probability distribution func-

tion, which ensure the exponential convergence. An application of GPC

was performed in the simulation of a stochastic incompressible channel

flow and flow around cylinder [124]. Lacor and Smirnov [125] investigated

the influence of stochastic boundary on 1D supersonic nozzle flow and the

influence of stochastic of viscosity coefficient on 2D cavity flow through cou-

pling PC method with compressible Navier-Stokes equation. Wu [126] also

simulated the cavity flow and back facing step flow with stochastic velocity

boundary, but the ap-plication was limited to Stokes equation. Wang and

Kang [127] solved the stochastic Burgers equation using PC method, and

validated the results with analytical solution and results of Monte Carlo

simulation. The comparison results showed the efficient of PC method is

much higher than that of the MC method at the same accuracy level.

For a conventional deterministic problem, the flow parameters are

real-valued functions of time t and space x. The support domain of these

functions is denoted by D(x, t). If the flow problem has a stochastic process

with some parameters being random variables, then the flow parameters

will be functions of stochastic variables. Hence, these flow parameters are

64


also stochastic variables. To avoid confusion, the variables introducing un-

certainty into the flow fields are named random variables, which are usu-

ally regarded as input parameters of the stochastic flow model. Other flow

parameters, regarded as output variables of the flow model, are named

response variables.

The stochastic property of random variables and response variables

can be represented by introducing an extra probability dimension into the

flow model. Consider a probability space denoted by (Ω, F, P ), where Ωdenotes the sample space composed of basic outcomes, called the sample

space. F is the σ − algebra of Ω, and P is the probability measure defined

on the measurable space (Ω, F ). The basic event in Ω is denoted by θ which

doesn’t depend on the specific physical property. The random variables and

response variables are the events in Ω, which are the set of basic events.

A stochastic flow process is a square-integrable function defined on

the product space D × Ω since any random physical problem should be

the process with finite-energy. The function space mapping D × Ω to Ris denoted by Θ, each map Ω → R is a random variable. Then, Θ is a

Hilbert space with respect to L2 norm. Let ξ = ξ1(θ), ξ2(θ), . . . , ξn(θ) be a

set of orthogonal and uncorrelated random variables in Θ, and φk(ξ) be a

polynomial in Φ. Hp denotes the linear space composed of all polynomials

φk(ξ) with order less than p. Let Hp denote the closure of Hp in Θ, and Φp

be the set of all polynomials in Hp and be orthogonal to Hp−1. Then, the

space spanned by Φp, written in Hp, is named homogenous chaos of order p, which is a completed subspace of Θ. Φp is a orthogonal basis of Hp, called

polynomial chaos of order p. Then, any stochastic variable φ(x, t, θ) in Θcan be approximated with the following representation:

φ(x, t, θ) = φ0 +n∑

k=1

φk(x, t)Ψ1(ξk) +n∑

k=1

k∑

j=1

φkj(x, t)Ψ2(ξk, ξj)

+

n∑

k=1

k∑

j=1

j∑

l=1

φkjl(x, t)Ψ3(ξk, ξj , ξl) + · · · . (3.105)

Rewrite it compactly as:

φ =

NPC∑

k=1

φk(x, t)Ψk(ξ). (3.106)

Then the stochastic variable is divided into two parts, the deterministic co-

efficient φk and the stochastic polynomials Ψk(ξ). NPC is the total number

of polynomials in the expansions, which can be calculated by the following

65


formula:

NPC =(p + n)!

p!n!− 1, (3.107)

where, p is the highest order of polynomials, n is the number of random

variables. The cumulative distribution function or the probability density

function of random variables is settled by the physical problem, therefore

the coefficients of polynomials are known. For response variables, these

coefficients are unknowns to be solved. The orthogonal property of polyno-

mial in Eq. 3.106 is defined by inner product with respect to a weighting

function ω(ξ1, . . . , ξn):

〈Ψi, Ψj〉 =

∫

Ψi(ξ1, . . . , ξn)Ψj(ξ1, . . . , ξn)w(ξ1, . . . , ξn)dξ1 . . . dξn , δij .

(3.108)

Selection of the polynomials in Eq. 3.106 depends on the probability den-

sity function of random variable. According to the Askey principle, an

optimal polynomial chaos exists which is corresponding to the distribution

function of random variable. Here, “optimal” means that the polynomial

chaos will keep the exponential convergence rate to the uncertain vari-

ables to be approximated if the weighting function is same as the probabil-

ity density function [123]. For random variable of Gaussian distribution,

the probability density function is given by

fξ(ξ) =1√2π

e−ξ2

2 . (3.109)

Thus, the optimal polynomial is Hermite polynomials with a weighting

function of

ω(ξ1, . . . , ξn) =1√2π

e−ξ21+...+ξ2

n2 (3.110)

The one dimensional Hermite polynomials are given as follows:

Ψ0 = 1,

Ψ1 = ξ, (3.111)

Ψp+1 = 2ξΨp(ξ) − 2pΨp−1(ξ), p = 1, 2, . . . .

To simplify the description, the principle of IPCM is emphasized for a

general stochastic differential equation:

L(a(θ))φ(x, t, θ) = S(x, t), (3.112)

where, L(a(θ)) is a differential operator which contains space and time

differentiation and can be nonlinear and depends on a random parameter

a(θ), S(x, t) is a space and time dependent source term.

66


Substituting the polynomial chaos Expansion 3.106 into the differen-

tial Equation 3.112 results in

L(a(θ))

NP C∑

k=1

φk(x, t)Ψk(ξ) ≈ S(x, t). (3.113)

A Galerkin projection on each basis Ψk(ξ) is applied which leads to

〈L(a(θ))

NP C∑

k=1

φk(x, t)Ψk(ξ), Ψj〉 = 〈S, Ψj〉, j = 0, 1, . . . , NPC , (3.114)

where, 〈 · , · 〉 denotes inner product of Eq. 3.108. Eq. 3.114 gives a system

of (NPC + 1) coupled deterministic differential equations with unknowns

φk(x, t). Solving this set of equations by FVM or other method can ob-

tain the deterministic coefficients in Expansion 3.106. Then the statistical

properties of the stochastic variable φ, such as the mean value, deviation

and other high order moments, can be easily obtained using the following

formula:

φ = φ0, (3.115)

σ2 =

NPC∑

k=1

φ2k〈Ψ2

k〉. (3.116)

The details about the derivation of Eq. 3.116 can be found in [128] or

[129]. The coupling process of IPCM and N-S equations also can be found

in these two references, which are omitted here. Note that, the existing

deterministic solver has to be modified since the governing equations are

replaced by the PC expansion equations.

3.7.2 Non-intrusive Polynomial Chaos Method

To avoiding the modification of the existing deterministic solver which

may introduce the potential risk of errors, the polynomial chaos method

can be coupled in a non-intrusive way which results in the Non-Intrusive

Polynomial Chaos Method, (NIPOCM). The idea of the non-intrusive ap-

proaches is to estimate the coefficients in Expansion 3.106 based on few

deterministic solutions. Then, a pre-process has to be implemented for

distribution of deterministic solutions. While, it can be regarded as the

coupling of polynomial chaos method and traditional sampling methods.

Walter [130] and Hosder et al. [131] choose (NPC + 1) vectors of ξi =(ξ1(θ), ξ2(θ), . . . , ξn(θ)i) , (i = 0, 1, 2, . . . , NPC) in random space as samples

67


for a PC of order NPC , while Reagan et al. [132] use Latin Hypercube

Sampling to generate the samples. For each sample, a deterministic cal-

culation is performed. The the polynomial coefficients of Expansion 3.106

are obtained by solving the following linear system:

Ψ0(ξ0) Ψ1(ξ0) . . . ΨNPC(ξ0)

Ψ0(ξ1) Ψ1(ξ1) . . . ΨNPC(ξ1)

......

. . ....

Ψ0(ξn) Ψ1(ξn) . . . ΨNPC(ξn)

φ0(x, t)φ1(x, t)

...

φNP C(x, t)

=

φ(x, t, ξ0)φ(x, t, ξ1)

...

φ(x, t, ξn)

.

(3.117)

However, the coefficients obtained yields approximation of the coefficients

in Expansion 3.106 which is different to that of intrusive polynomial chaos

method where the coefficients are exact. Onorato et al. [133] simulated

the uncertainty bypass flow around the RAE2822 airfoil using the IPCM

and NIPCM. The compared results shows that the mean values obtained

by these two methods are the same, while large difference exists in the

deviation. However, since a structured mesh and an unstructured mesh

are employed in the IPCM and NIPCM, respectively, the results still need

to be verified in following work.

3.7.3 Non-intrusive Probabilistic Collocation Method

The Non-intrusive Probabilistic Collocation Method (NIPRCM) is similar

to the NIPOCM in basis of few deterministic solutions. However, the sam-

pling method is replaced by points collocation method. The NIPRCM de-

veloped by Loeven et al. [134] is followed in the preset paper, where the

Lagrange Interpolating Polynomials are employed to construct the poly-

nomials chaos of stochastic variable. The NIPRCM expansion of variable

φ(x, t, θ) is represented in:

φ =

NP∑

k=1

φk(x, t)hk(ξ), (3.118)

where φk(x, t) is the value of variable φ(x, t, θ) at the kth collocation point

and NP is the number of collocation points. Note that, the collocation oper-

ator is implemented on the stochastic space of θ. One has the relationship

between ξ and θ as below:

fξ(ξ)dξ = fθ(θ(θ)) = dθ, (3.119)

68


where, fξ and fθ denote the Probability Density Function (PDF), and fθ = 1since θ is distributed uniformly in [0, 1]. After integration, one has

Fξ(ξ(θ)) = θ, (3.120)

where, Fξ is the Cumulative Distribution Function (CDF). The colloca-

tion points θi, (i = 1, 2, . . . , NP ) are chosen corresponding to the Guassian

quadrature points used to integrate the variable φ(x, t, θ) in stochastic

space of θ. Many algorithms exist to compute the Gaussian quadrature

points, while a powerful algorithm of Golub-Welsch algorithm is employed

in NIPRCM with details can be found in [134].

Then, in Expansion 3.118, φk(x, t) is the value of φ(x, t, θ) at the col-

location point θk, and hk denotes the Lagrange interpolation polynomial

chaos corresponding to the same point. The Lagrange interpolating poly-

nomial chaos is the polynomial chaos hk(ξ(θ)) of order NP − 1 that passes

through the NP collocation points, which is given by

hk((ξ(θ)) =

Np∏

i=1,i6=k

ξ(θ) − ξ(θi)

ξ(θk) − ξ(θi), (3.121)

with hk(ξ(θi)) = δki.

Submitting the NIPRCM Expansion 3.118 into into the differential

Equation 3.112 and applying a Galerkin projection on each collocation

point result in

〈L(a(θ))

NP∑

k=1

φk(x, t)hk, hi〉 = 〈S, hi〉, i = 1, . . . , NP . (3.122)

Then, a system of NP uncoupled deterministic equations is built, which can

be solved with existing deterministic solver. After getting the coefficients of

the NIPRCM expansion, the mean value and variance of random variable

φ can be calculated by:

φ =

NP∑

k=1

wkφk(x, t) (3.123)

σ2 =

NP∑

k=1

wkφ2k(x, t) − φ

2. (3.124)

Dinescu et al. [135] compared the NIPRCM and the IPCM by testing case

of rotor 37. The comparison shows a good agreement between the results

of these two methods.

69


70

Chapter 4

Numerical Simulations on

Film Cooling

As mentioned in Chapter 2, the localized cooling flow nearby a cooling hole

is a typical jets-in-crossflow (JICF) problem. The flow structure nearby the

cooling hole has great impact on the cooling effect. Within this chapter, a

simplified model of a flat plate with a single cooling hole is employed for the

investigation of the flow structure nearby the cooling hole. Based on the

validation with experimental results, the discussion mainly focus on the

effect of blowing ratio on the flow structure. Different from the local view

of flat plate case, the interactions between the cooling flow and the main

flow are investigated in a practical planar cascade with different cooling

hole geometry at the leading edge. The effect of configuration of cooling

hole and the additional losses to the main flow will be discussed in detail.

4.1 Jets in Crossflow on A Flat Plate

4.1.1 Review of Experiment

The numerical simulations performed in this section are based on the ex-

periment implemented by Ajersch et al. [64] in a low speed wind tunnel.

A row of six rectangular jets injected at 90 was placed on a flat plate with

width of 406 mm. The square cooling holes with diameter D of 12.7 mm are

distributed with 3D interval between the hole centers. To ensure a fully

71

CHAPTER 4. NUMERICAL SIMULATIONS ON FILM COOLING

turbulent boundary layer be present in the test section, a 2.4 mm rob was

affixed to the tunnel floor at the test section entry. A plenum is located

under the flat plate to supply the cooling air, whose inlet has a distance of

500 mm to the flat plate. The dimensions of the test section configuration

are shown in Figure 4.1.

Figure 4.1: Test section geometry of JICF (Ajersch, 1997 [64]).

The flow structure nearby the cooling hole is mainly affected by the

blowing ratio, Reynolds number of jets, the configuration of the cooling

hole, the curvature of the plate etc. Whereas, for the specific geometry of

square cooling hole on the flat plate and the inlet condition of jets, the flow

field mainly characterized by the blowing ratio which is usually defined as

the momentum ratio J :

J =ρjetV

2jet

ρcfV 2cf

, (4.1)

where ρ and V denote the density and velocity, respectively. The subscript

“jet” and “cf” stand for the cooling jets and the crossflow, respectively. In

the experiment, the same fluid with the same inlet temperature is used

for both the main flow and the cooling jets, then J can be simplified into

the velocity ratio Vjet/Vcf , denoted by M . Three cases with different Mof 0.5, 1.0 and 1.5 were measured in this experiment. The velocity of the

cooling jets was remained as 5.5 m/s, while the velocity of the cross flow

were 11 m/s, 5.5 m/s and 3.67 m/s corresponding to three blowing ratios,

respectively. The Reynolds number Re based on the hole diameter and

the velocity of cooling jets is approximately 4700. The flow field for this

experiment was measured by Laser Doppler Velocimetry (LDV).

72

4.1. JETS IN CROSSFLOW ON A FLAT PLATE

4.1.2 Numerical Model

In numerical simulations, a simplified model with a single hole is used.

Therefore, the width of computational domain is set to 3D in Y direction

with periodic boundaries on both sides. The X direction is consistent with

the streamwise of the main flow. The inlet of the main flow is imposed at

10D upstream the cooling hole, where a velocity profile with the boundary

layer thickness of 2D is predicted using an approximating exponential law:

Vz

Vcf=(z

δ

)1/7

. (4.2)

The outlet of computation domain is located at 20D downstream to the

exit of the cooling hole, where a uniform static pressure of 101325Pa is

settled. The inlet of jets is set to 5D beneath the flat plate, where a uniform

velocity of 5.5m/s is imposed. An external condition is settled to the upper

boundary which is 15D above the flat plate, where the velocity is the same

as the main flow velocity. Figure 4.2 illustrates the whole computational

domain.

Figure 4.2: Computational domain of JICF.

A “H” type structured mesh with 2.23 million cells is built by IGG—a

mesh generator of the software package FineTM/Turbo, as shown in Figure

4.3. The distribution of nodes above the flat plate is 257 × 65 × 121 (X,

Y, Z), and the distribution of nodes in the cooling hole is 45 × 45 × 49.

The mesh close to solid walls and the exit of the cooling hole is refined to

ensure the Y+ near the solid wall is lower than 1. The blowing ratio in

73


simulations is from 0.5 to 1.0 with an interval of 0.1. RANS simulation

is the main analytical method, in which the one equation SA turbulence

model is employed. URANS and DES simulations are also performed to

represent the unsteady properties of the mixing process only for M=0.5

and M=1.0, in which the physical time step is set to 0.0001 s and 0.0002

for two blowing ratios, respectively. During each physical time step, 100

pseudo time steps are implemented to ensure the global residual reduce by

at least 4 orders.

Figure 4.3: Mesh for simulations on JICF.

For steady simulations, the converged results can be obtained usually

after 1000 steps. Using 5 parallel AMD Opteron 8387 processors (2.9GHz

512kB cache/core), around 2.2 hours CPU time is needed. Figure 4.4 shows

the convergence history of steady simulations for different blowing ratios.

As can be seen, the convergence is quite good with residuals below -6. The

convergence history of the DES simulation is shown in Figure 4.5. The

left figure shows the residual curve for the first 1600 physical steps. Note

that, the first 1200 steps belongs to the steady computation whose result is

taken as the initial solution for the unsteady simulation which starts from

the 1200th step. The right figure shows the residual curve from physical

step of 1600 to 2000. Note that, this computation is the continued compu-

tation based on the first 1600 steps. So the value of residual starts with

the residual of the previous computation. As can be seen, the residual fluc-

tuates with a high frequency below -4. However, the amplitude of the fluc-

tuation is small, and the mean value is kept around -4.2 in the left figure.

In the right figure, the mean value is around -1.45. Using 5 AMD Opteron

8387 processors (2.9GHz 512kB cache/core), around 116 hours CPU time

is needed. The convergence of URANS simulation is similar to that of the

DES simulation, which will not be presented here. Noth that, using the

same time step setting and computational power, the CPU time costed by

74


the URANS simulation is about 10% less than that of the DES simulation,

which is about 101 hours for 2000 physical steps.

(a) M=0.5 (b) M=1.0

Figure 4.4: Residual convergence curves of steady simulations.

(a) Iterations of 0∼1600 (The computation

starts from

(b) Iterations of 1600∼2000

Figure 4.5: Residual convergence curves of DES simulations.

4.1.3 Results of Steady Simulations

In this subsection, the calculated results of steady simulations are pre-

sented. To explain in convenience, some items are noted firstly and which

will be followed afterward. The inlet boundary layer and cooling bound-

ary layer denote the boundary layer of the main flow and the boundary

layer developed in the cooling hole, respectively; the cooling jets denotes

the flows issue from the cooling hole, including the cooling flow and the

cooling boundary layer.

75


In Figure 4.6, profiles of the velocity component in X direction, Vx,

on the central plane (Y=0) are illustrated. The simulated results labeled

with “CFD” are depicted by lines, while the experimental data labeled with

“EXP” are denoted by discrete dots. Figures 4.6(a)∼4.6(c) show the com-

parisons between the simulated results and the experimental data at three

downstream sections with distances of 1D, 3D and 5D to the exit of the

cooling hole. The simulated results from Ajersch et al. [64] (RANS, k − εmodel), Tyagi and Acharya [66] (LES ) are also presented here. The veloc-

ity and coordinate Y are non-dimensionalized by the inlet velocity of jets

and the hole diameter D. As can be seen, at the section of X = 1D, the

simulated result has an excellent agreement with the experimental data,

which shows a triple peaked profile. The first peak is close to the flat plate,

which is corresponding to the part of the main flow enters into the region

beneath the jets. However, due to the measurement limitation, the velocity

information adjacent to the flat plate is missed. The second peak is located

in the main flow region where Vx almost is constant. A third small peak

is observed in the mixing region, and the position is close to the main flow

region where the jests have deflected totally streamwisely under the press

of the main flow. Between the first peak and the third peak, there exists a

sharp velocity gradient. A large recirculation region is found under 0.5D,

which is due to the blocking effect of the cooling jets on the main flow.

Tyagi’s results show a double peaked profile with the third peak missed.

While, the first and the third peaks are absent in Acharya’s results. The re-

circulation regions are much smaller for all blowing ratios. Moreover, both

Tyagi and Acharya’s results are a little overpredicted in the upper mixing

region. Note that, Acharya only used about 20 thousands cells for a larger

computational domain of 51D×3D×30D (X, Y, Z). So the mesh density is

much coarser than that in present thesis, which could be a large influence

factor. In Tyagi’s simulations, they also used about 20 thousands cells for a

smaller computational domain of 12D×3D×5D. However, the mesh is still

quite coarse for LES simulations, even for RANS simulations. And in his

simulation, the channel of cooling hole is not included in the computational

domain. At the exit of the cooling hole, a specified velocity is settled. While,

this boundary is not fully correctly settled since the velocity profile is not

always the same to the specified one which will be presented in following

sections. So, it also could be a error source of Tyagi’s simulations.

At sections at further downstream, X = 3D and 5D, good agreements

between the simulated results and the experimental data are presented in

the main flow region. In the shear flow region, the simulated results are

also satisfactory. In Figure 4.6(b), a double peaked profile is shown at the

section of X = 3D, while the third peak in Figure 4.6(a) is absent here

76


and the velocity gradient in the mixing region is smaller. The position of

the firs peak is higher than that at the section of X = 1D, which indicates

the blocking effect of cooling jets becomes weak with the increase of the

distance to the cooling hole. The recirculation region close to the flat plate

is also vanished. The profile at the section of X = 5D shown in Figure

4.6(c) is similar. Tyagi’s results also capture the double peaked profiles,

while the overpredicted problems at about Y = 1D are still obvious. The

Ajersch’s results failed to predict the peak in the mixing region again and

have big differences to the experimental data. The possible reasons have

been discussed above.

(a) X=1D (M=0.5) (b) X=3D (M=0.5)

(c) X=5D (M=0.5) (d) M=0.5, 0.7, 0.8 and 1.0 (X=1D)

Figure 4.6: Profile of Vx at the central plane (Y=0).

Figure 4.6(d) shows the profile of Vx at the section X = 1D with blow

ratio for 0.5 to 1.0. The third peak for M=0.7, 0.8 and 1.0 are found locating

in the range from Z = 1.0D to Z = 1.5D. For M=1.0, a fourth inconspicu-

77


ous peak appears at about Z = 0.5D, which indicates the flow structure be-

comes more complicated when the blowing ratio is large. The recirculation

region increases with the increase of the blowing ratio. While, when the

blowing ratio is over 0.8, the recirculation region starts to decrease again.

The reason can be found in Figure 4.7, in which the contour patterns of Vx

at the central plane for M=0.5 and 1.0 are shown. As can be seen, a recir-

culation region exits behind the jets flow. With the increase of the blowing

ratio, the impact of the jets to the main flow increases, then the recir-

culation region is stretched spanwisely and shranked streamwisely. The

curved line of jets flow for M=1.0 is obvious higher than that for M=0.5.

In particular, for M=1.0, there is a high Vx region just locating above the

curved line, which is corresponding to the third peak on the Vx profile in

Figure 4.6(d).

(a) M=0.5 (b) M=1.0

Figure 4.7: Contour pattern of Vx at the central plane.

Figure 4.8 shows the contour patterns of the normalized vertical ve-

locity Vz at the outlet of the cooling hole. Symmetric patterns respecting

to the central plane are shown with steep velocity gradients for all blow-

ing ratios. A small recirculation region is found close to the hole upstream

edge for all blowing ratios, where a part of crossflow enters into the cooling

hole because of the deflection of jets. With the increase of the blowing ratio,

the jet strength also increases which leads to the increase of the gradient

of Vz. Thus, the recirculation region is decreased. It is easy to understand,

when the jets are getting strong, for instance M=1.0, the jets in the cooling

hole are less affected by the main flow. On the contrary, for M=0.5, the jets

in cooling hole are weak, which will be easily to be affected.

Figure 4.9 shows the limiting streamline patterns on the flat plate un-

der different blowing ratios. The flow patterns of M=0.6 and M=0.9 are

omitted here since they are between the patterns of M=0.5, 0.7 and the

patterns of M=0.8, 1.0, respectively. As can be seen, the flow patterns can

78


Figure 4.8: Contour pattern of Vz at the outlet of the cooling hole

(a M=0.5; b M=0.7; c M=0.8; d M=1.0).

be separated into two distinct regions: the region before the downstream

exiting edge of cooling hole and the region after this edge. In the first re-

gion, the flow patterns with different blowing ratios are quite the same.

All of them show a typical bypass flow pattern which is symmetric to the

central plane. The crossflow denoted by the blue lines is blocked by the

cooling jets denoted by the red lines resulting in a stagnation region and

the horseshoe vortex. With the increase of the blowing ratio, the stagna-

tion point and horseshoe vortex is shifted upstream. However, the flow

patterns in second region of M=0.5 and 0.7 are completely different to that

of M=0.8 and 1.0. The patterns of M=0.5 and 0.7 are still symmetric to

the central plane, while the patterns of last two blowing ratios are oblique

to one side of the central plane. There exists a critical blowing ratio. For

the current specific configuration of cooling hole, it is 0.7. When the blow-

ing ratio is equal to or below 0.7, the flow structure after the cooling hole is

symmetric to the central plane. For instance when M=0.5, as shown in Fig-

ure 4.9(a), the cooling jets are bent streamwisely in a short distance after

issuing out of the cooling hole, and attach to the surface of the flat plate at

far downstream. Due to the friction between the main flow and the cooling

jets, a dominant vortex system, CVP, is developed at downstream of the

cooling hole, which pushes the legs of the horseshoe vortex to the sides of

the domain. With the increase of the blowing ratio, the penetrability of the

79


jets increases. The CVP is lifted from the flat plate, which results in a low

pressure region below the jets. The legs of the horseshoe vortex enter this

region and flow towards to the center plane. While the stability of the flow

structure decrease due to the crash of the horseshoe vortex legs. When the

blowing ratio rises to 0.8, a asymmetric flow structure is presented with

the CVP deviating to one side of the central plane. And it is seen that the

deviation of M=1.0 is larger than that of M=0.8.

Figure 4.10 shows a local view of the limiting streamline pattern close

to the cooling hole and the corresponding topological structure of M=0.5

and 1.0. For M=0.5, there exist three saddle nodes, S1, S2 and S3; one de-

generation nodes, N1; two spiral nodes, N2 and N3; four separation lines

Ls1∼Ls4 and one attachment line, Lr1, which are shown in Figure 4.10(a).

The saddle node S1 is related to the horseshoe vortex which locates just

at front of the cooling hole. Two separation lines Ls1 and Ls2 which in-

dicate the traces of two legs of the horseshoe vortex separated from two

sides of the saddle node S1. The saddle node S2 locates at downstream

of the exit of the cooling hole. Between these two saddle nodes, there is

a attachment node N1 which is connected with two spiral nodes N2 and

N3 located at the downstream of S2. The generation of these nodes can

be explained by analogizing the rigid bypass flow around a cylinder. The

pair of spiral nodes, N2 and N3, is generated by shear actions between

the crossflow and the cooling jets, which are the early origin of the CVP.

On the other side of the spiral nodes pair, there is the third saddle node

S3. Two separation lines Ls3 and Ls4 are derived from this saddle node.

Between these two separation lines, a reattachment line Lr1 is observed

exactly locating at the central plane. These three lines indicate the traces

of the CVP and second pair of counter rotating vortices, called secondary

CVP, with an opposite rotation sense to that of the main CVP. This limiting

streamlines pattern is quite similar to the pattern presented in Kang [63]

where the JICF nearby a circular hole on a flat plate was simulated. The

upstream half part of the limiting streamlines pattern of M=1.0, shown in

Figure 4.10(b), is consistent with that of M=0.5. The positions of S1 and

N1 are moved upstream due to the increase of the blocking effect of jets.

However, the downstream half part of the patter of M=1.0 is quite differ-

ent to that of M=0.5, where the pair of spiral nodes N2 and N3 in Figure

4.10(a) degenerates into a single node N2 which deviates obviously apart

the central plane. The same deviation also happens to the saddle node S3

and the attachment line Lr1. Different to the case of M=0.5, two legs of

the horseshoe vortex are related to the node N2 and two separation lines

Ls3 and Ls4 which indicates the legs of horseshoe vortex are warped into

the CVP when M=1.0.

80


(a) M=0.5

(b) M=0.7

(c) M=0.8

(d) M=1.0

Figure 4.9: Limiting streamline pattern on the flat plate. The limiting streamlines

in blue show the trace of the horseshoe vortex. The red lines show the trace of the

CVP.

81


(a) M=0.5 (b) M=1.0

Figure 4.10: Detailed view of Limiting streamline pattern on flat plate. N and S

denote the node and the saddle node, respectively. Ls and Lr denote the separation

line and reattached line, respectively. The description of lines is same to that in

Figure 4.9.

Figure 4.11 shows the position differences of the horseshoe vortex legs

clearly through a top view and a lateral view of 3D spatial streamlines un-

der different blowing ratios, where the horseshoe vortex and the CVP are

denoted by the blue lines and the pink lines separately. As can be seen,

when M is small (M=0.5) the blue lines bypass the cooling hole and flow

to downstream straightly from two sides of the dimain. They always af-

fix to the flat plate. However, when M is increased to 1.0, the blue lines

are wrapped up by the pink lines at about 1D downstream the cooling

hole. This difference can be explained by analogizing the flow over a slen-

der body with high attack angles. When the attack angle is small, a pair

of symmetric stable vortices exists behind the slender body. With the in-

crease of the attack angle, the stability of the vortex pair is weakened and

become quite sensitive to small disturbances, however, which always ex-

ist in practical problems. When the attack angle is larger than a critical

value, the vortices will generate an unstable shedding which changes the

topology of the flow structure, then a new stable asymmetric vortex struc-

ture is formed [136]. The flow property of JICF is similar to the bypass

flow around a slender body. The most influential factor is the blowing ra-

tio, whose effect can be analogized to the attack angle. With the increase of

the blowing ratio, the blocking effect of cooling jets is strong, which leads

to a large curve angles of the jets. When the blowing ratio reaches 0.8,

the vortex system becomes unstable, which results in a significant change

under a small disturbance. That is called bifurcation of a dynamical sys-

82


tem. Note that, the bias of the CVP is not necessary to the certain side

of the central plane, which depends on the property and the value of the

small disturbance. The disturbance can be the tolerance of geometry or

the nonuniform of the inlet flow. However, in numerical simulations, the

disturbance usually comes from the numerical errors of the mesh or algo-

rithms. Such kinds of small tiny error would little impact on the stable

dynamical system, for instance the JICF with low blowing ratios. While,

for the dynamical system in a critical state, a tiny error could lead to sig-

nificant changes, for instance the JICF with high blowing ratios. It has to

be mentioned that the analysis of stability of dynamical system is beyond

the scope of the present thesis, which will be investigated in future work.

(a) M=0.5 (b) M=1.0

Figure 4.11: Spatial streamline pattern of Horseshoe vortex. The blue lines depict

the horseshoe vortex and the pink lines issued from the cooling hole depict the

CVP.

Figure 4.12 shows the 3D view of the spatial streamline pattern nearby

the cooling hole for M=0.5 and 1.0. A symmetric vortex system of JICF at a

small blowing ratio (M=0.5), including the horseshoe vortex and the CVP,

is displayed in the left Figure 4.12(a). The horseshoe vortex denoted by

the blue streamlines locates at just front of the cooling hole, which is gen-

erated by the inlet boundary layer. It is separated into two legs by the

cooling jets, and flows to downstream straightly along two sides of the do-

main. Adjacent to the exit of cooling hole, there are two stagnation regions

generated by a pair of vortices, denoted by the red spatial lines, which are

related to nodes N2 and N3 in Figure 4.10(a) and considered as the early

origin of the CVP. This pair of vertex is mainly composed of the cooling

boundary layer which is rolled up vertically firstly under the shear action

83


of the inlet main flow, and then deflected under the press of the main flow.

Finally, they are wrapped into the CVP. Another reason for the generation

of the CVP comes from the interaction between the main flow boundary

layer and the cooling flow. The jets flow is inclined by the press of the main

flow, which results in a low speed region within the cooling hole. Note

that, this region is outside of the cooling boundary layer, which also can

be seen in Figure 4.8. Therefore, a part of inlet boundary layer enters in

this region. Under the impaction of the undisturbed cooling flow, this part

of flow are pushed out from two sides of the cooling hole, generating the

pair of vortices denoted by the pink lines. This pair of vortices and the red

vortex pair warp with each other, finally they develop into the dominated

CVP. Different to the symmetric flow pattern of M=0.5, the flow structure

of M=1.0, shown in Figure 4.12(b), present two distinct parts again, a sym-

metric 3D streamline pattern before the cooling hole and an asymmetric

3D streamline pattern after the cooling hole. The horseshoe vortex before

the cooling hole is still symmetric. However, its legs are wrapped into the

CVP rather than flowing downstream straightly as in Figure 4.12(a). The

pink vortex pair is generated by the inlet boundary layer, and forms one

part of the CVP. The cooling boundary layer only forms a single vortex

denoted by the red lines behind the exit of cooling hole, and it warps the

two legs of the horseshoe vortex denoted by the blue lines and forms into

another part of the CVP.

(a) M=0.5 (b) M=1.0

Figure 4.12: Spatial streamline pattern of vortices system nearby the cooling hole.

The red spatial lines issue from nodes N2 and N3, while one part of the pink spatial

lines issue from the cooling hole, another part of that come from the inlet boundary

layer. The blue limiting streamlines depict the traces of the horseshoe vortex, while

the red limiting streamlines issue from the cooling hole.

A detailed view and a top view of the 3D spatial streamline patterns

are shown in Figure 4.13 and Figure 4.14 respectively. It also can be ob-

84


served that the contribution of the inlet boundary layer to the CVP will

increase with the increase of the blowing ratio. The reason can be easily

get from Figure 4.15, in which the streamlines on the central plane of the

horseshoe vortex are presented. It is seen that the horseshoe vortex of

M=1.0 is much stronger than that of M=0.5.

(a) M=0.5 (b) M=1.0

Figure 4.13: Detailed view of vortices system. The description of the lines is the

same as that in Figure 4.12.

Figure 4.16 shows the surface streamline patterns and isoline con-

tours of Vx at three sections of X = 1D, 3D and 5D. As can be seen, for

M=0.5, a symmetric streamline pattern is presented at each section. At the

section close to the exit of cooling hole, X = 1D, two pairs of counter rotat-

ing vortices locate symmetrically in respect to the central plane. The inner

vortex pair are corresponding to the red stagnation vortex pair shown in

Figure 4.12(a). The outer vortex pair has the same scale of the inner vor-

tex pair, which is corresponding to the pink vortex pair in Figure 4.12(a).

These two vortex pairs merge together at far downstream to develop into

the dominant CVP which is displayed in the streamline pattern at the sec-

tion of X = 3D where the legs of the horseshoe vortex are observed at

the corners of the domain. A third pair of small vortices is observed locat-

ing beneath the CVP, whose existence has been inferred from the previous

topological analysis. This pair of vortices is called the secondary CVP to be-

ing distinguished from the primary (dominant) CVP. The secondary CVP

is firstly observed by Andreopoulos and Rodi [137] in their experiments.

Then it is often reported and identified in later literatures, while its origin

is still under arguments. Morton and Ibbeston [138] speculated them to

be the legs of the horseshoe vortex. Yuan et al. [62] clarified this pair of

vortex is not the horseshoe vortex with LES simulation where it is named

wall wake vortex. However, no clear explains for the origin are presented

85


(a) M=0.5

(b) M=1.0

Figure 4.14: Top view of Spatial streamline pattern of CVP. The description of the

lines is the same as that in Figure 4.12.

(a) M=0.5 (b) M=1.0

Figure 4.15: Velocity vectors and streamline patterns of the horseshoe vortex at

the central plane.

86


yet. Hale et al. [61] argued that the secondary CVP is generated by the

crossflow fluid sweeping around and under jet in their experiment. In the

present thesis, it is revealed further that the secondary CVP is certainly

generated by sweeping of the inlet boundary layer, as shown in Figure 4.17

which will be explained later. The streamline patterns at X = 5D is quite

similar to that at X = 3D, while the scale of the CVP increases. Being con-

sistent with the previous results, the streamline patterns of M=1.0 shown

in Figure 4.12(b) are also asymmetric. Since the legs of the horseshoe vor-

tex are warped into the CVP, which has been shown in Figure 4.12, no

vortex is observed in the corner. At the section of X = 1D, the scale of the

outer pair of counter rotating vortex is larger than that of M=0.5. While

the left vortex of the inner vortex pair is stretched to close to the flat plate.

At the downstream sections, X = 3D and 5D, the inner pair of vortex has

been merged with the outer pair of vortex to develop to the primary CVP.

A asymmetric secondary CVP is induced by the primary CVP and located

in a small region beneath the primary one. The scale of the secondary CVP

is much larger than that of M=0.5.

(a) M=0.5 (X=1D, 3D, 5D, from left to right)

(b) M=1.0 (X=1D, 3D, 5D, from left to right)

Figure 4.16: Streamline patterns (black lines) of vortices system and isolines (col-

ored lines) of Vx at streamwise sections of 1D, 3D and 5D.

Figure 4.17 illustrates the spatial streamlines of the vortex system

which reveals the development of the secondary CVP clearly. In Figure

4.17(a), the secondary CVP is depicted by the green streamlines, which

87


is developed from the upstream inlet boundary layer locating above the

horseshoe vortex. A part of inlet boundary layer bypasses the cooling hole

and sweeps to the central plane. Under the shear action of the primary

CVP denoted by the pink lines, this part of inlet boundary layer curls up

and develops into the secondary CVP. It is clearly shown that the sec-

ondary CVP is different to the horseshoe vortex which is denoted by the

blue streamlines. While, in Figure 4.17(b) which shows the related stream-

lines of M=1.0, the secondary CVP is mainly composed of the inlet bound-

ary layer comes only from one side.

(a) M=0.5 (b) M=1.0

Figure 4.17: Spatial streamline pattern of the secondary CVP. The blue lines show

the horseshoe vortex. The pink lines and green lines show the primary CVP and

the secondary CVP, respectively. The black streamline lines are located at the

streamwise section of X=3D.

4.1.4 Results of Unsteady Simulations

In the previous subsection, the calculated results of steady simulations

have been presented and discussed in detail. In this subsection, the un-

steady simulations results are presented. The comparisons of the profile

of Vx among the experimental data, the steady simulated results and the

time averaged results of the unsteady simulations are shown in Figure

4.18. The averaging is performed over 100 physical time steps. In these

figures, the RANS results and that of time averaged URANS results are

almost exactly the same. The time averaged DES results gives a better

prediction in the shear layer region than other simulations at X = 3Dsection.

Figure 4.19 show the time averaged contour pattern and isolines of

the velocity adjacent to the flat plate (1.27mm, 1/10 of the hole diame-

ter). The steady simulated results are omitted here since they are exactly

88


(a) X=1D (b) X=3D (c) X=5D

Figure 4.18: Profile of the time averaged Vx at the central plane.

the same as the time averaged results of URANS simulations. From the

comparison of DES simulated results and URANS simulated results, it is

seen that the former provides much more details of the turbulent property

than the latter. For M=0.5, the time averaged velocity of the DES simula-

tion is approximately symmetric to the central plane. Note that, the small

asymmetric error is caused by the limitation of the averaging time range.

Increasing the total number of physical time steps averaged, the result is

getting more symmetric, as shown in Figure 4.20(a). Two low speed regions

are located behinds the exit of cooling hole, which are associated to the two

stagnation vorticists in Figure 4.12(a). These two regions joint together at

about 1D distance downstream to the hole exit. A low speed region at the

central plane spreads to the outlet without mixing with the high speed

flow at two sides. At the two sides of the outer bounds of the flat plate, two

low speed regions are generated by the two legs of the horseshoe vortex.

The time averaged result of URANS simulations shows a symmetric flow

pattern, where the velocity shows a uniform distribution of velocity due to

the absence of turbulence information. A low speed region is present at the

central plane with a high velocity region locating at each side and contin-

ued to the outlet. For M=1.0, both the time averaged velocity of both the

DES simulation and the URANS simulation show asymmetric patterns

again, although the former is more nonuniform distributed. The middle

low speed region mixed with the high speed regions at two sides resulting

in a interval distributed contour pattern. The mixing flow region diffuses

toward two edges of the flat plate along with flowing downstream. How-

ever, close to the two edges, no low speed regions exist due to the absence of

the horseshoe vortex legs. Moreover, the velocity gradient of M=1.0 nearby

the cooling hole is visibly larger than that of M=0.5. Figure 4.20 shows the

time averaged contour pattern and isolines of the velocity in 400 physical

time steps range. It is found the difference between Figure 4.20 and Figure

89


4.19 is quite small, which means the time range used for averaging is long

enough. Since there is no changes between the URANS results averaged

in 100 physical time steps and in 400 physical time steps, the patterns for

the latter will not be presented here.

The instantaneous velocity contour patterns of DES and URANS sim-

ulations at the same position of Z=1.27mm are presented in Figures 4.21∼4.24.

For both M=0.5 and M=1.0, five patterns are presented with an interval of

0.004 s (20 physical time steps for M=0.5, 10 physical time steps for M=1.0),

where the time of the first pattern is set to 0. In Figures 4.22 and 4.24, the

results of URANS show little difference in different time indices, which

indicates the vortex systems for both M=0.5 and M=1.0 are stable and the

unsteady property of the flow mainly comes from the turbulence. However,

in Figure 4.21 and Figure 4.23, the DES results show the processes of vor-

tex shedding under different blowing ratios clearly. For M=0.5, two rows

of vortex on two sides of the central plane shed almost instantaneously,

generating two rows of neatly arranged vortex street. While, for M=1.0,

two rows of vortex shed alternatively from the exit of the cooling hole, and

break into many small scales vorticities in a short distance.

Figures 4.25∼4.28 show the spatial views of the instantaneous vor-

ticity isosurface patterns of M=0.5 and 1.0, the color on the surface de-

notes the amplitude of the velocity. From the results of the DES, it is seen

that the intact vortex surface generated at the exit of the cooling hole can

flow downstream to about 7D distance under M=0.5. When M is 1.0, the

vortex surface raptures in a quite short distance. The differences of the

RANS simulation results at different time indices are quite small, which

indicates that the RANS simulation fails to capture the generation and

rapture of the vortices.

4.2 Cooling Flow in Turbine Cascades

The simulations of the flows nearby the cooling hole presented in the last

section reveal the vortices system in JICF clearly. However, due to the

impact of the blade geometry, the streamwise pressure gradient and the

secondary flows in the blade passage, the flow structure of film cooling

jets in a turbine blade may not the same as that on a flat plate. The 3D

flow structures at different cooling configurations and the effect of them on

the loss production are still remained unclear. Steady RANS simulations

are performed on the experimental model of Fotter et al. [78] to get the

answers of these questions. The simulated results and validations with

experimental data are present below. Part of the simulated results has

90

4.2. COOLING FLOW IN TURBINE CASCADES

(a) M=0.5 DES

(b) M=0.5 URANS

(c) M=1.0 DES

(d) M=1.0 URANS

Figure 4.19: Time averaged contour pattern of velocity on the plane with 0.1D

distance from the flat plate. The total number of physical steps used in averaging

is 100.

91


(a) M=0.5 DES

(b) M=1.0 DES

Figure 4.20: Time averaged contour pattern of velocity on the plane with 0.1D

distance to the flat plate. The total number of physical steps used in averaging is

400.

been published in [82, 83].

4.2.1 Review of the Experiment

The experiments are performed in a high speed cascade wind tunnel. A

high pressure planar turbine cascade, named as AGTB, is used which con-

sists of three large scale blades (300 mm in span and 250 mm of chord) to

keep the periodicity (178.5 mm in pitch). Coordinates of the blade profile

can be found in reference [78]. The angle between the chord and the pitch

line is 73 degree. The inlet flow angle and outlet flow angle are 133 and

28.3 degrees, respectively. Three kinds of cooling hole models located at

the blade leading edge were tested. The first one is named as AGTB-B1,

with two rows of 20 straight cylindrical holes with 3 mm in diameter, dis-

tributed uniformly along the span. The second one is named as AGTB-B2,

with two rows of 19 cylindrical holes inclined spanwisely downwards with

45. The hole diameter is same to that of AGTB-B1. To keep the same in-

terval distance between holes in spanwise direction as AGTB-B1, each row

has only 19 holes. The last one is named AGTB-S, with two rows of dis-

crete slots. The slot width is 2.545 mm, and the interval distance between

92


(a) t=0.000 s

(b) t=0.004 s

(c) t=0.008 s

(d) t=0.012 s

(e) t=0.016 s

Figure 4.21: Instantaneous velocity

of DES (M=0.5) on the plane with

0.1D distance from the flat plate. The

time interval between two figures is

20 physical time steps.

(a) t=0.000 s

(b) t=0.004 s

(c) t=0.008 s

(d) t=0.012 s

(e) t=0.016 s

Figure 4.22: Instantaneous velocity

of URANS (M=0.5) on the plane with

0.1D distance from the flat plate. The

time interval between two figures is

20 physical time steps.

93


(a) t=0.000 s

(b) t=0.004 s

(c) t=0.008 s

(d) t=0.012 s

(e) t=0.016 s

Figure 4.23: Instantaneous velocity of

DES (M=1.0) on the plane with 0.1D

distance from the flat plate. The time

interval between two figures is 10 phys-

ical time steps.

(a) t=0.000 s

(b) t=0.004 s

(c) t=0.008 s

(d) t=0.012 s

(e) t=0.016 s

Figure 4.24: Instantaneous velocity of

URANS (M=1.0) on the plane with 0.1D

distance from the flat plate. The time

interval between two figures is 10 phys-

ical time steps.

94


(a) t=0.000 s

(b) t=0.004 s

(c) t=0.008 s

(d) t=0.012 s

(e) t=0.016 s

Figure 4.25: Instantaneous vorticity

isosurface of DES (M=0.5). The colored

contours show the value of the velocity.

The time interval between two figures

is 20 physical time steps.

(a) t=0.000 s

(b) t=0.004 s

(c) t=0.008 s

(d) t=0.012 s

(e) t=0.016 s


isosurface of URANS (M=0.5). The col-

ored contours show the value of the ve-

locity. The time interval between two

figures is 20 physical time steps.95


(a) t=0.000 s

(b) t=0.004 s

(c) t=0.008 s

(d) t=0.012 s

(e) t=0.016 s


isosurface of DES (M=1.0). The colored

contours show the value of the velocity.

The time interval between two figures

is 10 physical time steps.

(a) t=0.000 s

(b) t=0.004 s

(c) t=0.008 s

(d) t=0.012 s

(e) t=0.016 s


isosurface of URANS (M=1.0). The col-

ored contours show the value of the ve-

locity. The time interval between two

figures is 10 physical time steps.96


slots is 20 mm. Figure 4.29 shows the blade geometry and the profiles of

cooling configurations at the leading edge.

(a) Geometry of the blade (b) Cooling configurations

Figure 4.29: Geometry of AGTB cascade and the cooling configurations [78].

Five flow conditions with different blowing ratios were studied, as

listed in Table 4.1, where the blowing ratio is equal to the velocity ratio

as defined in Section 4.1.1.

4.2.2 Computational Model and Mesh

RANS simulations using SA turbulence model are performed. A structured

multiblock body-fitted mesh is built in IGG. Since the geometries of AGTB,

AGTB-S and AGTB-B1 are symmetric to the middle span, only half of the

blade is meshed. The cells number in a flow passage for AGTB, AGTB-

B1 and AGTB-S is about 1.8 millions in total, with about 4,000 cells for

each hole and 30,000 cells for each slot. For AGTB-B2, however, the whole

blade is meshed leading to about 3.6 million cells in total. Butterfly gird

is designed within the cooling holes and connected in a full non-matching

manner to the blade boundaries of the blade passage block which is meshed

with H-Grid topology. Y+ of the first grid away from the solid boundaries

is controlled to be less than 5 from the leading to trailing edges. As an

example, Figure 4.30 shows the mesh of AGTB-B1 and the detailed view

of the butterfly mesh in a cooling hole.

Total pressure and temperature with flow directions are imposed at

97


Table 4.1: boundary conditions of AGTB experiments

Models Blowing ratio (M) Pt1 (Pa) Pt2 (Pa) P2 (Pa) V1 (m/s)

AGTB - 19880 19700 14070 130.1

0 20490 20280 14630 142.1

AGTB-S 0.5 20170 19830 14740 133.1

0.7 20140 19670 14540 132.2

1.1 20170 19450 14100 132.3

0 19720 19560 14920 128.2

AGTB-B1 0.7 19620 19440 14710 128.1

1.1 19650 19450 14640 128.4

1.5 19620 19400 14560 128.5

0 19590 19450 15140 128.2

AGTB-B2 0.7 19730 19570 15190 127.8

1.1 19670 19490 15050 128.3

1.5 19630 19420 14930 128.1

both the cooling plenum (Pt2) and main flow inlets (Pt1). The latter is set

to 95 mm upstream of the blade leading edge with a velocity profile of 1/7

power law. Then a boundary layer with thickness of 16.5 mm at 30 mm up-

stream of the cascade is obtained as the experimental data suggested. The

temperatures of inlet flow and cooling jets are set to 303K. Adiabatic wall

is imposed for all the solid boundaries. At outlet, the static pressure P2

is given. Full non-matching periodic connection is applied to the periodic

boundaries. Mirror boundary condition is set on the middle span surface in

AGTB, AGTB-B1 and AGTB-S models. The detail information of boundary

conditions under different blowing ratios is listed in Table 4.1.

The simulations is performed on a PC with single Pentium4 HT530

(3.0GHz) processor. The converged results usually can be got in less than

1000 iterations. The CPU time is around 15 hours for simulations on

AGTB-S and AGTB-B1, 30 hours for simulations on AGTB-B2. Figure

4.31 shows the convergence history of the simulation on AGTB-B1 with

the blowing ratio is 0.7. The convergence history of other simulations are

similar.

98


Figure 4.30: Mesh for AGTB-B1 and de-

tailed view of the butterfly mesh in holes.

A: perspective of the configuration close

to the leading edge. B: top view of the

butterfly mesh.

Figure 4.31: Convergence history of

steady simulation on AGTB-B1 (M=0.7).

4.2.3 Simulation Results and Analysis

The simulated results are presented and discussed in this subsection. Dif-

ferent to the JICF in the last section, the main interests of this subsection

is in the influence of the different cooling configurations on the 3D passage

flow. As the flow fields in blade passage at different blowing ratios remain

essentially the same, the results to be shown is focused on a blowing ratio

of 0.7.

Static Pressure at Midspan

Firstly, the simulated results are validated by comparison with the exper-

imental data. Figure 4.32(a) shows the distribution of the static pressure

on the blade surface at midspan for all models (for AGTB, M=0). As can be

seen, an excellent agreement between the numerical results and the exper-

imental data is obtained over a wide chord range for all models, excepting

a small range on the pressure side of AGTB-S. At the leading edge, a static

pressure peak locates at the stagnation point. A sharp drop away from this

point is observed due to the high curvature of the blade surface close to the

leading edge. On the pressure side, the static pressure increases rapidly

in a short distance away from the stagnation point resulting in a so-called

“suction peak”, which is a well-known feature of the AGTB blade profile

[80]. On the suction side, the flow is smoothly accelerated away from the

leading edge up to the throat section which locates at about 3/5 chord. It is

also observed, by comparing the results of AGTB to that of AGTB-B1 and

AGTB-B2, the blade surface pressure in a wide range of the suction side

is greatly different. However, it should be noted that these differences are

not only attributed to the difference of hole configurations, but also to the

99


difference of inlet flow conditions which are shown in Table 4.1

(a) Full range view (b) Detailed view at the leading edge

Figure 4.32: Static pressure distributions on the blade surface at midspan of dif-

ferent models (M=0.7).

Figure 4.33 presents the computed static pressure for the four mod-

els with the same inlet boundary condition, which shows that except for

AGTB-S model the static pressure distribution of other model are quite

the same. For case of AGTB-S, the distribution of static pressure is greatly

different to that of others, especially on the pressure surface. Figure 4.34

shows the comparison between the computed static pressure distribution

at the midspan plane for AGTB-B1 under different blowing ratios and that

of experimental data. It is seen again that the numerical results agree very

well with the experimental data for all blowing ratios. With the increase of

the blowing ratio, the static pressure at both the pressure and suction sur-

faces are decreased, while the blade loading remains nearly unchanged.

For other two cases, the variation of static pressure under different blow-

ing ratio is quite the same.

Figure 4.35 shows a close observation of the static pressure at the lead-

ing edge range. Figure 4.36 shows the streamline pattern at midspan for

AGTB-S at M=0.7. In both these figures, five typical positions are marked

as a, b, c, d and e. Point b is related to the stagnant point. Points a and c

locate at the upstream edges of the slots, while points d and e locate at the

downstream edges. Away from the stagnation point b, flow is accelerated

under a positive pressure gradient which is slightly larger on the pressure

side than that on the suction side. Due to the blockage of the jet flow, the

local main flow is decelerated under a local negative pressure gradient op-

posite the upstream of the slot. In the slot exit range, from point a to point

100


Figure 4.33: Static pressure distributions

at midspan under the same inlet condi-

tion.


at midspan of AGTB-B1 with different

blowing ratios.

d and from point c to point e, the static pressure reduce rapidly at first and

recover again gradually which resulting in a local maximum at points d

and e. At the further downstream, the flow separates from the blade sur-

face due to the negative pressure gradient and forms a separation bubble

at each side, as illustrated in Figure 4.36. However, the separation bubble

on the pressure side is much larger than that on the suction side, which

results in a local pressure plateau in a very short distance downstream

to the point e. For models of AGTB-B1 and AGTB-B2, the static pressure

distributions at the leading edge are quite the same to that of AGTB, as

shown in Figure 4.32(b), with a very small separation bubble locating im-

mediate downstream of the cooling hole exits. It is clear that the visible

difference of the static pressure downstream of point e in model of AGTB-S

from other models is associated with the large scale separation bubble.

Figures 4.37∼4.39 show streamline patterns at three critical sections

of 50%H, 47.5%H and 45% in spanwise for three cooling models, where

H stands for the blade height. Then, the sections of 50%H and 47.5%H

are located at midspan and holes’ exit, respectively, while the section of

45%H is located between two holes. As can be seen in Figure 4.37, for

AGTB-S model, large recirculation regions are present at downstream of

the cooling slot on both the pressure side and suction side. The scale of

the recirculation region on the pressure side decreases from midspan to

endwall, while the recirculation region on the suction side is much smaller

than that on the pressure side, which changes little along spanwise. For

AGTB-B1, as shown in Figure 4.38, a small recirculation is observed at

downstream of the cooling hole on the pressure side at the section of 47.5%

101



at the leading edge. PS: pressure surface.

SS: suction surface.

Figure 4.36: Streamline pattern at

midspan of AGTB-S (M=0.7).

where is exact the exit of the cooling hole. At other two sections, there

is no recirculation region present. However, for AGTB-B2, at all three

critical sections, no recirculation region is observed. It can be concluded

that the cooling model of AGTB-B2 with inclined cooling holes provided

the best cooling effect, where the cooling flows attached to the blade surface

more closely and distributed uniformly. While the cooling effect of the slots

cooling model is the worst, in which the hot gas will be wrapped into the

large recirculation region to damage the blade surface.

(a) 50%H (midspan) (b) 47.5%H (c) 45%H

Figure 4.37: Streamline patterns of AGTB-S at different spanwise sections close to

the midspan.

Figure 4.40 shows the streamline patterns of AGTB-S at midspan un-

der different blowing ratios. Here, M=0 means there is no cooling, there-

fore a part of main flow issues into the plenum forming a cavity flow. For

M=0.5, a large recirculation region is generated at the pressure side, down-

stream the cooling hole exit. The reattached point is located about 1/3

chord. When M is increased to 0.7, this recirculation region is enlarged

102


(a) 50%H (midspan) (b) 47.5%H (hole exit) (c) 45%H

Figure 4.38: Streamline patterns of AGTB-B1 at different spanwise sections close

to the midspan.

(a) 50%H (midspan) (b) 47.5%H (hole exit) (c) 45%H

Figure 4.39: Streamline patterns of AGTB-B2 at different spanwise sections close

to the midspan.

to about 2/5 chord, and another recirculation region with smaller scale is

present at the suction side. When M is increased to 1.1, these two recircu-

lation regions are enlarged significantly.

Surface Flow Visualization and 3D Flow Structures

To reveal the 3D flow structure within the cascade with cooling injec-

tions, the blade surface flow visualizations (limiting streamline patterns

of simulated results and oil film pictures of the experiments), 3D stream-

line patterns and isolines of total pressure coefficient in cross-sections are

presented below. Figure 4.41 shows the computational limiting stream-

line pattern on the blade surface of AGTB, while the computation limit-

ing streamline pattern for other models and the comparison with oil film

pictures are shown in Figures 4.42∼4.45. Figure 4.46 presents the 3D

streamline pattern of the jets, and Figures 4.48 and Figure 4.49 show the

isolines of total pressure coefficient in several cross-sections for all models.

All of these figures are for blowing ratio of 0.7 since the flow fields remain

basically the same under other blowing ratios.

Since the oil film visualization for model of AGTB is absent in experi-

103


Figure 4.40: Streamline patterns of AGTB-S at midspan under different blowing

ratios.

ment, only the computed limiting streamline patterns on blade surface are

shown in Figure 4.41. As can be seen, the flow field on pressure surface is

uniform in a wide range except a small region at the leading edge corner

between the blade and endwall. A separation line is observed there, which

is associated with the pressure side leg of the horseshoe vortex. On the suc-

tion side, a separation line of the passage vortex merged together with the

pressure side leg of the horseshoe vortex can be observed near the endwall.

Close to the trailing edge, a small low pressure region is present at the 2/5

position in spanwise, including a locally reversed flow and a saddle point.

This shows the existence of wall vortex which is located above the passage

vortex. Adjacent to the endwall, another separation line which is related

to the corner vortex is observed, and is well consistent with the secondary

flow model from Want et al. which has been presented in Section 2.1.3.

It is seen from Figure 4.49(a) that the isolines of the total pressure coeffi-

cient on the pressure side show a uniform boundary layer in the spanwise

direction, except for those lines near the endwall. On the suction side, a

high loss region is visible at the end of passage, which is associated with

the passage vortex.

Figure 4.42 shows the computed limiting streamline pattern of the

blade surface for AGTB-S, compared to the Oil flow visualization picture,

over half span range. In this figure, characters N, S, Ls and Lr represent

node, saddle point, separation line and reattachment line, respectively. It

104


(a) Pressure side (b) Suction side

Figure 4.41: Limiting streamline patterns on the blade surface of AGTB. LE: lead-

ing edge. TE: trailing edge.

is seen that the computation results well reproduce the experimental ob-

servations, such as the locations of the saddle points S1 and S2, nodes N1,

N2 and N3, lines Ls1, Ls2, Lr1 and Lr2 on the pressure surface and Ls6,

Ls7 and Ls8 on the suction surface. From Figure 4.42(a), it can be seen

that the flow structure near the pressure surface is very complicated and

far different from that without the cooling jets shown in Figure 4.41(a).

Downstream of the slot exit, three spiral nodes, N1, N2 and N3, are formed

by the interaction between the jets and the main flow boundary layer. Fig-

ure 4.43 shows the 3D streamline pattern near the blade surface over the

half span which illustrated the interaction process clearly. In this figure,

the inlet boundary layer is denoted by red lines, while the yellow lines and

blue lines stand for the flow issues from the end of the slots and other re-

gion of slots, respectively. The latter shows the traces of the cooling jets. It

can be observed that due to the injection from two slots, the inlet bound-

ary layer on the pressure surface is divided into two distinct branches.

One wrapped upwards in spanwise and the other downwards at the down-

stream of the leading edge, which results in three vortices. The circulation

of the yellow lines near the mid span is associated with the separation bub-

ble which has been shown in Figure 4.36, leading to one vortex motion with

one end at the midspan and the other end at the spiral node N1. While the

yellow lines close the endwall form a half vortex ring which motion with

its two ends at the spiral nodes N2 and N3 respectively. All these three

vortices will engulf the coolant away from the blade surface resulting in a

reduction of cooling efficiency and a increase of aerodynamic losses. As one

can see from Figure 4.49(b), two high losses regions close to suction side

105


are generated by these vortices in blade passage. In addition, traces of the

jets on both the pressure (blue lines) and suction (red lines) side slots can

be viewed in Figure 4.46(a). The red lines near the endwall are curled due

to the passage vortex motion.

(a) Pressure side

(b) Suction side

Figure 4.42: Limiting streamline patterns on the blade surface of AGTB-S com-

pared with the Oil visualization [78] (M=0.7).

Figure 4.44 and Figure 4.45 show the limiting streamline patterns on

the blade surfaces of AGTB-B1 and AGTB-B2 respectively, and the compar-

ison with the Oil film visualization. Due to the hole inclination, the con-

figuration of AGTB-B2 model is asymmetric with respect to the midspan,

therefore the flow pattern over the whole span is presented. The simulated

results have again reproduced the experimental observations, such as the

position of the separation line on the suction surfaces. When comparing

these two figures with Figure 4.42, it is observed that the flow structures

are nearly the same on the suction surface, while greatly different on the

pressure surface. A wave shaped separation line extending along the span-

wise direction downstream of the cooling holes is observed. This is caused

106


Figure 4.43: Streamline pattern near the cooling slot (M=0.7), front view.

by the CVP which has been analyzed a lot in the last section.

Figure 4.47 shows a detailed view of the 3D flow nearby a cooling hole

on the pressure side near 1/4 span of AGTB-B1 model. The pink lines de-

note the jets issue from the cooling hole. The blue lines on cross-sections 1

and 2 (section 2 is located slightly downstream to the section 1) show the

CVP clearly viewed, though the localized mesh is coarser than that in sim-

ulations of JICF. The limiting streamline (red lines) show two spiral nodes

which are related to the origin of the CVP. Although the horseshoe vortex

is not visible due to the curvature of the leading edge, this flow structure

is still qualitatively the same as that shown in Section 4.1.3 and Kang

[63]. Due to the CVP, isolines of total pressure in the pressure side of the

cross-sections in Figures 4.49(c) and 4.49(d) show a wave shape along the

spanwise direction except for those near the endwall. In addition, traces

of the hole injections from both the pressure (blue lines) and suction (red

lines) side holes near the endwall can be viewed from Figures 4.46(b) and

4.46(c). The red lines near the endwall are curled due to the passage vor-

tex motion, while the blues lines are along the main flow except for the one

closest to the endwall where secondary flow from the pressure side to the

suction side is dominated.

Additional Losses

The additional losses caused by the cooling injections can be quantified

by variations of the total pressure. The definition of the total pressure

107


(a) Pressure side

(b) Suction side

Figure 4.44: Streamline patterns on the blade surface of AGTB-B1 compared with

the Oil visualization [78] (M=0.7).

108


(a) Pressure side

(b) Suction side

Figure 4.45: Streamline patterns on the blade surface of AGTB-B2 (M=0.7) com-

pared with the Oil visualization [78] (M=0.7).

109


(a) AGTB-S

(b) AGTB-B1

(c) AGTB-B2

Figure 4.46: Spatial streamline patterns of cooling flow for different models

(M=0.7).

(a) Section1 (b) Section2

Figure 4.47: Flow pattern in two cross-sections (blue) with streamlines issued

from cooling holes (pink) and limiting streamlines (red) on the pressure surface

of AGTB-B1.

110


coefficient (TPC) is as follows:

CPt =Pt1 − Pt

12ρV 12 (4.3)

where, Pt1 is the inlet total pressure, Pt is the local total pressure, ρ is the

density and V1 is the inlet velocity. Since only two blowing ratio M (0.7

and 1.1) are available for all three cooling models, two groups of the total

pressure coefficient are compared here. Table 4.2 and Table 4.3 show the

surface averaged inlet boundary conditions obtained by numerical simula-

tions under the blowing ratios of 0.7 and 1.1, respectively.

Table 4.2: Boundary conditions for M=0.7

Model Pt1 (Pa) ρ (kg/m3) V1 (m/s)

AGTB 19844.0 0.20517 133.9

AGTB-B1 19586.2 0.21063 125.3

AGTB-B2 19714.6 0.21273 122.1

AGTB-S 20131.1 0.21027 123.7

Table 4.3: Boundary conditions for M=1.1

Model Pt1 (Pa) ρ (kg/m3) V1 (m/s)

AGTB 198434.0 0.20517 133.9

AGTB-B1 19623.7 0.21098 125.4

AGTB-B2 19633.6 0.21210 122.4

AGTB-S 20227.3 0.21159 123.9

Eight YZ sections numbered 1-8 are settled from the leading edge to

the trailing edge, as shown in Figure 4.48(a). The section 1 is located at

10% axial chord length before leading edge and the section 2 is located just

at the leading edge. However, the section 3 is located at 10% axial chord

length behind leading edge. Sections 4, 5, 6 and 7 are located behind the

section 3 with an interval of 20% axial chord length. The section 8 is lo-

cated at 10% axial chord length behind the section 7. Figure 4.48(b) shows

the TPC for all models under the blowing ratios of 0.7 and 1.1. The black

line denotes the basic pressure losses of AGTB without cooling injections,

111


which has the smallest value in the figure. Being consistent with the flow

analysis in the last section, the additional losses caused by the slots cool-

ing model are the biggest, which are much higher than that of other cool-

ing models. The additional losses of AGTB-B2 are a little bit higher than

that of AGTB-B1 since the incline of the cooling jets increases the endwall

loss. Increasing the blowing ratio, the additional losses of AGTB-B1 and

AGTB-B2 increase a little, while a large increase of that is observed for

AGTB-S. The main additional losses are located on sections of 2, 3, 4 and

5 downstream to the cooling hole, where the large separation flow occurs.

It is obvious that the flow separation is the main reason to the additional

losses.

(a) Positions of cutting sections (b) Losses curves

Figure 4.48: Losses evolution through the blade passage of different models under

M=0.7 and 1.1

4.3 Summary

The blowing ratio has large effect on the flow structure nearby a cooling

hole. For a square hole on a flat plate, there exists a critical blowing ra-

tio. Lower than this blowing ratio, the flow structure presents a symmetric

pattern, where the legs of the horseshoe vortex are not involved with the

CVP; higher than this blowing ratio, the flow structure presents an asym-

metric pattern, where the legs of the horseshoe vortex are wrapped into the

CVP. The time averaged results of DES simulations show a better agree-

ment with the experimental data and provide the much more details of the

turbulent flow than that of RANS simulations.

112

4.3. SUMMARY

(a) AGTB (b) AGTB-S

(c) AGTB-B1 (d) AGTB-B2

Figure 4.49: Isolines patterns of total pressure losses of different models (M=0.7)

113


The slots cooling at the leading edge of the blade in cascades results in

a large separation flow downstream of the slot exit, which causes large ad-

ditional losses to the main flow. The inclined cooling can provide more uni-

form cooling effect than the straight cooling. However, the former causes a

little bit higher losses than the latter.

114

Chapter 5

Numerical Simulations on

Three Dimensional Flows

in Axial Turbine

In the present chapter, the flow field in a low speed axial turbine is inves-

tigated using full 3D unsteady simulations which are based on the exper-

iments implemented by Matsunuma [56]. The simulations are performed

on the platform of FineTM/Turbo and the Non-linear Harmonic Method

(NLH) is employed for the R/S treatment. Based on the validation with

the experimental data, the discussions mainly focus on the effectiveness of

NLH method and the effect of harmonic parameters. Parts of the related

results has been published in [139].

5.1 Review of Experiment

The turbine simulated in this chapter is composed of a single stage with

28 stator blades and 31 rotor blades which are designed using a free vortex

method to attain radial equilibrium. The measurements are performed in

a low speed suction-typed wind tunnel of National Institute of Advanced

Industrial Science and Technology (AIST, Japan). The detailed specifica-

tions of the turbine are listed in Table 5.1.

The geometries of the blade at three key sections are shown in Fig-

ure5.1. As can be seen, a negative inlet flow angle is found at tip of the

115

CHAPTER 5. NUMERICAL SIMULATIONS ON THREEDIMENSIONAL FLOWS IN AXIAL TURBINE

Table 5.1: Specifications of turbine cascades

Unit stator rotor

tip mid hub tip mid hub

number of blades, N 28 31

chord, C mm 69.2 67.6 66.2 58.6 57.5 58.7

axial chord, Cax mm 45.3 42.8 40.2 33.1 41.0 49.4

blade span, H mm 75.0 74.0

blade pitch, S mm 56.1 47.7 39.3 50.5 43.1 35.5

aspect ratio, H/C 1.08 1.11 1.13 1.26 1.29 1.26

solidity, C/S 1.23 1.42 1.68 1.16 1.33 1.65

inlet flow angle, α1 0.0 0.0 0.0 -16.2 22.1 51.9

exit flow angle, α2 63.9 67.4 71.1 66.9 63.4 58.7

turning angle, α1 − α2 63.9 67.4 71.1 50.7 85.5 110.6

stagger angle, ξ 49.3 51.0 52.8 55.9 44.8 33.4

inside diameter, D1 mm 350 350

outside diameter, D2 mm 500 500

hubtip ratio, D1/D2 0.7 0.7

tip clearance, k mm 0.0 1.0

upstream axial gap, g mm — 31.0 27.7 24.2

flow coefficient, Φ — 0.43 0.5 0.61

loading coefficient, Ψ — 0.88 1.20 1.77

reaction, Λ — 0.56 0.40 0.11

116

5.1. REVIEW OF EXPERIMENT

rotor. At hub of the rotor, the inlet flow angle has a large positive value.

While the outflow angles at both tip and hub of the rotor are near 60 de-

gree, which leads to large turning angles in the rotor, especially at hub.

(a) Stator (b) Rotor

Figure 5.1: Profiles of blades of the AIST turbine [56].

The turbulence intensity of the free stream at the inlet of the test sec-

tion is 0.5%. The Reynolds number used during the experiment is 20000

based on the chord length and inlet velocity of the stator. The axial in-

let velocity V0 at the test section is 4.47m/s and the rotation speed is set

at 402 rpm. The stator inlet flow condition is measured using a three-

hole pressure probe at the axial position of ZST /Cax,ST = −0.706, 30mmupstream from the stator leading edge at midspan, where the potential

influence of the stator is negligible. The wake traverse of the stator is car-

ried out at a distance of 6.6 mm axially from the stator trailing edge, i.e.,

ZST/Cax,ST = 1.154. The distribution of total pressure, velocity, flow an-

gle, etc, are obtained using a five-hole pressure probe. The internal flow

field within the rotor is measured by a LDV system. Comprehensive ex-

perimental data and flow patterns are provided, which gives a good CFD

validation case.

The Mach numbers based on the mass-averaged velocities at the sta-

tor inlet, stator outlet, rotor inlet and rotor outlet are 0.013, 0.031, 0.014

and 0.027, respectively. The flow in this experiment was considered to be

117


incompressible since the Mach number is very low.

5.2 Numerical Model

5.2.1 Geometry Model and Mesh

Figure 5.2 shows the domain of the computational model. The inlet is set to

about 30 mm (axial position Z/Cax = −0.706) upstream to the leading edge

of the stator, where a profile of absolute velocity is imposed. The outlet is

set to 3 axial chords downstream of the trailing edge of the rotor. A multi-

block body fitted mesh with the total grid number of 2.13 million is built for

a single blade passage with 133 nodes in pitchwise, 65 nodes in spanwise

and 9 nodes in the tip clearance. The number of nodes for stator and rotor

are around 0.93 million and 1.2 million respectively. The grid close to the

solid wall is clustered to ensure the Y+ is lower than 5. The mesh is in

high quality with minimum skewness of 37.85 degree, maximum aspect

ratio of 434.3 and maximum expansion ratio of 3.41. Figure 5.3 illustrates

the generated mesh.

Figure 5.2: Domain of the computational

model.

Figure 5.3: Mesh used simulations on the

AIST turbine.

5.2.2 Boundary Conditions

A uniform distribution of the static temperature of 300K is imposed at the

inlet of the stator. A spanwisely distributed profile of the absolute velocity

as shown in Figure 5.4 is also imposed there. The displacement thick-

nesses of the inlet boundary layer at tip and hub are 1.85mm and 1.68mm,

while the momentum thicknesses are 0.832mm and 0.730 mm, respectively.

An averaged static pressure of 101100 Pa is imposed at the outlet of the

118

5.2. NUMERICAL MODEL

domain. Periodic boundaries are set to the both sides of the domain with

a non-matching connection manner. The values at the periodic boundaries

are related via the inter-blade phase angle [116], i.e., the spatial phase

shift between adjacent blade passages. The R/S interface is set to 1D lo-

cal non-reflective boundary type with a full non-matching connection. The

endwall and the blade surfaces are set to no sliding adiabatic walls.

Figure 5.4: Spanwise distribution of velocity at stator inlet.

5.2.3 Numerical Method

According to Matsunuma [56],the maximum velocity of the flow in this

low speed turbine is around 10.7 m/s, the flow is considered to be incom-

pressible as suggested by the author. However, only the compressible fluid

model can be used in NLH method at the moment. In order to show the

influence of compressibility of the fluid model on the simulated results,

steady simulations using the incompressible fluid are also performed. Also

suggested by the author that most of the flow in the passage of the stator

is laminar flow, therefore different simulations solving the laminar N-S

equation and turbulence N-S equations are implemented separately. The

One-equation turbulence SA model is employed in turbulence simulations.

Since the number of harmonic used in the non-linear harmonic method

is finite, the continuity of the unsteady flow across the R/S interface cannot

be rigorously reproduced. It is expected that the observed discontinuity in

the numerical results would decrease with a higher number of harmonics

[119]. In order to reveal this effect more clearly, the number of frequencies

per perturbation, i.e. the number of harmonics, is set from 1 to 3 for dif-

ferent simulations. Since only one R/S interface exists, the max number of

perturbations per blade row is set to 1.

119


The steady simulation is performed on a PC with single Pentium4

HT530 (3.0GHz) processor. The total number of iterations is 1600 which

needs 32 CPU hours. Figure 5.5(a) shows the convergence history of steady

laminar simulation. It is seen the residual is decreased to -3.4 after about

700 iterations, then keep high frequency oscillation at around -3.0. Note

that, the solver employed in the present thesis is based on the density.

Then, a preconditioning method is employed to handle incompressible flows.

The detailed information about the preconditioning method can be found

in reference [89], which will not be extended here. Due to the very low

speed in the whole flow range, even using the preconditioning method, it

is still difficulty to get the well converged results. However, the amplitude

of the oscillation is quite small, and the mass flow rate at the inlet and

the outlet are kept the same after about 700 iterations. So the convergece

criteria is acceptable, which is also supported by comparsions between the

simulated results using incompressible fluid and the experimental data

presented in following sections. The NLH simulation is performed on a PC

with four Core2 Q6600 (2.4GHz) processors. The total number of iterations

is 1500 which needs 96 CPU hours for 3 harmonics computation. Figure

5.5(b) shows the convergence history of the NLH simulation on the lami-

nar flow where the compressible fluid is used. Getting benefits from the

Fourier transform, the convergence history of the NLH simulation is quite

the same as that of the steady simulation.

5.3 Simulation Results and Analysis

The simulated results and the validation with the experimental data are

presented in this section. Figure 5.6∼Figure 5.8 show the time-averaged

results, and the time-resolved results are shown in Figure 5.9∼Figure

5.14. All velocity presented below are nondimensional. The time-averaged,

pitchwise-averaged, and spanwise-averaged absolute velocity V1 at the sta-

tor exit is used to make the absolute velocity dimensionless. Similarly, the

time-averaged, pitchwise-averaged, and spanwise-averaged relative veloc-

ity W2 at the rotor exit is used to make the relative velocity dimensionless.

5.3.1 Time-averaged Simulation Results

In this subsection, the time-averaged results of unsteady simulations us-

ing the NLN method and the results of steady simulations are compared

with the experimental data in detail. Note that, the steady simulated re-

120

5.3. SIMULATION RESULTS AND ANALYSIS

(a) Convergence history of steady simula-

tion on laminar flow

(b) Convergence history of harmonic simu-

lation on laminar flow

Figure 5.5: Convergence history of simulations on flows in AIST turbine (Incom-

pressible fluid).

Figure 5.6: Steady simulated spanwise distribution of the velocity at the stator

outlet (Axial position: ZST /Cax,ST = 1.154. Pitchwise position: 0.1SST distance to

the suction side surface) The Blue dashed line denotes the time-averaged results

of the NLH simulation.

121


sults using the compressible fluid will not be presented here since they

are the same as the time-averaged results using the NLH method. So,

hereafter, all the results using the compressible laminar flow are the time-

averaged results of NLH simulations with harmonic number is 3.

Figure 5.6 shows the comparison of the spanwise profile of the abso-

lute velocity at the outlet of the stator (axial position ZST /Cax,ST = 1.154)

between the experimental data and simulated results. Note that the dis-

tribution is located at the 10% pitchwise distance to the axial extension of

the trailing edge of the stator. In this figure, the black circles denote the ex-

perimental data. The red dashed line and the blue dashed line denote the

steady turbulent simulation and the time-averaged laminar simulation re-

sults which use the incompressible fluid. The black solid line denotes the

steady simulations which use the compressible fluid. As can be seen, a

good agreement is obtained between the results of the incompressible lam-

inar flow simulation denoted by the black line and the experimental data

denoted by the black circles in most of the region, except in a small low-

velocity region close to the tip. This low-velocity region is related to the

tip passage vortex, where a low peak exists and the simulated results are

smaller than that of the experimental data. The simulated result of the

compressible laminar flow denoted by the blue dot line are more oblique

than the experimental data, which gives thicker boundary layers at both

tip and hub. The velocity gradient from tip to hub of the compressible tur-

bulent laminar flow denoted by the red dash line is obviously smaller than

any other results. This comparison verifies the conclusion that the flows

in most region on blade surface of the stator are laminar flows. Therefore,

the following discussions mainly focus on the results of the laminar flow

simulations. In addition, it is seen that the compressibility of the fluid has

great influence to the simulated results for the low speed flow.

Figure 5.7 shows the comparison between the oil film visualization

and the limiting streamline patterns of the suction surface flow of the sta-

tor. In the Figure 5.7(a) of the oil visualization, the suction surface flow

can be separated into four regions. The region close the leading edge,

marked A in the figure, indicates a laminar boundary layer on the sta-

tor surface because of the very low Reynolds number flow [56], where a

radially inward-directed surface flow is apparent. A laminar separation,

marked B, is formed behind the laminar flow region A. Within this region,

a strong radial flow is observed. A reverse flow region, marked C, is ob-

served to pass near the blade trailing edge from the edge of region B to

the hub endwall. The traces of the passage vortices of the stator are ob-

served near the tip and the hub endwalls. The strong confliction between

the passage vortex at the hub and the separated inward flow results in

122


a mixing flow region, marked D. Figure 5.7(b) shows the suction surface

flow of the incompressible laminar flow simulation. This pattern is quite

the same as that of the experimental data, which presents the laminar flow

region, separation region, reverse flow region and the mixing region on the

hub clearly. The trace of the passage vortex at the hub is observed clearly,

while at the tip no obvious trace of the passage vortex is presented which

is different to the experiment. The suction surface flow of the compressible

laminar flow simulation, presented in Figure 5.7(c), shows a separation re-

gion with the separation line being closer to the trailing edge. The scale of

the passage vortex at the hub is much smaller. Due to the absence of the

reverse flow region, there is no mixing region observed.

Figure 5.8 shows the velocity contour pattern at the stator exit (axial

position ZST /Cax,ST = 1.154). The simulated results of the incompress-

ible laminar flow in Figure 5.8(b) obtains again a good agreement with

the experimental data in Figure 5.8(a). The wake and secondary vortices

generate several low-velocity regions which form an inclined high turbu-

lent zone from tip to hub. The two low-velocity regions at tip and hub

are related to the passage vortices. The top region is extended spanwisely,

while the bottom region is extended pitchwisely. Between these two re-

gions, there exists a narrow low-velocity region which is caused by the

reversed flow in the separation region. However, the simulated scales of

the passage vortices are smaller than that of the experimental data. In

Figure 5.8(c) where the simulated results of the compressible laminar flow

shown, a strong passage vortex at tip is presented. While, there is no ob-

vious low-velocity region shown at hub.

5.3.2 Time-resolved Simulation Results

In this subsection, the time-resolved results of unsteady simulations are

presented and validated with the experimental data. Figure 5.9 shows the

spanwise velocity profile of the NLH simulation at the axial position of

ZST/Cax,ST = 1.154, where the time-averaged experimental data are de-

noted by black circles. As can be seen, the simulated results at different

time indices fluctuate with a small amplitude around the time-averaged

results. The velocity gradient from tip to hub is a little bit larger than the

experimental data, which is due to the influence of the fluid compressibil-

ity.

Figure 5.10 shows the time-resolved absolute velocity contour pattern

at the stator exit (axial position, ZST /Cax,ST = 1.435). The unsteady con-

tour patterns at four different time-indexes (t=0.2T, t=0.4T, t=0.6T and

t=0.8T) are shown in these figures. The red lines at hub indicate the blades

123


(a) Oil film visualization [56]

(b) Incompressible lam-

inar flow

(c) Compressible

laminar flow (Time-

averaged results of the

NLH simulation)

Figure 5.7: Comparison of the flow patterns on the stator suction surface.

124


(a) Experimental results (time-averaged, [56])

(b) Incompressible fluid

(c) Compressible fluid

Figure 5.8: Steady simulated velocity contour pattern at the stator exit

(ZST /Cax,ST = 1.154), computed with incompressible and compressible laminar

flows. 125


Figure 5.9: Time-resolved spanwise distribution of velocity at stator outlet (axial

position ZST /Cax,ST = 1.154) at different time indices. Pitchwise position: 0.1SST

distance to the suction side surface.

of the rotor. From the downstream observation of the leading edge of the

rotor, the absolute velocity at the right side (rotor suction side) is higher

than that at the left side (rotor pressure side). Between these two high-

velocity regions, a low-velocity region generated by the wake of the stator

is observed. Due to the R/S interaction, the high-velocity region and the

wake region fluctuate periodically.

Figure 5.11 shows the time-resolved entropy increase isoline patterns

at the midspan for four different time indices. It is seen that within the

blade passage of the stator the flow is quite uniform. After the exit of

the stator, a high losses region is generated by the wake flow, which is

distorted by the leading edge of the rotor blade to generate many small

distortion regions within the blade passage of the rotor.

The entropy contour patterns at three spanwise sections of 0.22H (hub),

0.5H (midspan) and 0.87H (tip) are presented in Figure 5.12. In each fig-

ure, behind the stator the high losses regions generated by the stator wake

and secondary vortices are distorted by the rotor. At the hub section shown

in Figure 5.12(a), the high losses region generated by the stator blade N1

enters the passage between rotor blades R1 and R2. However, at the tip

section shown in Figure 5.12(c), the same high losses region of N1 enters

the passage between rotor R2 and R3. At the midspan section shown in

Figure 5.12(b), the high losses region of N1 is chopped by the leading edge

of the rotor blade R2. It is easily to understand the distortion of the stator

wake in the rotor passage that the flow near the rotor suction surface side

moves faster than that near the rotor pressure surface side. The stator

wake flow passes through the blade passage of the rotor and is merged

126


(a) t = 0.2 T (b) t = 0.4 T

(c) t = 0.6 T (d) t = 0.8 T

Figure 5.10: Time-resolved absolute velocity at the stator exit (axial position:

ZST /Cax,ST = 1.435).

127


(a) t =0.2 T (b) t =0.4 T

(c) t =0.6 T (d) t =0.8 T

Figure 5.11: Time-resolved entropy increase isolines pattern at midspan.

128


with the rotor wake flow and secondary flows (passage vortex and tip leak-

age vortex).

(a) Y/H = 0.22 (b) Y/H = 0.5 (c) Y/H = 0.87

Figure 5.12: Time-resolved entropy contour patterns at three spanwise sections (t=

0 T).

Figure 5.13 shows the time-resolved relative velocity contour pattern

at the upstream of the rotor exit (axial position ZRT /Cax,RT = 0.853). Note

that, Figure 5.13(a) and Figure 5.13(c) show the results at t=0T of the ex-

periment and NLH method, respectively, while the result of the steady

simulation is shown in Figure 5.13(b) just for comparison. It is seen again

that the simulated results using the incompressible fluid is closer to the

experimental data than that of the simulated results using compressible

fluid. A spanwise low-velocity region is adjacent to the suction surface

of the blade. Close to the tip and hub endwall, two low-velocity regions

are generated by the tip leakage vortex and the passage vortex, respec-

tively. Between these two regions, a low-velocity region is generated by

the boundary layer which developed at the suction surface. Although a

low-velocity region is also observed in the same position in Figure 5.13(c),

the traces of secondary flows at hub and tip are not shown clearly.

Figure 5.14 shows the time-resolved relative velocity contour patterns

at upstream the outlet of the rotor (axial position, ZRT /Cax,RT = 0.853)

in different time indices. The difference between these patterns are not

as large as the patterns shown in Figure 5.10 where the absolute velocity

contour behind the stator are shown. A high-velocity region close to the

tip and suction surface is observed and enlarged pitchwisely in different

time indices. With the rotor moving, the spanwise velocity gradient in-

creases firstly from t=0.2T to t=0.6T and then decreases again from t=0.6T

to t=0.2T.

129


(a) Experiment [56] (b) Steady (Incompressible

flow simulation)

(c) Harmonic=3 (Compressible

flow simulation)

Figure 5.13: Time-resolved relative velocity contour pattern at t = 0 T (axial posi-

tion ZRT /Cax,RT = 0.853). The middle figure presents the steady simulation result

using incompressible fluid for comparison.

(a) t =0.2 T (b) t =0.4 T

(c) t =0.6 T (d) t =0.8 T

Figure 5.14: Time-resolved relative velocity contour patterns (axial position

ZRT /Cax,RT = 0.853).

130


5.3.3 Effect of the Harmonic Number

According to the asymptotic theory, increasing the number of harmon-

ics can improve the continuity of the unsteady flow cross the R/S inter-

face. However, the computational cost will also increase rapidly. Actually,

for the turbomachinery with multi stage, the influence on the flow in a

blade row mainly comes from the adjacent blade rows upstream and down-

stream. Normally, they have difference frequencies. Therefore, using two

harmonics can present the basic property of the R/S interaction for most

of cases. Figure 5.15 shows the amplitudes of the 1, 2 and 3 order compo-

nents of the static pressure on the blade surface at midspan. It can be seen

from Figure 5.15(a) that the first order component dominates the fluctua-

tion, whose amplitude is much larger than that of the second and the third

order components. Figure 5.15(b) and Figure 5.15(c) show the detailed

view of the distribution curves of the second and third order components.

Comparison of the amplitudes shows the first order harmonic is about 30

times larger than that of the first order harmonic, about 300 times larger

than that of the third order harmonic. There is no doubt that the third or-

der harmonic can be ignored totally during the simulation, which indicates

that using two harmonics is fine enough for one R/S interaction treatment.

It is also seen that the distribution curve of the first order harmonic is rel-

atively smooth to the curves of the second and the third order harmonics,

which shows the frequency property.

(a) 1,2,3 order harmonics (b) Detailed view of the second

order harmonic

(c) Detailed view of the third

order harmonic

Figure 5.15: Harmonic pressure amplitude on the blade surface of rotor at

midspan.

Figure 5.16 shows the isoline patterns of the entropy increase at the

midspan using different numbers of harmonics. All the patterns show the

propagation of the wake flows of the stator in blade passages of the rotor.

With the increase of the number of harmonics, more details are presented

which indicates more small scale perturbations are simulated which usu-

131


ally to the high order harmonics. However, the differences between Figure

5.16(b) and Figure 5.16(c) are quite small, which suggests again the same

conclusion that using two harmonics can present the general property of

the R/S interaction in a single stage.

(a) 1 harmonic (b) 2 harmonics (c) 3 harmonics

Figure 5.16: Isolines of Entropy increase at midspan.

5.4 Summary

The unsteady effect of R/S interactions in a low speed axial turbine is simu-

lated by the NLH method. The comparisons between the simulated results

and the experimental data show that the compressibility of the fluid has

great influence on the simulated results for low speed flows. Under a very

low Reynolds number, the surface flow on the stator blade is mainly in

laminar. The steady simulation using the incompressible fluid represents

exactly the laminar separation on the suction blade surface of the stator.

The secondary flow and the wake flow behind the stator are also shown

clearly. The simulated results using the NLH method represent the peri-

odic fluctuation of the wake flow of the stator, and the passage process of

the wake flow in the passage of the rotor. The amplitude analysis indicates

that using 2 harmonics in the NLH method can obtain good prediction of

the interactive effect for one R/S interface.

132

Chapter 6

Development of

Multi-objective

Aerodynamic

Optimization Framework

The second part of the present thesis, including the present chapter and

Chapter 7, focuses on the numerical methodologies of aerodynamic opti-

mization of turbomachinery. In this chapter, the most widely used nu-

merical optimization method Genetic Algorithm (GA) is investigated. Most

research efforts are contributed to the development of the coupled method

combining multi-objective genetic algorithm (MOGA) and artificial neural

network. The effect of the coupled method is validated by standard math-

ematical model problems.

6.1 General Introduction

Optimization is the process of adjusting the inputs or characteristics of a

device, mathematical process, or experiment to find the minimum or max-

imum output or result [140]. In a simple word, optimization is the process

making something better. That is the most common natural feature, which

133

CHAPTER 6. DEVELOPMENT OF MULTI-OBJECTIVEAERODYNAMIC OPTIMIZATION FRAMEWORK

can be easily found in manifold fields: the physical systems tent to be the

state with minimal energy; Engineers aim for the best performance of their

designs, and so on. All these problems can be represented into optimiza-

tion problems.

A general optimization problem could be expressed in following equa-

tions:

min fi(x), i = 1, 2, . . . , Isubject to

gj(x) ≤ 0, j = 1, 2, . . . , Jhk(x) = 0, k = 1, 2, . . . , Kx = [x1, x2, . . . , xn]T ,x ∈ X

where x = [x1, x2, . . . , xn]T is the vector of design variables with total num-

ber of n, X denotes the Design Space. Functions of fi(x), gj(x) and hk(x)stand for objective functions, inequality and equality constrain functions

with total numbers of I, J and K, respectively. The set of all possible val-

ues of objectives is called Objective Space, denoted by Y.

Note that, I should be larger than or at least equal to 1, which sug-

gests the multi-objective or single objective optimal problem. Most of the

complex problems are multi-objective problems which can be solved us-

ing multi-objective genetic algorithm, or sometimes can be transformed to

single objective problems and solved using the single objective genetic al-

gorithm. Constrains should be handled carefully since the optimization

often fails due to rigid and unpractical constrains. Hence, J and K are

not necessary to be larger than 0. Sometimes, constrains can be treated

as objectives using penalty function for analysis or programming conve-

nience. In this sense, let M denote the total number of objectives, then

M = I + J + K.

The functions fi(x), gj(x) and hk(x) are usually presented as explicit

equations for mathematical problems. However, for practical engineering

problems, they could be expressed in an implicit manner, like black-box

models. In these cases, the mathematical optimization, for instance the

steepest descent method cannot be used. Therefore, using numerical algo-

rithm is the only way available for general optimization problems.

Genetic algorithm (GA) is an efficiency numerical optimization method-

ology which mimics the natural behavior in terms of biological evolution to

reach the best possible solution to a given problem. GA allows a popula-

tion composed of many individuals to evolve under specified selection rules.

Weak individuals tent to die before reproducing, while stronger ones live

longer and bear many new offspring which often inherit the good qualities

that enabled their parents to survive.

134

6.2. GENERAL PRINCIPLES OF GA

The method was developed by Holland over the course of the 1960s

and 1970s. The name of “Genetic Algorithm” is proposed by Bagley [141]

in 1967. In 1975, the basic principles of GAs were described rigorously by

Holland in his book [142], which was widely considered as the symbol of

establishment of GA. In this book, Holland proposed the Schemata Theory

and Intrinsic Parallelism which reveals that the number of elite individ-

uals in population will increase with exponential ratio during evolution.

De Jong [143] performed lots of functional optimizations using GA and

proposed a set of mathematical model problems, which contributes the de-

velopment of GA greatly. A comprehensive exposition on the theory and

applications of GA was given by Goldberg [144] in his well known book Ge-

netic Algorithms in Search, Optimization, and Machine Learning in 1989.

This work could be considered a sign of the maturity of GA.

Some of the advantages of a GA to other optimization methods are

summarized by Haupt [140]:

• Optimizes with continuous or discrete variables;

• Doesn’t require derivative information;

• Simultaneously searches from a wide sampling of the design space;

• Deals with a large number of variables;

• Is well suited for Parallel computers;

• Optimizes variables with extremely complex design space;

• Provides a list of optimum variables, not just a single solution;

• May encode the variables so that the optimization is done with the

encode variables;

• Works with numerically generated data, experimental data, or ana-

lytical functions.

6.2 General Principles of GA

The optimization process of GA starts from a set of initial samples of the

design vector. Each sample is called an Individual which is corresponding

to a solution of design, while the set of individuals is called Population.

The number of the individuals, n, is called population size. Usually, the

initial population, or called the first generation individuals, is generated

randomly. Then, several genetic operators, such as Selection Operator,

Crossover Operator and Mutation Operator, are implemented on the ini-

tial population (Parent Population) to generate a new population (Child

Population), in which the individuals might have better fitness. Then, the

second run starts with this new population and repeats the above pro-

cesses. The process of iterations is named Evolution, while the index of

135


each iteration is named Generation. With the evolution, the individuals in

each generation are approaching the optimal range in objective space.

A general GA is composed of the following procedures: encoding and

decoding, initial population generation, fitness evaluation, selection, crossover

and mutation.

6.2.1 Encoding and Decoding

Encoding is the process transforming the variable to the string structure

which mimics the structure of chromosomes, and then the genetic opera-

tors can be easily implemented. The opposite process is decoding which

transforms the string back to real value. Many encoding ways have been

developed since the basic GA was proposed, including binary encoding,

denary encoding, real encoding and hybrid encoding.

Binary encoding is the primitive encoding way which is still widely

used today since it is simple and easy for programming. Each individual

is represented in a string of binary values which is named chromosome. A

chromosome is divided to several genes. Each gene is corresponding to a

design variable, which is composed of a number of bits, 0 or 1. The length

of bits has a great effect on the precision of the encoding value. Eq.6.1

presents an example of a 8-bit binary encoded chromosome:

chromosome =

11110000︸︷︷︸

gene1

11001100︸︷︷︸

gene2

. . . 00011101︸︷︷︸

geneN

(6.1)

Increasing the number of bits will reduce the encoding error to the real

value. However, the memory and computational cost also increase to han-

dle the encoding and decoding processes. When the length of bits is chosen,

the maximum and minimum individuals can be related to the bounds of bi-

nary chromosomes. Then, a single mapping between individual value and

encoded chromosome can be built using interpolation.

The encoding principle of the denary encoding and the real encoding

are similar to that of the binary encoding with replacing the binary genes

with denary and real genes. The denary encoding and real encoding have

been found superior to the binary encoding in some applications, especially

for engineering problems. The real encoding can build the mapping di-

rectly using the maximum and minimum individual value, which saves

lots of memory and computational cost.

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6.2.2 Fitness Function

The survival ability of an individual is measured by the fitness function

which usually is a function of the objectives. The fitness function has great

influence on the finial optimization results and the evolution speed.

6.2.3 Genetic Operators

Selection

Selection is the key operation of genetic algorithm, in which stronger in-

dividuals are selected to generate the new individuals in next generation

through genetic operations, such as crossover, mutation, and weaker indi-

viduals are eliminated. Three selection schemes are widely used: Roulette

wheel selection, Elitist selection and Ranking selection. The roulette wheel

selection is the basic selection scheme in which the selected probability

of an individual equals to the proportion of this individual’s fitness func-

tion in the summation of all individuals’ fitness function. In elitist selec-

tion, the individual or a couple of individuals with highest fitness function

values are copied to the next generation directly, which prevents the opti-

mal solution from being wrecked. However, in some cases, the elitist indi-

viduals might dominate the whole population leading to a local optimum

which is usually called Precocious. In ranking selection, all individuals

are ranked according to the fitness function value and imposed a selected

probability specified in advance.

In recent years, some new selection schemes are proposed for multi-

object genetic algorithms, for instance, the most widely used crowding se-

lection which is used in the present thesis and will be introduced in the

following section.

Crossover Operator

Crossover is an operator to create one or more offsprings from the par-

ents selected through exchange some bits on specification positions. For bi-

nary encoding method, three basic crossover operators are often used:One-

point crossover, Two-point crossover and Uniform crossover. For One-point

crossover operator, one random number related to a specific point of the

chromosome is generated randomly. All bits beyond that point in either

chromosome are swapped between the two parents organisms. Eq. 6.2

shows a simple example of the One-point crossover of 8-bits chromosomes.

[00 | 001111] + [11 | 110000] ⇒ [00 | 110000] + [11 | 001111] (6.2)

137


The Two-point crossover is similar to the One-point crossover excepting

the swapping points is two. The same example using Two-point crossover

is shown in Eq. 6.3:

[00 | 0011 | 11] + [11 | 1100 | 00] ⇒ [00 | 1100 | 11] + [11 | 0011 | 00] (6.3)

In the uniform crossover, the swapping of bits in parent chromosomes de-

pends on the possibility. A pseudo chromosome with same length is gener-

ated randomly. If the bit in pseudo chromosome is 1, then the correspond-

ing position bit in parents chromosomes is swapped. If the bit in pseudo

chromosome is 0, then is not swapped. Eq. 6.4 shows the same example

using a uniform crossover.

[00001111] + [11110000] → [10011100] ⇒ [10010011] + [01111100] (6.4)

The crossover operator of real encoding is a little complicated than that

of binary encoding. In the present thesis, the simulated binary crossover

(SBX) is used which will be introduced in detail in section 6.3.2.

Mutation Operator

Mutation operator is used to keep the diversity of individuals in population

and enhance the local searching ability of genetic algorithm. For binary en-

coding, the basic mutation operator is to take the inverse value in random

generated position of the chromosome. Only one parent individual is used

generating one new child individual. For example:

[00001111] ⇒ [00101101] (6.5)

For real encoding, a similar mutation operator can performed as follows.

For individual xi, i = 1, . . . , n, the component xk ∈ [ak, bk] is mutated.

Then the mutated component yk for the new individual can be calculated

by

yk =(xk − ak)ak + (bk − xk)bk

bk − ak(6.6)

6.2.4 Control Parameters

The control parameters usually includes the population size, crossover

rate, mutation rate and the maximum generation number, which have

great effect on the optimization results and the evolution speed.

138


Population Size

Population size is the number of individuals in one generation. Large

population size provides high diversity which gives more opportunities to

converge to the global optimal. However, the number of evaluations and

genetic operations needed are also large resulting in large computational

cost. Thus, in practical application, a moderate population size is neces-

sary. For the optimal problem which needs large computational cost, like

aerodynamic optimization, the suitable population size could be around

100.

Crossover Rate

The crossover rate controls the probability of the crossover operation. On

the one hand, a large crossover rate can enhance the searching ability of

genetic algorithm; on the other hand, it also decrease the possibility of

good qualities inherits from parent generation. However, the evolution

process will become very slow with a small crossover rate. In general, the

crossover rate could be in range of [0.8, 1.0].

Mutation Rate

The mutation rate controls the probability of the mutation operation. Large

mutation rate can benefit the diversity of population, but might also de-

crease the evolution efficiency. While, if the mutation rate is too small, the

local search ability of genetic algorithm is weaken. Usually, the mutation

rate is set to [0.001, 0.01].

Maximum Generation Number

The maximum generation number is an important evolution controlling

parameter. For practical applications, it is quite difficult to determine the

rigid convergence in a traditional way since the information of the optimal

solution cannot be known beforehand. Then, imposing a maximum gener-

ation number is the most widely used stop criteria of evolution. This num-

ber heavily depends on the genetic algorithm and the property of the opti-

mization problem, which is usually set to a large number if there is no any

experience. In order to save computational cost for some fast converged

cases, an additional stop criterion can be used, in which the evolution is

stopped if the solution has no change in several successive generations.

Note that, this additional criterion only can be launched after a certain

number generations.

139


6.3 Multi-objective Genetic Algorithm

6.3.1 Pareto Optimal Concept

In most cases, the objectives are conflicting with each other, thus there is

no “best” solution for which all objectives are optimal simultaneously. The

increase of one objective will lead to the decrease of other objectives. Then,

there should exist by nature a set of solutions, the so-called Pareto-optimal

set or Pareto front, in which one solution can not be “dominated” by any

other member of this set. The “domination” is defined as follows [145]:

Definition 1: For minimal problem, a solution a ∈ X dominates a solu-

tion b ∈ X , (a ≻ b) if and only if it is superior or equal in all objectives and

at least superior in one objective. This can be expressed as:

a ≻ b, if

∀i ∈ 1, 2, . . . , m : fi(a) ≤ fi(b)∧∃j ∈ 1, 2, . . . , m : fj(a) < fj(b)

Definition 2: The solution a is indifferent to a solution c, if and only if

neither solution is dominating the other.

Each solution of the Pareto optimal set is called a feasible solution or

a non-inferior solution, which is corresponding to one point on the Pareto

front. A general example of the Pareto front with two objectives is illus-

trated in Figure 6.1. In this example, the Pareto front is composed of five

points from A to E which are indifferent to each other. While, points Fand G do not belong to the Pareto front since they are dominated by point

B and points C and D, respectively. Note that, theoretically the Pareto

Figure 6.1: Pareto front with two objectives.

front should be a continuous curve (two objectives) or several segments of

140

6.3. MULTI-OBJECTIVE GENETIC ALGORITHM

a curve. However, in practice, only a finite number of discrete points on

the Pareto front can be obtained by numerical optimizations in one run.

The Pareto optimal origins in economics, proposed by Vilfredo Pareto

for the first time in 1896 for the problem of income distribution. After that,

this concept is developed to engineering application gradually. In 1985,

Vector-evaluated Genetic Algorithm (VEGA, [146]) was proposed by Schaf-

fer is the pioneering research on multi-objective genetic algorithms (MO-

GAs). After that, many multi-objectives optimization genetic algorithms

have been developed, such as Multi-objective Genetic Algorithm(MOGA,

[147]) by Fonseca and Fleming in 1993, Non-dominated Sorting Genetic

Algorithm (NSGA, [148]) by Srinivas and Deb in 1994, Niched Pareto Ge-

netic Algorithm (NPGA, [149]) by Horn and Naphiotis in 1994, etc. Gong

et al. [150] classified these MOGAs listed above into the first generation

MOGAs which use fitness sharing scheme to keep the diversity of popula-

tion. The second generation MOGAs include Strength Pareto Evolutionary

Algorithm (SPEA, [145]) by Zitzler and Thiele in 1999 and SPEA2 [151]

by Zitzler et al. in 2001; Pareto Archived Evolution Strategy (PAES, [152])

by Knowles and Corne in 2000, Pareto Envelope-based Selection Algorithm

(PESA, [153]) and PESA2 [154] by Corne et al. in 2000 and 2001 respec-

tively; NSGA-II [155] by Deb et al. in 2002. Coello et al.[156] summarized

all these MOGAs in their book Evolutionary Algorithms for Solving Multi-

Objective Problems, more details about these MOGAs can found there.

Most of these MOGAs are applied successfully in a number of math-

ematical and engineering applications. However, NSGA-II and SPEA2

gained much more attentions than any other MOGAs due to the outstand-

ing performance in lots of mathematical model problems tested [157–160].

In following subsections, these two algorithms are introduced and com-

pared in detail.

6.3.2 NSGA-II

Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is proposed by Deb

et al. [155] in 2002, which is the improved version of NSGA proposed by

Srinivas and Deb in 1994 [148]. In this improved version, three shortcom-

ings of NSGA are overcome:

1 The computational complexity is reduced from O(MN3

)to O

(MN2

),

N is the population size, M is the number of objectives;

2 Introducing Elite keeping strategy;

3 No need for specifying a sharing parameter;

141


The evolutionary procedure starts from an initial parent population

P0 of size N which is generated randomly in the design space. The popu-

lation P0 is sorted using Fast Non-dominated Sorting Scheme firstly. Each

individual is assigned a value of fitness, or called rank, which equals the

non-domination level of this individual. Level 1 stands for the best level,

which will be the Pareto front when the evolutionary procedure is finished;

level 2 is the secondary level win which individuals are dominated and

only dominated by some individuals in level 1, and so on. Then binary

tournament selection operator, crossover operator and mutation operator

are implemented on P0 to generate the first generation of the child popu-

lation Q0 of size N . Afterwards, P0 and Q0 are combined to a whole set

C0 of size 2N and C0 is sorted. A new generation of parent population

P1 of size N is generated by filling with the first N individuals of set C0.

The same procedure can be repeated with P1 till the maximum number of

generations is reached.

In genetic operations, the selection operation is based on two criteria:

rank and crowding distance. The first one indicates the non-dominated

level which is obtained by sorting process. The second one suggests the

distributed uniformity of individuals in the same level. Large crowding

distance ensures the population diversity to prevent the individuals from

being stuck into a local optimum.

Fast Non-dominated Sorting Scheme

In NSGA, in order to identify if an individual belongs to the first non-

dominated front in a population, each individual has to be compared in

pairs with all other individuals of the same population. This requires

O(MN) comparisons for each individual. When this process is continued

to find all members of the first non-dominated front in the population, the

total computational complexity is O(MN2). In order to find the individuals

in the next level front, the same procedure is repeated without discounting

the individuals of the first front temporarily. In the worst case, the task

of finding the second front also requires O(MN2) computations. If this ar-

gument is also true for finding third and higher levels of non-domination,

then the worst case requires overall O(MN3) computations when there are

fronts and there exists only one individual in each front. Note that, O(N)storage is required for this procedure.

In order to reduce the computational complexity, a fast non-dominated

sorting approach is suggested by Deb et al. [155] in NSGA-II. Firstly, for

each individual p, two entities are calculated. One is the domination count

np, the number of individuals which dominate the individual p. Another is

142


Sp, the set of individuals dominated by individual p. This requires O(MN2)comparisons. The domination count of all individuals in the first non-

dominated front will be zero. For each individual with nq = 0, visiting

each member q of Sp and reduce its domination count nq by one. During

the process, if for any member whose domination count becomes zero, it is

put in a separate list Q. After this process, all individuals in Q constitute

the secondary non-dominated front. For each individual in second front,

repeating the above procedures, then the third front can be identified. All

the rest fronts can also be identified by continuing the same procedures.

For each individual p in the second or higher level of non-dominated

front, the domination count np can be at most N −1. Thus, each individual

p will be visited at most N − 1 times before its domination count becomes

zero. Since at most N − 1 times such procedure are needed, then the total

complexity is O(MN2) which is lower in one order than that of NSGA.

However, the storage requirement is increased to O(N2), one order higher

than before.

The pseudo code of the fast non-dominated sorting operator can be

expressed in Algorithm 1 [155].

Crowding Distance

In difference to the NSGA using a sharing function to preserve the diver-

sity of individuals, a new criterion named Crowding Distance is employed

in NSGA-II, which estimates the density of individuals surrounding a par-

ticular individual in the population. The definition also can be expressed

in

Cj =

Nobj∑

i=1

F j+1i − F j−1

i

(F ji )max − (F j

i )min

(6.7)

For two objectives problem, the crowding distance of the jth solution is the

averaged length of the cuboid u + v as illustrated in Fig.6.1. For boundary

points, the crowding distance is set to the maximum value of the system

in order to ensure these points can survive to the next generation. In

the same rank level, the individuals which have larger crowding distances

also have more opportunities to be selected. A pseudo code of the crowding

distance selection operator is presented in Algorithm 2, which is also can

be found in [155].

Simulated Binary Crossover

In NSGA-II, a simulated binary crossover (SBX) operator is used, which

simulates the working principle of the single point crossover operator on

143


Algorithm 1 Fast non-dominated sorting approach

for p ∈ P do

Sp = ∅np = 0

for q ∈ P do

if p ≺ q then

Sp = Sp ∪ qend if

if p ≻ q then

np = np + 1

end if

end for

if Np = 0 then

prank = 1

F1 = F1 ∪ pend if

end for

i = 1

while Fi 6= 0 do

Q = ∅for p ∈ Fi do

for q ∈ Sp do

nq = nq − 1

if nq = 0 then

qrank = i + 1

Q = Q ∪ qend if

end for

end for

i = i + 1

Fi = Q

end while

144


Algorithm 2 Crowding distance assignment

l = |I|for i ∈ I do

I[i]distance = 0

end for

for m ≤ M do

I = sort(I, m)

I[1]distance = I[l]distance = ∞for i = 2 to (l − 1) do

I[i]distance = I[i]distance + (I[i + 1].m − I[i − 1].m) /(fmax

m − fminm

)

end for

end for

binary strings [161, 162]. Let x1,ti and x2,t

i denote two parent individuals

for ith selection in generation t. The procedure of computing the children

individuals x1,t+1i and x2,t+1

i from parent individuals x1,ti and x2,t

i is de-

scribed below.

A spread factor βi is defined as the ratio of the absolute difference in

children values to that of parent values:

βi =

∣∣∣∣∣

x2,t+1i − x1,t+1

i

x2,ti − x1,t

i

∣∣∣∣∣

(6.8)

Firstly, a random number ui ∈ [0, 1) is generated, whereafter, from a spec-

ified probability distribution function, the ordinate βqi is found so that the

area under the probability curve from 0 to βqi is equal to the chosen ran-

dom number ui. The probability distribution used to create a child indi-

vidual is derived to have a similar search power as that in a single-point

crossover in binary-coded GAs and is given as follows:

P (βi) =

0.5(η + 1)βiη, if βi ≤ 1

0.5(η + 1)/βiη+2, otherwise

(6.9)

where, η is the distribution index which can be any nonnegative real num-

ber. A lager value of η gives a higher probability for creating near parent

individuals and a small value of η allows distant individuals to be selected

as children individuals. Usually, the value of η is set to 15. Using Eq.6.9,

145


the βqi can be calculated as follows:

βqi =

(2ui)1

η+1 , if ui ≤ 0.5(

12(1−ui)

) 1η+1

, otherwise(6.10)

Thereafter, two children individuals can be obtained by:

x1,t+1i = 0.5

[

(1 + βqi)x1,ti + (1 − βqi)x

2,ti

]

x2,t+1i = 0.5

[

(1 − βqi)x1,ti + (1 + βqi)x

2,ti

] (6.11)

Note that, these two children individuals are symmetric about the parent

individuals. This is deliberately used to avoid any bias towards any par-

ticular parent individual in a single crossover operation.

Mutation Operator

Let xk be the component of a individual xi, which is going to be mutated.

And xuk and xl

k stand for the maximum and minimum value of this com-

ponent in all individuals, respectively. The mutated individual yk can be

calculated as follows:

yk = xk + δq

(xu

k − xlk

)(6.12)

Here δq is a mutation parameter, which stands for:

δq =

(2r + (1 − 2r)(1 − δ1)

ηm+1)1/(ηm+1) − 1, if r ≤ 0.5

1 −(2r(1 − r) + 2(r − 0.5)(1 − δ2)

ηm+1)1/(ηm+1)

otherwise

(6.13)

where ηm is the mutation index which is set to 20 in general, and r is a

random number. The intermediate variables δ1 and δ2 stand for:

δ1 =xk − xl

k

xuk − xl

k

(6.14)

δ2 =xu

k − xk

xuk − xl

k

(6.15)

6.3.3 SPEA2

The Strength Pareto Evolutionary Algorithm 2 (SPEA2) is proposed by

Zitzler et al. [151] in 2001 based on the previous version SPEA [145] in

146


1999. In contrast to its predecessor, SPEA2 incorporates a fine-grained fit-

ness assignment strategy, a density estimation technique and an enhanced

archive truncation method. The procedures of SPEA2 are composed of fol-

lowing steps [151]:

1 Initialization: Generate an initial population P0 and create the empty

archive (external set) P0 with size of N . Set P0 = ∅ and the generation

index t = 0.

2 Fitness assignment: Calculate fitness values of individuals in Pt and

the archive Pt.

3 Environmental selection: Copy all non-dominated individuals in Pt

and Pt to Pt+1. If size of Pt+1 exceeds N then reduce Pt+1 by means of

the truncation operator, otherwise if size of Pt+1 is less than N then

fill Pt+1 with dominated individuals in Pt and Pt.

4 Termination: If t ≥ T or another stopping criterion is satisfied then

set A to the set of decision vectors represented by the non-dominated

individuals in Pt. Stop.

5 Mating selection: Perform binary tournament selection with replace-

ment on Pt in order to fill the mating pool.

6 Variation: Apply recombination and mutation operators to the mat-

ing pool and set Pt to the resulting population. Increment generation

counter (t = t + 1) and go to step 2.

Fitness Assignment

In step 2 of the main loop, each individual i in the archive Pt and the

population Pt is assigned a strength value S(i), representing the number

of individuals it dominates.

S(i) = |j|j ∈ Pt + Pt, ∧i ≻ j|

where, | · | denotes the cardinality of a set. Based on the value of strength,

the raw fitness R(i) of an individual i is calculated:

R(i) =∑

S(j), j ∈ Pt + Pt and j ≻ i

Note that the fitness is to be minimized here.

Density Estimation

The density estimation technique used in SPEA2 is an adaptation of the k-

th nearest neighbor method. The inverse of the distance to the k-th neigh-

bor, denoted as σki is taken as the density estimated. As suggested in [151],

147


k is set to the square root of the summation of the population size and the

archive size, k =√

N + N . Then the density D(i) of the individual i can be

defined by

D(i) =1

σki + 2

In the denominator, 2 is added to ensure that its value is greater than zero

and that D(i) < 1. Finally, the fitness of individual i is composed of the

sum of R(i) and D(i).From the description above, the procedures of SPEA2 are quite similar

to that of NSGA-II, while two differences exist in:

1 In SPEA2, an external archive is used to keep the elite individuals of

all generations, while in NSGA-II the elite individuals only in current

generation are stored. That makes SPEA2 has more opportunities to

get better diversity of population. However, the computational com-

plexity is O(MN3) which is higher than that of NSGA-II. And more

storage is also needed for external archive.

2 In SPEA2, the k-th nearest neighbor method is used to estimate the

density information. While in NSGA-II only the basic nearest neigh-

bors is used. Certainly, the same k-th nearest neighbor method is

also can be adopted in NSGA-II. Whereas, there is no obvious supe-

riority of the former over the latter be found. In addition, the former

needs more computational effort on sorting procedures.

The flow charts of NSGA-II and SPEA2 are illustrated in Figure 6.2,

which also illustrate the difference between these two algorithms.

6.3.4 Performance Evaluation Criteria

The Convergence Metric proposed by Deb et al. [158] and the spacing cri-

terion by Schott [163] are widely accepted as the performance evaluation

criteria for MOGAs. The former is for the convergence towards a reference

set, while the latter is for the diversity of solutions.

Let P ∗ =(p1, p2, · · · , p|P∗|

)denote a reference set and A =

(a1, a2, · · · , a|A|

)

denote the non-dominated set of the t-th population P t during the evolu-

tion. The convergence metric can be calculated in the following manner:

1 For each point ai in A, calculate the smallest normalized Euclidean

distance to P ∗ as follows:

di =|P∗|

minj=1

√√√√

M∑

k=1

(fk(ai) − fk(pj)

fmaxk − fmin

k

)2

(6.16)

148


(a) NSGA-II (b) SPEA2

Figure 6.2: Procedures of NSGA-II and SPEA2.

where, fmaxk and fmin

k are the maximum and the minimum function

values of k-th objective function in P ∗.

2 Calculate the convergence metric by averaging the normalized dis-

tance for all points in A:

C(P t) =

∑|A|i=1 di

|A| (6.17)

The value of C(P t) is smaller, the optimized results are closer to the

reference set.

The convergence metric represents the distance between the current

non-dominated set and the reference set; therefore the convergence is bet-

ter when this value is smaller.

The spacing metric S(P t), or called diversity metric, can be calculated

as:

S(P t) =

√√√√ 1

|A| − 1

|A|∑

i=1

(d − di

)2(6.18)

149


where

di =|A|

minj=1

(k∑

m=1

|fm(ai) − fm(aj)|)

, ai, aj ∈ A, i, j = 1, 2, . . . , |A|

Here, d denotes the averaging value of di which stands for the distance

between point pi and the nearest adjacent point. In this sense, the spacing

metric indicates the normalized variation of this distance. If the spac-

ing metric equals to 0, which means the points on the Pareto front are

uniformly distributed. Normally, the distribution is acceptable when the

value of d is lower than 0.05.

6.3.5 Comparison between NSGA-II and SPEA2

Mathematical Model Problems

In order to test the performance of different multi-objective genetic algo-

rithms, various mathematic optimization problems were proposed, such

as SCH problem [146], KUR problem [164], POL problem [165], ZDT prob-

lems [157], etc. Among them, the ZDT problems proposed by Zitzler (the

proposer of SPEA2) and Deb (the proposer of NSGA-II) in 2000 are the

most widely accepted test problems, which includes 5 well-designed highly

nonlinear problems. Since the last one is designed especially for the ef-

fect of binary encoding, it will not be adopted in the present thesis where

real encoding is used. The definition of the functions are listed in Table

6.1. In this table, the dimension n stands for the number of input vari-

ables, i.e. the number of design variables. The dimension of 30 for the first

three problems and 10 for the last problem indicate these problems are all

high dimensional and complicated. All these problems are defined as min-

imization problems with 2 objectives. According to the definitions of F2(x)in both ZDT1 and ZDT2 problems, F2(x) is an increasing function of the

intermediate functiong(x), which has the minimum value when g(x) = 1.

Therefore, the real Pareto front can be obtained by setting g(x) = 1..

Test Results

The evolution starts with a randomly generated initial population with

size of 100. The generation number of evolution is set to 200 which can en-

sure the results fully converged according to the experiences. The crossover

rate and the mutation rate are set to 0.9 and 0.033, respectively. At the

moment, open source codes form Deb and Zitzler are used in this compari-

son.

150


Table 6.1: Mathematic test problems for MOGA

Name Dimension Variable Objective Function

(n) range (minimized)

ZDT1 30 [0,1]

F1(x) = x1

F2(x) = g(x)[

1 −

√

F1/g(x)]

g(x) = 1 + 9

(n∑

i=2

xi

)

/(n − 1)

ZDT2 30 [0,1]

F1(x) = x1

F2(x) = g(x)[1 − (F1/g(x))2

]

g(x) = 1 + 9

(n∑

i=2

xi

)

/(n − 1)

ZDT3 30 [0,1]

F1(x) = x1

F2(x) = g(x)

[

1 −

√

F1/g(x) −x1

g(x)sin(10πF1)

]

g(x) = 1 + 9

(n∑

i=2

xi

)

/(n − 1)

ZDT4 10

x1 ∈ [0, 1]

xi ∈ [−5, 5]

i = 2, . . . , n

F1(x) = x1

F2(x) = g(x)[1 − (F1/g(x))2

]

g(x) = 1 + 10(n − 1) +n∑

i=2

xi

[x2

i − 10 cos(4πxi)]

151


The test results are illustrated in Figure 6.3, where the red circles

and blue squires denote the results of NSGA-II and SPEA2, respectively.

As can be seen, both the NSGA-II results and SPEA2 results can converge

to the real Pareto Front for all test problems. However, the computation

of NSGA-II is about 3 times faster than that of SPEA2 in our practical

implementations. The same conclusion is also indicated by Hiroyasu et al.

[166] in their application. Bui et al. [167] tested the performance of NSGA-

II and SPEA using ZDT1-ZDT3 problems with small noises, and concluded

that NSGA-II usually got better results during the evolution. Moreover, it

is found that NSGA-II algorithm gains more attentions in public released

documents and is also employed in some commercial robust optimization

software, such as iSIGHT and modeFRONTIER. Based on summarizing

the comparison results and information above, NSGA-II is chosen as the

optimization method in the present thesis.

(a) Pareto front of ZDT1 (b) Pareto front of ZDT2

(c) Pareto front of ZDT3 (d) Pareto front of ZDT4

Figure 6.3: The real Pareto fronts of ZDTs problems.

152

6.4. COUPLED METHOD WITH APPROXIMATION MODEL

6.4 Coupled Method with Approximation Model

During the optimization process, the computational cost is determined

mainly by the evaluations of fi(x), gj(x) and hk(x). For mathematical

problems, these values can be easily obtained from a formula. However,

for practical engineering problems, they are usually evaluated by numeri-

cal simulations which are computationally quite expensive. A typical way

to reduce the cost is using some approximation models to replace numeri-

cal simulations to evaluate the objectives. The mostly often used approxi-

mation methods in engineering includes the Response Surface Method, the

Kriging Method, Artificial Neural Network, etc. [168]. The approxima-

tion quality of response surface and Kriging model heavily depend on the

property of the function being approximated. They often fail for strong

non-linear problems. On the other hand, artificial neural networks show

an excellent capability of non-linear mapping in numerous applications,

which leads to their use in various engineering fields.

6.4.1 Artificial Neural Network

An artificial neural network is a system based on the operation of biologi-

cal neural networks. In other words, it is an emulation of biological neural

system. One of the most widely used artificial neural networks, namely

Back Propagation Neural Network (BPNN), is employed in the present re-

search. The basic structure of BPNN, schematically depicted in Figure

6.4, usually consists of a series of neural layers which simulate the inner

connections between variables. Each layer is composed of several elemen-

tary processing units, called nodes. The first layer connects all the inputs,

whereas the last layer connects the outputs. Between these two layers, one

or several “hidden” layers are generated.

Before an ANN can be used for prediction, it has to be feeded firstly

with some information about the real physical model. This feeding process

is called training. For BPNN, the training process basically involves two

phases: forward phase and backward phase [169]. In the forward phase,

all the inputs of a layer are connected to each node of this layer through a

weighting factor. The summation of all the contributions with in addition

a bias value is introduced in a sigmoidal transfer function. The output of

this function will be treated as the inputs for the nodes in the next layer.

Then the signal is propagated in the same way through hidden layers up

to the output layer. The output of the kth node in the (t + 1)th layer can be

153


calculated as follows:

yk(t + 1) = fk(xk(t + 1)) = fk

Nt∑

j=1

wjk(t)yj(t) + θk(t + 1)

(6.19)

where, yk(t) is the output of the kth node in the tth layer, xk(t) is the input

of the same node. Nt is the number of nodes in layer t, j = 1, . . . , Nt. θk(t)denotes the bias function of the kth node in the tth layer. fk is the transfer

function. The choice of transfer function may strongly influence complexity

Figure 6.4: Example of the configuration of BPNN.

and performance of neural networks. Duch and Jankowski [170] gave an

excellent survey of more than 40 exiting neural transfer functions, for in-

stance, the most widely used linear function (Figure 6.5(a)), piecewise lin-

ear function (Figure 6.5(b)), threshold function (Figure 6.5(c)), sigmoidal

function (Figure 6.5(d)), etc. Although, in various applications, some well-

designed transfer functions can benefit the performance of ANN, the sig-

moidal transfer functions is still the most common one which is found suit-

able for most applications. The standard sigmoidal transfer function, de-

fined as Eq. 6.21, is adopted in the present research.

fk(x) =1

1 + e−x(6.20)

f ′k(x) = fk(x) (1 − fk(x)) (6.21)

In the backward phase, the error between the output of the neural

network yik and the real output di

k is propagated through the output layer

backward up to the input layer. The connecting weights and the value of

154


(a) Linear function (b) Piecewise linear function

(c) Threshold function (d) Sigmoid function

Figure 6.5: Transfer functions of ANN.

155


bias functions of each layer are adjusted according to the error signals.

∆wjk(t) = αyk(t + 1)eik, i = 1, . . . , Ns (6.22)

∆θk(t + 1) = αeik (6.23)

where, α ∈ (0, 1) denotes the learning ratio, Ns denotes the number of

samples. eik is the adjusted error, which stands for:

eik = yk(t) (1 − yk(t)) (di

k − yik) (6.24)

where dik is the real value of the kth output of the ith sample, yi

k is the

corresponding approximated value of the same output. As can be seen,

the adjust value ∆wjk(t) of connecting weights depend on three values:

α, ejk and yk(t). ∆wjk(t) is proportional to the error ej

k and output value

yk(t), which means the adjustment is larger where the error is bigger and

the node is more active. Learning ratio α is used to adjust the training

speed, which is usually set to 0.25∼0.75. A large α can accelerate the

learning process, while a small α is beneficial to the convergence stability.

Therefore, α can be put a large initial value at the early training process,

then it is decreased gradually along with the training.

The learning process of the neural network consists of iterations be-

tween these two phases to find the components of the weight matrices and

bias vectors that lead to an output vector that coincides with the prescribed

output vector. Least mean-square error of the output is used to evaluate

the convergence:

E =

Ns∑

i=1

No∑

k=1

(dik − yi

k)2 (6.25)

where Ns is the number of samples, No is the number of outputs.

The approximation accuracy of BPNN not only depends on the num-

ber and quality of the training samples, but also depends on the topology

of the neural network. The present thesis will not extent to the details

of the accuracy investigation of BPNN since it has been discussed exten-

sively. For instance, the related discussion can be found in Rojas [169]. In

theory, a back propagation neural network with only one hidden layer can

approximated to any continuous functions. However, a full convergence of

the learning process does not guarantee that the network correctly mimics

the true function. In reference [27], it has been shown the neural network

with a topology of 2 hidden layers can give a much better approximation to

a certain non-linear function than that with 1 hidden layer. In the present

work, a topology with 2 hidden layers is therefore chosen. The number

156


of nodes in the hidden layers N1h , N2

h are set to equilibrate the number of

unknowns Nunk and the number of available equations Neq:

Neq = NoNs (6.26)

Nunk = N1hNi + N1

h + N1hN2

h + N2h + N2

hNo + No (6.27)

where, Ni, No denote the number of nodes in input layer and output layer,

respectively.

6.4.2 Integration of NSGA-II into Design3D

In the present thesis, NSGA-II is implemented through programming in

C++ language and coupled with the artificial neural network. The main

body of NSGA-II is defined as a class. All genetic operations are defined

as subclasses subjected to the class of NSGA-II. In order to use to existing

parametric modeler, all the development is based on the Design3D, the op-

timization module of Fine software package. The code of Design3D is mod-

ified to link the developed optimization framework with the CFD solver

and the parametric modeler. Hereafter, all optimal calculations is imple-

mented in Design3D environment. Following the framework of Design3D

by Pierret [171] and Hirsch et al. [27], a general sequence of optimizations

is given below.

1 A parametric model based on the 3D geometry of turbomachinery is

built in a versatile geometry modeler, named Autoblade, where the

design variables are settled.

2 A database containing the results of all N-S computations performed

during the previous and present design processes is generated us-

ing Design of Experiments (DoE) method. This database contains the

parameters defining the geometry, fluid field properties and total per-

formance of turbomachinery.

3 An ANN is trained using the samples stored in the database, which

connects the aerodynamic performance of turbomachinery with de-

sign variables.

4 Finding a new design using an optimization procedure where the

aerodynamic performance being evaluated by means of the trained

neural network.

5 The new geometry provided by the optimization is evaluated by means

of the 3D N-S flow solver (EURANUS) and this new sample is added

157


to the database. Then, go to step 3 to start next iteration till the

performance of new design evaluated by CFD is consistent with that

predicted by ANN.

The 3D parametric model of the blade of turbomachinery usually is

stacked by several 2D sections. There are two design modes for blade sec-

tions are integrated in Autoblade. One mode constructs the blade by in-

dependent suction and pressure side surfaces, which is typically adapted

to the treatment of turbine blades. The other one uses a camber line and

thickness distribution to obtain the suction and the pressure side surfaces.

However, for mathematical model problems, there is no geometry exist and

CFD simulations performed. Then, all the default geometry parameter

are fixed and some pseudo variables (x1, x2 . . .) are generated. Two pseudo

derived quantities (F1 and F2) as the objectives are evaluated using the

formula in Table 6.1 to replace CFD solver during generation of database.

Latin hypercube sampling method is employed to generate the initial sam-

ples in the database, whose introduction can be found in reference [172].

6.4.3 Test Results of the Coupled Method

In this subsection, the coupled optimization method of NSGA-II and BPNN

is verified using ZDT problems. For simplification, only ZDT1 and ZDT2

problems are tested. The same setting for NSGA-II in section 6.3.5 is

adopted. A BPNN with 2 hidden layers is used. The input and output

nodes are 30 and 2, which equal the numbers of input and output vari-

ables, respectively. The numbers of nodes in hidden layers are defined by

equation 6.27. Latin hypercube sampling is used to arrange the initial

training samples. The number of training iterations is set to 1 × 105 to en-

sure the global training error be lower than 0.0001. The definition of the

global training error Eg is as follows:

Eg =

√∑No

k=1 Ek

No(6.28)

where Ek is the training error for the kth output, which is defined by:

Ek =

∑Ns

i=1 (dik − yi

k)2

Ns(6.29)

As mentioned in last section, the prediction accuracy of the approxi-

mation model depends largely the number of training samples. In order to

show this influence clearly, several computations with different number of

158


initial training samples are implemented. The results of ZDT1 and ZDT2

are illustrated in Figure 6.6 and Figure 6.7, respectively. As can be seen,

for ZDT1 problem, the results cannot convergence to the real Pareto front,

even using 480 initial training samples. Increasing the number of samples,

the result is getting closer to a local optimum which is still far away from

the real Pareto front in the whole range. The results of the ZDT2 problem

are worse compared to that of ZDT1. Figure 6.6(b) and Figure 6.8(b) show

the detailed views of the results in the range [-0.15, 0.05]. It is clearly seen

that many points cluster in a small range close to the boundary of F1 = 0,

and most of them are out of the correct lower bound F1 = 0.

(a) Full view (b) Detailed view of the local zone

Figure 6.6: Pareto front of ZDT1 problem using coupled optimization method.

Pareto front: the real Pareto front calculated using formula. NSGA2: the results

are calculated using coupled optimization method. DoE: the DoE method used to

generate the initial samples.

Figure 6.8 shows the convergence and spacing metrics of ZDT prob-

lems. The convergence curve shows the effect of the samples number

on the convergence. Increasing samples number, the convergence is im-

proved. However, when the samples number is over 120, the decrease of

residual is small with the increase of the samples number.

There are two reasons for this problem. The first one is the predic-

tion error of the neural network, which is due to insufficient information

feeded to the neural network. This prediction error, also called general-

ization error, exits in any approximation models. Note that, it should be

distinguished from the training error which only indicates the approxi-

mation level to the training samples. Each training sample covers only

a part of the real physical problem. If the predicted points are outside

159



Figure 6.7: Pareto front of ZDT2 problem using ANN.

(a) Zdt1 (b) Zdt2

Figure 6.8: Convergence and spacing metrics.

160


of that part covered by the set of training samples, the generalization er-

ror may be very large. That is the reason why many points in Figure 6.6

and Figure 6.7 have negatives values of F1. According to the famous “no

free lunch theory” [173], the more information training samples can pro-

vide about the real problem, the more accurate will approximation model

become. However, for practical engineering optimization based on CFD

simulations, it is impossible to use a huge number of training samples.

Moreover, if these samples are far away of the Pareto front, then little in-

formation about the Pareto front can be fed to the neural network. Hence,

the key issues are how to take the fully advantage of useful information

from a reasonable number of samples and how to improve the distribution

of samples in the design space. The second reason is the defect of the selec-

tion principle of crowding distance which is based on the cubic distance to

the adjacent points. However, this cubic distance is quite sensitive to the

prediction error for strong non-linear problems. Taking ZDT problems for

instance, the value of F2 increases quite fast when F1 is close to 0. Then,

the points close to the boundary, which have a larger crowding distance,

will get more opportunity to survive to the next generation. Along with

the evolution, more and more points will cluster in a small range. The

diversity of individuals is lost which leading to a local optimum.

In order to expose these two factors more clearly, two modified ZDT

problems with lower dimensions of 3 in design space, named ZDT∗ prob-

lems, are introduced. The definitions of the objectives are still kept the

same as that in the original ZDT problems. Figure 6.9 shows the real

Pareto front of ZDT∗ problems where the red circles denote the results us-

ing exact evaluations by formula. It can be seen that the Pareto fronts of

the ZDT∗ problems are same to that of ZDT problems. It is no doubt that

using the exact evaluation, NSGA-II can obtain the real Pareto front.

Since the number of input variables of ZDT∗ problems are small, the

full factorial factor method can be used to generate the initial training

samples. Four discrete levels of 2, 3, 4 and 5 are used for each variables

leading to 8, 27, 64, and 125 samples, respectively. Figure 6.10 and Figure

6.11 show the results of ZDT∗ using the coupled method under different

number of training samples. As can be seen, the results of ZDT∗ problems

are much closer to their Pareto fronts than that of ZDT problems even

with a small number training samples for ANN. If using only 8 samples,

the difference between the optimized results and the real Pareto front are

quite large. It can be easily explain that two discrete levels only can repre-

sent a linear function of the variable. The non-linear information of ZDT∗

problems is not included in the training samples. Increasing the samples

number to 27, the real Pareto front can be reached, which indicates that

161


(a) Pareto front of ZDT1∗ (b) Pareto front of ZDT2∗

Figure 6.9: Pareto front of ZDT∗ problems.

a certain number of samples is the basis of the convergence of optimiza-

tion. Hence, the first reason discussed above is confirmed. However, it still

can be found that many points still locate at a small range of F1, which

suggests the influence of the second reason.


Figure 6.10: Pareto front of ZDT1∗.

Figure 6.12 and Figure 6.17 show the effect of DoE methods on the op-

timized results with same samples number of 125, which actually present

the influence of the distribution of initial training samples on the optimiza-

tion results. Five DoE methods are compared, including Random waking,

Latin Hypercube, Full factorial factor, D-optimal and Continuous method.

The detail description of these methods can be found in [172] and [89]. As

162



Figure 6.11: Pareto front of ZDT2∗.

can be seen, the difference between the optimization results using differ-

ent DoE methods are quite small, excepting in the small range close to

the boundary. The full factorial factor method and D-optimal give the best

results, while the performance of Latin hypercube method and continuous

method are not satisfied. It should be noted that this conclusion is only

for the problems with few input variables. And the computational cost of

D-optimal method is much higher than that of other methods. Therefore,

it is suggested to use full factorial factor method when the number of de-

sign variables is small. For the problem with moderate number of design

variables (5∼10), D-optimal method can be considered as an alternative

method. While, for the problem with larger number of design variables,

only Latin hypercube method or the basic random walk method can be

used.

Figure 6.14 shows the effects of DoE methods on the convergence and

spacing metrics. It is seen from this figure that the results using Full Fac-

torial method, Latin Hypercube method and D-optimal method didn’t show

any advantage to the result using Random method. However, it should be

noted that few extreme points have great influence on the convergence

and spacing metrics due to the domination of F2 value in metrics’ mea-

surement. Therefore, when the noise of prediction error is large, the con-

vergence and spacing metrics cannot represent the real performance of

optimizations.

To summarize the test results shown above, the number of samples

has great influence on the prediction accuracy of ANN, which must be large

enough to insure the accuracy of ANN prediction. The distribution of the

initial training samples also has influence on the results. If there is no

163



Figure 6.12: Effect of DoE on optimized results of ZDT1∗.


Figure 6.13: Effect of DoE on optimized results of ZDT2∗.

164

6.5. IMPROVING STRATEGIES

(a) ZDT1∗ (b) ZDT2∗

Figure 6.14: Effect of DoE on convergence and spacing metrics of ZDT∗.

any prior knowledge about the location of the optimal set, usually the uni-

form distribution generated by some DoE methods is preferred. When the

samples number and the distribution have been determined, the robust-

ness of NSGA-II to the prediction error is the key influential factor to the

optimization results. Therefore, it is must be improved to eliminate the in-

fluence of ANN prediction error. The last but not the least, the predication

of approximation model still needs to be improved. Some new approxima-

tion models based on statistical theory have been developed and applied

successfully in some applications in recent years. For instance, the Sup-

port Vector Machines (SVM) has been found superior to ANN in some cases

with small number of samples [174, 175]. However, the effect of SVM on

the approximation of aerodynamic design still need lots of investigations

to verify than will be the future work in following projects.

6.5 Improving Strategies

In this section, several improving strategies are proposed and implemented

on the optimization code developed. The effects of these strategies on the

optimization results are presented below.

6.5.1 Improved Crowding Distance

In NSGA-II, as illustrated in Figure 6.1 and equation 6.7, the crowding

distance of individual j is the cuboid length between objective values of the

two adjacent individuals. In order to show the drawback of this selection

principle clearly, a simplified example of the ZDT1 problem is presented

165


below. Figure 6.15 shows fifteen points extracted from the current Pareto

front after several generations. Each point corresponds to a individual in

the population. As can be seen, several points cluster in a small range

close to F1 = 0. Supposing the population size is 11, then 5 individuals

will be removed and the remaining 11 individuals will survive to the next

generation.

Figure 6.15: Example of ith generation of population. The points are indexed as

1:16 from left to right.

In Figure 6.16(a), the crowding distance of each point is calculated ac-

cording to the original definition. The selection operator is implemented,

then the next generation of the Pareto front is illustrated in Figure 6.16(b)

where the points removed are denoted by red squires. While, for minimiza-

tion problem, the aim of optimization is to find the value of all objectives

as small as possible. It can be seen that for points in the red dashed box F2increases greatly whereas F1 is almost constant, while F1 of these points

almost keep constant with respect to the variation of F2. For the points in

the blue dashed box, the variations of F1 and F2 are of comparable size.

For a minimization problem, if an objective is constant with respect to the

other in a part of the Pareto front, points in these areas are less interest-

ing. However, due to the rapid increase of F2, the crowding distances of

the points in red dashed box are larger than that of points in blue dashed

box. As a result, the less interesting points are kept, while the interesting

points are abandoned.

The fundamental reason to this drawback is the original crowding dis-

tance ignores the existing internal relationship among objectives. In order

to eliminate this drawback, a profile factor is imposed to the crowding dis-

tance to balance the contribution of each objective to the crowding distance,

166


(a) Histogram of original crowding distances (b) Selected resutls using original crowding

distance

Figure 6.16: Selection process using the original crowding distance. The points in

Figure b is indexed of 1:16 from left to right. The red squares denote the points are

removed. The crowding distances of the boundary points are set to infinity.

which is defined as below:

C∗j = σjCj (6.30)

σj =

M∏

i=1

F j+1i − F j−1

i

(F ji )max − (F j

i )min

(6.31)

where, C∗j is the improved crowding distance of the jth individual, σj and

Cj denote the corresponding profile factor and the original crowding dis-

tance respectively. (F ji )max and (F j

i )min are the maximum and minimum

values of objective i of all individuals in the same front level with individ-

ual j. The value of improved crowding distance and the updated selected

results are illustrated in Figure 6.17(a) and Figure 6.17(b). As can be seen,

the results using improved crowding distance are more reasonable. The re-

lated spacing metric is decreased from 0.34 to 0.33.

This improving strategy is applied to the ZDT∗ and ZDT problems

with results shown in Figures 6.18∼6.21. For the low dimensional prob-

lems, great improvements has been found when comparing to Figure 6.10

and 6.11. Only few points are out of the correct bounds.

Figure 6.20 shows the comparison of convergence and spacing metrics

of ZDT∗ problems between the improved method and the original method.

It is clearly seen that the residual is decreased greatly by using the im-

proved method. However, due to the influence of few extreme points close

to the boundary of F1 = 0, the spacing metric using the improved method

167


(a) Histogram of improved crowding dis-

tances

(b) Improved selected results

Figure 6.17: Improved selection process using improved crowding distance. The

points in Figure b is indexed of 1:16 from left to right. The crowding distances of

the boundary points are set to infinity.


Figure 6.18: Optimized resutls of ZDT1∗ problems using the improved crowding

distance.

168



Figure 6.19: Optimized results of ZDT2∗ problems using the improved crowding

distance.

is increased greatly.

(a) ZDT1∗ (b) ZDT2∗

Figure 6.20: Convergence and spacing metrics of ZDT∗ problems.

For the high dimensional problems, the effect is weak so that the opti-

mized results still cannot converge to the Pareto front. The reason can be

understood from Figure 6.21. As can be seen, most of the points are far be-

yond the correct range, which indicates that the prediction of ANN is quite

poor for a high dimensional problem. It also indicates that the improved

crowding distance has effect only when the prediction error is not quite

large. So the prediction accuracy of ANN must be improved. However,

there are several points do escape from the local optimum, which gives the

potential possibility of further improvements.

169


(a) ZDT1 (b) ZDT2

Figure 6.21: Optimized results of ZDT problems using the improved crowding dis-

tance. Latin hypercube method is used to generated 120 initial training samples.

6.5.2 Coarse-to-fine Iteration

As mentioned in the last section, the second reason is that the insuffi-

cient quantity and the quality of training samples for ANN. Actually, the

quality has more improving potentials than the quantity since the compu-

tational labor is the shackles of the latter. Nain and Deb [176] proposed

a coarse-to-fine approaching model based on NSGA-II and artificial neu-

ral network. According to their strategies, the evolution process is sep-

arated into several segments which are composed of a certain number of

generations. In each segment, the individuals of the first few generations

are evaluated by CFD simulations to provide an initial approximation of

the Pareto front. Then an artificial neural network trained by the exist-

ing points is employed to predict objectives in the following generations of

each segment. The same sequence will be implemented for the following

segments. Then, with the proceeding of the segments the locations of the

training samples are approaching the Pareto front and the approximation

accuracy of the neural network is improved. Following this coarse-to-fine

strategy, the Pareto-optimal front can be reached. A practical application

of B-spline curve fitting problem with 40 input variables and 2 output

objectives was presented and more than 10000 exact simulated runs are

needed. Whereas, this strategy is not so efficient since only about 32% com-

putation labor is saved. Cunha et al [177, 178] improved Nain and Deb’s

method with a local approximation strategy reducing further computation

time with about 10% for ZDT problems [157]. However, these methods are

still computationally expensive for those engineering applications which

170


are based on CFD simulations.

In fact, there exists a noticeable problem that the model going to be ap-

proximated is not the real physical model in the whole objective space, but

the projection of the physical model on the Pareto front surface (or curve).

From this point of view, the information from any points not belonging to

the Pareto front is redundant information. However, as the Pareto front of

real practical problems cannot be known beforehand, it has to start from

the certain number of samples with random or specific distribution. In

most cases, these samples will not be close the optimal solutions by co-

incidence. And the situation is the same for the individuals of the first

few generations which take much local information rather than that of the

Pareto front. Hence, it would be inefficient if too much computational ef-

forts are spent on evaluations of these individuals. This is the reason why

the Deb’s strategy [176] still needs a large number of samples in their en-

gineering applications. In fact, they just approximated the real physical

model in the whole objective space indeed. In order to accelerate the con-

vergence to the Pareto front, an improving strategy is proposed, which is

similar to that used by Pierret [26] in single optimizations. Firstly, a coarse

approximation model is trained by some training samples with an uniform

distribution or other specific distributions. Then all evolutions during the

optimization will be performed based on the prediction of this model. Tak-

ing advantage of the evolution process of genetic algorithms, the individu-

als will approach quickly to the probable region where Pareto front maybe

located. Then these new points which take more information about Pareto

front could be used to refine the approximation model. Avoiding confusion

with the inner iteration of evolution, the refinement iteration of the ap-

proximation model is called global design iteration. After refinement, a

new optimization evolution is launched and all evolutions are performed

based on the refined model. With the proceeding of iterations between re-

finements and optimizations, the evolution results will converge to the real

Pareto front gradually. While, different to the single optimization, some

methods have to be used to chose some specific results to launch CFD sim-

ulations. A Multi-criteria decision making (MCDM) strategy is proposed

in the present thesis to handle this problem, which is described below in

detail.

Multi-criteria Decision Making

According to the iteration strategy described above, after each optimiza-

tion run, some specific solutions are chosen to be evaluated by CFD sim-

ulations and the related results will be added into the training database

171


to feed more information to BPNN. For single objective optimizations, the

iteration is easy to implement since only one solution is available after op-

timization. However, different to single objective optimization, the results

of multi-objective is a set of alternative solutions. How many solutions

should be chosen and which solutions should be chosen still are not clear.

It is obviously that the more points are chosen, the more information can

be obtained, but also the more computational cost is needed. There must

exist a “balanced” value of the number of samples chosen. Since this ques-

tion is a big issue which needs many more investigations, it will not be

discussed in the present thesis. For the present research, only one solu-

tion is chosen in optimization results which leads to a multi-criteria deci-

sion making (MCDM) problem. Theoretically, this point could be any point

of the current Pareto front and different MCDM strategies have different

effects. However, the aim of improving strategy is to make the points ap-

proach the Pareto front as quickly as possible. A natural idea is to choose

the point in the current Pareto front with the largest distance to the most

inferior solution. It should be noted that the most inferior solution is the

one with maximum values for all objectives in the whole population, which

usually is located far away from the Pareto front. Figure 6.22 illustrates an

example of the most inferior solution and the result of MCDM. In this fig-

ure, the maximum values of F1 and F2 are 1 and 1.2, and the most inferior

solution is thus (1, 1.2). In reference to the membership function concept

in fuzzy set theory [179], a partial order sorting can be implemented by

the value of membership:

µji =

Fi,max − F ji

Fi,max − Fi,min, Fi,min ≤ F j

i ≥ Fi,max

µj =

M∑

i=1

µji

N∑

j=1

M∑

i=1

µji

where, µj is the membership value of individual j, N and M denote the

population size and the number of objectives respectively, Fi is the value

of ith objectives, µj is the normalized membership value. This strategy is

also found being used in references [180] and [181].

The flow chart of the coupled optimization framework including global

design iterations is illustrated in Figure 6.23, and this framework is ver-

ified by ZDT problems again. The optimization results of ZDT2 problem

with 2 iterations are presented in Figure 6.24, with the convergence and

172


spacing metrics shown in Table 6.2. It is seen that from the figure and the

table, great improvements have been obtained for the results of both the

coupled methods using the original NSGA-II and the improved NSGA-II

after only two iterations. The advantage of the improved NSGA-II is obvi-

ous over the original NSGA-II. However, there are still several points out

of correct left boundary. An extra control strategy for boundary points will

be discussed in the next subsection.

Figure 6.22: Example of Fuzzy MCDM.

Table 6.2: Convergence and Spacing metrics

Methods Convergence metric Spacing metric

coupled method 2.23675 0.04056

coupled method 0.53206 0.05711

(2 global iterations)

improved coupled method 2.42919 0.05813

improved coupled method 0.21028 0.07652

(2 global iterations)

Boundary Control

According to the selection principle of NSGA-II, the boundary points of

each generation will be imposed the largest value of crowding distance to

ensure them to survive to the next generation, so the prediction of the

173


Figure 6.23: Global flow chart of the optimization framework.

174


Figure 6.24: Improved results of ZDT2 using coarse-to-fine iterations.

boundary is quite important. In order to improve the prediction accu-

racy of the boundary, all the extreme points of each global design iteration

will also be evaluated by CFD simulations and added into the training

database. Then, for the problem with n objectives, n + 1 CFD computa-

tions should be launched in one global design iteration. Figure 6.25 shows

the boundary points of the optimized results of ZDT1 problem in six global

iterations, where the old boundary points and new boundary points are

denoted by black dots and hollow star respectively. This figure shows the

convergence history of boundaries clearly. After 6 iterations, new boundary

points are quite close to the bounds of the real Pareto front. The difference

between the old boundary point and new boundary point for each bound-

ary can be used as the stop criterion. If this difference is lower than a

small value, for instance 0.5%, this boundary can be considered as con-

verged. Then the boundary point at this boundary will not be evaluated

by CFD simulations in following global iterations. All the points exceeding

this boundary during the evolution are unrealistic and abandoned, and the

same number of new points are generated randomly to keep the population

size.

It should be noted that, the aim of boundary control is to restrict all

solutions into the correct range, which can accelerate the convergence of

global iterations. However, this strategy has little effect on the approxi-

mation on the profile of Pareto front. Usually, if the real Pareto front is a

“smooth” curve (for two objectives), for instance a quadratic curve, it is easy

to get the converged solutions. However, if the real Pareto is far away from

a quadratic curve, for instance a multi-segment curve, it will need a large

number of global iterations to capture exactly the inner boundary of each

175


(a) Iteration 1 (b) Iteration 2

(c) Iteration 3 (d) Iteration 4

(e) Iteration 5 (f) Iteration 6

Figure 6.25: Convergence history of boundary points. Black dots denote the bound-

ary points obtained in last global iterations. Red stars denote the boundary points

obtained in the current global iterations.

176

6.6. SUMMARY

segment. A further improved boundary control strategy can be as follows.

When all boundary points have converged, the points located at the middle

positions between boundary points and the MCDM solution are evaluated

in following iterations, instead of the boundary points themselves. Then

the prediction accuracy at the middle range between boundary points and

the MCDM solution are enhanced.

For the ZDT1 problem, 10 global iterations are needed with 150 ex-

act evaluations performed in total including 120 evaluations are for initial

training database. For ZDT2, 20 global iterations are needed with 180 ex-

act evaluation performed. Figure 6.26 and Figure 6.27 show the results of

different number of iterations, where “BC” in the legend denotes boundary

control. As can be seen, the frame using improved NSGA-II and coarse-to-

fine iteration can get the well converged Pareto front. Figure 6.28 shows

the convergence and spacing metrics using coarse-to-fine iterations and

boundary control strategy. The convergence residuals reach to 0.013 and

0.007 for ZDT1 and ZDT2 problems respectively. Although, the spacing

metrics of ZDT1 and ZDT2 decreased to 0.03 and 0.048 respectively, they

are still lower than the acceptable value of 0.05.

(a) Full view (b) Detailed view in the local zone

Figure 6.26: Pareto front of ZDT1 problems using the improved crowding distance,

coarse-to-fine iterations and boundary control. BC: boundary control.

6.6 Summary

A coupled optimization method using NSGA-II and ANN is developed in

this chapter. The test results of standard mathematical model problems

177


(a) Dull view (b) Detailed view in the local zone

Figure 6.27: Pareto front of ZDT2 problems using the improved crowding distance,

coarse-to-fine iterations and boundary control. BC: boundary control.

(a) ZDT1 (b) ZDT2

Figure 6.28: Convergence and spacing metrics of ZDT problems using the improved

coupled method with boundary control

178

6.6. SUMMARY

show the prediction error of ANN and crowding distance selection princi-

ple of NSGA-II have great influences on the optimization results. An im-

proved crowding distance and a coarse-to-fine iteration strategy are pro-

posed to improve the coupled method. From the comparison of the test

results, it is seen that great improvements have been obtained. Based on

these validations, the improved robust optimization platform can be used

in practical design in the following chapter.

179


180

Chapter 7

Application of Robust

Optimization on An Axial

Compressor

In the previous chapter, a multi-objective optimization framework for the

aerodynamic design is developed. Three strategies are proposed to im-

prove the coupled effect of NSGA-II and BPNN. In the present chapter,

this optimization framework will be verified through a practical industrial

application: aerodynamic optimization on an axial transonic compressor—

NASA Rotor37. A multi-points optimization with three optimization objec-

tives is performed firstly as the verification. After that, a robust optimiza-

tion of rotor 37 under an uncertainty boundary condition is performed,

where a non-intrusive probabilistic collocation method is adopted to repre-

sent the stochastic properties of the fluid field. Several different optimiza-

tion strategies are adopted, including a single objective optimization us-

ing weighting functions, a single objective optimization using a combined

objective and a multi-objective optimization based on the Pareto optimal

concept. The discussions mainly focus on the difference of the results from

different optimization strategies.

181

CHAPTER 7. APPLICATION ON ROTOR37

7.1 Design Background of the NASA Rotro37

Rotor37 is a transonic axial compressor designed by NASA Lewis Research

Center and initially tested as a part of a four stages axial flow compressor.

It is a well documented test case and widely used for validations of flow

solvers. The rotor is composed of 36 Multiple-Circular-Arc blades with

the inlet hub-tip diameter ratio of 0.7, the blade aspect ratio of 1.19 and

the tip solidity of 1.29. A well known feature of rotor37 is that the mid-

span region is dominated by a strong shock attached at the blade leading

edge. This shock interacts strongly with the suction side boundary layer

which may separate either up to the trailing edge after the shock. A strong

radial movement is also observed in the separated area from the hub to

the tip wall [182]. Strong Interactions occur in blade passages, such as

the corner stall, the shock/boundary layer interaction, the tip vortex and

the tip leakage secondary interaction. Note that, since the experimental

weighting process neglects the high loss regions in the annulus boundary

layers, usually the predicted efficiencies based on the data from all grid

points of CFD simulations should be about 2% below the test data [183].

The main configurations of rotor37 are listed in Table 7.1. More details

can be found in reference [182].

Table 7.1: General configurations of rotor37

Parameters Value Unit

Rotation speed 17188 rpm

Hub radial 1752 cm

Shroud radial 2566 cm

inlet total pressure 101800 Pa

inlet total temperature 288 K

nominal mass flow rate 20.7 kg/s

nominal pressure ratio 2.012 -

7.1.1 Application Platform

All optimizations are performed in the FineTM/Design3D environment of

NUMECA, in which the developed optimization framework is integrated.

A brief description of the Design3D has been presented in the previous

section 6.4.2.

182

7.1. DESIGN BACKGROUND OF THE NASA ROTRO37

7.1.2 Parametric Model

A parametric model of the blade geometry is needed for optimization, which

is built using the parametric modeling module—Autoblade. Two cubic B-

splines with 5 control points are used to represent the hub and shroud

lines separately, as illustrated in Figure 7.1(a). Night sections stacking at

the center of gravity in spanwize are used for blade modeling. Each sec-

tion is defined by a central camber line with 5 control points to control the

turning angle, which is shown in Figure 7.1(b). The positions of the control

points on two Bezier lines, 5 for suctions side and 4 for pressure side, are

used to build the blade thickness, the leading edge, the trailing edge and

the wedge angles, as shown in Figure7.1(c).

(a) Hub and shroud (b) Central camber line

(c) Thickness control (d) 3D profile

Figure 7.1: Parametric model of rotor37 blade.

A simple Bezier model based on 2 angles on hub and tip respectively

is adopted for the blade sweep. The same Bezier model is used to model

the tangential curve of blades. Figure 7.2(a) and Figure 7.2(b) show the

configurations of the swept angles and curved angles. In total, there are

204 parameters in the parametric model.

183


(a) Swept angles (b) Curved angles

Figure 7.2: Swept model and curved model of blades. Z: axial direction. θ: circum-

ferential direction

7.1.3 Numerical Model and Mesh

In order to save computational time, a coarse mesh with 83907 cells is used

in CFD simulations during the optimization. Whereas, the performance of

the optimal result will be verified using a fine mesh with 615519 cells.

The full 3D RANS simulations are performed in the FineTM/Turbo soft-

ware package which solves the time dependent Reynolds averaged Navier-

Stokes equations through the finite volume method. The One-equation

low Reynolds number model of Spalart-Allmaras is employed. A second-

order centered scheme with second and fourth order artificial dissipation

is adopted for spatial discretization. The time marching is performed with

an explicit four-stage Runge-Kutta scheme, coupled with local time step-

ping and implicit residual smoothing technologies. The distribution of total

pressure and total temperature with flow direction are imposed at the in-

let. The outlet static pressure is 110000 Pa. The rotating speed is fixed on

17188 rpm.

The CFD simulation usually converges to -6 in around 800 iterations.

Using single Opteron 8387 (2.9GHz) processor, the CPU time is 8 minutes

for the coarse mesh, 64 minutes for the fine mesh. The CPU time spent on

the approximation model building and optimizations will be presented in

following sections.

184

7.2. MULTIPLE WORKING POINTS OPTIMIZATION

7.2 Multiple Working Points Optimization

7.2.1 Setting Statement

The main aim of this optimization is to increase the efficiencies of rotor37

at two working points while keeping the choked flow unchanged, which

leads to a three-objective optimization problem. Two swept angles and two

curved angles are chosen as design variables. The first point works with

98% choked mass flow (98%CMF) rate and another one works with 100%

choked mass flow (100%CMF) rate. The outlet pressure related to these

two working points are 114.2 kPa and 90 kPa, respectively. The population

size and the maximum generation number are set to 100 and 200, respec-

tively. The crossover rate and the mutation rate are kept the same values

to that in the last chapter, 0.9 and 0.033. An initial training database is

built using full factorial factor method [172], including 36 samples of dif-

ferent blade geometries which have been evaluated by CFD simulations

in advance. For each sample, 2 CFD simulations are performed since the

performance of two working points is needed. The design variables and

the corresponding discrete levels are listed in Table 7.2. In order to eval-

uate the performance of multi-objective optimization, a single objective

optimization using weighting functions is also performed for comparison.

Equal weights are set to all three objectives. To ensure the optimization

convergence, a number of 30 global optimization iterations have been per-

formed for the single objective optimization, which needs 60 extra CFD

simulations in total. While, for multi-objective optimization, 4 new sam-

ples are evaluated each time, then 8 CFD simulations are needed in one

global iteration. Therefore, 20 global optimization iterations have been

performed which require 160 extra CFD simulations.

Table 7.2: Discrete levels of design variables (Unit:degree).

Variables Range Discrete level Property

α1 [−5, +5] 3 meridional swept at hub

α2 [−5, +5] 3 meridional swept at tip

β1 [−3, +3] 2 tangential curved at hub

β2 [−3, +3] 2 tangential curved at tip

185


7.2.2 Optimization Results

Figure 7.3 shows the convergence curves and the spacing metrics. Note

that, since the real Pareto front cannot be known beforehand, the conver-

gence metric is modified as the distance between the ith iteration results

to the 20th iteration results. It is seen from the convergence curves that

the global residual is decreased from 0.17 of the first iteration to 0.04 of

the 18th and 19th iterations. The spacing metric fluctuates around 0.025

during the whole iteration process. Hence, it can be considered that the

optimization has been converged. The optimization results of the first it-

eration, the 18th iteration, the 19th iteration and the 20th iteration are

illustrated in Figure 7.4. As can be seen, the results of the first iteration

is still far away from the final results. While, after about 18 global iter-

ations, the optimization results changed little with more iterations, then

the iteration process is considered as being finished.

Figure 7.3: Convergence and spacing

metrics

Figure 7.4: Pareto front of multi-

objective optimization. CFM: choked

mass flow rate. Delta CMF: variation

of the choked mass flow rate. Effi-

ciency(98%CMF): the efficiency at the

working point with 98% choked mass

flow rate. Efficiency(100%CMF): the

efficiency at the working point with

100% choked mass flow rate.

Five typical solutions of the final results are extracted from the Pareto

front, including three extreme solutions and two trade-off solutions:

• the solution with maximum isentropic efficiency with 98% choked

186


mass flow (CMF)

• the solution with maximum isentropic efficiency with 100% choked

mass flow

• the solution with minimum difference of choked mass flow between

optimization and initial geometry

• trade-off solution 1 and 2 with balanced aerodynamic performance of

three objectives.

The comparison of the design variables among all designs mentioned above

are listed in Table.7.3. Two trade-off solutions are quite the same, ex-

cepting that the swept angles at hub are different. All the optimization

tent to increase the value of α2 to 5and decrease the value of β2 to -3,

which can improve the performances of all working points. Both of two

angles reached the bounds of the variation range. Note that, the varia-

tion range is moderately small due to the limitation of the mesh quality.

If α2 is over 5or β2 is lower than -3, it is found the quality of the mesh

generated automatically during the optimization became so poor that the

simulation blows up. The variations of α1 and β1 are not the same for

different optimizations. Increasing of α1 and β1 will improve the perfor-

mance of 98%CMF working point, but decrease the performance of 100%

CMF working point.

Table 7.3: Comparisons of design variables of the multi-objective optimization re-

sults (Unit:degree).

Designs α1 α2 β1 β2

initial geometry 0.753 0.312 0.260 0.654

max efficiency at 98% CMF 1.715 5.0 0.036 -3.0

max efficiency at 100% CMF -3.397 5.0 -3.0 -3.0

min ∆CMF -2.125 5.0 -0.426 1.368

trade-off 1 -0.215 4.998 -2.905 -2.993

trade-off 2 0.001 4.657 -2.262 -3.0

The performances of these five solutions are validated using the fine

mesh. And the same validations are also performed for the initial design

and the result of the single objective optimization. The aerodynamic per-

formance curves of the multi-objective optimization solutions are shown

in Figure 7.5. As can be seen, the performance curves of two trade-off

187


designs are quite close to each other which obtain the best aerodynamic

performance at all objectives. The averaged increase of efficiency is over

1.1% and that of the total pressure ratio is over 1%, while the choked mass

flows only decreased less than 0.04%. Although, the design of “max effi-

ciency 98%CMF” gives the highest efficiencies in most of the range, but

the choked mass flow is decreased significantly. The total pressure ra-

tio declined rapidly near the choked working point. On the contrary, the

choked mass flow of the “max efficiency 100%CMF” design has increased

too much, although it gives a good performance of the total pressure ratio

in most of the range. The choked mass flow of the design with minimum

∆CMF has increased 0.25% which is much higher than the expected value.

This error is due to the difference of the mesh density in optimizations and

validations. The optimizations using the fine mesh need a considerable

computation time, which will be performed in future work.

Figure 7.5: Aerodynamic performance of multi-objective optimization results.

From the comparison, it can be concluded that for a transonic axial

compressor, the backward swept and positive curve would improve the

aerodynamic performance. Based on this conclusion, and also the vari-

ables’ value in Table 7.3, a final design can be obtained with further im-

provements. The four angles can be valued by 0, 5, -3, -3 in turn. The per-

formance of the finial design is validated by the CFD computation using

fine mesh, and compared with the single objective result. The values of the

design variables of the single objective optimization are listed in Table 7.4.

From this table, it is seen that the design variables of the final design and

the single objective result is quite the same, except the small difference of

188


swept angles at hub. The performance curves of the finial design and that

of the single objective optimization result are shown in Figure 7.6. As can

be seen, great improvements of efficiency and total pressure ratio are also

obtained by the single objective optimization. However, the efficiency is a

little bit lower than that of the final design and the choked mass flow in-

creased 0.14% which is much larger than the variation of the final design.

Summarizing the comparison presented above leads to the conclusion that

the multi-objective optimization methods not only can find solutions better

than that of the single objective optimization method, but also can provide

more alternative solutions and more information to designers.

Figure 7.6: Comparison of the aerodynamic performance between the final design

and the single objective optimization result.

Table 7.4: Comparison of design variables between the final design and the single

objective optimization result (Unit:degree).


initial geometry 0.753 0.312 0.260 0.654

single objective -0.855 5.0 -3.0 -3.0

multi-objective (final) 0 5.0 -3.0 -3.0

To compare the changes of the blade geometry, 2D profiles at three

sections, hub, midspan and tip of the blade, are illustrated in Figure 7.7(a)

for the initial design and Figure 7.7(b) for the final design. As can be seen

189


from the comparison, the blade profiles are still kept the same since only

the curved angle and swept angle are adopted as design variables. And

the position of the hub section is unchanged, while the positions of the

midspan section and the tip section moved obviously towards the directions

of forward curve and backward sweep.

(a) Initial design (b) Final design

Figure 7.7: Comparison of the blade profiles at different spanwise sections.

Two well known features of Rotro37 are the inlet shock and the stall

at hub, which involve the most part of the losses. The contour patterns of

relative Mach number of 98% choked mass flow rate at three spanwise sec-

tions are shown in Figure 7.8 for the initial design and in Figure 7.9 for the

finial design. It is seen from the comparison that the shock at 35% span

is moved upstream a little, while at 90% span it is oblique towards down-

stream. And the intensity of the shock is weakened. Figure 7.10 shows

the limiting streamline patterns on the suction side of the blade surface.

A large stall is found at about half chord distance for the initial design.

Close to the outlet corner at the tip, a small spanwisely recirculation flow

region is observed. While, for the final design, this region is vanished and

the separation line is oblique towards the outlet.

The components of the CPU time spent on CFD simulations, the ap-

proximation and the optimization (A&O) in one global iteration are listed

in Table 7.5. In this table, 8 × 8 means one CFD simulation using the

coarse mesh costs 8 minutes and 8 CFD simulations are performed in one

global iteration. And that 64 means one CFD simulation using the fine

mesh costs 64 minutes. The last column of this table suggests the percent

190


(a) At 35% span

(b) At midspan

(c) At 90% span

Figure 7.8: Contour patterns and iso-

lines of the relative Mach number at

the working point with 98% choked

mass flow rate of the initial design.

(a) At 35% span

(b) At midspan

(c) At 90% span

Figure 7.9: Contour patterns and iso-

lines of relative the Mach number at

the working point with 98% choked

mass flow rate the optimized final de-

sign.

191


(a) Initial design (b) Final design

Figure 7.10: Limiting streamline patterns on the suction surface.

of the CPU time spend on the approximation and the optimization (A&O).

For the present work, 28.1% of total computational labor is spent on build-

ing the approximation model and optimization. However, if the fine mesh

is used, this percent will decrease to 4.7% steeply. If the popular size and

evolution generation number are reduced by half, these two percents will

be reduced to 9.9% and 1.3%, respectively. That means the computational

cost is determined mainly by CFD simulations. In other words, the com-

putational labor only depends on the number of training samples which is

related to the number of input variables, the number of output objectives

and the global iterations. The population size and the number of evolution

generations have small impacts on the computational cost, which is quite

different to the methods used by Nain and Deb [176] and Poloni et al. [29]

where the computational cost increases rapidly with the increase of the

population size and the generation number.

Table 7.5: Distribution of the CPU time in one global iteration.

Approximation and optimization(A&O) CFD simulation

population generation CPU time mesh CPU time percent

size number [min] [min] of A&O

50 100 7 83907 8 × 8 9.9%

100 200 25 83907 8 × 8 28.1%

50 100 7 615519 64 × 8 1.3%

100 200 25 615519 64 × 8 4.7%

Table 7.6 shows the comparison of the CPU time cost in optimizations

192

7.3. ROBUST OPTIMIZATION

with and without the approximation model. It can be seen that the CPU

time needed by optimizations with the approximation model is almost two

orders of magnitude lower than that of optimizations without the approxi-

mation model.

Table 7.6: Comparison of the CPU time cost in optimizations with and without the

approximation model

Population Generation Mesh CPU time [hour]

size number With Without

50 100 83907 24 1333

100 200 83907 30 5333

50 100 615519 173 10667

100 200 615519 179 42666

7.3 Robust Optimization

7.3.1 Uncertainty Statement

The outlet static pressure is supposed to be an uncertainty parameter in

a Gaussian distribution with the mean value of 110 kPa and the standard

deviation of 4400 Pa, 4% of the mean value. A second order non-intrusive

probabilistic collocation method is employed for uncertainty quantifica-

tion. According to the NIPRCM described in Section 3.7.3, the lower and

upper bounds of the outlet static pressure are imposed to 102379 Pa and

117621 Pa, respectively. The corresponding weights and collocation points

are listed in Table 7.7. Hence, three CFD simulations with different static

pressure at the outlet are needed for the evaluation of each individual in

the evolution.

Table 7.7: Collocation points and weights.

Collocation points [Pa] Weights

102379 0.166667

110000 0.666667

117621 0.166667

193


7.3.2 Optimization setting Statement

The aim of optimization is to increase the efficiency at design point and

minimize the variation of efficiency to the outlet static pressure, which

leads to a two-objective optimization problem. The settings of the multi-

objective genetic algorithm are same to that of the last section. Single

objective optimizations using different objective strategies are also per-

formed, which are described as below:

1 Deterministic optimization: the objective is to maximize the effi-

ciency of design point with 110000 Pa at outlet.

2 Single objective optimization using weighing functions: the

objective is the minimum value of the sum of penalties F :

F = P1 + P2 (7.1)

Here, P1 and P2 denote the penalties of the mean value of efficiencies

µ and the standard deviation σ. The penalty is defined as follows:

P = W

(Qimp − Q

Qref

)k

(7.2)

where W denotes the weight, Q stands for the values of objective,

Qimp and Qref are the imposed values of objectives and reference

values for normalization. The idea of this optimization strategy is

to transfer two objectives (maximum mean value of the efficiency

and the minimum standard deviation of the efficiency) into one ob-

jective through weighting functions. While, the weights is imposed

in a “soft” way by using a penalty function, not by setting a value

directly. The parameter k is used to adjust the value of penalty into

a reasonable range. The settings of these parameters are listed in

Table 7.8

Table 7.8: Settings of the penalty function

For P1 For P2

W 2 2

Qimp 1 1

Qref 0.5 0.02

k 2 2

194


3 Single objective optimization using a combined objective:the

objective is the maximum value of (µ/σ); µ is the mean efficiency; σis the standard deviation of the efficiency.

4 Multi-objective optimization: using multi-objective framework de-

veloped in last chapter. Two objectives, one is the mean value µ, the

other is the standard deviation σ.

The design variables are the same as that used in Section 7.2.2, whose

ranges and the corresponding discrete levels are listed in Table 7.2. 36

initial training samples are generated by the full factorial factor method.

7.3.3 Optimization Results

The optimization results are listed in Table 7.9, and 2D and 3D geome-

try profiles of the initial design and optimized design are shown in Figure

7.11∼ Figure 7.14. It can be found that the value of α2 and β2 in differ-

ent optimizations are almost the same, which are . The main differences

are located at α1 and beta1. Three sections of 2D profile of blade at hub,

midspan and tip are presented at left side with the corresponding whole

3D configurations of one blade passage presented at right side. Since only

curve and sweet of blades are used to control the shock structures and sec-

ondary flows, the profiles at each section are kept unchanged. The main

difference is found only in radial stacking. Consistent changes to α2 and

β2 are found in all optimization results, which are backward sweet and

positive curve. The changes of α1 and β1 are different for different opti-

mizations. However, their variation amplitudes are small compared with

other angles, which indicates the sweep at hub has a small effect on the

aerodynamic performance. Due to the positive curve and backward sweet,

the positions of the midspan section and the tip section shifted distinctly

towards lower right.

Table 7.9: Values of the design variables (Unit: degree).


initial design 0.753 0.312 0.26 0.654

deterministic optimization -1.03 5.0 -0.64 -3.0

single(weighting function) 0.635 5.0 -2.8 -3.0

single(combined objective) 0.005 5.0 -3.0 -3.0

multi(trade-off) -0.542 5.0 -1.291 -2.979

195


(a) Profiles of blade at hub, midspan and tip (b) 3D model of blade

Figure 7.11: Blade geometry of the initial design.


Figure 7.12: Blade geometry of the trade-off design obtained by the multi-objective

optimization.

196



Figure 7.13: Blade geometry of the single objective optimization result using

weighting functions.


Figure 7.14: Blade geometry of the single objective optimization result using a

combined objective.

197


The comparisons of the aerodynamic performances among the opti-

mized results and the initial design are illustrated in Figure 7.15. As can

be seen, all optimization results outperformed the initial design. Two sin-

gle objective optimizations give the best results, whose standard devia-

tions are smaller than that of the trade-off design and the deterministic

optimization result. While, the trade-off design is a little better than the

deterministic optimization result. It has to be mentioned that the trade-off

design depends on the decision making method. Never the less, different to

the situation in the last section, the same total numbers (90) of CFD sim-

ulations are performed for all optimizations. That means only 10 global

iterations are performed in the multi-objective optimization. Obviously,

this is relatively inadequate for the multi-objective optimization. Figure

7.16 shows the convergence and spacing metrics of the multi-objective op-

timization. From this figure, it is seen that the results of multi-objective

optimization are well converged. However, there exits better results which

is not reached by the multi-objective optimization. Hence, it still can be

concluded that the existing convergence metric should be improved.

Figure 7.15: Comparison of aerody-

namic performances of different opti-

mization designs.

Figure 7.16: Convergence and spacing

metrics for multi-objective optimiza-

tion.

The detailed comparisons of the total performances of all optimiza-

tions are listed in Table 7.10. The results of two single objective optimiza-

tions are quite the same. Although they provide high efficiencies and low

levels of the standard deviation, the total pressure ratios and the mass

flows have decreased compared to the original design. In this sense, the

trade-off solution of the multi-objective optimization provides the most fea-

sible result since it increases the efficiency 0.98% and the total pressure

ratio 0.002 respectively, while the mass flow only increases 0.09 kg/s and

the standard deviation decreases by half.

198


Table 7.10: Comparison of the total performances of different optimization designs.

MEAN STD CMF (kg/s) Pt2/Pt1

Initial 86.44% 0.165% 20.60 2.010

Deterministic 87.37% 0.126% 20.71 2.016

Single (weighting) 87.50% 0.063% 20.61 2.003

Single (combined) 87.48% 0.064% 20.64 2.007

Multi (trade-off) 87.42% 0.108% 20.69 2.012

Getting benefit from the Pareto optimal concept, some alternative so-

lutions are also obtained. The Pareto front obtained by the multi-objective

optimization is shown in Figure 7.17. Note that, the value in this figure

is extracted directly from the optimization process. Since a coarse mesh

is used during optimizations, the value of efficiency is a little bit lower

than that in Table 7.10. From this figure, it can be seen the result of the

first iteration is still far away from the real Pareto front. The standard

deviation of some points is out of the reasonable range due to the predic-

tion error. After 8 iterations, the results almost converged to the Pareto

front. Never the less, more evolution generations and global iterations are

needed for multi-objective which has been emphasized at the beginning of

this section.


Figure 7.17: Pareto front of the multi-objective optimization.

199


7.3.4 Effect of Penalty Setting

For the single objective optimization using weighting functions, the penalty

settings of objection functions shown in Eq. 7.2 maybe have influence on

the optimization results. In order to clarify this question, three single ob-

jective optimizations using different penalty settings are performed and

the optimization results are presented in this section.

Penalty setting 1: To keep the maximum value of penalties for all

objectives be in the same level (this setting is used in Section 7.3.3). The

parameters and the maximum value, the minimum values and the aver-

aged value of the penalty for 36 samples in the initial database are listed

in Table 7.11.

Table 7.11: Penalty setting 1.

W K Qref Penalty

min max averaged

Mean 2 2 0.5 0.1413899 0.1930342 0.1597975

STD 2 2 0.02 0.00022 0.1810029 0.0340682

Penalty setting 2: Using real physical values listed in Table 7.12 as

reference values. These values are obtained by Cristian et al. [135] using

a 4th order non-intrusive probabilistic collocation method.


W K Qref Penalty

min max averaged

Mean 2 2 1 0.035347 0.048258 0.039949

STD 2 2 0.0029 0.00022 0.1810029 0.0340682

Penalty setting 3: To keep the averaged value of the penalties for all

objectives be in the same level. The values of all parameters are listed in

Table 7.13.

The convergence history curves using different penalty settings are

shown in Figure 7.18∼Figure 7.20. As can be seen, the range of the objec-

tive value is influenced by the penalty setting. However, the convergence

behaviors for different settings are almost the same. The optimized solu-

tion of the first iteration is quite close to the final solution. It is seen from

200



W K Qref Penalty

min max averaged

Mean 2 2 1 0.035347 0.048258 0.039949

STD 2 2 0.02 0.00022 0.1810029 0.0340682

the detailed view in the right side figure that the changes of the objective

function are quite small in the following iterations. The finial optimized

values of the objectives are listed in Table 7.14. As can be seen, the dif-

ferences of optimization results using different penalty settings are quite

small.


Figure 7.18: Convergence history using the penalty setting1.

7.3.5 Effect of the Number of Initial Training Samples

According to the discussion in last chapter, the global convergence behavior

of optimizations mainly depends on the accuracy of prediction of artificial

neural network, especially the number and the quality of initial training

samples. Using more high-quality training samples, the more information

can be feed to ANN, and then the prediction is getting more accuracy. In

order to test the influence of the number of initial training samples on the

convergence behavior, three optimization processes using different num-

bers of initial training samples are performed. The settings of the penalty

are the same as that of the penalty setting 1. The convergence curves are

201






202

7.4. SUMMARY

illustrated in Figure 7.21. As can be seen, using 6 initial training sam-

ples, the residual decreased slowly with global iterations. Increasing the

number of initial samples to 10, the best solution will be reached after 4

iterations. While, if the number of initial training samples is increased to

36, the accuracy of prediction of artificial neural network increased quickly,

hence the optimization result at the first iteration is already quite close to

the best solution. The optimization results using different settings of the

(a) 6 samples (b) 10 samples (c) 36 samples

Figure 7.21: Convergence history using different number of initial training sam-

ples.

penalty and different numbers of the initial samples are listed in Table

7.14. As can be seen, using a large number of initial training samples can

improve the finial optimization result to some extent.

Table 7.14: Comparison of the optimization results using different penalty settings.

Penalty Number of Polytropic efficiency Isentropic efficiency

setting initial samples Mean STD Mean STD

6 0.8664 0.001444 0.8528 0.001609

setting 1 10 0.8663 0.000473 0.8528 0.000321

36 0.8671 0.000033 0.8536 0.000352

setting 2 36 0.8670 0.000027 0.8535 0.0003

setting 3 36 0.8670 0.000047 0.8535 0.000299

7.4 Summary

Practical applications of the multi-objective optimization framework on the

blade design of Rotor 37 are performed. The optimization results of the

203


multiple working point optimization and the robust optimization show the

effect of the multi-objective optimization framework. For the multi-point

optimization with three objectives, the multi-objective optimization result

is found better than that of the single objective optimization using weight-

ing functions. While, for the robust optimization with two objectives, the

result of multi-objective optimization shows no superiority than that of

the single objective optimization. The penalty setting has small influence

on the finial optimization result. While the number of the initial train-

ing samples has large influence on both the convergence behavior and the

optimization result.

204

Chapter 8

Conclusions and

Perspectives

8.1 Conclusions

The present thesis has been devoted to two research topics which are both

related to turbomachinery. In the first part, the complex internal flows

in turbomachinery have been investigated by CFD simulations, includ-

ing film cooling jets on a flat plate, film cooling flows in a turbine cas-

cade and 3D unsteady flows in a low speed axial turbine. In the second

part, an aerodynamic optimization framework of turbomachinery, based

on Multi-objective Genetic Algorithm, Artificial Neural Network and CFD

simulations, was developed and applied successfully to the aerodynamic

optimization of rotro 37.

8.1.1 Numerical Simulations of Complex Flows

The objectives of this part is to give a clear understanding of both the local

flow structure nearby the cooling hole and the interaction between cooling

jets and the main flow. A simplified model of jets-in-crossflow on a flat plate

and a planar turbine cascade model with cooling holes at the leading edge

were employed as simulated models, respectively. Both RANS and DES

simulations have been performed on the JICF on a flat plate to simulate

flows under different blowing ratios. For the film cooling flow in a turbine

cascade, RANS simulations were performed to investigate the influence

205

CHAPTER 8. CONCLUSIONS AND PERSPECTIVES

of blowing ratios and cooling hole configurations on the cooling effect and

additional losses. All the simulated results obtained in the present thesis

have been validated with experimental data. In most cases, the simulated

results have good agreements with experimental results. The scientific

contributions of this part research mainly are:

1. The 3D flow structure nearby a cooling hole is revealed clearly through

numerical simulations on JICF. The existence of a critical flow ratios

is found to be related to the stability of the vortex system. The sim-

ulated results show the differences of horseshoe vortex legs and the

CVP under different blowing ratios. The origin of the secondary CVP

is clarified through the flow structure analysis. The comparison be-

tween simulated results of DES simulations and URANS simulations

show the effect of DES on turbulence simulations. It is concluded that

the asymmetric structure is stable and the unsteadiness is mainly

caused by turbulence.

2. The detailed limiting streamline pattern on the blade surface for slot

cooling model is presented. The simulated results reveal the interac-

tion between the cooling jets issued from cooling slots and thy bound-

ary layer on the blade surface, resulting in a large scale separated

flow. The interaction has great influence on the pressure distribution

on the blade surface, especially in the leading edge region. The sim-

ulated results show the influence of blowing ratios and cooling hole

configurations on the separated flow. The additional losses caused

by the cooling jets under different blowing ratios for different cooling

models are quantified. These conclusions can be used to guide the

engineering design of cooling holes on gas turbines.

3. The NLH method is used to handle the treatment of R/S interactions

in a low speed axial turbine. The comparison between the simulated

results and experimental data shows the effectiveness of the NLH

method, which also indicates that the compressibility of the fluid

used in simulations has great influence on simulated results for the

low speed flow. The effect of harmonic number used in simulations

is presented in the thesis. In general, for a turbine with single R/S

interface, 2 harmonics are needed in the simulation.

206

8.1. CONCLUSIONS

8.1.2 Multi-objective Optimization of Aerodynamic De-

sign

The objective of the second part is to develop a coupled optimization frame-

work based on a multi-objective genetic algorithm—NSGA-II and the back

propagation neural network for the aerodynamic design of turbomachin-

ery. The optimization is based on the Pareto optimal concept to overcome

drawbacks of the traditional single objective optimization. The optimiza-

tion framework developed and several improving strategies are validated

by both standard mathematical model problems and practical engineering

applications. The scientific contributions of this part research mainly are:

1. The framework is programmed in C++ and integrated into the Desing3D

platform. Through the verification of typical mathematical model

problems, it is shown that the optimized results using original NSGA-

II and ANN cannot converge to the real Pareto front. Two reasons for

this problem are explained in the thesis. One is the prediction error

of BPNN, the other is that the crowding distance selection strategy

of NSGA-II is sensitive to small errors.

2. An improved crowding distance is proposed in to adjust the original

selection strategy. The simulated results show that the improved

selection strategy can obtain more reasonable results.

3. A coarse-to-fine iteration strategy based on a fuzzy multi-criteria de-

cision making method is proposed in the thesis to improve the pre-

diction accuracy of BPNN. In addition, a boundary control method

is proposed to improve further the ability to capture the boundary of

the Pareto front. The test results of mathematical problems show the

effectiveness of improving strategies.

4. The improved aerodynamic optimization framework is applied on the

multi working points aerodynamic optimization of rotor37. The Pareto

fronts of the multi-objective optimizations are obtained and better

results than that of the single objective optimization is found, which

indicates the effectiveness of the optimization framework developed.

5. A robust aerodynamic optimization based on non-deterministic CFD

simulations and multi-objective optimization is performed. The multi-

objective optimization results are compared with that of the original

design and single objective optimization in detail. The existing con-

vergence metric of multi-objective optimization is discussed. The con-

vergence analysis suggests that the penalty setting of the single ob-

jective optimization using weighting functions has little influence on

207


optimization results and the convergence behavior, while the number

of initial training samples has great influences on both of that.

8.2 Perspectives

Since the analysis of complex flows in turbomachinery is a large and com-

plex topic, only a small part of this topic can be covered in one thesis. Even

for the film cooling flow and 3D unsteady flow which are simulated in the

present thesis, there still are considerable extended analysis should be im-

plemented in future work. The following topics could be good choices:

1. For the JICF on a flat plate, a critical blowing ratio of 0.7 exists for

the model used in this thesis. An interesting question is that if the

critical blowing ratio exists for other hole configuration or not, for

instance, a single circular cooling hole on a flat plate. How is the in-

fluence of hole configurations on the critical flowing ratio? Therefore,

a large number of numerical simulations are needed to seek answers

to these questions. The stability analysis also should be performed

in the following work to provide theoretic supports to the simulated

results.

2. Although the NLH method has been proven to be an efficient nu-

merical simulation method for R/S interactions of turbomachinery, it

has to be improved to work with some preconditioning methods for

low speed flows. Other two unsteady treatments to R/S interactions,

the domain scaling method and the phase lagged method, have been

used to simulate the flow in the same low speed axial turbine. The

simulated results should be compared in detail to evaluate the per-

formances of all these three methods.

3. The prediction accuracy of the approximation model is an important

factor which influences the optimized results of multi-objective opti-

mizations. Efforts could be spent on finding new powerful approxima-

tion model. For instance, Support vector machine, which is developed

in recent years based on the statistical theory, has been proven to

be superior to ANN in considerable applications, especially for cases

with a small samples number. Then, the coupled method should be

modified to replace ANN with SVM and verified with standard test

problems.

4. Robust aerodynamic optimization methodology, based on the uncer-

tainty quantification methods, CFD simulations and multi-objective

208

8.2. PERSPECTIVES

optimization methods, is a state-of-the-art technology. At present,

few related documents and little research experiences can be found

and refereed to. However, due to the excellent application prospects,

considerable efforts should be spent on the exploratory investigation

of this methodology. Many interesting problems remain unclear, for

instance, the influence of objective definitions of the robust optimiza-

tion which is supposed to have great influences on the optimization

results judging from the present applications.

209


210

List of publications

Journal articles

1. X. D. Wang, C. Hirsch, Sh. Kang and C. Lacor. Multi-objective op-

timization of turbomachinery using improved NSGA-II and approxi-

mation model. Comput. Methods Appl. Mech. Engrg., revised version

submitted, 2010.

2. X. D. Wang, Sh. Kang. Application of polynomial chaos on numerical

simulation of stochastic cavity flow. Sci. China Ser. E-Tech. Sci.,

53(10):2853-2861, 2010.

3. X. D. Wang, Sh. Kang. Solving stochastic burgers equation using

polynomial chaos decomposition . J. Eng. Therm., 31(3):393-398,

2010. (In Chinese)

4. X. D. Wang, Sh. Kang. Nonlinear harmonic method in unsteady

numerical simulation on a low speed axial turbine. J. Eng. Therm.,

30(6):949-952, 2009. (In Chinese)

5. X. D. Wang, Sh. Kang. Numerical simulation on the slot film cool-

ing field at the leading edge of a turbine cascade. J. Eng. Therm.,

29(11):1835-1838, 2008. (In Chinese)

Conference proceedings

1. X. D. Wang, Sh. Kang. Numerical Investigation of the Flow Field in

a Turbine Cascade with Different Film Cooling Holes at the Leading

Edge. In Proceedings of ASME Turbo Expo 2008, Berlin, Germany,

June 2008. ASME-GT08-50027.

211

LIST OF PUBLICATIONS

2. X. D. Wang, Sh. Kang and K. Yang. Investigation of a Steam Tur-

bine with Leaned Blades by Through Flow Analysis and 3D CFD

Simulation. In International Conference on Power Engineering 2007,

Hangzhou, China, November 2007. ICOPE Paper 2007-C204.

212

List of projects

participated in

Research projects

1. EU project NODESIM-CFD financed by the European Commission’s

Sixth Framework Programme, AST-CF-2006-030959, 2008-2010.

2. Foundational research project of Credibility Analysis of the CFD Soft-

ware on Aerodynamics of Turbomachinery financed by National Nat-

ural Science Foundation of China, No.90718025, 2007-2010.

3. Foundational research project of Investigation on the design of film

cooling holes financed by National Natural Science Foundation of

China, No.50876028, 2008-2010

Engineering projects

1. The engineering project supported by Beijing Full Three Dimension

Power Engineering Company. Optimization of the low pressure sec-

tions of the 150MW stream turbine. 2006-2007

213

LIST OF PROJECTS

214

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Documents

CFD Simulation of Complex Flows in Turbomachinery and Robust Optimization of Blade Design