CEAN O ENGI N EE RI NG GROUP - cavity.caee.utexas.edu

ENGI N E E RI NG

GOR

Propulsor Blade Design

UP

Austin, TX 78712

CO E A N

ENVIRONMENTAL AND WATER RESOURCES ENGINEERING

DEPARTMENT OF CIVIL, ARCHITECTURAL

Report No. 05−2

and ENVIRONMENTAL ENGINEERING

THE UNIVERSITY OF TEXAS AT AUSTIN

Yumin Deng

December 2005

Performance Database Interpolation and

Constrained Nonlinear Optimization

Applied to

iii

Copyright

by

Yumin Deng

2005

Performance Database Interpolation andConstrained Nonlinear Optimization

Applied toPropulsor Blade Design

by

Yumin Deng, B.E.

Thesis

Presented to the Faculty of the Graduate School

of The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science in Engineering

The University of Texas at Austin

December 2005

Performance Database Interpolation andConstrained Nonlinear Optimization

Applied toPropulsor Blade Design

APPROVED BYSUPERVISING COMMITTEE:

Supervisor:Spyros A. Kinnas

Reader:Jonathan F. Bard

Reader:Hanseong Lee

To family and friends

Acknowledgements

I would like to express my respect and gratitude to my advisor, Prof. Spyros A.

Kinnas, for his valuable advice and continuous support during my master’s program.

He taught me how to be serious and insightful in study and research, which is a very

invaluable thing I have learned.

I would like to thank Dr Jonathan F. Bard for agreeing to be my thesis reader

in spite of his busy schedule. His detail comments and valuable suggestions were of

great help for me to revise my thesis.

I would also like to thank Dr Hanseong Lee for his kindly help. He has taught

me a lot of things in research. Without his instruction, it would be very difficult for

me to finish the thesis on time.

I also appreciate the kindly help from Prof. Justin. E. Kerwin. His instruction

helped me compare the results from the current design method with those from the

well known PVL method.

It is a pleasure for me to work with my CHL friends: Hua, Hong, Yi-Hsiang,

Vimal, Bikash, Fahad and Lei. We worked together and learned from each other.

Their kindly help worths appreciating.

I would also like to thank my family. They are the most important people

of mine. I appreciate the freedom they have given to me. Without their support, it

would be impossible for me to finish my master degree.

v

This work was supported by the Office of Naval Research under the Na-

tional Naval Responsibility for Naval Engineering (NNR-NE) program, through

Florida Atlantic University (Subagreement TRD67). Partial support of this work was

also provided by Phase IV “Consortium on Cavitation Performance of High Speed

Propulsors” with the following current members: AB Volvo Penta, American Bu-

reau of Shipping, Daewoo Shipbuilding and Marine Engineering Co.Ltd., Kawasaki

Heavy Industries, Naval Surface Warfare Center Carderock Division through the

Office of Naval Research (Contract N00014-04-1-0287), Rolls-Royce Marine AB,

Rolls-Royce Marine AS, VA Tech Escher Wyss GMBH, W�� rtsil

�� Propulsion Nether-

lands B.V., and W�� rtsil

�� Propulsion Norway AS.

vi

Performance Database Interpolation and

Constrained Nonlinear Optimization

Applied to

Propulsor Blade Design

by

Yumin Deng, M.S.E.

The University of Texas at Austin, 2005

SUPERVISOR: Spyros A. Kinnas

Abstract

Highest efficiency subject to prescribed requirements has always been the objective

of marine propeller design. As the loading of propeller is increased due to the grow-

ing demands of larger and faster ships, cavitation becomes an important issue requir-

ing special consideration. Furthermore, the complexity of modern propulsion sys-

tems and unsteady non-axisymmetric inflow have made the design procedure much

more complicated and challenging.

A new design method (named CAVOPT-BASE) is presented in this thesis. This

method couples a vortex/source lattice method (MPUF-3A), a database interpolation

method (least squares method called LSM or piecewise linear interpolation method

called LINTP) and a constrained nonlinear optimization technique to determine the

vii

optimum blade geometry under given operating conditions and constraints. At the

first stage, a blade geometry is selected as the base geometry to generate a propeller

family by multiplying the base geometric parameters with corresponding factors.

The performance database associated with the propeller family is constructed by

MPUF-3A. The objective and constraint functions are then approximated from the

database via LSM or LINTP, linking propeller performance with the design vari-

ables. The approximating functions are incorporated into the constrained nonlinear

optimization algorithm to solve for the designed blade geometry with the highest

efficiency. Constraints on the minimum pressure over the blade in the case of fully

wetted flows, and the maximum allowable cavity extent in the case of cavitating

flows are imposed. In the case of ducted/podded propellers design, the duct/pod

geometry is taken to be given and remains unchanged during the design procedure.

When a propeller is designed subject to non-axisymmetric inflow, the effective wake

is approximated by GBFLOW, which is a flow field solver based on finite volume

method.

Several applications of the described design method to the design of open,

ducted, inside tunnel, podded and contra-rotating propellers are presented.

viii

Table of Contents

Acknowledgements v

Abstract vii

List of Tables xiii

List of Figures xvii

Nomenclature xxiii

Chapter 1. Introduction and Literature Review 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Previous propeller analysis and design methods . . . . . . . . 4

1.2.2 Present design method . . . . . . . . . . . . . . . . . . . . . 10

Chapter 2. Numerical Methods of Constrained Nonlinear Programming 162.1 The penalty method . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Augmented Lagrangian penalty function . . . . . . . . . . . 18

2.1.2 Update schemes for Lagrangian multipliers and penalty coef-ficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Quasi-Newton method . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Descent direction . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.2 Line search method . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Scaling and stopping . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Outer iteration and inner iteration . . . . . . . . . . . . . . . . . . . 25

2.5 Optimization validation . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.1 Unconstrained optimization . . . . . . . . . . . . . . . . . . 27

2.5.2 Constrained optimization . . . . . . . . . . . . . . . . . . . 30

ix

Chapter 3. Formulation of Propeller Blade Design 333.1 Blade geometric parameters . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Propeller family generation . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Base geometry and propeller family . . . . . . . . . . . . . . 37

3.2.2 Linearly distributed parameter multipliers . . . . . . . . . . . 38

3.2.3 General base propeller geometries . . . . . . . . . . . . . . . 39

3.3 Database interpolation for propeller performance . . . . . . . . . . . 40

3.3.1 Database generation . . . . . . . . . . . . . . . . . . . . . . 42

3.3.2 Least squares method . . . . . . . . . . . . . . . . . . . . . 43

3.3.3 Piecewise linear interpolation . . . . . . . . . . . . . . . . . 45

3.4 Formulation for optimization . . . . . . . . . . . . . . . . . . . . . 46

3.4.1 Minimum pressure constraint . . . . . . . . . . . . . . . . . 47

3.4.2 Cavity constraint . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 4. Applications of Blade Design Methods to Open Propellers 514.1 Steady fully wetted case subject to axisymmetric inflow . . . . . . . 51

4.1.1 Operating conditions and constraints . . . . . . . . . . . . . 52

4.1.2 Design results . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.3 Comparison of designs from CAVOPT-BASE and CAVOPT-3D for hubless case . . . . . . . . . . . . . . . . . . . . . . 70

4.1.4 Comparison of designs from CAVOPT-BASE and PVL sub-ject to uniform inflow . . . . . . . . . . . . . . . . . . . . . 74

4.1.5 Optimization with linear multipliers for��

. . . . . . . . . 81

4.2 Unsteady fully wetted case subject to non-axisymmetric inflow . . . 84


4.2.2 Design results . . . . . . . . . . . . . . . . . . . . . . . . . 87


. . . . . . . . . 99

4.3 Unsteady cavitating case subject to non-axisymmetric inflow withonly �� inequality constraint . . . . . . . . . . . . . . . . . . . . . 103


4.3.2 Design results . . . . . . . . . . . . . . . . . . . . . . . . . 104


. . . . . . . . . 116

x

4.4 Unsteady cavitating case subject to non-axisymmetric inflow withboth �� constraint and �� constraint . . . . . . . . . . . . . . . 119


4.4.2 Design results . . . . . . . . . . . . . . . . . . . . . . . . . 120

Chapter 5. Applications of Blade Design Methods to Ducted Propellers 1255.1 Steady fully wetted propeller inside tunnel subject to axisymmetric

inflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126


5.1.2 Design results . . . . . . . . . . . . . . . . . . . . . . . . . 128


. . . . . . . . . 137

5.2 Unsteady fully wetted ducted propeller subject to non-axisymmetric inflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141


5.2.2 Design results . . . . . . . . . . . . . . . . . . . . . . . . . 145


. . . . . . . . . 152

5.3 Unsteady cavitating propeller inside tunnel subject to non-axisymmetricinflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156


5.3.2 Design results . . . . . . . . . . . . . . . . . . . . . . . . . 157


. . . . . . . . . 167

Chapter 6. Applications of Blade Design Methods to Other PropulsionSystems 170

6.1 Unsteady fully wetted pushing type podded propeller subject to non-axisymmetric inflow . . . . . . . . . . . . . . . . . . . . . . . . . . 170


6.1.2 Design results . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.2 Steady fully wetted contra-rotating propellers subject to uniform inflow182


6.2.2 Design results . . . . . . . . . . . . . . . . . . . . . . . . . 184

Chapter 7. Conclusions and Recommendations 1957.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

xi

Appendix A 200

Bibliography 213

Vita 221

xii

List of Tables

2.1 Number of iterations and solutions, optimal function values for dif-ferent initial guess (Problem No.1). . . . . . . . . . . . . . . . . . . 27






3.1 Format of the database. . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1 Solutions from LINTP method and LSM method for �� database (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Performance characteristics of the designed propellers from LINTPmethod and LSM method (N4148-based). . . . . . . . . . . . . . . 55

4.3 Relative errors for LSM methods in approximating the �� database (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Relative errors of 4th-order LSM method for different database sizes(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5 Optimal solutions from LINTP method and 4th-order LSM methodfor different database sizes (N4148-based). . . . . . . . . . . . . . 60

4.6 Design results of LINTP method and 4th-order LSM method fordifferent database sizes (N4148-based). . . . . . . . . . . . . . . . 60

4.7 Optimal solutions from 4th-order LSM for different initial guesses(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.8 Optimization solutions from different blade grid sizes solved byLINTP (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . 61

4.9 Optimal efficiency of different �� constraints solved by LINTPand rechecked by MPUF3A. (N4148-based). . . . . . . . . . . . . . 63

xiii

4.10 Optimal efficiencies from CAVOPT-BASE and the contours for dif-ferent �!�� constraints (N4148-based). . . . . . . . . . . . . . . . 65

4.11 Design results based on different base geometries. . . . . . . . . . . 69

4.12 Optimal solutions from CAVOPT-BASE for open propeller designwithout hub. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.13 Design results from CAVOPT-BASE and CAVOPT-3D. . . . . . . . 71

4.14 Optimum efficiencies from CAVOPT-BASE and PVL for "$#&%(')�+*,� - . 76

4.15 Optimum efficiencies from CAVOPT-BASE and PVL for "$# % '.�/*10 . 77

4.16 Optimum efficiencies from CAVOPT-BASE and PVL for "$# % '�+* 2 . (’-’ means there is no feasible point in the database. i.e. thedatabase needs to be extended to provide a solution.) . . . . . . . . 77

4.17 Solutions from LINTP method and LSM method for the � �3�4� �3�� 5�4�� database (N4148-based). . . . . . . . . . . . . . . . . . . . 82

4.18 Performance characteristics of the designed propellers from LINTPmethod and LSM method (N4148-based). . . . . . . . . . . . . . . 82

4.19 Relative errors for LSM method for the � �6�7� �6�8� �9�8� � database(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.20 Iteration results between MPUF3A and GBFLOW for different :<;(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.21 Optimal solutions for open propeller design subject to non-axisymmetricinflow (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . 90

4.22 Design results for open propeller design subject to non-axisymmetricinflow (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . 90

4.23 Approximation errors of the =?>A@ order LSM (N4148-based). . . . . . 90

4.24 Optimal efficiency for different �� constraints determined by LINTP(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


4.26 Optimal solutions for the design problem with four variables. . . . . 100

4.27 Design results for the design problem with four variables. . . . . . . 100

4.28 Optimal solutions of open cavitating blade design (N4148-based). . 107

4.29 Design results of open cavitating blade design (N4148-based). . . . 107

4.30 Approximation errors of the =?>A@ order LSM in open cavitating bladedesign (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . 107

4.31 Design results from different base geometries. . . . . . . . . . . . . 111

4.32 Optimal efficiency for different CA constraints (N4148-based). . . . 113

xiv

4.33 Optimal efficiency and solutions for two design variables generatedby LSM (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . 114

4.34 Optimal solutions of open cavitating blade design with four variables. 117

4.35 Design results of open cavitating blade design with four variables. . 117

4.36 Optimal solutions of open cavitating blade design involving both�� and �( �� constraints. . . . . . . . . . . . . . . . . . . . . . . 122

4.37 Design results of open cavitating blade design involving both �B�and �( �� constraints. . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.1 Solutions from LINTP and LSM for ��7�C��D�C� � database and�E-9�4� -F�D�E- database (N3745-based). . . . . . . . . . . . . . . . . 129

5.2 Performance characteristics of the designed propellers from LINTPand LSM (N3745-based, � �F�4� �5�4�� database). . . . . . . . . . . 130

5.3 Performance characteristics of the designed propellers from LINTPand LSM (N3745-based, �E-5�4�E-9�4� - database). . . . . . . . . . . 130

5.4 Approximation errors for � �3��$�G� � database and �E-3�H�E-$�H�E-database by =I>A@ order LSM (N3745-based). . . . . . . . . . . . . . 130

5.5 Effect of different �� constraints. . . . . . . . . . . . . . . . . . 130


5.7 Optimal solutions of the design problem with four variables. . . . . 138

5.8 Design results of the design problem with four variables. . . . . . . 138

5.9 Iteration results between MPUF3A and GBFLOW for different :<;(N3745-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.10 Optimal solutions of ducted propeller subject to non-axisymmetricinflow solved by LINTP (N3745-based). . . . . . . . . . . . . . . . 148

5.11 Design results of ducted propeller subject to non-axisymmetric in-flow solved by LINTP (N3745-based). . . . . . . . . . . . . . . . . 148

5.12 Performance characteristics generated from the designed geometryby coupling MPUF3A and GBFLOW, together with the requiredcharacteristics from interpolation. . . . . . . . . . . . . . . . . . . 148

5.13 Optimal solutions for the design problem with four variables. . . . . 153

5.14 Design results for the design problem with four variables. . . . . . . 153

5.15 Optimal solutions for cavitating blade design inside tunnel (N3745-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.16 Design results for cavitating blade design inside tunnel (N3745-based).159

5.17 Optimal efficiency for different �� constraints (N3745-based). . . . 162

xv

5.18 Optimal efficiency for different �� constraints (N3745-based). . . . 163

5.19 Optimal solutions for cavitating propeller inside tunnel with fourvariables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.20 Design results for cavitating propeller inside tunnel with four andthree variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.1 "F#KJ LMJ NPO , "F#/ and "F#RQSLUT from different :V; by coupling MPUF3A andGBFLOW (N4148-based). . . . . . . . . . . . . . . . . . . . . . . 175

6.2 Optimal solutions of podded propeller design with respect to "�#/solved by LINTP (N4148-based). . . . . . . . . . . . . . . . . . . . 178

6.3 Design results of the podded propeller design with respect to "�#/solved by LINTP (N4148-based). . . . . . . . . . . . . . . . . . . . 178

6.4 Propeller performance solved by coupling MPUF3A and GBFLOWwith the designed effective wake. . . . . . . . . . . . . . . . . . . . 178

6.5 Thrust and torque coefficients for different combinations of :XWK; and:ZYR; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.6 Interpolation results for "3# J LMJ NPO '.�+*[-\� and "�] J[L^J[NPO ')� . . . . . . . . 188

6.7 Optimal solutions for the CRP propeller design. . . . . . . . . . . . 190

6.8 Designed results from CAVOPT-BASE and rechecked by MPUF3A. 190

1 Geometry Parameters for N4148 propeller [39]. . . . . . . . . . . . 200




xvi

List of Figures

2.1 Armijo’s rule of line search for step size. . . . . . . . . . . . . . . . 24

2.2 Flow chart for constrained nonlinear optimization. . . . . . . . . . 26

2.3 Convergence rates of variables of constrained optimization problemNo.5 starting from (0.0,0.0,0.0). . . . . . . . . . . . . . . . . . . . 32

2.4 Convergence rates of variables of constrained optimization problemNo.6 starting from (0.0,0.0,0.0,0.0). . . . . . . . . . . . . . . . . . 32

3.1 Propeller-fixed coordinate system and propeller geometric parame-ters [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Nondimensionalized propeller geometric parameters [39]. . . . . . . 36

3.3 General base geometry 1. . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 General base geometry 2. . . . . . . . . . . . . . . . . . . . . . . . 41

3.5 Computational cell of the database. . . . . . . . . . . . . . . . . . . 45

3.6 Minimum pressure constraint [16]. . . . . . . . . . . . . . . . . . . 49

4.1 Convergence rate of the optimization problem solved by 0I_�` orderLSM (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Convergence rate of the optimization problem solved by 2baP` orderLSM (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Convergence rate of the optimization problem solved by = >A@ orderLSM (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4 Designed propeller geometry solved by LINTP method (N4148-based). 57

4.5 Pressure coefficients distribution from the designed propeller solvedby LINTP (N4148-based). . . . . . . . . . . . . . . . . . . . . . . 57

4.6 Geometric parameters distribution from LINTP and 4th-order LSM(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.7 Mean circulation distribution of the designed propellers by LINTPand 4th-order LSM (N4148-based). . . . . . . . . . . . . . . . . . 58

4.8 Effect of the �c�� constraints (N4148-based). . . . . . . . . . . . . 62

4.9 Mean circulation distribution of different database sizes from LINTPand LSM (N4148 based). . . . . . . . . . . . . . . . . . . . . . . . 62

xvii

4.10 Contour plot for :ed�'f�I*[� generated by LSM (N4148-based). . . . 66

4.11 Mean circulation distribution from different base geometries. . . . . 68

4.12 Geometric parameters distributions from CAVOPT-BASE and CAVOPT-3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.13 Mean circulation distributions from CAVOPT-BASE and CAVOPT-3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.14 Pressure coefficients distribution from LINTP. . . . . . . . . . . . . 73

4.15 Pressure coefficients distribution from LSM. . . . . . . . . . . . . . 73

4.16 Mean circulation distribution from CAVOPT-BASE and PVL forCB-1 based propellers (Left) and CB-2 based propellers (Right). . . 78

4.17 Mean circulation distribution from CAVOPT-BASE and PVL forN4148 based propellers (Left) and N4119 based propellers (Right). . 79

4.18 Mean circulation distribution from CAVOPT-BASE and PVL forGeneral-1 based propellers (Left) and General-2 based propellers(Right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.19 Convergence rate of the optimization problem solved by 0I_�` orderLSM (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.20 Convergence rate of the optimization problem solved by 2 aP` orderLSM (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.21 Convergence rate of the optimization problem solved by =b>A@ orderLSM (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.22 Iteration scheme between MPUF3A and GBFLOW. . . . . . . . . . 84

4.23 Nominal wake contour for open propeller [43]. . . . . . . . . . . . 85

4.24 Interpolation for "3#�'.�/*,�E- . . . . . . . . . . . . . . . . . . . . . . 89

4.25 Approximate effective wake contour for "�#g'h�+*i�E- . . . . . . . . . 89

4.26 Convergence rates of the design variables solved by =b>A@ order LSM(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.27 Equality residual of the optimization problem solved by =b>A@ orderLSM (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.28 Geometric parameters distributions from LINTP and the = >A@ orderLSM (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.29 Mean circulation distributions from LINTP and the =b>A@ order LSM(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.30 Designed propeller geometry solved by LINTP (N4148-based). . . . 93

4.31 Pressure coefficients distribution from the designed geometry solvedby LINTP (N4148-based). . . . . . . . . . . . . . . . . . . . . . . 93

xviii

4.32 Effective wake rechecked by MPUF3A and GBFLOW from the de-signed propeller (N4148-based). . . . . . . . . . . . . . . . . . . . 95

4.33 Optimal efficiency for different �� constraints determined by LINTP(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.34 Contour plot generated by LSM (N4148-based). . . . . . . . . . . . 98

4.35 Comparison of geometric parameters distributions from three andfour design variables solved by LINTP (N4148-based). . . . . . . . 101

4.36 Comparison of mean circulation distributions from three and fourdesign variables solved by LINTP (N4148-based). . . . . . . . . . . 101

4.37 Pressure coefficient distribution from three design variables solvedby LINTP (N4148-based). . . . . . . . . . . . . . . . . . . . . . . 102

4.38 Pressure coefficient distribution from four design variables solvedby LINTP (N4148-based). . . . . . . . . . . . . . . . . . . . . . . 102

4.39 Wake contours for open cavitating blade design [43]. . . . . . . . . 105

4.40 Convergence rates of the design variables solved by =b>A@ order LSM(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.41 Equality residual of the optimization problem solved by = >A@ orderLSM (N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.42 Geometric parameters distributions from the design results solvedby LINTP and the =I>A@ order LSM (N4148-based). . . . . . . . . . . 109

4.43 Mean circulation distributions from the design results solved by LINTPand the = >A@ order LSM (N4148-based). . . . . . . . . . . . . . . . . 109

4.44 Blade geometry designed solved by LINTP (N4148-based). . . . . . 110

4.45 Cavity shape designed solved by LINTP (N4148-based). . . . . . . 110

4.46 Mean circulation distributions from different base geometries solvedby LINTP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.47 Optimal efficiency for different CA constraints (N4148-based). . . . 112

4.48 Contours plot generated by LSM at :jd�'k�\*[� (N4148-based). . . . . 115

4.49 Geometric parameters distribution from three and four design vari-ables solved by LINTP (N4148-based). . . . . . . . . . . . . . . . 118

4.50 Mean circulation distribution from three and four design variablessolved by LINTP (N4148-based). . . . . . . . . . . . . . . . . . . . 118

4.51 Wake geometry with wave fraction l > 'm�+* n+�E0 for open cavitatingblade design with both �B� and �� constraints. . . . . . . . . . . 121

4.52 Distributions of mean circulation, pressure coefficient and cavitypattern from CAVOPT-BASE for CB-1 based propellers (Left) andCB-2 based propellers (Right). . . . . . . . . . . . . . . . . . . . . 123

xix

4.53 Distributions of mean circulation, pressure coefficient and cavitypattern from CAVOPT-BASE for N4148 based propellers (Left) andN4119 based propellers (Right). . . . . . . . . . . . . . . . . . . . 124

5.1 Convergence rate of the three design variables from = >A@ order LSM(N4148-based, � -9�4�E-5�o�E- database). . . . . . . . . . . . . . . . 131

5.2 Equality residual of the optimization problem solved by =b>A@ orderLSM (N4148-based, �E-9�4�E-5�o�E- database). . . . . . . . . . . . . 131

5.3 Geometric parameters distribution from LINTP and LSM (N3745-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.4 Mean circulation distribution from LINTP and LSM (N3745-based). 132

5.5 Designed propeller geometry from LINTP method (N3745-based). . 133

5.6 Pressure coefficients distribution from LINTP method (N3745-based).133

5.7 Pressure coefficients distribution from LSM (N3745-based). . . . . 134

5.8 Optimal efficiency for different �� constraints (N3745-based). . 134

5.9 Contour plot at :edC' � for different �!�p�+� � constraints (N3745-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.10 Comparison of geometric parameters from three and four designvariables from LINTP (N3745-based). . . . . . . . . . . . . . . . . 139

5.11 Comparison of mean circulation distribution from three and four de-sign variables from LINTP (N3745-based). . . . . . . . . . . . . . 139

5.12 Pressure coefficient distribution from four design variables from LINTP(N3745-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.13 Initial inflow for ducted propeller design [43]. . . . . . . . . . . . . 143

5.14 Duct geometry [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.15 Duct three dimensional geometry [18]. . . . . . . . . . . . . . . . . 144

5.16 Interpolation for "3#KJ LMJ NPOe'.�+*[-\� . . . . . . . . . . . . . . . . . . . . 146

5.17 Effective wake geometry for ducted fully wetted propeller blade de-sign. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.18 Effective wake from the designed propeller (N3745-based). . . . . . 149

5.19 Geometry parameters distribution solved by LINTP (N3745-based). 150

5.20 Mean circulation distribution solved by LINTP (N3745-based). . . . 150

5.21 Propeller blade geometry solved by LINTP (N3745-based). . . . . . 151

5.22 Pressure coefficient distribution solved by LINTP (N3745-based). . 151

5.23 Comparison of geometric parameters from three and four designvariables by LINTP (N3745-based). . . . . . . . . . . . . . . . . . 154

xx

5.24 Comparison of mean circulation distribution from three and four de-sign variables by LINTP (N3745-based). . . . . . . . . . . . . . . . 154

5.25 Pressure coefficients distribution from three design variables by LINTP(N3745-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.26 Pressure coefficients distribution from four design variables by LINTP(N3745-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.27 Inflow for cavitating propeller design inside tunnel [43]. . . . . . . 158

5.28 Geometry Parameters solved by LINTP and = >A@ order LSM (N3745-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.29 Mean circulation distribution solved by LINTP and =b>A@ order LSM(N3745-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.30 Designed Propeller geometry solved by LINTP (N3745-based). . . . 161

5.31 Cavity shape of the designed geometry by LINTP (N3745-based). . 161

5.32 Effect of the �� constraints (N3745-based). . . . . . . . . . . . . . 163

5.33 Cavity shape for different �B� constraints. . . . . . . . . . . . . . . 164

5.34 Cavity shape for different �B� constraints with :rq('k�I* � . . . . . . . 165

5.35 Contour plot generated by LINTP with :jq('k�I*[� (N3745-based). . . 166

5.36 Geometric parameters distribution from three and four design vari-ables solved by LINTP (N3745-based). . . . . . . . . . . . . . . . 169

5.37 Mean circulation distribution from three and four design variablessolved by LINTP (N3745-based). . . . . . . . . . . . . . . . . . . . 169

6.1 Nominal wake geometry for the podded propulsor design with onecomponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.2 Pod and strut 2D geometries of the podded propulsor design withone component. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172



6.5 "F#KJ LMJ NPO , "F#/ and "F#RQSLUT from different :V; by coupling MPUF3A andGBFLOW (N4148-based). . . . . . . . . . . . . . . . . . . . . . . 176

6.6 Effective wake of the podded propeller design (N4148-based). . . . 176

6.7 Effective wake generated from the designed propeller (N4148-based). 179

6.8 Designed podded propeller geometry solved by LINTP (N4148-based).180

6.9 Pressure coefficients distribution from the designed podded propeller(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

xxi

6.10 Designed podded propeller geometric parameters distribution solvedby LINTP (N4148-based). . . . . . . . . . . . . . . . . . . . . . . 181

6.11 Mean circulation distribution from the designed podded propeller(N4148-based). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.12 Base geometric parameters for the design of contra-rotating propellers.185

6.13 Thrust and torque coefficients from For propeller. . . . . . . . . . . 187

6.14 Thrust and torque coefficients from Aft propeller. . . . . . . . . . . 187

6.15 Total thrust and torque coefficients from the propulsion system. . . . 187

6.16 Approximate effective wake for For propeller. . . . . . . . . . . . . 189

6.17 Approximate effective wake for Aft propeller. . . . . . . . . . . . . 189

6.18 Geometric parameters distributions of For and Aft propellers. . . . . 191

6.19 Designed CRP propeller geometry. . . . . . . . . . . . . . . . . . . 191

6.20 Pressure coefficients distribution of For propeller. . . . . . . . . . . 192

6.21 Pressure coefficients distribution of Aft propeller. . . . . . . . . . . 192

6.22 Mean circulation distribution of For propeller. . . . . . . . . . . . . 193

6.23 Mean circulation distribution of Aft propeller. . . . . . . . . . . . . 193

6.24 Effective wake to For propeller generated from designed geometries. 194

6.25 Effective wake to Aft propeller generated from design geometries. . 194

1 Geometry parameters distribution of N4148 propeller. . . . . . . . . 201



4 Pith, chord and camber distribution of N4990 propeller. . . . . . . . 207

5 Rake distribution of N4990 propeller. . . . . . . . . . . . . . . . . 208

6 Skew distribution of N4990 propeller. . . . . . . . . . . . . . . . . 208

7 Pitch, chord camber distribution of CB-1 propeller. . . . . . . . . . 209

8 Rake distribution of CB-1 propeller. . . . . . . . . . . . . . . . . . 210

9 Skew distribution of CB-1 propeller. . . . . . . . . . . . . . . . . . 210

10 Pitch, chord camber distribution of CB-2 propeller. . . . . . . . . . 211

11 Rake distribution of CB-2 propeller. . . . . . . . . . . . . . . . . . 212

12 Skew distribution of CB-2 propeller. . . . . . . . . . . . . . . . . . 212

xxii

Nomenclature

�ts area of the actuator disku blade section chord lengthu �\�non-dimensional chord length distribution along spanwise directionv u �\�gwyx Y{z^| non-dimensional chord length of the base geometryv u �\�gw ` |Sz^}[~ _ non-dimensional chord length of the designed geometryu @} penalty coefficient corresponding to the � -th equality constraintu @} updated penalty coefficient corresponding to the � -th equality

constraintu ~} penalty coefficient corresponding to the � -th inequality constraintu ~} updated penalty coefficient corresponding to the � -th inequality

constraint

��# thrust coefficient; ��#�')� � v �/*1-��?��R� q� w�?�

descent direction for the Quasi-Newton method in � -th iteration�propeller diameter� �positive definite matrix for the Quasi-Newton method in � -th iteration� v : wobjective function for the constrained nonlinear optimization� v : wdatabase approximation function�E�maximum camber distribution�E� � u non-dimensional maximum camber distribution along spanwise directionv � � � u wyx Y{z^| non-dimensional maximum camber distribution of the base geometryv � � � u w ` |Sz^}[~ _ non-dimensional maximum camber distribution of the designed geometry�objective function for the unconstrained nonlinear optimization

xxiii

� _ Froude number based on propeller diameter�

,� _ ')� q �g��

�gravitational acceleration� } v : w � -th inequality constraint for the constrained nonlinear optimization�non-dimensional circulation;

� 'h� � v 0��V��rz w� } v : w � -th equality constraint for the constrained nonlinear optimization�7� ; v : w

inverse Hessian matrix at location :�identity matrix� � advance coefficient based on the ship speed;

� � 'C�Zz � v � ��w� iteration index for the optimization problem

"F# propeller thrust coefficient; "3#g')� � v �?� q �3�pw"F# J[L^J[NSO total thrust coefficient of the propulsion system

"F#/s required thrust coefficient

"F#/ thrust coefficient of the propeller

"F#+� thrust coefficient of the duct

"F#RQ�LUT non-dimensional drag forces of the pod

"�] propeller torque coefficient; "�]�')� � v �?� q ��Rw�

augmented Lagrangian penalty function number of inequality constraints

� number of design variables

� propeller rotational speed¡ number of equality constraints¡ pressure on the propeller blade¡ z @ Y¢W > pressure at the propeller shaft far upstream¡�£ vapor pressure¡Z¤ pressure at far upstream�propeller pitch�¥�\�non-dimensional propeller pitch along spanwise direction

xxiv

v �¥�\��w¦x Y¢zM| non-dimensional pitch of the base geometryv �¥�\��w ` |SzU}1~ _ non-dimensional pitch of the designed geometry� @} v : wpenalty function corresponding to � -th equality

constraint� } v : w

� ~} v : wpenalty function corresponding to � -th inequality

constraint� } v : w

� propeller torque§ radial coordinate§ @ non-dimensional propeller hub radius; §�¨!©�ª � �� propeller radius

�t_ n-dimensional Euclidean space

� � «m-dimensional positive Euclidean space¬ non-dimensional chordwise coordinate¬ } slack variable corresponding to the � -th inequality constraintspecified domain for the design variables®propeller blade section thickness® �maximum thickness distribution® � �\�non-dimensional maximum thickness distribution along

spanwise direction

� propeller thrust

�j thrust due to the propeller

�r� thrust due to the duct¯ ~} Lagrangian multiplier corresponding to the � -thinequality constraint¯ ~} updated Lagrangian multiplier corresponding to the � -thinequality constraint

xxv

° ¤ inflow velocity at far up stream± @} Lagrangian multiplier corresponding to the � -thequality constraint± @} updated Lagrangian multiplier corresponding to the � -thequality constraint

� � ship speed

l²# Taylor wake fraction

: �propeller rake

:Z³ ª}µ´ : ©�ª} lower bound and upper bound of the � -th design variable

: axial coordinate of the propeller-fixed coordinate system

: �solution of the Quasi-Newton method in � -th iteration

:e¶ optimal solution of the constrained nonlinear optimization

:·} �/¸ +# � -th optimal solution of the constrained nonlinear optimization

:·} � -th design variable¹feasible region for the design variables

XR non-dimensional propeller radius; XR ' § � �XPI non-dimensional pitch; XPI 'º� § ® � �V» v § w{� �XCHD non-dimensional chord length; XCHD ' u v § w¢�\�XSKEW non-dimensional propeller skew; XSKEW '.¼ � v § w � � 2I½I�XRAKE non-dimensional propeller rake; XRAKE 'º: � v § w{�\�XCI non-dimensional camber; XCI ' � v § ´ ¬ w¢� u v § wXTI non-dimensional thickness; XTI ' ® v § ´ ¬ w{�\�¾

number of propeller blades¿ �step size of Quasi-Newton method in � -th iteration

� circulation strengthÀ tolerance for convergenceÁ propeller efficiency; Á ' v �bÂr� 0�� w v "5# � "�] wxxvi

¼ �propeller skew distribution

¼ circumferential coordinate of the propeller-fixed system

� fluid densityÃ cavity number based on inflow; Ã ' v ¡ }ZÄ ¡�£ w¢� v �+*[-�� ° q} wÃ _ cavity number based on propeller rotational speed;Ã _ ' v ¡ z @ Y¢W > Ä ¡�£ w¢� v �+*[-��?� q � q w» propeller pitch angle

LINTP linear interpolation method

LSM least squares approximation method

BFGS Broyden-Fletcher-Goldfarb-Shanno method to

approximate the inverse Hessian matrix

�� backside sheet cavity area ratio

CAMAX maximum allowable backside sheet cavity area ratio

�!� pressure coefficient

CPMIN maximum negative pressure coefficient

CAVOPT-3D CAVitating propeller design OPTimization program

CAVOPT-BASE CAVitating propeller design OPTimization program base

on selected BASE propeller geometry

CRP contra-rotating propellers

GBFLOW General Body FLOW solved by finite volume method

MPUF-3A Mit cavitating Propeller Unsteady Force analysis program

(with hub effect) solved by a vortex/source lattice method

RMS relative errors; RMS 'CÅ _}AÆj; v � v :·} w Ä � v :·} w¢w q � Å _}AÆj; � v :·} wwhere

� v : wis the approximating function to

� v : w

xxvii

Chapter 1

Introduction and Literature Review

1.1 Introduction

Cavitation has always been a major concern in propeller design. The pressure on a

blade surface decreases due to the acceleration of the fluid by the propeller. Cavita-

tion occurs when the pressure is below the vapor pressure of fluid and its presence

can lead to propeller erosion. Furthermore, the periodic occurrence and collapse

of cavities due to an unsteady non-axisymmetric inflow leads to periodic pressure

fluctuations and can cause vibrations of the hull and the shaft that may lead to fa-

tigue and failure of these structures. In the past design methods, a propeller was

designed in a way such that the cavitation was completely avoided. Although such

methods have been well applied to lightly loaded propellers, the application to heav-

ily loaded and/or high speed propellers is not appropriate since cavitation is more

likely to occur at these conditions. Additionally, the efficiency of a cavitating blade

may be higher than that of a non-cavitating blade because of less frictional losses at

the parts of the blade which are covered by sheet cavitation. The method presented

in this thesis accounts for the sheet cavitation coverage through an inequality con-

straint. The maximum allowable sheet cavitation coverage is controllable for various

design purposes.

1

As the demands for heavily loaded propellers increase, modern propellers have

been devised with complicated configurations such as ducted, podded and contra-

rotating propellers and waterjet systems. A propeller operating inside an acceler-

ating duct is a desirable configuration in high loading propeller design for higher

efficiency [6]. Both the thrusts from the propeller and the duct contribute to the total

thrust of the propulsion system. The presence of an accelerating duct can accelerate

the inflow to the propeller plane such that cavity is much more likely to occur. The

duct wall effect also increases the circulation strength at the tip and that may lead to

higher efficiencies.

Podded propellers have been another popular type of modern propulsion system

in the last decade. More than one components can be installed onto the pod to pro-

duce larger thrust at higher efficiencies. Furthermore, unlike traditional propellers

which work upstream of a rudder, the thrust of a podded propeller can be easily

directed into any direction by only rotating the pod, and that greatly improves the

maneuverability and does not require the presence of a rudder. Podded propellers

are the favorite in the cases of low ship speed, high loading, especially when con-

trollable thrust direction and precise positioning, as in the case of navigation in ice

and low speed ferries, is required. Another propulsion system to save energy is

contra-rotating propellers (named CRP) having two components rotating in oppo-

site directions around dual shafts. The trailing wakes of the two components in a

CRP have opposite tangential velocities (also called swirl) which, if are designed to

cancel each other, can lead to higher propeller efficiency.

The often complex configuration of modern propulsion systems makes the de-

sign procedure more challenging. Furthermore, the complexity of non-axisymmetric

inflow makes the design much more difficult such that conventional methods based

2

on experimental data, like series charts, are no longer suitable for modern propeller

design. In the present design method, the nominal non-axisymmetric inflow is trans-

formed to the appropriate effective wake which is used during the design procedure.

The method presented in this thesis uses constrained nonlinear optimization

to search for the optimum blade geometry within a performance database. The

database is constructed by performing cavitating propeller analysis using a code

(described later) MPUF-3A for each geometry of a propeller family generated from

a base-propeller by changing the selected geometric parameters. The thrust re-

quirement and the constraints, such as cavity coverage constraint, are incorporated

into the constrained nonlinear optimization. The blade geometry corresponding to

the optimal solution achieves the highest efficiency satisfying the performance re-

quirements under given operating conditions. The method is first applied to non-

cavitating and cavitating open propeller design subject to uniform or non-axisymmetric

inflows. The application is then extended to ducted, inside tunnel, podded and

contra-rotating propulsion systems subject to uniform or non-axisymmetric inflows.

1.2 Literature review

A number of researchers and engineers have contributed their valuable work to the

improvement of propeller blade design. This section reviews the previous literature

for both propeller analysis methods and propeller design methods.

3

1.2.1 Previous propeller analysis and design methods

Possibly the actuator disc is one of the earliest and simplest model for propeller

analysis [21]. The inflow is supposed to be uniform. The propeller is reduced to an

actuator disc with uniform pressure jump at the propeller plane accelerating fluid.

The performance characteristics of the actuator disc are solved by applying mass,

momentum, and energy conservation. The actuator disc is the limit case of a pro-

peller with the highest efficiency at a specified thrust, and corresponds to a hubless

propeller with infinite number of blades and infinitesimally small advance coeffi-

cient and chord length [21].

In the case of a planar wing, Munk [21] gave a theorem stating that adding an

increment of bound vorticity onto a lifting line is the same as adding the identical in-

crement far downstream, which is known as Munk’s theorem. The theorem resulted

in the reciprocity relation [21] in the wing theory.

For uniform inflow, Betz derived a condition for optimal propeller efficiency

which is known as Betz condition [3]. He increased the propeller loading by adding

the increment of bound vorticity far downstream according to Munk’s theorem [21].

Betz’s condition required that the pitch of the resulting helical flow should be radi-

ally constant for optimal radial circulation distribution.

Lerbs [40] derived the Lerbs criterion which was similar to Betz condition for

optimal radial circulation distribution when the inflow was axisymmetric. The pitch

of the resulting flow was found to be proportional to the square root of the inflow

velocity at the same radius.

Prandtl derived an equation for the Prandtl tip factor by solving the problem of

uniform vertical flow stream passing a series of horizontal parallel semi-infinite flat

plates rather than the flow problem induced by helical trailing vortices. The factor

4

was a circulation reduction factor which was the ratio of the circumferential mean

tangential velocity far downstream and the local tangential velocity at a lifting line.

Once the local tangential velocity is known, the optimal circulation distribution can

be determined by Betz’s condition [3].

Goldstein [13] followed Prandtl’s work, and solved the flow problem induced by

infinite helical vorticity surface translating with uniform axial velocity. He solved

the problem for the circulation reduction factor, which is also known as the Gold-

stein factor.

Kramer [35] calculated the ideal propeller efficiency accurately and systemati-

cally for various combinations of number of blades, advance coefficients, and thrust

coefficients. He organized the combinations and made a concise chart, which was

known as Kramer diagram, for convenient application. The designer used the ideal

efficiency to calculate the Goldstein factor and to determine the corresponding opti-

mal circulation distribution.

Following Goldstein’s work, Tachmindji and Milam [53] used numerical meth-

ods to generate tables and graphs for accurate Goldstein factors when computer was

available. Wrench [39] also worked on numerical computation of Goldstein factors

and achieved high-precision values.

Kerwin [20] solved the lifting surface problem by vortex lattice method. The lift-

ing surface was represented by a set of vortex panels placed onto the mean camber

surface. The strengths of vortex panels were solved by satisfying the kinematic and

dynamic boundary conditions. Greeley and Kerwin [14] solved for a general blade

geometry, represented by radial distributions of pitch, chord length, rake, skew, and

thickness. The blade loading and vorticity in the trailing wake were represented by

vortex lattices distributed on the mean camber surface while the blade thickness was

5

accounted for by adding thickness source lattices.

Lee [38] developed a method analyzing non-cavitating and cavitating propellers

by numerical lifting surface theory, which was the original version of PUF-3A de-

veloped at MIT. The effect of cavity was considered by placing cavity source lattices

throughout the cavity extent onto the mean camber surface.

Kinnas [27] improved PUF-3A by adding nonlinear leading-edge correction to

the linear theory of partial cavitating hydrofoils. Kinnas and Fine [30] further im-

proved the vortex/source lattice method for cavity detachment at some specified lo-

cations. To account for the effects of hub and duct in propeller analysis, Kinnas and

Coney [29] developed a generalized image model such that the effects of hub and

duct were included directly into the numerical scheme by adding the corresponding

vorticity images.

The vortex/source lattice method, PUF-3A, has been improved significantly in

the last decade. The method was extended to include general blade thickness sec-

tions, including those of supercaviting propellers, by the work of Kudo and Kinnas

[37]. In addition to traditional blade geometry representation, Mishima and Kinnas

[45] extended the method by using B-Splines to represent a blade geometry. Fur-

thermore, Pyo and Kinnas [48], and Kinnas and Pyo [49] updated the method by

including a wake alignment model for inclined inflow.

Coney [6, 7] developed a design method for the optimal circulation distribution

based on variational optimization. He represented a propeller by concentrated lifting

lines which he discreitzed into a finite number of vortex horseshoes, and described

the thrust and torque as functions of the horseshoes strengths which were solved for

by constrained optimization. The method was easily extended to include the effects

of hub and duct by using the generalized image model [29], and furthermore was ap-

6

plied to multi-component propulsion systems such as propeller-stator combinations

and contra-rotating propellers.

In general, a marine propeller is designed by using either series charts based

on experimental results, like the B-series of MARIN (van Lammeren et al. 1969),

or numerical tools such as the lifting line method and the lifting surface method,

like those developed by Prof. Kerwin at MIT [25, 20]. Although the series charts

have been well applied to conventional propeller design, their application to non-

traditional propellers like ducted or podded propellers is not well established. Con-

ventional design methods design a propeller in three stages. At the first stage, the

optimum circulation distribution is determined by using lifting line theory satisfying

the loading constraints. At this stage, the circumferential averaged inflow to the pro-

peller is utilized and constraints on allowable amount of cavitation are considered

via the Burrill diagram [41]. In the second step, the blade geometry is determined

by using lifting surface method to produce the required loading distribution subject

to the circumferential averaged inflow. The chord length distribution is determined

using the ”cavitation buckets” [54]. In the third stage, the propeller is analyzed in

the actual non-axisymmetric inflow, including the effects of sheet cavitation, and the

design is adjusted, via trial and error, until the required mean thrust is achieved with

acceptable amount of cavitation.

Kinnas in 1993, while at MIT, proposed a new approach for propeller blade

design based on the following philosophy:

Ç Design the shape of a propeller blade via optimization techniques to provide a

specified mean thrust with minimum mean torque when subject to the actual

non-axisymmetric inflow.

7

Ç Couple the optimization techniques with MPUF-3A, the most recent version

of PUF-3A, and impose the appropriate physical constraints for wetted or cav-

itating conditions.

The main advantage of the above approach is that the blade is designed to have its

best efficiency when subject to the actual non-axisymmetic inflow. In addition, this

approach does not require the use of ”cavitation buckets” [54], which are based on

2-D flow around hydrofoils, or the use of the semi-empirical Burrill diagrams [41],

which are based on cavitation extents for old propeller geometries.

Mishima [43] and Mishima and Kinnas [47] developed a numerical method

named (CAVOPT-3D) to determine the blade geometry with best efficiency for spec-

ified thrust and cavity size constraints. A blade geometry was represented by B-

splines and the movements of B-splines vertices were considered to be the design

variables. The propeller performance was described as a function of design vari-

ables whose combination determined a blade geometry. The method consisted of

two steps: linear optimization at the first step and quadratic optimization at the sec-

ond step. In each step, a limited updated database, based on predictions of PUF-3A,

was constructed to generate approximate polynomial functions linking the propeller

performance with the design variables. PUF-3A, the unsteady cavitating propeller

analysis algorithm was then coupled with a constrained nonlinear optimization in

the design method to keep updating the database by adding new solutions until all

the constraints were satisfied. The method was applied to non-cavitating and cavi-

tating blades design subject to uniform or non-axisymmetric inflow.

Griffin [16] and Griffin and Kinnas [17] further improved the propeller analysis

and design methods. In particular, the analysis method was improved such that the

8

cavity detachment location could be searched for across the blade surface automat-

ically. At that time the new version of PUF-3A was renamed the MPUF-3A. The

design method was extended to include one-variable quadratic skew distribution and

three dimensional minimum pressure constraint.

Many numerical methods have been developed for ducted propeller design since

the 1960s. Sparenberg [51, 52] determined the optimum loading distribution on a

propeller surrounded by a duct with finite chord length, and his results showed the

finite loading at the blade tip in the case of zero gap.

George [10] modelled a duct by ring vortices whose strengths varied in both ax-

ial and circumferential directions, and determined the optimum loading distribution

based on Lerb’s criteria for a ducted propeller with zero gap.

Kinnas and Coney [28] developed a lifting surface method to determine the op-

timum radial loading distribution for ducted propellers with a surface panel method

representing the duct geometry, and later their method was extended to account for

the effects of hub and duct using a generalized image model [29].

The coupled axisymmetric RANS (Reynolds Averaged Navier-Stokes) calcula-

tion and the vortex/source lattice method were developed to design ducted propellers

by Kerwin et al [22, 23].

Numerical methods for the prediction of ducted propeller performance have been

developed based on potential theory as well. Gibson and Lewis [11] developed a sur-

face vorticity method coupled with the actuator model to predict the performance of

a ducted propeller, in which the singularities were distributed on the actual duct sur-

faces to represent the duct more exactly and thus to introduce the nonlinear thickness

effect into the calculation of hydrodynamic characteristics.

Glover and Ryan [12] represented the propeller by lifting line and the duct by

9

distributing surface vorticity on the actual duct surface.

The coupled actuator disk model with an Euler solver was developed by Falcao

de Campos [8], and in his method the axisymmetric shear flow effect of a radially

varying inflow field was considered.

The lifting surface and panel methods have been applied to account for the com-

plex blade geometries more accurately by Van Houten [55] and Kerwin et al. [24] in

steady, and by Feng and Dong [9] and Kinnas et al. [32] in unsteady flow.

Recently, Kinnas et al. [31] have coupled the lifting surface method with the

finite volume method to analyze the performance of ducted propellers.

Viscous flow methods have been applied to properly analyze the gap flow at the

propeller tip inside a duct. Kerwin et al. [23, 26] and Warren et al. [56] have devel-

oped the coupled RANS method for the duct with a lifting surface method for the

propeller. Recently, the RANS equations with the k- À turbulence model were applied

to solve the incompressible viscous flow around ducted propeller by Sanchez-Caja

et al. [50]. Abdel-Maksoud and Heinke [1] investigated the scale effect between

model and full scale ducted propeller by solving RANS equations.

1.2.2 Present design method

This thesis presents a new design method which applies to open, ducted, inside

tunnel, podded and contra-rotating propellers subject to uniform or unsteady non-

axisymmetric inflow, including the effect of the blade sheet cavitation. The design

method (named CAVOPT-BASE) is performed by using constrained nonlinear op-

timization based on a database constructed from the performance analysis of a pro-

peller family by using a vortex/source lattice method (named MPUF-3A). As a first

step, a base-propeller is selected arbitrarily from well-known propellers, and then a

10

family of the base-propeller is generated by changing the geometric parameters of

the base-propeller such as pitch, chord, and camber. The process which generates

a family of a base-propeller is automated by multiplying the geometric parameters

by factors within bounded ranges. Once a family is generated, MPUF-3A is run to

predict the propulsive performance of each propeller in the family, and constructs a

database of the performance characteristics. A database includes the information of

geometric parameters, operating conditions (advance ratio and cavitation number),

propeller forces (thrust and torque), minimum pressure (searched over the blade sur-

face and all blade angles) on the back or face of the blade, and the cavity area to

blade area ratio. The constructed database is approximated by polynomials based on

least squares method (LSM) or the piecewise linear interpolation method (LINTP).

Once the coefficients of the polynomial functions are determined, the objective and

constraints functions of the constrained nonlinear optimization are then expressed as

functions of geometric parameters. This technique is an extension of the method of

Mishima (PhD thesis [43]) andd Mishima and Kinnas [46] which was first applied

to the design of cavitating hydrofoils. CAVOPT-BASE searches for the optimum

design parameters under the given operating conditions and loading constraints, in

which the optimal solutions are determined to minimize the torque at a given thrust

and with a maximum allowable sheet cavity area to blade area ratio. In the case of

ducted/podded propellers design, the geometry of duct/pod is assumed to be given,

and remains unchanged during the design procedure. When the inflow to propeller

is non-axisymmetric, the effective wake is approximated at the beginning of the de-

sign procedure. It should be noted that the method of Mishima [43] and Mishima

and Kinnas [47] assumed that the inflow was the effective inflow.

Once the propeller blade is designed for a given inflow, the performance of open,

11

inside tunnel, ducted, podded or contra-rotating propeller is analyzed by MPUF-3A

which models a propeller by a lifting surface method and a duct/pod by image model,

coupled with GBFLOW which uses a finite volume method [4, 5] to analyze the

entire flow field including the effect of duct/pod [33]. MPUF-3A solves the poten-

tial flow around the propeller subject to a non-axisymmetric inflow by distributing

the line vortices and line sources on the blade camber surface, and as a result the

unsteady forces and cavity patterns are determined. In the case of ducted/podded

propellers, a generalized image model is applied to include the effect of the duct or

pod [33]. When the gap between the blade tip and the duct inner surface is small,

the flow discharge model is applied to include the viscous effect [18]. GBFLOW

solves the Euler equations with body force terms converted from the pressure distri-

bution on the propeller blade evaluated by MPUF-3A, and with boundary conditions

of zero normal velocity on the duct/pod surface. The solution of GBFLOW gives the

total velocity inside the flow field, and determines the effective wake by subtracting

the propeller induced velocity from the total velocity. The computed effective wake

is used by MPUF-3A to update the pressure on the blade surface. The iterations

between MPUF-3A and GBFLOW continue until the forces converge.

When a propeller subject to a non-axisymmetric inflow is designed, the nominal

wake needs to be transformed to the effective wake which is updated by coupling

MPUF-3A and GBFLOW. At the beginning of the design procedure, the pitch is

considered to be the only geometric parameter. A subset of the propeller family

is generated by only changing the pitch of the base-propeller. For each propeller

of the family, the unsteady forces (thrust and torque) are determined and updated

by coupling MPUF-3A and GBFLOW until the forces converge. A database corre-

sponding to the propeller family is established including the pitch, operating condi-

12

tions and performance characteristics, and then interpolated to evaluate the desired

pitch which corresponds to the required design loading. Once the desired pitch

is obtained, the propeller geometry can be determined, for which MPUF-3A and

GBFLOW are coupled to obtain the loading and effective wake until the unsteady

forces converge. The effective wake remains unchanged for the rest of the design

procedure. In the second step, a more complete database is constructed by consid-

ering more geometric parameters such as pitch, chord and camber, based on which

CAVOPT-BASE searches for the optimal blade geometry satisfying the operating

conditions, the requirements and the constraints.

The most attractive advantage of the present method is the adaptability to include

different design constraints. Once the database is constructed, the new design blade

geometry can be obtained in minutes when the control values of the constraints are

updated. Additionally, CAVOPT-BASE is designed into a modular structure such

that it can be easily decoupled from MPUF-3A. This means that CAVOPT-BASE

can be used with the most recent version of MPUF-3A, which is expected to predict

the propeller flow and forces, and the cavity shapes more accurately.

The method is first applied to open propeller design both for non-cavitating and

cavitating conditions. Uniform and non-axisymmetric inflows are included in the

cases. For the cases subject to non-axisymmetric inflow, the propeller pitch is con-

sidered to be the only geometric parameter to determine the effective wake by cou-

pling with GBFLOW. Once the effective wake is determined, it is considered to be

unchanged for the rest of the procedure. The design results are compared extensively

to the results from other blade design methods.

The method is then applied to ducted propeller and inside tunnel propeller design

both for non-cavitating and cavitating blades subject to uniform and non-axisymmetric

13

inflow. The effect of the duct/tunnel is accounted for by the simplified image model

[33]. When the gap between the blade tip and the duct/tunnel inner surface is small,

the viscous effect is considered by using the flow discharge model. For the cases

subject to non-axisymmetric inflow, the pitch multiplier is considered to be the only

design variable at the first stage. The function of the loading versus the pitch mul-

tiplier is determined numerically and the propeller pitch for the given loading is

evaluated. The effective wake is obtained for the resulting geometry, and is consid-

ered to be unchanged for the rest of the procedure.

Finally, the method is extended to podded and contra-rotating propellers design

subject to non-axisymmetric inflow. The podded propellers are designed to include

one component. The effect of the pod is accounted for by the generalized image

model. The drag forces of the pod and strut are computed by GBFLOW. Since the

inflow is non-axisymmetric, the effective wake needs to be determined at the begin-

ning of the design procedure by considering the propeller pitch as the only geometric

parameter. The same scheme is used in contra-rotating propeller design to determine

the effective wake. The objective of podded/contra-rotating propellers design is to

maximize the overall efficiency of the propulsion system while satisfying the total

thrust requirement and other constraints.

The current design procedure can be summarized by the following steps:

Ç Select a base geometry and determine the ranges of the factors.

Ç If the inflow is non-axisymmetric or the propeller is ducted, podded or contra-

rotating, the following steps are used to determine the effective wake.

– Consider the pitch multiplier to be the only design variable. For different

pitch multiplier values, compute the corresponding loading by coupling

14

MPUF-3A and GBFLOW.

– Determine the relationship between the loading and the different pitch

multiplier values numerically.

– Compute the pitch multiplier corresponding to the required loading by

interpolation.

– Determine the effective wake using the propeller geometry with the com-

puted pitch multiplier by coupling MPUF-3A and GBFLOW.

Ç Construct the performance database by running MPUF-3A.

Ç Use the least squares method or the piecewise linear interpolation method to

approximate the objective and constraints functions based on the performance

database.

Ç Solve the constrained nonlinear optimization problem for the optimal solu-

tions and the corresponding blade geometry.

Ç Use MPUF-3A to solve for the operating performance using the designed ge-

ometry, and verify that the performance satisfies the requirements and con-

straints.

15

Chapter 2

Numerical Methods of Constrained NonlinearProgramming

A number of numerical methods like primal methods, penalty and barrier meth-

ods, and dual and cutting plane methods [42] have been proposed for constrained

nonlinear optimization problem. The penalty method, one of the most widely used

methods, converts a constrained nonlinear problem to an unconstrained nonlinear

problem by adding penalty terms. As the penalty terms tend to infinity, the solution

to the unconstrained nonlinear problem tends toward the solution of the constrained

nonlinear problem. Even though the penalty method is easy to program, its conver-

gence rate is quite slow due to the ill-conditioned Hessian matrix. To remedy this

disadvantage, the augmented Lagrangian penalty function is used instead of the orig-

inal penalty method. This chapter discusses the numerical method for constrained

nonlinear optimization by augmented Lagrangian and penalty method. The numer-

ical method in this chapter is divided into two parts: the outer iteration and inner

iteration. The outer iteration includes the augmented Lagrangian penalty function

and the update schemes for Lagrangian multipliers and penalty coefficients, which

can be obtained from the well known Karush-Kuhn-Tucker conditions [42]. The

inner iteration solves the unconstrained nonlinear optimization problem by quasi-

16

Newton method. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) [2] method for

approximation to the inverse Hessian matrix and the inaccurate Armijo’s rule line

search method [42] for the step size are utilized in the Quasi-Newton method. In the

last section, some test cases including both unconstrained and constrained problems

have been provided for validation. The algorithm was first developed by Mishima

[43] and incorporated in CAVOPT-2D (for the design of cavitating hydrofoil) and

later in CAVOPT-3D (for the design of cavitating propeller blades). The method is

summarized in the next sections for completeness.

2.1 The penalty method

The constrained nonlinear optimization problem discussed in this chapter has the

following mathematic form:

minimize� v : w

(2.1)

subject to� } v : w '.� �È'É� ´ 0 ´ *i*,* ´ ¡ (2.2)� } v : w Ê � �È'k� ´ 0 ´ *i*,* ´ (2.3)

: Ë (2.4)

The functions� v : w

,� } v : w

and� } v : w

are supposed to be convex and second order

differentiable. The function� v : w

is called the objective function while the functions� } v : ware equality constraints and

� } v : ware inequality constraints. The vector of

the design variables : is defined in

which is supposed to be convex. The set

,

together with the equality and inequality constraints, defines the feasible region for

: , which is denoted by¹

.

¹ 'fÌ :�Ë�� _FÍ :�Ë ´ � } v : w 'h� ´ � } v : w(Ê �/Î (2.5)

17

If¹ Ï')Ð , the optimization problem is feasible.

The general form of the penalty method is to convert the constrained optimization

problem to unconstrained problem by adding the penalty terms.

minimize� v : wVÑ �Ò}AÆj; u @} � @} v : wVÑ �Ò }ÓÆj; u ~} � ~} v : w

(2.6)

subject to :�Ë (2.7)

The functions� @} and

� ~} are the penalty functions corresponding to� } v : w

and� } v : w

.

The coefficients u @} and u ~} are the corresponding penalty coefficients.

u @} Ô � ´ �È'É� ´ 0 ´ *,*i* ´ ¡ (2.8)u ~} Ô � ´ �È'É� ´ 0 ´ *,*i* ´ (2.9)

Generally, the quadratic penalty functions are used.� @} v : w ' �0 � q} v : w(2.10)� ~} v : w ' �0tÕ$Ö�× Ì�� ´ � } v : w Î q(2.11)

2.1.1 Augmented Lagrangian penalty function

To improve the convergence rate of penalty method, the augmented Lagrangian

penalty function is used. As the first step, the inequality constraints can be con-

verted to equality constraints by introducing the vector of slack variables:

¬ ' Ø ¬ ; ´ ¬ q ´ *,*i* ´ ¬ ��Ù #¬ } Ô � ´ �È'f� ´ 0 ´ *i*i* ´ (2.12)� } v : w(Ê � Ú � } v : wVÑ ¬ }r'h� (2.13)

The augmented Lagrangian penalty function is constructed as follows:� v : ´ ¯ ´ ± ´ u @ ´ u ~ ´ ¬ w ' � v : wVÑ �Ò}AÆj; ± } � } v : wVÑ �Ò }AÆj; ¯ }SØ � } v : wÛÑ ¬ } Ù

18

Ñ �Ò }AÆj; �0 u @} � q} v : wÛÑ �Ò }ÓÆj; �0 u ~} Ø � } v : w<Ñ ¬ } Ù q' � v : wVÑ �Ò}AÆj; ± } � } v : wVÑ �Ò }AÆj; �0 u @} � q} v : w

Ñ �Ò }AÆj; �0 u ~} Ø � } v : wVÑ ¬ } Ñ ¯ }u ~} Ù q Ä�Ò}AÆj;

¯ q}0 u ~}where ¯ } and ± } are Lagrangian multipliers associated with the inequality constraints� } v : w

and equality constraints� } v : w

, respectively. The original problem is converted

to minimizing function�

.

Õ3ÜÞÝ Ì � v : w Í :ßË ¹ ÎÚ Õ3ÜÞÝ Ì � v :Ûà ¯ ´ ± ´ u @ ´ u ~ w Í :�Ëá�t_ ´ ¯ Ë8� � ´ ± Ëá� � ´u @ Ëá� � « ´ u ~ Ëá� �« Î

(2.14)

For terms including the slack variables:

Õ3ÜÞÝ Ì�Ò }ÓÆj; �0 u ~} Ø � } v : wVÑ ¬ } Ñ ¯ }u ~} Ù q Î

'�Ò }AÆj; �0 u ~} Õ3ÜÞÝ Ì/Ø � } v : wVÑ ¬ } Ñ ¯ }u ~} Ù q Î (2.15)

which implies the two following situations.

If� } v : wÛÑfâ �ãåä� Ô � æ ¬ }ç'h�

Else if� } v : wÛÑfâ �ã ä �6è � æ ¬ }ç'ÉÄ6Ø � } v : wÛÑfâ �ã ä � Ù (2.16)

so that

Õ$Ü,Ý Ì Å �}AÆj; ;q u ~} Ø � } v : wÛÑ ¬ } Ñfâ �ãåä� Ù q Î ' éê ë Ø � } v : wÛÑfâ �ã ä � Ù q Í � } v : wVÑfâ �ã ä � Ô �� Í � } v : wVÑfâ �ãåä� è �' Õ$Ö�× q Ì � } v : wVÑìâ �ãåä� ´ �/Î

(2.17)

The optimization problem thus will not include the slack variables ¬ . Define the

augmented function which does not include ¬ :

� v :Ûà ¯ ´ ± ´ u @ ´ u ~ w ' � v : wVÑ �Ò}AÆj; ± } � } v : wVÑ �Ò }AÆj; �0 u @} � q} v : w

19

Ñ �Ò}AÆj; �0 u ~} qÕ�Ö�× Ì � } v : wÛÑ ¯ }u ~} ´ �íÎ�Ä�Ò}AÆj;

¯ q}0 u ~} (2.18)

so that for some specifiedv ¯ ´ ± ´ u @ ´ u ~ w

Õ3ÜÞÝ Ì � v : w Í :�Ë�� _ Î�Ú Õ3ÜÞÝ Ì � v : w Í :�Ëá� _ Î (2.19)

The original problem (2.1) thus changes to

Õ3ÜÞÝ Ì � v : w Í :�Ë ¹ ÎÚ Õ3ÜÞÝî�ï â ï £ ï ã^ðpï ã ä Ì � v :Ûà ¯ ´ ± ´ u @ ´ u ~ w Í :ßËá�t_ ´ ¯ Ëá� � ´ ± Ëá� � ´u @ñË�� « ´ u ~ Ëá� �« Î

(2.20)

2.1.2 Update schemes for Lagrangian multipliers and penalty coefficients

Note that the minimum value of� v : w

depends on the values ofv ¯ ´ ± ´ u @ ´ u ~ w

. Chang-

ing the values ofv ¯ ´ ± ´ u @ ´ u ~ w

may further decrease� v : w

. Thus the update schemes

forv ¯ ´ ± ´ u @ ´ u ~ w

are needed in the algorithm. The gradient vector of augmented func-

tion� v : w

is:ò3� v :Ûà ¯ ´ ± ´ u @ ´ u ~ w ' ò � v : wóÑ �Å}AÆj; ± } ò � } v : wÛÑ �Å}AÆj; u @} � } v : w{ò � } v : wÑ �Å}AÆj; u ~} Õ$Ö�× Ì � } v : wVÑìâ �ãUä� ´ �íÎ òF� } v : w

' ò � v : wóÑ �Å}AÆj; Ø ± } Ñ u @} � } v : w Ù ò � } v : wÑ �Å}AÆj; Ø ¯ } Ñ Õ$Ö�× Ì u ~} � } v : w ´ Ä ¯ }�Î Ù òF� } v : w

(2.21)

The Karush-Kuhn-Tucker necessary conditions require that

ò � v : ¶ wVÑ �Ò}AÆj;Vô± } ò � } v : ¶ wVÑ �Ò}AÆj;Èô¯ } ò5� } v : ¶ w 'h� (2.22)

ô¯ } � } v : ¶ w 'h� (2.23)

ô¯ } Ô � (2.24)

20

where :e¶ is the minimum point of the original problem. The update schemes for the

Lagrangian multipliers are as follows when comparing (2.21) with (2.22):

ô± }õ' ± } Ñ u @} � } v : w(2.25)

ô¯ }ç' ¯ } Ñ Õ$Ö�× Ì u ~} � } v : w ´ Ä ¯ }MÎ (2.26)

For the penalty multipliers, the values should be increased to impose the penalty

functions to be zero. In general, the largest penalty should be put to the term with

the maximum violation. Let ö be the index of the most violated constraint:

öV' aug Õ$Ö�× Ì Õ$Ö�× Ì � } v : w ´ �íÎ ´\÷ �?ø v : w ÷1´ �È'É� ´ 0 ´ *i*,* ´ à ù5'k� ´ 0 ´ *i*,* ´ ¡ Î (2.27)

The update schemes for the penalty coefficients are as follows:

ôu @} ')= u @} if�·ú v : w

is the most violated term

ôu ~} '.= u ~} if� ú v : w

is the most violated term

ôu @} 'h0 u @} for � Ï'höôu ~} 'C0 u ~} for � Ï'hö

(2.28)

For summary, the update schemes for Lagrangian multipliers and penalty coeffi-

cients are described in Eqns. (2.25)-(2.28).

2.2 Quasi-Newton method

For fixed Lagrangian multipliers and penalty coefficients, the original problem is

reduced to an unconstrained nonlinear optimization.

minimize� v : w ' � v : wóÑ �Å}ÓÆj; ± } � } v : wÛÑ �Å}AÆj; ;q u @} � q} v : w

Ñ �Å}ÓÆj; ;q u ~} Õ$Ö�× q Ì � } v : wÛÑfâ �ãåä� ´ �/Î¥Ä �Å}AÆj; â&û�q ãUä�subject to :áËá�ü_

(2.29)

21

The unconstrained optimization problem is solved by Quasi-Newton method. The

iteration scheme has the common formulation as follows:

: � « ;�'º: � Ñ ¿ ��?�(2.30)

where�b�

is the descent direction and ¿ �is the step size along the descent direction.

2.2.1 Descent direction

Generally, for Newton methods, the descent direction is computed from the inverse

Hessian matrix. �?� 'ÉÄ � � ; v : w¢ò$� v : w(2.31)

However, the Hessian matrix is not always available and computing the inverse ma-

trix is costly. Instead of computing it directly, the inverse Hessian matrix is usually

approximated with Newton-based algorithms. In this thesis, the Broyden-Fletcher-

Goldfarb-Shanno (BFGS) method is used:

�b� ' Ä � � v : w{ò$� v : � w¡ � ' : � « ;!ÄD: �ý � ' ò$� v : � Ñ � w Ä ò$� v : � wþ � ' ý #� � � ý �¡ # � ý �± � ' ¡ � Ä �þ � � � ý �

� � « ;ÿ' � � Ñ ¡ � ¡ # �¡ # � ý ��Ä � � ý � ý #� � �ý #� � � ý � Ñ þ � ± � ± >� (2.32)

where the initial matrix� s is any positive definite matrix. For simplicity,

� s²' �is

used in the algorithm.

22

2.2.2 Line search method

Once the descent direction is computed by the BFGS method, the step size along the

direction is determined by solving the following optimization problem:

Õ$Ü,Ý� Ì � v : � Ñ ¿ �?� w ´ ¿ Ô �/Î (2.33)

A number of algorithms have been proposed to solve the line search problem, in-

cluding some accurate algorithms such as Fibonacci search and curve fitting meth-

ods, and some inaccurate but fast algorithms such as Armijo’s Rule. For the overall

efficiency of the algorithm, Armijo’s Rule is used to determine the step size with

some sacrifice in accuracy.

Define a function » v ¿ wtogether with the first Taylor’s expansion as follows.

» v ¿ w ' � v : � Ñ ¿ �?� w� » v � wrÑ » � v � w ¿ (2.34)

and » � v � w ' ò3� v : � w # �b� Ê � as required by the property of descent direction.

Armijo’s Rule requires that the step size ¿ should not be too large:

» v ¿ w²Ê » v � wVÑ Ã » � v � w ¿ with � è Ã è � (2.35)

nor too small:

» v Á+¿ w�� » v � wVÑ Ã » � v � w Á+¿ with Á � � (2.36)

The acceptable region of step size ¿ according to Armijo’s Rule is illustrated in

Figure 2.1. Numerically, ¿ is set to be ¿ � Y î in the algorithm. Let be the first

non-negative integer such that: ¿ � '�� » v � � w�Ê » v � wVÑ Ã » � v � w � � (2.37)

23

φ

α

acceptable range

σφ’(0)

σφ’(0)η

0 < σ < 1η > 1

αmin αmax

Figure 2.1: Armijo’s rule of line search for step size.

which is equivalent to: ¿ � '�� v : � w Ä � v : � Ñ � � �?� w Ô Ä Ã � � ò3� v : � w # �?� (2.38)

� ' �+*[- and Ã ' �+*[0 are used in the program. is set to be zero initially and

increased by 1 until the above inequality is satisfied.

2.3 Scaling and stopping

The convergence rate of the algorithm can be affected by the scaling of the vari-

ables : , the scaling of the objective function and constraints functions. Appropriate

scaling can reduce the number of iterations greatly. However, the scaling scheme

is directly related to the optimization problem. One general idea for scaling is to

normalize all the variables and functions by their typical values.

24

The stopping criterion for the quasi-Newton method is provided by Dennis and

Schnabel [19]:

÷ : � « ;ÈÄo: � ÷ Ê �(2.39)

÷ ò$� v : � w ÷ Ê �(2.40)

where�

is a specified tolerance.

For the penalty coefficients, the stopping criterion requires that the summation of

the violated constraints should not exceed the tolerance.�Ò }ÓÆj; ÷ Õ�ÖE× Ì � } v : w ´ �/Î ÷ Ñ �Ò}AÆj; ÷ � } v : w ÷ Ê��(2.41)

2.4 Outer iteration and inner iteration

The overall algorithm for constrained nonlinear optimization presented in this chap-

ter can be divided into two parts: the outer iterations for updating the penalty co-

efficients and Lagrangian multipliers, and the inner iterations for Quasi-Newton

method. The flow chart illustrating the implementation of the algorithm is shown

in Figure 2.2.

2.5 Optimization validation

The nonlinear optimization algorithm is validated in this section. Both uncon-

strained and constrained nonlinear problems are tested. The value of objective func-

tion and number of iterations are investigated for different initial guesses. Please

note that some of the following tests were also presented in Mishima’s PhD the-

sis [43]. However, since some minor changes were made in updating the penalty

25

START

Initialization

END

Convergent ?N

Y

UpdateLagrangianmultipliers andpenalty coefficients

descent direction dk

step sizeαk

new solutionsxk+1 = xk + αk dk

Quasi-Newton Method

Inner Iteration

Outer Iteration

Figure 2.2: Flow chart for constrained nonlinear optimization.

26

coefficients and the stopping criteria, results of the tests are presented here as well.

2.5.1 Unconstrained optimization

The outer iteration is not activated since the problems are unconstrained. The quasi-

Newton method for the inner iteration is tested for the unconstrained problems.

Problem No.1:

Minimize� v : w ' v :V; Ä � w q Ñ v :Zq Ä�0 w q

subject to :�Ëá� q (2.42)

Table 2.1: Number of iterations and solutions, optimal function values for differentinitial guess (Problem No.1).

Initial guess (0.0,0.0) (5.0,5.0) (100.0,100.0) (-100.0,-100.0)Number of inner iterations 3 9 111 115Number of outer iterations 1 1 1 1

Solutions (1.0,2.0) (1.0,2.0) (1.0,2.0) (1.0,2.0)Objective function values 0.0 0.0 0.0 0.0

Problem No.2

27

Minimize� v : w ' ¯ q ; Ñ ¯ qq Ñ ¯ qd

lt� ® � ¯ ;c'k�\*1-üÄ4:X; v ��Ä4:Zq w¯ q²'h0í*10I-üÄo:X; v � Ä4: qq w¯ d²'h0í*[½?0\-�ÄD:X; v � Ä4: dq w

subject to :ßËá� q(2.43)


Initial guess (0.0,0.0) (2.0,2.0) (5.0,5.0)Number of inner iterations 50 13 61Number of outer iterations 1 1 1

Solutions (3.0,0.5) (3.0,0.5) (3.0,0.5)Objective function values 0.0 0.0 0.0

Problem No.3

Minimize� v : w ' Å}AÆj; v ��½Äo� w v :·}eÄ � w q

subject to :�Ë�� (2.44)

Problem No.4Minimize

� v : w 'ÉØ Å ;^s}AÆj; � d v :·}ZÄ � w q Ù�� subject to :�Ë�� ;^s (2.45)

28


Initial guess (0.0,0.0,0.0,0.0,0.0,0.0) (5.0,5.0,5.0,5.0,5.0,5.0)Number of inner iterations 14 23Number of outer iterations 1 1

Solutions (1.0,1.0,1.0,1.0,1.0,1.0) (1.0,1.0,1.0,1.0,1.0,1.0)Objective function values 0.0 0.0


Initial guess (0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0)Number of inner iterations 130Number of outer iterations 1

Solutions (1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0)Objective function values 0.0

Initial guess (5.0,5.0,5.0,5.0,5.0,5.0,5.0,5.0,5.0,5.0)Number of inner iterations 1298Number of outer iterations 1

Solutions (1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0)Objective function values 0.0

29

2.5.2 Constrained optimization

The outer iterations for updating the Lagrangian multipliers and penalty coefficients

are activated in the constrained problems. The convergence rates of the variables are

investigated, as shown in Figures 2.3 and 2.4.

Problem No.5

Minimize� v : w '.0�: q ; Ñ 0E: qq Ñ : qd Ñ 0�:X;y:Zq Ñ 0�:X;P:ZdÄ�n�:X; ÄG½�:Zq ÄH=\:Zd Ñ��

subject to :V; Ñ :Zq Ñ 0E:Zd Ê 2:·} Ê � ´ � 'f� ´ 0 ´ 2:ßËá� d

(2.46)


Initial guess (0.0000,0.0000,0.0000)Number of outer iterations 23

Solutions (1.0000,0.8889,0.5556)Objective function values 0.2222

Initial guess (0.5000,0.5000,0.5000)Number of outer iterations 14

Solutions (1.0000,0.8889,0.5556)Objective function values 0.2222

30

Problem No.6

Minimize� v : w ' v :X;!Ä � w q Ñ v :Zq ÄG0 w q Ñ v :Zd ÄH2 w q Ñ v : � ÄH= w q

subject to :V; ÄG0B'.�: qd Ñ : q� ÄG0B'.�:ßËá� d

(2.47)


Initial guess (0.0000,0.0000,0.0000,0.0000)Number of outer iterations 18

Solutions (2.0000,2.0000,0.8485,1.1314)Objective function values 13.8579

Initial guess (0.5000,0.5000,0.5000,0.5000)Number of outer iterations 20

Solutions (2.0000,2.0000,0.8485,1.1314)Objective function values 13.8579

31

Outer iteration number

x 1,x 2,

x 3

0 5 10 15 200

0.5

1

1.5

x1

x2

x3

Figure 2.3: Convergence rates of variables of constrained optimization problemNo.5 starting from (0.0,0.0,0.0).


x 1,x 2,

x 3,x 4

0 5 10 150

1

2

3

4x1

x2

x3

x4

Figure 2.4: Convergence rates of variables of constrained optimization problemNo.6 starting from (0.0,0.0,0.0,0.0).

32

Chapter 3

Formulation of Propeller Blade Design

In general, most of the blade design methods search for the optimal circulation dis-

tribution such as the methods based on the well known Betz’s condition [3], Lerb’s

criterion [40] and the variational optimization method developed by Coney [6, 7].

The blade geometry is then designed to develop this circulation distribution [14, 15].

In this work, instead of the optimal circulation distribution method, we follow the

blade optimization method developed by Mishima [43], and Mishima and Kinnas

[47] and further improved by Griffin [16] and Griffin and Kinnas [17]. The blade

design problem can be considered as an optimization problem aimed at maximizing

the propeller efficiency while satisfying the constraints. Such constraints include

the equality constraint for thrust coefficient "$# , and the inequality constraints for

minimum pressure CPMIN and maximum cavity area ratio CAMAX. The meth-

ods solve the optimization problem to obtain the optimal blade geometry. Then the

performance characteristics, such as circulation distribution, are evaluated from the

designed propeller blade as a by-product.

This chapter discusses the fundamental idea of the blade design method. As a

first step, a so-called base geometry is selected to generate a propeller family by

multiplying the geometric parameters of the base propeller with design variables. A

33

vortex/source lattice method, MPUF-3A [39], is applied over the propeller family at

the specified inflow conditions to construct a performance database, which is then

approximated by polynomial functions. Two methods for database interpolation,

the least squares method and the piecewise linear interpolation method, are used

in order to evaluate the coefficients of the polynomial functions and describe the

performance characteristics, such as the thrust coefficient "�# and torque coefficient

"�] , as functions of the design variables. Thus the blade design problem is changed

to a constrained nonlinear optimization which is solved by the algorithm described

in the previous chapter. The performance, such as mean and unsteady forces and

moments on the blade or shaft, maximum cavity area to blade area ratio, the mean

circulation distribution and unsteady pressure distributions over the blade are easily

computed for each given geometry by the vortex/source lattice method (MPUF-3A).

3.1 Blade geometric parameters

A three dimensional coordinate system is used to describe the propeller geometry.

As shown in Figure 3.1, the : axis is positive along the downstream direction. The� axis is normal to the : axis and attached to the key blade. The � axis follows the

definition of the right-handed system. The propeller-fixed cartesian system is used

with the following corresponding cylindrical coordinates.

: ' :§ ' � � q Ñ � q¼ ' tan

� ; �� (3.1)

As shown in Figure 3.1, the propeller geometry can be described by the follow-

ing parameters: pitch angle » v § w, chord length u v § w

, skew ¼ � v § w, rake : � v § w

, camber

34

Figure 3.1: Propeller-fixed coordinate system and propeller geometric parameters[14].

distribution� v § ´ ¬ w

and thickness distribution® v § ´ ¬ w

, where § and ¬ denote the radial

position and the chordwise position, respectively.

Figure 3.2 shows the non-dimensional geometric parameters of a propeller. The

geometric parameters are nondimensionalized in the following ways [39]:

XR ' § � �XPI ' � § tan » v § w{� �XCHD ' u v § w{�\�XSKEW ' ¼ � v § w � � 2I½I�XRAKE ' : � v § w¢�\�XCI ' �E� v § w{� u v § wXTI ' ® � v § w¢�\�

(3.2)

where � is the radius of the propeller and�

is the diameter:� ' 0\� ;

�� v § wand

35

Figure 3.2: Nondimensionalized propeller geometric parameters [39].

® � v § ware the maximum camber and thickness distributions in the spanwise direction,

respectively.

The number of blades, denoted by¾

, is assumed to be known in this work.

3.2 Propeller family generation

This section describes the procedure to generate a propeller family from the selected

base geometry. In this thesis, the N4148, N4149 and N3745 propellers, the geome-

36

tries of which are given in Appendix A, have been selected as the base geometries

for open and ducted propellers. In addition, two other general base geometries have

been used in the thesis and are described in this section. The design variables (fac-

tors which multiply the characteristics of the base geometry) can be either constant

or linearly distributed along the spanwise direction.

3.2.1 Base geometry and propeller family

In order to generate a propeller family, a proper propeller geometry must be selected

as a base geometry. The propeller family is generated by changing the parameters

of the base geometry. To simplify the situation, three parameters are considered in

this procedure: XPI, XCHD and XCI. The propeller family can be generated by

multiplying the three parameters of the base geometry with different factors. For

convenience, XPI, XCHD and XCI are also denoted by��

, u �\�and

�\� � u , re-

spectively.

v �¥�\�gw ` |�z^}1~ _ ' v �¥�\��w¦x Y¢zM| � :X;v u ��gw ` |SzU}1~ _ ' v u �\�gwyx Y{z^| � :Zqv �E� � u w ` |SzU}1~ _ ' v � � � u wyx Y{z^| � :Zd (3.3)

where :V; , :Zq and :ed are the multipliers corresponding to�¥�\�

, u �\�and

�� u , re-

spectively. The bounds for the designed�¥�\�

, u �\�and

�\� � u should be specified:

v �¥�\�gw ³ ª Ê v �¥�\�gw ` |�z^}1~ _ Ê v �¥�\��w ©�ªv u ��gw ³ ª Ê v u �\�gw ` |�z^}1~ _ Ê v u �\�gw ©�ªv �E� � u w ³ ª Ê v �E� � u w ` |�z^}1~ _ Ê v � � � u w ©�ª (3.4)

The corresponding multipliers :Û; , :Zq and :Zd should be within the domain:

: ³ ª; Ê :X; Ê : ©�ª;37

: ³ ªq Ê :Zq Ê : ©�ªq: ³ ªd Ê :Zd Ê : ©�ªd (3.5)

where superscript LB and UB denote the lower and the upper bounds of each multi-

plier.

Once the base geometry is selected, each combination of :<; , :Zq and :Zd cor-

responds to a propeller geometry within this family. Thus the geometry of each

propeller can be viewed as a function of the three multipliers :<; , :Zq and :Zd , which

can be described as follows:

Designed Geometry ' � v :V; ´ :Zq ´ :Zd w(3.6)

3.2.2 Linearly distributed parameter multipliers

To introduce more flexibility, linearly distributed parameter multipliers are also used

to generate the propeller family. Suppose the non-dimensional pitch distributions are� v § w{�\�at each radius from hub (with non-dimensional radius § @ ) to tip (with non-

dimensional radius 1.0). The design variable :<; is the multiplier corresponding to

the the pitch ratio at § @ while :eq is the multiplier corresponding to the pitch ratio at

the blade tip. The linearly distributed multipliers are given by the following expres-

sion:

:Z��P� v § w '):X; Ñ § � �ºÄ § @� Ä § @v :Zq�Äo:X; w

(3.7)

where : ��P� are the multipliers for the pitch; � is the propeller radius and § @ is the

hub ratio non-dimensionalized by R.

The designed�¥�\�

can be generated as follows:

38

v ��gw ` |SzU}1~ _ ' v �¥�\�gw¦x Y¢zM|²�ß:Z��P� (3.8)

v �¥�\��w ` |SzU}1~ _ should be within the prescribed bounds:

v �¥�\�gw ³ ª Ê v �¥�\�gw ` |�z^}1~ _ Ê v �¥�\�gw ©�ª(3.9)

as well as the design variables:

:Z³ ª; Ê :X; Ê : ©�ª;: ³ ªq Ê :Zq Ê : ©�ªq (3.10)

To use linearly distributed multipliers, two design variables are required for each ge-

ometric parameter. The linearly distributed multipliers can generate larger propeller

family, including the one generated by constant multipliers.

3.2.3 General base propeller geometries

The selection of the base geometry is an important part in the design procedure since

the propeller family is constructed from the base propeller geometry. Different base

geometries will generate different propeller families. Two general base geometries

are considered in this thesis with simplified parameters distribution. One of these

has uniform��

, linearly distributed�� u , and elliptic distributed u �\�

over the

blade radius :

�¥�\� ' �I* �

39

u �\� ' �+*[0I-�� ¥Ä v § � �)Ä § @ w qv � Ä § @ w q� � � u ' �+* �+� � Ñ �/*[�I2I0 § � �ºÄ § @� Ä § @ (3.11)

Another general base geometry has uniform�¥�\�

and�� u , elliptically distributedu �\�

over the blade radius:

�¥�\� ' �I* �u �\� ' �+*[0I-�� Ä v § � �ºÄ § @ w qv � Ä § @ w q�E� � u ' �+* �?0I- (3.12)

where � is the propeller radius and § @ is hub ratio.

The geometric parameters of the two general base geometries are shown in Fig-

ures 3.3 and 3.4. The rake and skew of the blade are assumed to be zero for both

cases. The maximum thickness distributions® � �\�

are the same. NACA66 thickness

form and � '.�+*[n meanline are used.

3.3 Database interpolation for propeller performance

This section discusses the database generation and interpolation. At first, a database

associated with a propeller family is generated by the vortex/source lattice method

MPUF-3A, including the design variables and the propeller performance coeffi-

cients. The database is the core of the design procedure. Based on this database, the

unknown coefficients of the approximation functions are calculated to link the per-

formance coefficients with the design variables. One popular interpolation method

is the least squares method, which minimizes the errors in a global sense. The other

method used in this thesis is the piecewise linear interpolation method. As presented

40

r/R

P/D

,c/D

f/c,t

/D

0.2 0.4 0.6 0.8 10.00

0.25

0.50

0.75

1.00

1.25

1.50

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10P/Dc/Df/ct/D

Figure 3.3: General base geometry 1.

r/R

P/D

,c/D

f/c,t

/D

0.2 0.4 0.6 0.8 10.00

0.25

0.50

0.75

1.00

1.25

1.50

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10P/Dc/Df/ct/D

Figure 3.4: General base geometry 2.

41

in the following chapters, both methods work well for the database interpolation.

3.3.1 Database generation

As discussed in the previous section, each combination of the design variables corre-

sponds to one design propeller geometry. A propeller family is a collection of these

design geometries. For each geometry of the family, the propeller performance such

as the mean value of the thrust coefficient "$# , the mean value of the torque coef-

ficient "$] , the minimum pressure coefficient ( � �� ), and the maximum cavity to

blade area ratio ( �B� ), can be computed by the vortex/source lattice method MPUF-

3A. When the operating conditions are specified, the propeller performance can be

viewed as functions of the propeller geometry:

Propeller performance ' � vPropeller geometry

w(3.13)

Consider Equations (3.6) and (3.13), the propeller performance can be considered

as functions of the design variables:

Propeller performance ' � vdesign variables

w(3.14)

A database can be constructed by computing the propeller performance for all of

the combinations of the design variables. The format of each item of the database

is shown in Table 3.1. Note that the functions in Equation (3.14) are implicit and

unknown. It is necessary to approximate the performance database to find the rela-

tionship between performance and the design variables.

42

Table 3.1: Format of the database.

Propeller performances Design variables"F# ��I"�] �!�� CA :X; :Zq :Zd3.3.2 Least squares method

The least-squares method [42] is one of the most popular methods in approximating

a database. The basic idea is to approximate the database by polynomial functions.

The unknown coefficients of the polynomial functions are computed by minimizing

the sum of the squares of the errors.

For example, suppose the values of some function� v : w

are known at the series

of points :V; ´ :Zq ´ *i*i* ´ : �. The function values are denoted by

� v :Û; w ´ � v :Zq w ´ *,*i* ´ � v : � w.

The polynomial function used to approximate� v : w

can be set up as:

� v : w ' � _ : _ Ñ � _ � ;P: _ � ; Ñ *,*i* Ñ � s (3.15)

where � è , and the coefficient vector ��' v � s ´ � ; ´ *,*i* ´ � _ w.

The errors due to approximation can be expressed as:

� � ' � v : � w Ä � v : � w ´ �$'f� ´ 0 ´ *,*i* ´ (3.16)

The objective is to minimize the summation of the squares of the errors:

Õ$Ü,Ý�Ò� Æj; v � q� w

(3.17)

Define the objective function� v

aw:

� va

w '�Ò� Æj; v � q� w

'�Ò� Æj; Ø � v : � w Ä v � _ : _ � Ñ � _ � ;P: _ � ;� Ñ *,*i* Ñ � s w Ù q

(3.18)

43

Define

ý } ø '�Ò� Æj; v : � w } « ø

�yø '�Ò� Æj; � v : � w v : � w ø

u '�Ò� Æj; � v : � w q

The objective function can be simplified as follows:

� va

w ' a # Qa ÄG0 b # aÑ u (3.19)

where Q ' Ø ý } ø Ù, b ' v � ; ´ � q ´ *i*,* ´ � _ w

.

The minimization problem turns out to be a quadratic minimization. The solutions

are solved by using the first-order necessary condition:

Qa ' b (3.20)

Since Q and b are known, the coefficients a can be obtained by solving the above

linear system of equations.

The least squares method is quite efficient in approximating the database. The

main advantage is that the method minimizes the errors in a global sense. Even

though there are some unreliable data points observed inside the database, the ap-

proximation results will not be affected as much. The accuracy of the least-squares

method is related to the order of the polynomial function used for approximation.

Higher order polynomial functions can always approximate the data better than the

lower order functions. In this thesis, up to = >A@ order polynomial functions are used.

The accuracy is also affected by the data properties. The approximation is satis-

factory if the data distribution is smooth. However, if the data distribution is not

smooth, the approximation error turns out to be quite big. The issue of accuracy will

be further discussed in the following sections where design examples are presented.

44

1

2

3

4

5

6

7

8

Figure 3.5: Computational cell of the database.

3.3.3 Piecewise linear interpolation

Piecewise linear interpolation is another basic and simple approximating scheme.

Suppose that only the multipliers of three geometric parameters are considered as

design variables: :V; , :Zq and :Zd . The interval of the three variables are �g:Û; , �g:Zq and

�g:Zd . The cell formed by the eight pointsv :Û; ´ :Zq ´ :Zd w

,v :X; Ñ ��:X; ´ :Zq ´ :Zd w

,v :X; ´ :Zq Ñ

�g:Zq ´ :Zd w,

v :X; ´ :Zq ´ :Zd Ñ �g:Zd w,

v :X; Ñ �g:X; ´ :Zq Ñ �g:Zq ´ :Zd w,

v :V; Ñ �g:X; ´ :Zq ´ :Zd Ñ �g:Zd w,v :X; ´ :Zq Ñ �g:Zq ´ :Zd Ñ �g:Zd w

andv :X; Ñ �g:X; ´ :Zq Ñ �g:Zq ´ :Zd Ñ �g:Zd w

is shown in Figure 3.5.

At the vertices of the cell, the function values are known as� } v :X;U} ´ :ZqS} ´ :ZdS} w ´ �$'

� ´ *i*i* ´ n . The function value at a pointv :Û; ´ :Zq ´ :Zd w

inside the cell can be calculated by

linear interpolation:� v :X; ´ :Zq ´ :Zd w ' � ;y:X;y:Zq :Zd Ñ � q :X;P:Zq Ñ � d :Zq :Zd Ñ � � :Zd¢:X;Ñ � � :X; Ñ � :Zq Ñ �� :Zd Ñ � � (3.21)

where � ; ´ � q ´ *i*,* ´ � � are the unknown coefficients, which can be calculated using the

values at the eight vertices of the cell.

45

The accuracy of the piecewise linear interpolation method is related to the size

of the cell. If the cell is too big, the method will not be satisfactory. For a reasonably

small cell, the error due to the approximation by piecewise linear interpolation can

be much smaller than that obtained by least-squares method. The approximation

accuracy is also affected by the function values at the vertices. The piecewise linear

interpolation method treats all values at the vertices as “true” values. If the values at

some vertices are not reliable, the method can give false results.

The piecewise linear interpolation method is easy to expand for more design

variables. The terms of the function can be derived from the following expression:

v :X; ´ � w"! v :Zq ´ � w"! *i*,* ! v : _ ´ � w(3.22)

where the operation!

means component-wise multiplication. For example,v : ; ´ � w#!

v :Zq ´ � wgenerates the terms:

v :V;y:Zq ´ :X; ´ :Zq ´ � w.

In the present method, when the piecewise linear interpolation is used, the

method automatically performs nonlinear optimization for each cell within the database,

and gets solutions from that cell. Among these available solutions, the final optimal

solution, ”the best of the best”, is selected with the highest efficiency. Thus the op-

timal solution is globally optimal.

3.4 Formulation for optimization

The objective of the optimization is to achieve the highest propeller efficiency. When

the propeller is required to produce a specified thrust, the optimization problem is

46

the same as minimizing the torque coefficient according to the following equation:

Á ' � �0�� "5#"�] (3.23)

Õ$Ö�× Á Ú Õ3ÜÞÝ "$]when "F# ' "5#+s (3.24)

where "F#/s is the required thrust coefficient. "3#�')"F#/s is the equality constraint of

the optimization problem.

As discussed in the previous sections, the performance coefficients can be ex-

pressed as functions of the design variables. The expression of the functions can be

approximated by least-squares method or piecewise linear interpolation method.

"F#g' �%$"& v :X; ´ :Zq ´ *i* ´ : _ w(3.25)

"�]ß' �'$)( v :X; ´ :Zq ´ *i* ´ : _ w(3.26)

3.4.1 Minimum pressure constraint

The minimum pressure on the propeller blades is an important parameter to distin-

guish whether the propeller is operating with cavitation or not. The pressure on the

blade surface drops due to high velocities on the surface. If the pressure is lower

than the vapor pressure of the fluid, the fluid will be vaporized and cavity will occur.

The minimum pressure needs to be constrained if the design propeller is required

to operate under fully wetted condition [16]. The minimum pressure is described

by �c�� which is closely related to the cavitation number Ã _ defined as the non-

dimensional vapor pressure. The cavitation number Ã _ and the pressure coefficient

�!� are defined as follows:

Ã _ ' ¡ z @ Y¢W > Ä ¡�£�/*1-��?� q � q (3.27)

47

(3.28)

�!� ' ¡ Ä ¡ z @ Y¢W >�?� q � q (3.29)

where ¡·£ is the vapor pressure and ¡ z @ Y¢W > is the pressure at the shaft, taken to be the

reference pressure.

Cavity will occur when the pressure at the blade surface ¡ is below the vapor

pressure� £ , i.e.,

Ät�!� Ô Ã _0 (3.30)

Equation (3.30) provides a criterion for cavitation inception, as is also shown in

Figure 3.6, with �c�� defined as the maximum value of Ä�� :

�!��' Õ$Ö�× ÌbÄ��!�IÎ (3.31)

The propeller will be operating without cavitation if

�!�� Ê Ã _0 Ä PTOL (3.32)

where PTOL is the prescribed tolerance.

The upper bound of �� can be specified with the information of the fluid prop-

erties. Equation (3.32) serves as an inequality constraint for the optimization prob-

lem to ensure that the designed propeller operates without cavitation under given

fluid properties.

Like "F# and "�] , �!�p�+� � can also be expressed as a function of the design vari-

ables through database approximation:

�!��' �%* Q �+� � v :X; ´ :Zq ´ *,*i* ´ : _ w(3.33)

48

Chord

-CP

σn / 2

PTOL

Cpmin = max { -Cp }

Figure 3.6: Minimum pressure constraint [16].

3.4.2 Cavity constraint

Cavitation will occur if the criterion (3.32) is not satisfied. The presentation of

cavitation can cause erosion of the propeller blade and lead to excessive hull pressure

fluctuations. The collapse of the bubbles can increase the instability of the propeller

and may even damage the blades. In the old days, propeller was designed to avoid

cavitation due to its undesirable effects. In modern propeller design, however, some

specified amount of cavitation is allowed to produce higher loading and reduce the

frictional force over the blade surface. Thus the propeller efficiency can increase if

cavitation is present.

The parameter �B� is introduced to characterize the cavitation, which is defined

as the ratio of cavity area over blade area [43]:

�B�.' maximum cavity areablade area

(3.34)

49

�� is also a function of the design variables:

��)' �'* Â v :X; ´ :Zq ´ *,* ´ : _ w(3.35)

The cavity constraint requires that �� is under the specified value:

�� ÊCAMAX (3.36)

where CAMAX is the maximum allowed cavity area ratio.

3.4.3 Formulation

The optimization problem for blade design can be summarized as follows:

Õ$Ü,Ý "�] v : w¬ * ® *k"F# v : w ')"F#/s

�!�p�+� � v : w(ÊCPMIN

�B� v : w(ÊCAMAX

:Z³ ª} Ê :·} Ê : ©�ª} �È'É� ´ 0 ´ *,*i* ´ �where :·} ´ �6' � ´ 0 ´ *,*i* ´ � are the design variables and +f' v :Û; ´ :Zq ´ *i*,* ´ : _ w

. "F#/s is

the required thrust coefficient. CPMIN and CAMAX are the maximum allowed

�!�� and �B� , respectively. The inequality constraint on ��p�+� � guarantees that the

propeller always operates under fully wetted condition. The inequality constraint on

�� allows some specified partial cavitation during propeller operation. For fully

wetted cases, the �� constraint is not utilized. For cavitating cases, however, in

addition to the �� constraint for the maximum back side cavity area over blade area

ratio, the �� constraint is also activated to avoid face cavitation.

50

Chapter 4

Applications of Blade Design Methods to OpenPropellers

This chapter describes how to apply the present method (the corresponding code

is called CAVOPT-BASE) to open propeller blades design for the following cases:

steady fully wetted case subject to axisymmetric inflow; unsteady fully wetted case

subject to non-axisymmetric inflow; unsteady cavitating case subject to non-axisymmetric

inflow. Infinite hub is included in all of the design cases. The blade designs are car-

ried out with different base geometries, design variables and approximating func-

tions. The design results for hubless case are compared to those from CAVOPT3D

[34, 43]. The method is also compared to the well known PVL ; method developed

by Professor Justin E. Kerwin of MIT [21].

4.1 Steady fully wetted case subject to axisymmetric inflow

CAVOPT-BASE is applied to design the blade of open fully wetted propeller subject

to axisymmetric inflow. The operating conditions and constraints are specified at the

beginning. The designed results are rechecked by MPUF3A.

,Propeller Vortex Lattice Lifting Line Method

51

4.1.1 Operating conditions and constraints

The open propeller is supposed to operate subject to uniform inflow under non-

cavitating conditions. The design conditions are described as follows.

� � ' �\*[� (4.1)

Ã _ ' ��I�I� (4.2)� _ ' ��I�I� (4.3)¾ ' 2 (4.4)

The maximum thickness distribution (® � �\�

) depends on the required strength of

the blade, and remains unchanged for all members of the propeller family in the

database. The three geometric parameters are bounded as follows (see Eqns (3.3)

and (3.4)):

�+* n Ê v �¥�\�gwyx Y{z^| Ê �\*1-�+*[0 Ê v u �\�gw¦x Y¢zM| Ê �/*1-�+* � Ê v � � � u w¦x Y¢zM| Ê �/*[�In (4.5)

The optimization problem is:

Minimize "�]subject to "F#�'.�+*,� -

�!�p�+� � Ê �I*[-:Z³ ª; Ê :X; Ê : ©�ª;:Z³ ªq Ê :Zq Ê : ©�ªq: ³ ªd Ê :Zd Ê : ©�ªd

(4.6)

52

The effect of the infinite hub is included. The hub radius ratio is § @ 'C�+*10 . Both the

cavitation number Ã _ and Froude number� _ are set to be large values to avoid cav-

itation. The required thrust coefficient is "$#�'h�+*i�E- , which is an equality constraint

for the optimization problem. The inequality constraint � �� Ê �I*[- ensures that the

run is a fully wetted run in the case Ã _ 'C2+* � or higher. The N4148, N4119 and the

two general propellers are used as the base propellers. The geometric parameters

considered are�¥�\�

, u ��and

�� u , with the corresponding design variables :Û; , :Zqand :ed , respectively. Currently, skew is not considered in the design procedure. The

bounds of design variables are determined by the selected base geometry satisfying

the limits of the three parameters.

4.1.2 Design results

N4148-based geometry

To satisfy the bounds of the three geometric parameters, the domain of the design

multipliers can be determined by Eqns (3.3) and (3.4):

�+*[n Ê :X; Ê �I*1-�+*[n Ê :Zq Ê 0/*[��+*[� Ê :Zd Ê 2+*[� (4.7)

The geometric parameters of N4148 propeller are shown in the Appendix A. The

database size is � �� , namely 10 equally distributed computational points

for�¥�\�

, u �\�and

�E� � u , respectively.

53

Effect of database interpolation schemes

Piecewise linear interpolation method (LINTP) and three kinds of least squares

methods (LSM) are used for database interpolation inside CAVOPT-BASE. The op-

timization solutions are shown in Table 4.1. Note that in Table 4.1, :rq �/¸ +#o'm�+* nhits the lower bound of :eq , which means that the efficiency may further increase if

chord length u �\�decreases.

The design results are shown in Table 4.2. The values from MPUF3A are the pre-

dicted performance of the propeller designed by CAVOPT-BASE. Both the equality

constraint "F#G' �+*i�E- and the inequality constraint �c�� Ê �I*[- are satisfied. The

design efficiencies are very close to each other. Both piecewise linear interpolation

method and least squares methods work very well in "$# and ��I"�] interpolation.

For �!�p�+� � , however, piecewise linear interpolation method works better.

The approximation errors RMS of the three least squares methods are shown in

Table 4.3. The =\>A@ order LSM works better than lower orders and has smaller relative

approximation errors. The convergence rates at each polynomial order are shown in

Figures 4.1, 4.2 and 4.3.

The designed propeller geometry solved by linear interpolation method and the

corresponding pressure distribution are shown in Figures 4.4 and 4.5, respectively.

From Figure 4.5, the maximum pressure coefficient is 1.2103, as shown in Table 4.2.

The geometric parameters distributions and mean circulation distribution from

different interpolation methods are compared in Figures 4.6 and 4.7, respectively.

The distributions are close to each other as expected since the design variables from

different interpolation methods are also close to each other, as shown in Table 4.1.

54

Table 4.1: Solutions from LINTP method and LSM method for � �!�� !�� database(N4148-based).

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# min � �I"�]LINTP 1.2667 0.8000 2.1708 0.29940E_�` 1.2402 0.8000 2.3709 0.2925

LSM 2�aP` 1.2569 0.8000 2.2670 0.2982= >A@ 1.2604 0.8000 2.2414 0.2991

Table 4.2: Performance characteristics of the designed propellers from LINTPmethod and LSM method (N4148-based).

CAVOPT-BASE Recheck by MPUF3A"F# � �I"$] �!�� "F# � �I"�] Á �!��LINTP 0.1500 0.2994 1.2101 0.1500 0.2993 79.8% 1.21030�_�` 0.1500 0.2925 1.5000 0.1469 0.2923 80.0% 1.1329

LSM 2�aP` 0.1500 0.2982 1.5000 0.1495 0.2981 79.8% 1.1320=\>A@ 0.1500 0.2991 1.2925 0.1500 0.2991 79.8% 1.1576

Table 4.3: Relative errors for LSM methods in approximating the � �á�.��á�.� �database (N4148-based).

RMS Errors"F# � �I"�] �!��0�_�` 4.43E-04 5.11E-04 7.53E-03LSM 2 aP` 8.40E-05 7.25E-05 6.23E-03= >A@ 1.42E-05 1.37E-05 3.95E-03

55


x 1,x 2,

x 3

0 2 4 6 8 10-3

-2

-1

0

1

2

3

4

5

x1

x2

x3


Equ

ality

resi

dual

2 4 6 8 100

1

2

3

4

5

6

7

Figure 4.1: Convergence rate of the optimization problem solved by 0I_�` order LSM(N4148-based).


x 1,x 2,

x 3

0 1 2 3 4 5 6 7 8 9-1

0

1

2

3

x1

x2

x3


Equ

ality

resi

dual

1 2 3 4 5 6 7 8 90.0

0.5

1.0

1.5

2.0

Figure 4.2: Convergence rate of the optimization problem solved by 2 aP` order LSM(N4148-based).


x 1,x 2,

x 3

0 1 2 3 4 5 6 7 8 9-1

0

1

2

3

x1

x2

x3


Equ

ality

resi

dual

1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

Figure 4.3: Convergence rate of the optimization problem solved by =í>A@ order LSM(N4148-based). 56

X

Y

Z

Figure 4.4: Designed propeller geometry solved by LINTP method (N4148-based).

x / c

-CP

0.25 0.5 0.75 1-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00CPMIN = 1.5

Figure 4.5: Pressure coefficients distribution from the designed propeller solved byLINTP (N4148-based).

57

r / R

P/D

,c/D

f/c

0.2 0.4 0.6 0.8 1

0.50

1.00

1.50

2.00

0.00

0.05

0.10LINTP2nd order LSM3rd order LSM4th order LSM

P/D

c/D

f/c

Figure 4.6: Geometric parameters distribution from LINTP and 4th-order LSM(N4148-based).

r / R

Γ/(2

πRV

S)

0.20 0.40 0.60 0.80 1.000.000

0.005

0.010

0.015

0.020

0.025

0.030

LINTP2nd order LSM3rd order LSM4th order LSM

Figure 4.7: Mean circulation distribution of the designed propellers by LINTP and4th-order LSM (N4148-based).

58

Effect of size of database

The size of database is also essential to database interpolation method. Table 4.4

shows the RMS errors of the =I>A@ order LSM for different database sizes. The RMS

errors are reasonably small for approximation of "$# and � �I"�] when the database

size increases. For �c�� , however, the errors are still relatively big even though they

decrease slowly as the database size increases.

Tables 4.5 and 4.6 show the optimal solutions and the design results from LINTP

method and the =I>A@ order LSM. For the two methods, the optimal solutions and

design results do not change much even though the size increases. LINTP method

is more sensitive to the database size since it is dependent directly on the cell size.

However, in Table 4.5, the optimal solution from � �F�o��3�4�� database is closer to

the optimal solution from 0\�g�40��g�H0\� database than that from the �E-�� E-��E-database. The reason has not been discovered yet and further investigation is needed.

The mean circulation distributions of different database sizes solved by LINTP and

LSM are compared in Figure 4.9.

Effect of initial guess

For unique solution, the optimization results should not be affected by the initial

guess. Table 4.7 shows the different initial guesses and the resulting optimal solu-

tions using the = >A@ order LSM. The optimal solutions remain almost the same for

different initial guesses.

Effect of blade grid size

The blade grid size used in MPUF3A is also considered as a factor in database con-

59

Table 4.4: Relative errors of 4th-order LSM method for different database sizes(N4148-based).

Size RMS"F# "�] �!�� F�o� �F�4� � 1.42E-05 1.37E-05 3.95E-03�E-5�o�E-5�4�E- 4.01E-06 6.81E-06 2.31E-030\�F�á0\�F�80\� 4.47E-06 4.23E-06 1.51E-03

Table 4.5: Optimal solutions from LINTP method and 4th-order LSM method fordifferent database sizes (N4148-based).

Size Optimal SolutionsLINTP = >A@ LSM:X; :Zq :Zd :X; :Zq :Zd� �5�4� �F�o�� 1.2667 0.8000 2.1708 1.2604 0.8000 2.2414�E-9�4�E-5�o� - 1.2510 0.8000 2.3571 1.2605 0.8000 2.24040\�5�80\�F�á0�� 1.2634 0.8000 2.2105 1.2603 0.8000 2.2415

Table 4.6: Design results of LINTP method and 4th-order LSM method for differentdatabase sizes (N4148-based).

Size Design Results from MPUF3ALINTP =\>A@ LSM"F# 10 "�] �!�p�+� � Á "F# � �I"�] �!�� Á� �F�o� �F�o�� 0.1500 0.2993 1.2103 79.8% 0.1500 0.2991 1.1576 79.8%�E-5�o�E-5�o� - 0.1500 0.2994 1.1387 79.7% 0.1499 0.2990 1.1582 79.8%0\�F�á0\�F�á0�� 0.1500 0.2994 1.1821 79.8% 0.1499 0.2990 1.1571 79.8%

Table 4.7: Optimal solutions from 4th-order LSM for different initial guesses(N4148-based).

Initial Guess Optimal solutions:X;^s :ZqPs :ZdPs :X; :Zq :Zd0.8000 0.8000 0.0000 1.2604 0.8000 2.24141.3500 1.4000 1.5000 1.2605 0.8000 2.24071.5000 2.0000 2.0000 1.2605 0.8000 2.2409

60

Table 4.8: Optimization solutions from different blade grid sizes solved by LINTP(N4148-based).

Blade Grid Sizes :V; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# min � �I"�]0\�F� �1.2667 0.8000 2.1708 0.29940��3�D� n 1.2667 0.8000 2.2049 0.2993

struction since the grid size can affect the accuracy of the numerical values from

MPUF3A. Two kinds of grid sizes are used in Table 4.8. The results show that the

optimal solutions from CAVOPT-BASE are only affected slightly.

Effect of �!�p�+� � constraints

As discussed above, the constraint on ��p�+� � is used to guarantee that the propeller

operates under fully wetted conditions. Table 4.9 and Figure 4.8 show the effect of

different �!�� constraints for different thrust coefficient requirements. The values

shown in Table 4.9 are the optimal efficiencies computed by MPUF3A from the de-

signed propeller by CAVOPT-BASE. Note that for some � �� constraints, there are

no optimal solutions since the constraints can not be satisfied for all of the points

of the database. As expected, the optimal efficiency increases as the constraint is

relaxed since the feasible region is enlarged.

61

CPMIN

η

0.50 1.00 1.50 2.00 2.500.60

0.65

0.70

0.75

0.80

0.85

KT = 0.10KT = 0.15KT = 0.20KT = 0.25

Figure 4.8: Effect of the �� constraints (N4148-based).

r / R

Γ/(2

πRV

S)

0.2 0.4 0.6 0.8 10.00

0.01

0.02

0.03

10x10x10 LINTP10x10x10 LSM15x15x15 LINTP15x15x15 LSM20x20x20 LINTP20x20x20 LSM

Figure 4.9: Mean circulation distribution of different database sizes from LINTPand LSM (N4148 based).

62

Table 4.9: Optimal efficiency of different �� constraints solved by LINTP andrechecked by MPUF3A. (N4148-based).

�!�� constraints"F# constraintsÊ �+*[½ Ê �+* n Ê �I* � Ê �\*1- Ê 0/* � Ê 0/*1-"5#�'h�+*i� � 82.1% 83.7% 83.9% 83.9% 83.9% 83.9%"5#�'h�+*i�E- - 78.7% 79.8% 79.8% 79.8% 79.8%"5#�'h�+*[0\� - 73.2% 74.4% 75.4% 75.4% 75.4%"5#�'h�+*[0I- - - 69.4% 70.9% 71.2% 71.2%

63

Optimization with two parameters

To show the effect of �c�� constraint clearly, only two geometric parameters�¥�\�

and u �\�are selected for the design procedure. The following design problem in two

dimensions is considered:

Minimize "�]subject to "F#g'h�+*i�E-

�!�p�+� � ÊCPMIN

�I*,� Ê :X; Ê �I*1-�+*[n Ê :Zq Ê 0/*[�

(4.8)

N4148 propeller is selected as the base geometry. The design variables : ; and :Zqare associated with

��and u �\�

, respectively. The maximum camber distribution�E� � u remains the same as the base geometry.

The contours for constant "3# , �!�� and efficiency Á are shown in Figure 4.10,

generated by LSM. Table 4.10 shows the optimal efficiencies for different �� con-

straints for "F#�'h�/*,�E- . The values from CAVOPT-BASE are the interpolated values

based on LSM scheme. The solutions :Û; and :eq are the results from CAVOPT-

BASE. The values from the contours figure are obtained from Figure 4.10 directly

at the corresponding solution points. Ideally, the optimal efficiency from CAVOPT-

BASE should be the same as that from the contours figure. However, there are

differences between these two kinds of values since there are approximation errors

in generating the contours by LSM.

Table 4.10 also shows that the optimal efficiency increases to some degree as the

64

Table 4.10: Optimal efficiencies from CAVOPT-BASE and the contours for different�!�� constraints (N4148-based).

"5#+�'h�/*,�E- �!�� constraintsÊ �\*[n Ê �I* � Ê 0/* � Ê 0/*[- Ê 2+*[�CAVOPT-BASE 77.9% 78.0% 78.2% 78.2% 78.2%

Optimal ÁContour Figure 78.7% 78.7% 78.7% 78.7% 78.7%:X; 1.2641 1.2657 1.2697 1.2697 1.2697:Zq 1.8468 1.8004 1.7003 1.7003 1.7003

allowable CPMIN increases. The increment, however, is not very much.

Note that for "F#�' �+*,� - and �!�� Ê 0/*[- , the optimal efficiency in Table 4.9

(79.8%) is higher than the efficiency in Table 4.10 (78.2%) since the feasible region

of two dimensional design is a subspace of the feasible region of three dimensional

design.

Figure 4.10 also shows the relationship between the propeller performances and

the geometric parameters. The thrust coefficient "$# is very sensitive to the�¥�\�

factor :X; . �!�� decreases when :V; decreases and :eq increases. Thus for small cav-

itation number Ã _ , the designed propeller has small pitch and wide chord to operate

under fully wetted conditions.

Other base geometries

In order to investigate the effect of the base geometry, other base geometries are

used to generate the propeller families. The bounds of design variables are specified

such that the limits of�¥�\�

, u ��and

�� u are satisfied. The design results from

these propeller families are compared to those from the N4148-based propeller fam-

65

x1

x 2

1.1 1.2 1.3 1.4 1.50.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.90

2.00

KT =

0.15

KT =

0.20

KT =

0.10

η=

75%

η=

87%

η=

86%

η=

85%

η=

84%η

=83%

η=

82%η

=81%

η=

80%

η=

79% η=

78%η

=77%

η=

76% CPM

INM

AX

=7.

0

CPM

INM

AX

=1.

0

CPM

INM

AX

=1.

5

CPM

INM

AX

=2.

0

CPM

INM

AX

=2.

5C

PMIN

MA

X=

3.0

CPM

INM

AX

=3.

5C

PMIN

MA

X=

4.0

CPM

INM

AX

=4.

5C

PMIN

MA

X=

5.0

CPM

INM

AX

=5.

5

CPM

INM

AX

=6.

0C

PMIN

MA

X=

6.5

Figure 4.10: Contour plot for :jd²'É�I* � generated by LSM (N4148-based).

66

ily. The base geometries described in the previous section used in this comparison

include N4119 propeller and the two general propellers described in the previous

section. The database size is � �$�� . To approximate the behavior of � �� better, LINTP method is used.

Table 4.11 shows the design results using different base geometries. The interpo-

lation method works well for "3# , � �I"�] and �c�� for all the base geometries. The

efficiencies are very close to each other for different base geometries. For �� ,

however, the design values are quite sensitive to base geometries.

The mean circulation distributions generated from different base geometries are

different, as shown in Figure 4.11.

Note that all of the base geometries in this section have simple parameter distri-

butions with zero skew.

67

r / R

Γ/(2

πRV

S)

0.20 0.40 0.60 0.80 1.000.00

0.01

0.02

0.03

N4148N4149General-1General-2

Figure 4.11: Mean circulation distribution from different base geometries.

Some conclusions can be drawn by investigation:

Ç For least squares method, higher order polynomial functions can approximate

the database better than lower order polynomial functions in a global sense.

Ç Both linear interpolation method and least squares method can approximate

"5# and ��I"�] satisfactory. However, linear interpolation method works better

in approximating the behavior of �� .

Ç The approximation errors can be decreased by increasing the size of database.

However, from the viewpoint of overall efficiency of the algorithm, � �� database size is acceptable.

Ç The solution from LSM is not affected by the initial guess in this case. How-

68

Table 4.11: Design results based on different base geometries.

CAVOPT-BASE Recheck by MPUF3ABase geometry "3# � �I"�] �!�� "F# ��I"�] Á �!��

N4148 0.1500 0.2993 1.2103 0.1500 0.2994 79.8% 1.2101N4119 0.1500 0.3057 1.5000 0.1509 0.3069 78.3% 1.4862

General-1 0.1500 0.3027 1.4929 0.1505 0.3020 79.3% 1.4805General-2 0.1500 0.3031 1.5000 0.1504 0.3026 79.1% 1.3676

ever, it is recommended that the initial guess is located in the center of the

domain for fast convergence.

Ç The base geometry can affect the design results greatly, including the geomet-

ric parameters distributions and mean circulation distributions. However, the

efficiency did not seem to be affected appreciably.

69

4.1.3 Comparison of designs from CAVOPT-BASE and CAVOPT-3D for hub-less case

The objective is to design an open propeller without hub under fully wetted condi-

tions. The design problem is as follows.

Minimize "�]subject to "3#�'h�+*i�E-

�!�� Ê2.5

�+* n Ê :X; Ê �I*�=�+* n Ê :Zq Ê 0/* ��+* � Ê :Zd Ê 2+* �

(4.9)

Both CAVOPT-BASE and CAVOPT-3D q are used for this design problem. The

design geometry from CAVOPT-3D is selected as the base geometry for CAVOPT-

BASE except for pitch distribution. The pitch distribution is constant radially for

the base geometry. Thus for reasonable consistency, the optimal values of design

variables should be close to 1.0.

The optimal solutions from CAVOPT-BASE are shown in Table 4.12. As ex-

pected, the values of design variables are all close to 1.0. The design results from

CAVOPT-BASE and CAVOPT-3D are compared in Table 4.13, which are reason-

ably close to each other.

The geometric parameters distributions from the two methods are compared in

Figure 4.12. The mean circulation distributions are compared in Figure 4.13. The

-Comparison with Professor Justin.E. Kerwin’s design method, PBD-10, were also prescribed in

Mishima’s thesis. And please note that in Mishima’s thesis, the .0/�1�243 constraint was not appliedactually.

70

results are well compared to each other. The pressure coefficients distributions from

LINTP and LSM are shown in Figures 4.14 and 4.15, which satisfy the �� con-

straint.

Table 4.12: Optimal solutions from CAVOPT-BASE for open propeller design with-out hub.

Present methodCAVOPT-BASE)LSM LINTP:X; �/¸ +# 0.9837 0.9970:Zq �/¸ +# 1.0023 0.9333:Zd �/¸ +# 1.0115 1.0000�!�� 2.2053 2.3302

Table 4.13: Design results from CAVOPT-BASE and CAVOPT-3D.

Present method CAVOPT-3DCAVOPT-BASE) (Mishima, 1996)LSM LINTP"5# 0.1500 0.1500 0.1499� �I"�] 0.3027 0.3032 0.3030Á 78.9% 78.8% 78.7%

71

r / R

P/D

,c/D

f/c

0.40 0.60 0.80 1.000.00

0.50

1.00

1.50

2.00

0.00

0.05

0.10LSMLINTPCAVOPT-3D

P/D

f/c

c/D

Figure 4.12: Geometric parameters distributions from CAVOPT-BASE andCAVOPT-3D.

r / R

Γ/(2

πRV

S)

0.2 0.4 0.6 0.8 10.00

0.01

0.02

0.03

LSMLINTPCAVOPT-3D

Figure 4.13: Mean circulation distributions from CAVOPT-BASE and CAVOPT-3D.

72

x / c

-CP

0 0.25 0.5 0.75 1-5

-4

-3

-2

-1

0

1

2

3 CPMIN = 2.5

Figure 4.14: Pressure coefficients distribution from LINTP.

x / c

-CP

0 0.25 0.5 0.75 1-5

-4

-3

-2

-1

0

1

2

3 CPMIN = 2.5

Figure 4.15: Pressure coefficients distribution from LSM.

73

4.1.4 Comparison of designs from CAVOPT-BASE and PVL subject to uni-form inflow

PVL is a well known propeller design method based on propeller vortex lattice lifting

line method developed by J. E. Kerwin [21]. In order to test the current method, the

optimum efficiencies from CAVOPT-BASE are compared to those from PVL, under

the following operating conditions.

� � ' �\*[� (4.10)

Ã _ ' ��I�I� (4.11)� _ ' ��I�I� (4.12)¾ ' 2 (4.13)





and (3.4)):


74


Minimize "�]subject to "F#�'."F#&%

�!�p�+� � Ê 0/*[-:Z³ ª; Ê :X; Ê : ©�ª;:Z³ ªq Ê :Zq Ê : ©�ªq: ³ ªd Ê :Zd Ê : ©�ªd

(4.15)

N4148, N4119, General-1, General-2 and the other two container propellers CB-1

and CB-2 (see Appendix A) are used as the base geometries. The bounds of the

three design variables are specified according to the corresponding base geometry.

"F# % is the required thrust coefficient. The design results should satisfy the pressure

coefficient constraint � �+� � Ê 0í*1- . The frictional coefficient for CAVOPT-BASE is

��Wü')�+* �I�\= .

For the PVL method, the pitch is not unloaded both at the hub and at the tip.

Hub vortex is not included such that the hub vortex drag is not considered. The

frictional coefficient is � ` ' �+* �I�In , which is double of �¥W in CAVOPT-BASE. To

start PVL method, the chord length distribution has to be provided. For the results

to be comparable, the chord distribution of the design results from CAVOPT-BASE

is provided as the input chord distribution for PVL.

For different thrust requirements "3# % (0.15, 0.2 and 0.3) the optimal design ef-

ficiencies are shown in Tables 4.14, 4.15, and 4.16, respectively. The comparison

shows that the design results from CAVOPT-BASE and PVL are close to each other.

The difference may due to the pressure coefficient constraint: CAVOPT-BASE re-

quires �� Ê 0í*1- , however, PVL does not have any requirement on �¥ �+� � . An-

75

Table 4.14: Optimum efficiencies from CAVOPT-BASE and PVL for "�# % 'h�+*i�E- .

"F# % = 0.15 Design methodsCAVOPT-BASE PVL

CB-1 0.7838 0.7722CB-2 0.7897 0.7821

N4148 0.7982 0.8015N4119 0.7850 0.8011

General-1 0.7939 0.7971General-2 0.7955 0.8029

other reason for the difference, especially in the case of higher loading, is the differ-

ence in the propeller flow models, which are a moderately loaded lifting line model

in PVL, and lifting surface model in MPUF-3A.

The mean circulation distributions are also compared in Figures 4.16, 4.17 and

4.18.

76

Table 4.15: Optimum efficiencies from CAVOPT-BASE and PVL for "$# % '.�/*10 .

"5# % = 0.2 Design methodsCAVOPT-BASE PVL

CB-1 0.7323 0.7326CB-2 0.7434 0.7448N4148 0.7280 0.7545N4119 0.7413 0.7580

General-1 0.7511 0.7660General-2 0.7518 0.7700

Table 4.16: Optimum efficiencies from CAVOPT-BASE and PVL for "�# % ' �+* 2 .(’-’ means there is no feasible point in the database. i.e. the database needs to beextended to provide a solution.)

"F#&% = 0.3 Design methodsCAVOPT-BASE PVL

CB-1 0.6452 0.6518CB-2 - -

N4148 - -N4119 0.6611 0.6899

General-1 0.6727 0.6963General-2 0.6664 0.6987

77

r / R

Γ/(

πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

PVLCAVOPT-BASE

Propeller CB-1 KT = 0.15

r / R

Γ/(

2π

RV

S)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

PVLCAVOPT-BASE


r / R

Γ/(

2πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

PVLCAVOPT-BASE


r / R

Γ/(

2πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

PVLCAVOPT-BASE


r / R

Γ/(

2πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

PVLCAVOPT-BASE


Figure 4.16: Mean circulation distribution from CAVOPT-BASE and PVL for CB-1based propellers (Left) and CB-2 based propellers (Right).

78

r / R

Γ/(

2πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

PVLCAVOPT-BASE

Propeller N4148 KT = 0.15

r / R

Γ/(

2πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

PVLCAVOPT-BASE

Propeller 4119 KT = 0.15

r / R

Γ/(

2πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

PVLCAVOPT-BASE

Propeller N4148 KT = 0.2

r / R

Γ/(

2πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

PVLCAVOPT-BASE


r / R

Γ/(

2πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

PVLCAVOPT-BASE


Figure 4.17: Mean circulation distribution from CAVOPT-BASE and PVL forN4148 based propellers (Left) and N4119 based propellers (Right).

79

r / R

Γ/(

2πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

PVLCAVOPT-BASE

Propeller General-1 KT = 0.15

r / R

Γ/(

2πR

VS

)

0.4 0.6 0.80

0.01

0.02

0.03

PVLCAVOPT-BASE


r / R

Γ/(

2πR

VS

)

0.4 0.6 0.80

0.01

0.02

0.03

0.04

PVLCAVOPT-BASE


r / R

Γ/(

2πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

PVLCAVOPT-BASE


r / R

Γ/(

2πR

VS

)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

PVLCAVOPT-BASE


r / R

Γ/(

2πR

VS

)

0.4 0.6 0.80

0.01

0.02

0.03

0.04

0.05

0.06

PVLCAVOPT-BASE


Figure 4.18: Mean circulation distribution from CAVOPT-BASE and PVL forGeneral-1 based propellers (Left) and General-2 based propellers (Right).

80

4.1.5 Optimization with linear multipliers for�¥�\�

Three geometric parameters��

, u �\�and

�� u are considered in the design pro-

cedure. Linearly distributed multipliers are used for�¥�\�

, while constant multipliers

are used for u �\�and

�E� � u . The optimization problem is as follows:


�!�� Ê �I*1-�+* n Ê :X; Ê �I*[-�+* n Ê :Zq Ê �I*[-�+* n Ê :Zd Ê 0/* ��+* � Ê : � Ê 2+* �

(4.16)

N4148 propeller is selected as the base geometry. The design variables : ; and :Zqare the multipliers assigned to

�¥�\�. The factor :Û; is the multiplier corresponding to

the pitch ratio at the root while :eq is the one corresponding to the pitch ratio at the

tip. The design variables :ed and : � are the constant multipliers assigned to u ��and�E� � u , respectively. There are 10 equally distributed computational points for each

design variable. Thus the size of the generated database is � ��3� ��$� ��3� � . LINTP

method and three kinds of least-square methods are used for database interpolation

inside CAVOPT-BASE.

The optimization solutions are shown in Table 4.17, which are feasible points of

the domain. Note that the solution :jd �/¸ +#G'm�+* n hits the lower bound of :ed . The

efficiency may further increase if chord u �\�decreases.

The design results are shown in Table 4.18. For LINTP method and the =b>A@ order

81

Table 4.17: Solutions from LINTP method and LSM method for the �� È�� database (N4148-based).:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# : � �/¸ +# min � �I"�]

LINTP 1.1889 1.2955 0.8000 2.3333 0.29940 _�` 1.2870 1.2280 0.8000 2.2210 0.2924LSM 2�aP` 1.2099 1.2767 0.8000 2.3464 0.2985=\>A@ 1.2007 1.2991 0.8000 2.2465 0.2990

Table 4.18: Performance characteristics of the designed propellers from LINTPmethod and LSM method (N4148-based).

CAVOPT-BASE Recheck by MPUF3A"F# � �I"$] �!�� "F# � �I"�] Á �!��LINTP 0.1500 0.2994 1.2328 0.1501 0.2992 79.8% 1.18770�_�` 0.1500 0.2924 1.5000 0.1464 0.2913 80.0% 1.1019

LSM 2�aP` 0.1500 0.2985 1.5000 0.1495 0.2979 79.8% 1.1410=\>A@ 0.1500 0.2990 1.5000 0.1500 0.2989 79.8% 1.2374

Table 4.19: Relative errors for LSM method for the � ��)�� )�� database(N4148-based).

Relative Error"F# � �I"�] �!��0 _�` 1.29E-04 1.47E-04 2.08E-03LSM 2�aP` 2.46E-05 2.06E-05 1.65E-03=�>A@ 4.08E-06 3.83E-06 1.06E-03

LSM method, both the equality constraint "3#7' �+*i�E- and the inequality constraint

�!�� Ê �I*[- are satisfied.

Table 4.18 also reflects the interpolation from different methods. Both LINTP

method and the = >A@ order LSM work well for "3# and � �I"�] . For �!�p�+� � , LINTP

method is a better choice.

The approximation errors for the second, third and fourth order LSM are shown

in Table 4.19. The convergence rates are shown in Figures 4.19, 4.20 and 4.21.

82


x 1,x 2,

x 3

0 2 4 6 8 10-3

-2

-1

0

1

2

3

4

5

6

x1

x2

x3

x4


Equ

ality

resi

dual

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

11

12

Figure 4.19: Convergence rate of the optimization problem solved by 0?_�` order LSM(N4148-based).


x 1,x 2,

x 3,x 4

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2

3

x1

x2

x3

x4


Equ

ality

resi

dual

2 4 6 8 100.0

0.5

1.0

1.5

2.0

Figure 4.20: Convergence rate of the optimization problem solved by 2baS` order LSM(N4148-based).


x 1,x 2,

x 3,x 4

0 1 2 3 4 5 6 7 8 9 10-1.0

0.0

1.0

2.0

3.0

x1

x2

x3

x4


Equ

ality

resi

dual

1 2 3 4 5 6 7 8 9 100.0

0.5

1.0

1.5

2.0

Figure 4.21: Convergence rate of the optimization problem solved by =í>A@ order LSM(N4148-based). 83

Initial inflow

MPUF3A

GBFLOW

bodyforces

effective wake

Figure 4.22: Iteration scheme between MPUF3A and GBFLOW.

4.2 Unsteady fully wetted case subject to non-axisymmetric in-flow

This section describes the application of the CAVOPT-BASE to open propeller blade

design, which operates under fully wetted conditions. The initial nominal wake is

shown in Figure 4.23. For the given inflow, MPUF3A is used to compute the body

forces produced by the propeller. GBFLOW is used to compute the flow field in-

cluding the effect of the body forces. The flow field is not the same as the initial one

any more due to the existence of the body forces. Thus the nominal wake should

be updated iteratively. The iteration scheme between MPUF3A and GBFLOW to

predict the effective wake is described in Figure 4.22.

The iteration between MPUF3A and GBFLOW including three geometric pa-

rameters will cost too much computational time. Currently, only one geometric

parameter�¥�\�

is considered in the iteration to obtain the effective wake.

84

UX1.10

1.06

1.02

0.98

0.94

0.90

0.86

0.82

0.78

0.74

0.70

Figure 4.23: Nominal wake contour for open propeller [43].

85


The initial nominal wake is given in Figure 4.23. The designed propeller operates

without cavitation. The operating conditions and constraints are as follows:

� � ' �\*[� (4.17)

Ã _ ' ��I�I� (4.18)� _ ' ��I�I� (4.19)¾ ' 2 (4.20)





and (3.4)):

�I* � Ê v �¥�\�gwyx Y{z^| Ê 0í*[��+*[0 Ê v u �\�gw¦x Y¢zM| Ê �/*1-�+* � Ê v � � � u w¦x Y¢zM| Ê �/*[�In (4.21)


Minimize "�]subject to "F#�'.�+*,� -

�!�p�+� � Ê 2+*[-:Z³ ª; Ê :X; Ê : ©�ª;:Z³ ªq Ê :Zq Ê : ©�ªq: ³ ªd Ê :Zd Ê : ©�ªd

(4.22)

86

The designed propeller is required to be three-bladed. The advance ratio,� � , is

1.0. Both Ã _ and� _ are set to be large value to avoid cavitation on the blade. The

required thrust coefficient is 0.15. The inequality constraint on �� ensures that the

designed propeller operates under fully wetted conditions when Ã _ Ô�5 * � . The three

parameters,�¥�\�

, u ��and

�� u , are considered in the design. As discussed above,

however, only�¥�\�

is considered to solve for the effective wake by the iteration

scheme. The bounds of the design variables :<; , :eq and :Zd are determined by the

selected base geometry.



In MPUF3A, the propeller blade is discretized by 20 panels along chordwise and 9

panels along spanwise direction. The three design variables are bounded as follows:

�I*[� Ê :X; Ê 0/*[��+*[n Ê :Zq Ê 0/*[��+*[� Ê :Zd Ê 2+*[� (4.23)

To generate the effective wake, several discrete :<; values are used within the above

range. Table 4.20 shows the predicted "$# values for the selected :Û; . Figure 4.24

shows the determination of : ¶ ; ' �I* 2?0\½ which produces the design "3#�'C�+*i�E- . The

corresponding effective wake is shown in Figure 4.25.

The effective wake is used to generate a � �á�.� �á�h� � database. Linear in-

terpolation method and the = >A@ order least squares method are used for database

interpolation inside CAVOPT-BASE.

87

Table 4.20: Iteration results between MPUF3A and GBFLOW for different :È;(N4148-based).

Selected :X; values:X; 'f�I*10 :X;�'É�\*[2 :X; 'k�\* ="F# 0.1056 0.1414 0.1741

The optimization solutions from LINTP and LSM are shown in Table 4.21,

which satisfy the constraints for the design variables. There are some differences

between the solutions from both methods due to different interpolation methods.

The design results are shown in Table 4.22. Both the equality constraint for "�#and the inequality constraint for �� are satisfied approximately. It is obvious that

LINTP method is a better choice to approximate � �� . The approximation errors

of the =\>A@ order LSM is shown in Table 4.23. The convergence rates of the design

variables and the equality residuals of the optimization problem are shown in Fig-

ures 4.26 and 4.27.

The geometric parameters distributions and mean circulation distributions from

the two methods are compared in Figures 4.28 and 4.29. The design propeller ge-

ometry solved by LINTP is shown in Figure 4.30. The pressure coefficients �²� at

different radius solved by LINTP are shown in Figure 4.31 with the maximum value

3.5037, which satisfies the constraint ��p�+� � Ê 2+*[- approximately.

Since the designed geometry is available, MPUF3A and GBFLOW are coupled

to recheck the corresponding propeller thrust and effective wake. The propeller

thrust coefficient after the iterations is 0.1566, which is close to the required value

0.1500. The effective wake from the designed propeller is shown in Figure 4.32,

which is close to the wake shown in Figure 4.25.

88

x1

KT

1.2 1.25 1.3 1.35 1.40.00

0.05

0.10

0.15

0.20

x1* = 1.326

KT = 0.15

Figure 4.24: Interpolation for "3#g'h�+*i�E- .

UX1.101.061.020.980.940.900.860.820.780.740.70

Figure 4.25: Approximate effective wake contour for "�#g'h�+*i�E- .

89

Table 4.21: Optimal solutions for open propeller design subject to non-axisymmetricinflow (N4148-based).

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# Õ3ÜÞÝ � �\"�]LINTP 1.1951 1.8667 1.8512 0.3188=\>A@ LSM 1.2024 2.0000 1.6867 0.3147

Table 4.22: Design results for open propeller design subject to non-axisymmetricinflow (N4148-based).

CAVOPT-BASE Recheck by MPUF3A"F# ��I"�] �!�� "5# � �I"$] Á �!�p�+� �LINTP 0.1500 0.3188 3.5000 0.1505 0.3177 75.4% 3.5037= >A@ LSM 0.1500 0.3147 3.5000 0.1500 0.3151 75.8% 3.6644

Table 4.23: Approximation errors of the =?>A@ order LSM (N4148-based).

"5# � �I"$] �!��Relative errors 1.1928E-05 1.2110E-05 7.8999E-04

90


x 1,x 2,

x 3

0 5 10 15 20

-300

-200

-100

0

100

x1

x2

x3

Figure 4.26: Convergence rates of the design variables solved by = >A@ order LSM(N4148-based).


Equ

ality

resi

dual

5 10 15 200

100

200

300

400

500

600

Figure 4.27: Equality residual of the optimization problem solved by =í>A@ order LSM(N4148-based).

91

r / R

P/D

,c/D

f/c

0.2 0.4 0.6 0.8 10.00

0.50

1.00

1.50

2.00

0.00

0.05

0.10

LINTP4th order LSM

P/D

f/c

c/D

Figure 4.28: Geometric parameters distributions from LINTP and the =í>A@ order LSM(N4148-based).

r / R

Γ/(2

πRV

S)

0.2 0.4 0.6 0.8 10.00

0.01

0.02

0.03

LINTP4th order LSM

Figure 4.29: Mean circulation distributions from LINTP and the = >A@ order LSM(N4148-based).

92

X

Y

Z

Figure 4.30: Designed propeller geometry solved by LINTP (N4148-based).

x / c

-CP

0 0.2 0.4 0.6 0.8 1-4

-3

-2

-1

0

1

2

3

4 CPMIN = 3.5

Figure 4.31: Pressure coefficients distribution from the designed geometry solvedby LINTP (N4148-based).

93

Table 4.24: Optimal efficiency for different � �� constraints determined by LINTP(N4148-based).

�!�p�+� � constraintsÊ =�* � Ê =+*1- Ê -/* � Ê -/*1- Ê ½+* �"5#�')�+*,�� 83.1% 83.6% 84.0% 84.2% 84.3%"5#�')�+*,� - 78.7% 79.0% 79.6% 79.8% 80.0%"5#�')�+*10�� 72.5% 73.6% 74.5% 74.9% 75.2%

Effect of �c�� constraints

The optimal efficiency for different �� constraints are shown Tables 4.33 and

Figure 4.33. The values in Table 4.33 are computed by MPUF3A from the designed

propeller. As the �c�� constraint relaxed, the optimal efficiency increases for the

enlarged feasible domain. However, the increment in efficiency is not significant.

94

UX1.101.061.020.980.940.900.860.820.780.740.70

Figure 4.32: Effective wake rechecked by MPUF3A and GBFLOW from the de-signed propeller (N4148-based).

CPMIN

η

3.5 4 4.5 5 5.5 60.65

0.70

0.75

0.80

0.85

KT=0.10KT=0.15KT=0.20

Figure 4.33: Optimal efficiency for different �� constraints determined by LINTP(N4148-based).

95

Optimization with two geometric parameters

N4148 is selected as the base geometry. The maximum camber distribution is the

same as the base geometry. Two geometric parameters,�¥�\�

and u �\�, are con-

sidered in the blade design method. The corresponding optimization problem is as

follows:Minimize "�]subject to "F#g'h�+*i� �


�I*[� Ê :X; Ê �I*1-�+*[n Ê :Zq Ê 0/*[�

(4.24)

Figure 4.34 shows the contours of constant "$# , efficiency Á and �c�� generated

by LSM. Table 4.25 shows the optimal efficiencies from CAVOPT-BASE and from

the contours figure for different �c�p�+� � constraints. The solutions :Û; , :Zq and :Zd are

solutions from CAVOPT-BASE. The optimal Á from the contours figure are obtained

directly from Figure 4.34 corresponding to the solution points. The differences are

due to the approximating errors in generating the contours by LSM while the prob-

lem is solved by LINTP.

Table 4.25 also shows the effect of different � �� constraints. Same as the three

dimensional case, the optimal efficiency increases as the � �� constraint is relaxed.

Figure 4.34 also shows the relationship between performances and the geometric

parameters which is similar to the three parameters case. This is obvious since the

feasible region of the two parameters case is the subspace of the feasible region of

the three parameters case.

96


"F#/�'h�+*i� � �!�p�+� � constraintsÊ =�*[� Ê =�*[- Ê -/* � Ê -í*1- Ê ½+*[�CAVOPT-BASE (LINTP) 80.0% 81.1% 82.0% 82.6% 83.1%

Optimal ÁContour Figure (LSM) 77.6% 79.2% 80.6% 81.8% 83.0%:X; 1.1463 1.1512 1.1566 1.1623 1.1692:Zq 1.9032 1.6747 1.4921 1.3357 1.1956

97

x1

x 2

1.0 1.1 1.2 1.3 1.4 1.50.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.90

2.00

KT =

0.20

KT =

0.10

KT =

0.15

η=

80%

η=

77%

η=

84%η

=83%

η=

82%η

=81%

η =85

%

η=

79%η

=78

%

CPM

INM

AX

=7.

0

CPM

INM

AX

=2.

5C

PMIN

MA

X=

3.0

CPM

INM

AX

=3.

5

CPM

INM

AX

=4.

0C

PMIN

MA

X=

4.5

CPM

INM

AX

=5.

0

CPM

INM

AX

=5.

5C

PMIN

MA

X=

6.0

CPM

INM

AX

=6.

5

Figure 4.34: Contour plot generated by LSM (N4148-based).

98



, u �\�and



, while constant multipliers

are used for u �\�and

�E� � u . The optimization problem is as follows:


�!�� Ê 2+*1-�I* � Ê :X; Ê 0/* ��I* � Ê :Zq Ê 0/* ��+* n Ê :Zd Ê 0/* ��+* � Ê : � Ê 2+* �

(4.25)


��. The design variable :<; is the multiplier corre-

sponding to the pitch ratio at the root while :jq is the multiplier corresponding to the

pitch ratio at the tip. The design variables :jd and : � are the constant multipliers as-

signed to u �\�and

�� u , respectively. There are 5 equally distributed computational

points for :X; and :Zq while 10 equally distributed computational points for :rd and

: � , respectively. Thus the size of the generated database is -ñ��-9�D��9�D� � . LINTP

method is used for database interpolation inside CAVOPT-BASE.

Tables 4.26 and 4.27 show the optimization solutions and the design results

obtained by LINTP. The optimal efficiency, Á st' 5 n+*[-76 , is higher than the optimal

efficiency (75.4%) with three design variables. The linear multipliers for��

allow

99

Table 4.26: Optimal solutions for the design problem with four variables.

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# : � �/¸ +# Õ3ÜÞÝ � �\"�]LINTP 1.5000 1.0717 0.8000 2.5028 0.3070

Table 4.27: Design results for the design problem with four variables.

CAVOPT-BASE Recheck by MPUF3A"F# ��I"�] �!�p�+� � "5# � �I"$] �!�� ÁLINTP 0.1500 0.3070 3.5000 0.1505 0.3052 3.4570 78.5%

more flexibility in generating the propeller family. Thus the feasible region of the

optimization problem is enlarged and results in a better solution. It is worth noting

that in the case of four design variables an unloaded circulation at the tip was pro-

duced, but since it was driven by the optimization algorithm, the final design has

considerably higher efficiency than that using three design variables.

Figure 4.35 compares the three geometric parameters distribution for three de-

sign variables and four design variables. The design with four variables results in

bigger camber and smaller chord length. The result has bigger�¥�\�

for § � � Ê �+* nwhile smaller

�¥�\�for § � � Ô �+* n . The distribution of

�¥�\�, u �\�

and�� u leads

to the circulation distribution shown in Figure 4.36. For the design problem with

four variables, the circulation strength is bigger for § � � Ê �+* 5 while smaller for§ � � Ô �+* 5 . Bigger�¥�\�

and�� u , and smaller u �\�

reduce the minimum pressure,

and thus increase �c�� . As shown in Figures 4.37 and 4.38, the results from four

design variables have wider range of Ät�� across the chord.

100

r / R

P/D

,c/

D

f/c

0.2 0.4 0.6 0.8 10.0

0.5

1.0

1.5

2.0

0.00

0.05

0.103 design variables4 design variables

P/D

f/c

c/D

Figure 4.35: Comparison of geometric parameters distributions from three and fourdesign variables solved by LINTP (N4148-based).

r / R

Γ/(

2πR

VS)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

3 design variables4 design variables

Figure 4.36: Comparison of mean circulation distributions from three and four de-sign variables solved by LINTP (N4148-based).

101

x / c

-CP

0 0.2 0.4 0.6 0.8 1-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

CPMIN = 3.5

Figure 4.37: Pressure coefficient distribution from three design variables solved byLINTP (N4148-based).

x / c

-CP

0.00 0.20 0.40 0.60 0.80 1.00-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

CPMIN = 3.5

Figure 4.38: Pressure coefficient distribution from four design variables solved byLINTP (N4148-based).

102

4.3 Unsteady cavitating case subject to non-axisymmetric inflowwith only 8:9 inequality constraint

In this section, CAVOPT-BASE is applied to open cavitating propeller blade design.

The inflow, as shown in Figure 4.39, is non-axisymmetric. Partial sheet cavitation

is allowed. The design results are presented. The effect of the cavity area ratio is

also investigated. In this section, only inequality �� constraint is considered. The

design problems involving both �B� and � �� constraints will be discussed in the

next section.


The operating conditions are as follows:

� � ' �I*[0 (4.26)Ã _ ' 0/*[- (4.27)� _ ' -/* � (4.28)¾ ' 2 (4.29)





and (3.4)):


103


Minimize "�]subject to "F#�'.�+*10��

�B� Ê �+* 2:Z³ ª; Ê :X; Ê : ©�ª;:Z³ ªq Ê :Zq Ê : ©�ªq:Z³ ªd Ê :Zd Ê : ©�ªd

(4.31)

The design thrust coefficient is 0.20. The cavity constraint �B� Ê �+* 2 requires that

the maximum allowable partial cavity area ratio is 30%. Three geometric parame-

ters,�¥�\�

, u ��and

�E� � u , with their corresponding design variables :<; , :Zq and :Zd ,

are used in constructing the propeller family. The bounds of the design variables

are determined by the selected base geometry satisfying the limits of�¥�\�

, u �\�and�E� � u . The blade is discretized with 20 panels in chordwise and 9 panels in spanwise

for MPUF3A. The database size is � �F�o� �F�o�� .



The limits of�¥�\�

, u �\�and

�� u are satisfied for N4148 base geometry by setting

the design variables inside the following domain:

�+*[n Ê :X; Ê �I*1-�+*[n Ê :Zq Ê 0/*[��+*[� Ê :Zd Ê 2+*[� (4.32)

104

UX1.10

1.06

1.02

0.98

0.94

0.90

0.86

0.82

0.78

0.74

0.70

Figure 4.39: Wake contours for open cavitating blade design [43].

105

LINTP and the = >A@ order LSM are used in this case. The = >A@ order LSM can approxi-

mate "5# and "�] better than �B� . The approximation errors are shown in Table 4.30.

The convergence rates are shown in Figure 4.40 and 4.41.

The solutions of the optimization problem are shown in Table 4.28. They all

satisfy the bounds of the design variables. Note that :<; �/¸ +#�' �I*[- hits the upper

bound of :X; , which means that the efficiency may further increase if the pitch ratio�¥�\�increases.

The design results from LINTP and LSM are very close, as compared in Ta-

ble 4.29. The optimization results are close to the feedback results from MPUF3A

using the designed geometries. The equality constraint "�#D' �/*10\� is satisfied. the

design cavity ratio is 30.6%, which approximately satisfies the inequality constraint

�� Ê 2\�+*[�76 .

The design geometric parameters distributions of the two methods are compared

in Figure 4.42 and the mean circulation distributions are compared in Figure 4.43.

The distributions are close as expected since the design variables from the two meth-

ods are close. The designed blade geometry and the cavity shape solved by LINTP

are shown in Figures 4.44 and 4.45, respectively.

Effect of other base geometries

General-1 and General-2 base geometries are used to investigate the effect of the

selected base propeller. Table 4.31 shows the design results from different base ge-

ometries. The values in Table 4.31 are the feedback values from MPUF3A. The

required thrust coefficient 0.20 is satisfied for all the base geometries. The inequal-

ity constraint �B� Ê 2I�+* � 6 is also satisfied approximately. The optimal efficiencies

from the three different base geometries are all close to 77.0%. Thus the optimal

106

Table 4.28: Optimal solutions of open cavitating blade design (N4148-based).

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# Õ3ÜÞÝ � �\"�]LINTP 1.5000 1.7713 1.9745 0.4962= >A@ LSM 1.5000 1.7653 1.9707 0.4960

Table 4.29: Design results of open cavitating blade design (N4148-based).

CAVOPT-BASE Recheck by MPUF3A"F# � �I"$] �� "5# � �I"$] Á ��LINTP 0.2000 0.4962 30.0% 0.2000 0.4959 77.0% 30.6%= >A@ LSM 0.2000 0.4960 30.0% 0.1997 0.4950 77.0% 30.8%

Table 4.30: Approximation errors of the = >A@ order LSM in open cavitating bladedesign (N4148-based).

RMS"F# ��I"�] CA=\>A@ LSM 2.23E-04 2.03E-04 4.36E-03

107


x 1,x 2,

x 3

0 1 2 3 4 5 6 7 8 9 10 11-1

0

1

2

3

x1

x2

x3

Figure 4.40: Convergence rates of the design variables solved by =í>A@ order LSM(N4148-based).


Equ

ality

resi

dual

1 2 3 4 5 6 7 8 9 10 110.0

0.5

1.0

1.5

2.0

Figure 4.41: Equality residual of the optimization problem solved by =í>A@ order LSM(N4148-based).

108

r / R

P/D

,c/D

f/c

0.20 0.40 0.60 0.80 1.000.00

0.50

1.00

1.50

2.00

0.00

0.05

0.10LINTP4th order LSM

P/D

f/c

c/D

Figure 4.42: Geometric parameters distributions from the design results solved byLINTP and the = >A@ order LSM (N4148-based).

r / R

Γ/(2

πRV

S)

0.2 0.4 0.6 0.8 1.00

0.01

0.02

0.03

0.04

LINTP4th order LSM

Figure 4.43: Mean circulation distributions from the design results solved by LINTPand the =\>A@ order LSM (N4148-based).

109

X

Y

Z

Figure 4.44: Blade geometry designed solved by LINTP (N4148-based).

CA = 30.6%

Angle = 30o

Figure 4.45: Cavity shape designed solved by LINTP (N4148-based).

110

Table 4.31: Design results from different base geometries.

Design ResultsBase geometry "F# � �\"�] Á CA

N4148 0.2000 0.4959 77.0% 30.6%General-1 0.2005 0.4986 76.8% 29.4%General-2 0.2006 0.5000 76.6% 30.7%

efficiencies are not affected too much by the base geometry.

However, the base geometry can affect the mean circulation distributions. Fig-

ure 4.46 compares the circulation distributions from the three base geometries. The

circulation distributions are different due to different base geometries.

Effect of cavity constraints

For different required "3#/s , the optimal efficiencies of different cavity constraints

are shown in Table 4.32 and in Figure 4.47. N4148 propeller is used as the base

geometry. The size of database is � �F�H� �F�4� � . LINTP is used as the interpolation

method inside CAVOPT-BASE. The values in Table 4.32 are the feedback values

from MPUF3A using the designed geometries.

From Table 4.32, for specified "3# , the optimal efficiency increases as the maxi-

mum allowable cavity ratio increases. The optimal efficiency is very sensitive to the

cavity ratio at the beginning. However, the optimal efficiency does not change much

when the cavity ratio increases to some degree.

111

r / R

Γ/(2

πRV

S)

0.20 0.40 0.60 0.80 1.000.00

0.01

0.02

0.03

0.04

N4148N3745General-1General-2

Figure 4.46: Mean circulation distributions from different base geometries solvedby LINTP.

CAMAX

η

0 0.1 0.2 0.3 0.40.70

0.75

0.80

0.85

KT = 0.10KT = 0.15KT = 0.20

Figure 4.47: Optimal efficiency for different CA constraints (N4148-based).

112

Table 4.32: Optimal efficiency for different CA constraints (N4148-based).

Cavity constraintsÊ -76 Ê � � 6 Ê �E-;6 Ê 0�� 6 Ê 0I-76 Ê 2I� 6 Ê 2?-;6 Ê =I� 6"F#�'.�+*i� � 76.8% 79.6% 81.5% 82.7% 83.5% 83.9% 84.5% 84.9%"F#�'.�+*i�E- 75.1% 77.1% 78.8% 79.8% 80.4% 80.9% 81.2% 81.4%"F#�'.�+*[0\� 72.6% 74.2% 75.4% 76.2% 76.8% 77.0% 77.1% 77.3%

Optimization problem with two geometric parameters

To illustrate the design results clearly, only two parameters�¥�\�

and u �\�are used

in the design procedure.

The optimization problem is as follows:

Minimize "�]subject to "F#�'h�/*,� �

�� ÊCAMAX

�\*10 Ê :X; Ê �I*[-�\*[� Ê :Zq Ê 0/* �

(4.33)

N4148 propeller is selected as the base geometry. The camber distribution�I� � u

remains the same as the base geometry. The optimization results from CAVOPT-

BASE for "F#�'ç�/*,� � with different CA constraints are shown in Table 4.33. The

contours plot for constant "$# , �� and efficiency Á is generated by LSM and shown

in Figure 4.48.

Note that in Table 4.33, there are small differences between the values from

CAVOPT-BASE and the values obtained directly from the contours figure. The dif-

113

Table 4.33: Optimal efficiency and solutions for two design variables generated byLSM (N4148-based).

"5#�')�+*,�� Cavity constraintsÊ 0I-76 Ê 2I�76 Ê 2I-76 Ê =?� 6 Ê =b-76CAVOPT-BASE (LINTP) 83.5% 84.0% 84.4% 84.8% 85.1%


Ê -\� 6 Ê -I-;6 Ê ½\� 6 Ê ½?-76 Ê 5 � 6CAVOPT-BASE (LINTP) 85.3% 85.5% 85.6% 85.6% 85.6%


ferences are due to the LSM approximation errors in generating the contours plot.

From Table 4.33, the optimal efficiency increases to some degree as the maximum

allowable cavity ratio increases.

Figure 4.48 also illustrates the relationship between the propeller performance

and the geometric parameters. For this case, "$# is very sensitive to the pitch fac-

tor :X; . The cavity area ratio decreases when :Û; decreases while :eq increases. This

explains that the designed cavitating propeller always has wide blade to reduce the

cavity area.

114

x1

x 2

1.20 1.25 1.30 1.35 1.40 1.45 1.501.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.90

2.00

KT = 0.10

KT = 0.15

KT = 0.05

CA=

15%

CA=

20%

CA=

25%

CA=

30%

CA = 80%

CA=

35%

CA = 40%

CA = 45%

CA = 50%

CA = 55%

CA = 60%

CA = 65%

CA = 70%

CA = 75%

η=

80%

η=

86%η

=85%

η=

84%

η=

83%

η=

82%

η=

81%

Figure 4.48: Contours plot generated by LSM at :jd('É�\*[� (N4148-based).

115


Geometric parameters,�¥�\�

, u �\�and

�� u are considered in this case. The design

variables :V; and :Zq are assigned to��

. The multipliers for u �\�and

�� u are :Zdand : � , respectively. The cavitating open propeller design problem is as follows:

Minimize "�]subject to "3#�'h�+*[0

�� Ê 2\� 6�I* � Ê :X; Ê 0/* ��I* � Ê :Zq Ê 0/* ��+* n Ê :Zd Ê 0/* ��+* � Ê : � Ê 2+* �

(4.34)

N4148 propeller is selected as the base geometry. Ten equally distributed computa-

tional points are assigned to each design variable. Thus the size of the database is

� �F�o� �F�4� �5�4�� .

Table 4.34 shows the optimal solutions for the design problem while Table 4.35

shows the corresponding designed propeller performance. The optimal efficiency

(77.2%) from four design variables is only slightly higher than the efficiency (77.0%)

from three design variables. The increment is due to larger feasible region which al-

lows linearly distributed multipliers for��

.

The geometric parameters distribution from three and four design variables is

compared in Figure 4.49 while the mean circulation distributions are compared in

Figure 4.50.

116

Table 4.34: Optimal solutions of open cavitating blade design with four variables.

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# : � �/¸ +# Õ3ÜÞÝ � �\"�]LINTP 1.4487 1.5558 1.8666 1.7814 0.4952

Table 4.35: Design results of open cavitating blade design with four variables.

CAVOPT-BASE Recheck by MPUF3A"F# � �I"$] �� "5# � �I"$] �B� ÁLINTP 0.2000 0.4952 30.0% 0.1998 0.4946 30.5% 77.2%

117

r / R

P/D

,c/D

f/c

0.2 0.4 0.6 0.8 10

1

2

0.00

0.05

0.103 design variables4 design variables

P/D

f/c

c/D

Figure 4.49: Geometric parameters distribution from three and four design variablessolved by LINTP (N4148-based).

r / R

Γ/(

2πR

VS)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03


Figure 4.50: Mean circulation distribution from three and four design variablessolved by LINTP (N4148-based).

118

4.4 Unsteady cavitating case subject to non-axisymmetric inflowwith both 8:9 constraint and 8=<?>"@BA constraint

In this section, CAVOPT-BASE is applied to open cavitating propeller design with

both �B� constraint and �� constraints. The inflow wake is non-axisymmetric,

as shown in Figure 4.51. Partial back side sheet cavitation is allowed, however, face

side cavitation is not allowed. The back side cavitation is constrained by the ��constraint. The �� constraint is used to eliminate any face side cavitation.



� � ' �I*[0 (4.35)Ã _ ' 0/*[- (4.36)� _ ' -/* � (4.37)¾ ' 2 (4.38)





and (3.4)):


119


Minimize "�]subject to "F#g'h�+*i�E-

�B� Ê �+* 2 ( apply to back side )

�( �� Ê �I*10\- ( apply to face side )

:Z³ ª; Ê :X; Ê : ©�ª;: ³ ªq Ê :Zq Ê : ©�ªq: ³ ªd Ê :Zd Ê : ©�ªd

(4.40)

The thrust requirement is "3# % 'õ�/*,�E- . The cavity constraint �� Ê 2I�76 requires

that the back side cavity ratio should not exceed 30%. Since Ã _ 'h0/*1- , the constraint

�( �� Ê �I*[0I- eliminates any cavitation on the face side. The geometric parameters,�¥�\�, u �\�

and�E� � u , with their corresponding design variables :Û; , :Zq and :Zd are

used to generate the performance database. The bounds of the design variables are

determined by the selected base geometries.


The CB-1, CB-2, N4148 and N4119 propellers are selected as the base geometries

for the same design problem. The optimal solutions and designed propeller perfor-

mance are shown in Tables 4.36 and 4.37. The designed performances are well

compared to that rechecked by MPUF3A. The optimal efficiencies are different for

different base geometries. The designed circulation distribution, pressure coefficient

distribution and cavity patterns are shown Figures 4.52 and 4.53.

120

UX0.92940.90980.89030.87070.85110.83150.81200.79240.77280.75320.73360.71410.69450.67490.65530.63580.61620.59660.57700.55750.5379

Figure 4.51: Wake geometry with wave fraction l > ' �/*[n+� 0 for open cavitatingblade design with both �� and � �� constraints.

121

Table 4.36: Optimal solutions of open cavitating blade design involving both �B�and �( �� constraints.

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# Õ3ÜÞÝ � �I"$]CB-1 1.3011 0.7302 1.3186 0.2925CB-2 1.3282 0.8603 1.4483 0.2922

N4148 1.2641 1.8664 0.8374 0.2962N4119 1.1621 0.8567 0.8053 0.3083

Table 4.37: Design results of open cavitating blade design involving both �� and�( �� constraints.

CB-1 CB-2 N4148 N4119"5# 0.1500 0.1500 0.1500 0.1500CAVOPT � �I"�] 0.2925 0.2922 0.3037 0.3083

BASE �B� 30.0% 30.0% 30.0% 30.0%�( �+� � 1.2500 1.1297 1.2500 1.2500"5# 0.1503 0.1500 0.1499 0.1508Rechecked � �I"�] 0.2930 0.2922 0.3031 0.3084

by �B� 29.5% 34.5% 29.2% 27.92%MPUF3A �� +� � 1.2537 1.1322 1.2296 1.1280Á 79.5% 79.6% 76.7% 75.8%

122

r / R

Γ/(

2πR

VS)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

CB-1 based

r / R

Γ/(

2πR

VS)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

CB-2 based

x / c

CP

0 0.25 0.5 0.75 1

-2

-1

0

1

CPmin = 1.25

CB-1 based

x / c

CP

0 0.25 0.5 0.75 1

-2

-1

0

1

CPmin = 1.25

CB-2 based

CB-1 based

CA = 29.5%

CB-2 based

CA = 34.5%

Figure 4.52: Distributions of mean circulation, pressure coefficient and cavity pat-tern from CAVOPT-BASE for CB-1 based propellers (Left) and CB-2 based pro-pellers (Right).

123

r / R

Γ/(

2πR

VS)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

N4148 based

r / R

Γ/(

2πR

VS)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

N4119 based

x / c

CP

0 0.25 0.5 0.75 1

-2

-1

0

1

CPmin = 1.25

N4148 based

x / c

CP

0 0.25 0.5 0.75 1

-2

-1

0

1

CPmin = 1.25

N4119 based

N4148 based

CA = 29.5%

N4119 based

CA = 27.9%

Figure 4.53: Distributions of mean circulation, pressure coefficient and cavity pat-tern from CAVOPT-BASE for N4148 based propellers (Left) and N4119 based pro-pellers (Right).

124

Chapter 5

Applications of Blade Design Methods to DuctedPropellers

Propeller inside an accelerating duct has always been a viable choice for high-

loading propulsion system design. The efficiency of a ducted propeller under high

loading has been proven to be higher than the open propellers by Kuchemann and

Weber [36]. They modelled the ducted propulsor by actuator disc with constant

pressure jump at the propeller plane. The efficiency from this model, following the

work of Coney [6], can be expressed as:

Á;C ' 0� ÑED � Ñ þ ��# (5.1)

with � ' �r Ñ �r�þ ' �r � ��# ' 0�� v �?�tsp� q� w (5.2)

where þ is the thrust ratio; �üs is the area of the actuator disc; � � is the ship velocity;

� is the total thrust of the propulsion system; �X and �X� are the thrust due to the

propeller and the duct, respectively.

When þ 'k� , the propulsion system reduces to an open propeller. For accelerat-

ing ducted propeller with � è þ è � , larger �(# can be obtained by using larger �V�125

but smaller þ . The efficiency described gained is higher than the efficiency of the

corresponding open propeller.

This chapter applies the described blade design method to blade design prob-

lems of ducted propellers. The inflow is not restricted to be uniform. Both non-

cavitating and cavitating propeller blades can be designed by the presented method.

The situation considered in this chapter includes the following cases: steady fully

wetted case subject to axisymmetric inflow; unsteady fully wetted case subject to

non-axisymmetric inflow; unsteady fully wetted case subject to non-axisymmetric

inflow with skew constrained; unsteady cavitating case subject to non-axisymmetric

inflow. Infinite hub is included in all the cases. The design results are investigated

with different design variables and different approximating functions, and compared

with those from other design methods.

5.1 Steady fully wetted propeller inside tunnel subject to axisym-metric inflow

This section applies CAVOPT-BASE to propeller inside tunnel subject to axisym-

metric inflow. The tunnel is modelled by cylindrical surface with infinite length and

zero gap. The inflow is uniform. The effect of infinite hub is included with the hub

ratio § @ ' �+*[0 . The thrust produced by a ducted propeller can be divided into two

parts: the thrust due to the propeller "3#/ and the thrust due to the duct "3#+� . In this

case, only "3#/ is considered in the design procedure since "$#+� is much smaller

than "F#/ . The designed ducted propeller is required to produce a specified thrust

under fully wetted operating conditions.

126


The operating conditions and constraints are:� � ' �\*[� (5.3)Ã _ ' ��I�I� (5.4)� _ ' ��I�I� (5.5)¾ ' - (5.6)





and (3.4)):

�I* � Ê v �¥�\�gwyx Y{z^| Ê 0í*[��+*[0 Ê v u �\�gw¦x Y¢zM| Ê �/*[½�+* � Ê v � � � u w¦x Y¢zM| Ê �/*[�In (5.7)


Minimize "�]subject to "F#/�')�+*[-

�!�p�+� � Ê 0/*[-:Z³ ª; Ê :X; Ê : ©�ª;: ³ ªq Ê :Zq Ê : ©�ªq: ³ ªd Ê :Zd Ê : ©�ªd

(5.8)

The propeller is required to be a 5-bladed ducted propeller. Both the cavitation num-

ber Ã _ and Froude number� _ are set to be large values to avoid any cavitation. The

127

thrust requirement is "F#+�'m�+*[- , which is an equality constraint for the optimiza-

tion problem. The inequality constraint �� Ê 0/*1- ensures that the run satisfies

the fully wetted conditions. The N3745 propeller is used as the base geometry. In

this case, the geometric parameters considered are�¥�\�

, u �\�and

�\� � u , with the

corresponding design variables :Û; , :Zq and :Zd , respectively. The lower bounds and

upper bounds of the design variables are determined by the selected base geometry

satisfying the limits of the three parameters. The skew constraint is not included in

this design. For the vortex/source lattice method MPUF3A, the blade is discretized

with 20 panels in chordwise direction while 9 panels in spanwise direction.



To satisfy the limits for the three geometric parameters, the design variables are

within the domain:

�/* 5 - Ê :X; Ê �I*1-��/*[nI� Ê :Zq Ê 0/*[�\��/*[�I� Ê :Zd Ê 2+*[�\� (5.9)

The geometric parameters of N3745 propeller are shown in the appendix A. Both

� �á�C� �á�C�� database and �E-á�h�E-á�h�E- database are generated for the design

method.

The design results are shown in Tables 5.1, 5.2 and 5.3. All the constraints in

the optimization problem have been satisfied. The solution points from the two inter-

polation methods, LINTP and LSM, are close to each other. The interpolated values

are close to the feedback values from MPUF3A. For approximating �� , LINTP

128

Table 5.1: Solutions from LINTP and LSM for � � �� È�� database and �E-È��E-<�� -database (N3745-based).

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# min � �I"$]LINTP 0.9377 0.8765 2.3333 1.0901� �5�4��3�D� � = >A@ LSM 0.9156 0.8000 2.7304 1.0830LINTP 0.9370 0.8641 2.3571 1.0870�E-9�4� -F�D�E- =\>A@ LSM 0.9145 0.8132 2.8098 1.0930

works better than LSM. The results from LSM are more sensitive to the database

size than those from LINTP. This can also be discovered from the approximation

errors of LSM as shown in Table 5.4. The convergence rates of the optimization

problem solved by =I>A@ order LSM are shown in Figures 5.1 and 5.2.

For �E-F�G�E-$�H�E- database, the designed geometric parameters and circulation dis-

tribution from LINTP and =I>A@ order LSM are compared in Figures 5.3 and 5.4,

respectively. The designed propeller geometry from LSM is shown in Figure 5.5.

The pressure coefficients distribution at different radii is shown in Figure 5.6. It is

clear that the pressure coefficients satisfy the inequality constraint � �� Ê 0/*1- .

Effect of �c�� constraint

For �E-B�7�E-B�7� - database, the effect of different �� constraints onto the optimal

efficiency are investigated. The relationship is shown in Table 5.5 and Figure 5.8.

The values in Table 5.5 are the feedback values from MPUF3A.

129

Table 5.2: Performance characteristics of the designed propellers from LINTP andLSM (N3745-based, ��3�D� �F�4� � database).� �F�o� �F�4� � CAVOPT-BASE Recheck by MPUF3A"5#+ � �\"�] �!�� "F#/ � �I"$] Á �!��

LINTP 0.5000 1.0901 2.5000 0.4994 1.0860 73.2% 2.3093=\>A@ LSM 0.5000 1.0830 2.5000 0.4833 1.0555 72.9% 1.7861

Table 5.3: Performance characteristics of the designed propellers from LINTP andLSM (N3745-based, � -F�D�E-5�4�E- database).�E-5�o�E-5�4�E- CAVOPT-BASE Recheck by MPUF3A"5#+ � �\"�] �!�� "F#/ � �I"$] Á �!��

LINTP 0.5000 1.0870 2.5000 0.5007 1.0865 73.3% 2.3250=\>A@ LSM 0.5000 1.0930 2.5000 0.4936 1.0848 72.4% 2.0736

Table 5.4: Approximation errors for ��²�5� �²�5� � database and �E- �F� -��5�E- databaseby =\>A@ order LSM (N3745-based).

RMS"F#/ "�] �!�� F�o� �F�4� � 1.64E-03 9.77E-04 2.11E-03= >A@ LSM �E-5�o�E-5�4�E- 9.20E-04 5.70E-04 1.25E-03

Table 5.5: Effect of different �� constraints.�!�� Constraint2.5 3.0 3.5 4.0 4.5 5.0"5#�'h�+*[-\� 73.3% 73.7% 73.8% 74.0% 74.0% 74.0%"5#�'h�+*[-I- 71.6% 71.8% 72.0% 72.2% 72.2% 72.2%"5#�'h�+* ½I� 69.5% 69.9% 70.2% 70.4% 70.5% 70.5%

130


x 1,x 2,

x 3

0 1 2 3 4 5 6 7

-1

0

1

2

3

x1

x2

x3

Figure 5.1: Convergence rate of the three design variables from =b>A@ order LSM(N4148-based, �E-5�4�E-9�4� - database).


Equ

ality

resi

dual

1 2 3 4 5 6 70

1

2

3

4

5

Figure 5.2: Equality residual of the optimization problem solved by =b>A@ order LSM(N4148-based, �E-5�4�E-9�4� - database).

131

r / R

P/D

,c/

D

f/c

0.2 0.4 0.6 0.8 10.0

0.5

1.0

1.5

2.0

0

0.05

0.1

0.15

LINTPLSM

P/D

f/c

c/D

Figure 5.3: Geometric parameters distribution from LINTP and LSM (N3745-based).

r / R

Γ/(

2πR

VS)

0.20 0.40 0.60 0.80 1.000.00

0.01

0.02

0.03

0.04

0.05

0.06

LINTPLSM

Figure 5.4: Mean circulation distribution from LINTP and LSM (N3745-based).

132

X

Y

Z

Figure 5.5: Designed propeller geometry from LINTP method (N3745-based).

x / c

-C

P

0 0.2 0.4 0.6 0.8 1-5

-4

-3

-2

-1

0

1

2

3 CPMIN = 2.5

Figure 5.6: Pressure coefficients distribution from LINTP method (N3745-based).

133

x / c

-CP

0 0.2 0.4 0.6 0.8 1-5

-4

-3

-2

-1

0

1

2

3CPMIN = 2.5

Figure 5.7: Pressure coefficients distribution from LSM (N3745-based).

CPMIN

η

0.50 1.00 1.50 2.00 2.500.60

0.65

0.70

0.75

0.80

0.85

KT = 0.10KT = 0.15KT = 0.20KT = 0.25

Figure 5.8: Optimal efficiency for different � �� constraints (N3745-based).

134


"5#/ß'.�+*,�� !�� Constraints5.5 6.0 6.5 7.0 7.5 8.0

CAVOPT-BASE 73.8% 74.0% 74.0% 74.0% 74.0% 74.0%Optimal Á

Contour Figure 74.7% 74.8% 74.8% 74.8% 74.8% 74.8%:X; 0.9571 0.9566 0.9566 0.9566 0.9566 0.9566:Zq 1.7725 1.7238 1.7238 1.7238 1.7238 1.7238

Optimization with two geometric parameters

Two geometric parameters�¥�\�

and u ��are used in the blade design method. The

optimization problem is as follows:

Minimize "�]subject to "F#/�'.�+*[-


�+*[n\� Ê :X; Ê �I*[0\��+*[n\� Ê :Zq Ê 0/* �I�

(5.10)

N3745 propeller is the base geometry. The maximum camber distribution�I� � u

is the same as the base geometry. The contours of constant "�#/ , constant �!�� and constant efficiency Á are shown in Figure 5.9, which is generated by =í>A@ order

LSM. The solutions :V; , :eq and :Zd shown in Table 5.6 are the results from CAVOPT-

BASE. Table 5.6 compares the optimal efficiencies from the CAVOPT-BASE and

the contour plot corresponding to the solution points.

135

x1

x 2

0.90 1.00 1.10 1.200.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.90

2.00K TP

=0.7

5

K TP=

0.30

K TP=

0.35

K TP=

0.40

K TP=

0.45

K TP=

0.50

K TP=

0.55

K TP=

0.60

K TP=

0.65

K TP=

0.70

CPM

INM

AX

=10

.0

CPM

INM

AX

=3.0

CPM

INM

AX

=3.5

CPM

INM

AX

=4.0

CPM

INM

AX

=4.5

CPM

INM

AX

=5.0

CP

MIN

MA

X=

5.5

CP

MIN

MA

X=

6.0

CP

MIN

MA

X=

6.5

CP

MIN

MA

X=

7.0

CP

MIN

MA

X=

7.5

CP

MIN

MA

X=

8.0

CPM

INM

AX

=8.

5

CPM

INM

AX

=9.

0C

PMIN

MA

X=

9.5

η=

72%

η=

61%

η=

62%

η=

63%

η=

64%

η=

65%

η=

66%

η=

67%

η=

68%

η=

69%

η=

70%

η = 81%

η=

73%

η=

74%

η = 75%η = 76%

η=

71%

η = 76%η = 77%

η = 78%η = 79%η = 80%

Figure 5.9: Contour plot at :jd('k� for different �c�� constraints (N3745-based).

136



, u �\�and



, while one multiplier is

used for u �\�and one multipliers for

�� u . The image effects due to hub and duct

are included. The optimization problem is as follows:

Minimize "�]subject to "F#�'.�+*[-\�

�!�� Ê 0/*1-�+* 5 - Ê :X; Ê �I*[-�+* 5 - Ê :Zq Ê �I*[-�+* nI� Ê :Zd Ê 0/* ��+* �I� Ê : � Ê 2+* �

(5.11)

N3745 propeller is selected as the base geometry. The design variables :È; and

:Zq are the multipliers assigned to��

. The design variable :<; is the multiplier cor-

responding to the pitch ratio at the root while :jq is the multiplier corresponding to

the pitch ratio at the tip. The design variables :rd and : � are the multipliers assigned

to u ��and

�E� � u , respectively. There are 5 equally distributed computational points

for :V; and :Zq while 10 equally distributed computational points for :rd and : � , re-

spectively. Thus the size of the generated database is -3�7-F�4� �$�H� � . The LINTP


Tables 5.7 and 5.8 show the optimization solutions and the design results from

LINTP. The optimal efficiency, Á s�' 5 2+*[-76 , is only slightly higher than the optimal

137

Table 5.7: Optimal solutions of the design problem with four variables.

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# : � �/¸ +# Õ3ÜÞÝ � �\"�]LINTP 0.9550 0.9375 0.8276 2.3864 1.0845

Table 5.8: Design results of the design problem with four variables.


efficiency (73.2%) with three design variables. This is because the linear multipliers

for�¥�\�

allow more flexibility in generating the propeller family. Thus the feasible

region of the optimization problem is enlarged and results in a better solution.

Figures 5.10 and 5.11 compare the geometric parameters distributions and mean

circulation distributions from three design variables and four design variables from

LINTP. The distribution is well compared to each other. Figure 5.12 shows the pres-

sure coefficients distribution. The inequality constraint � �� Ê 0í*1- is satisfied.

138

r / R

P/D

,c/

D

f/c

0.2 0.4 0.6 0.8 10.0

0.5

1.0

1.5

2.0

0.00

0.05

0.10


Figure 5.10: Comparison of geometric parameters from three and four design vari-ables from LINTP (N3745-based).

r / R

Γ/(

2πR

VS)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.00

0.01

0.02

0.03

0.04

0.05

0.06


Figure 5.11: Comparison of mean circulation distribution from three and four designvariables from LINTP (N3745-based).

139

x / c

-CP

0.00 0.20 0.40 0.60 0.80 1.00-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

CPMIN = 2.5

Figure 5.12: Pressure coefficient distribution from four design variables from LINTP(N3745-based).

140

5.2 Unsteady fully wetted ducted propeller subject to non-axisymmetric inflow

This section applies CAVOPT-BASE to ducted propeller design under fully wetted

conditions subject to unsteady non-axisymmetric inflow. The non-axisymmetric ini-

tial nominal wake is shown in Figure 5.13. The duct geometry is DUCT 19A [18]

which is shown in Figures 5.14 and 5.15.

The length of the duct is from -0.5 to 0.5, nondimensionalized by propeller ra-

dius. The propeller is located at the center of the duct at :�'.�+*[� . The effective wake

is evaluated at :�'ÉÄ��+*[0I0 .

Since the inflow is unsteady non-axisymmetric, the nominal wake needs to be

transformed to effective wake to start the design procedure.


The operating conditions and constraints are:

� � ' �\*[� (5.12)

Ã _ ' ��I�I� (5.13)� _ ' ��I�I� (5.14)¾ ' - (5.15)





and (3.4)):

�I*�= Ê v �¥�\�gwyx Y{z^| Ê 0í*[�141

�+*[0 Ê v u �\�gw¦x Y¢zM| Ê �/*[½�+* � Ê v � � � u w¦x Y¢zM| Ê �/*[�In (5.16)


Minimize "�]subject to "F#&J[L^J[NPOe'h�/*1-

�!�p�+� � Ê =�*[-:Z³ ª; Ê :X; Ê : ©�ª;:Z³ ªq Ê :Zq Ê : ©�ªq:Z³ ªd Ê :Zd Ê : ©�ªd

(5.17)

The designed propeller is required to be 5-bladed. Both the cavity number Ã _ and

Froude number� _ are set to be large values to avoid any cavitation. There is one

equality constraint onto the total thrust coefficient "$# J LMJ NPO of the propulsion system,

which is the summation of the thrust coefficient due to the propeller and the thrust

coefficient due to the duct.

"5# J LMJ NPO '."5#+ Ñ "F#+� (5.18)

From the viewpoint of optimization, the total thrust coefficient should be expressed

as a function of the design variables. However, it costs too much in computational

time since the MPUF3A has to be coupled with the GBFLOW for the total thrust

coefficient in each iteration. A way to overcome this difficulty is to find the relation-

ship between "3# J[L^J[NSO and "F#/ , and design a ducted propeller with respect to "$#/which corresponds to "3# J[L^J[NPO ')�+*1-��+*

142

UX1.10

1.06

1.02

0.98

0.94

0.90

0.86

0.82

0.78

0.74

0.70

Figure 5.13: Initial inflow for ducted propeller design [43].

Figure 5.14: Duct geometry [18].

143

X

Y

Z

Figure 5.15: Duct three dimensional geometry [18].

144

Table 5.9: Iteration results between MPUF3A and GBFLOW for different :È;(N3745-based).

Selected :X; values:X; 'f�I*,� :X; 'É�\*[2 :X;�'É�\* ="F# � > � > Y ú0.3466 0.5592 0.6570"F# � 0.2997 0.4413 0.4993"F# � � 0.0469 0.1179 0.1577



N3745 propeller is selected as the base geometry. The lower bounds and upper

bounds of the design variables are:

�\*[�I� Ê :X; Ê �I*1-��/*[nI� Ê :Zq Ê 0/*[�\��/*[�I� Ê :Zd Ê 2+*[�\� (5.19)

Three geometric parameters XPI, XCHD and XCI are considered in the design pro-

cedure. The corresponding design variables are :<; , :Zq and :ed , respectively. Several

:X; values are selected to generate the effective wake. The iteration results are shown

in Table 5.9 and Figure 5.16.

From Figure 5.16, :Û;(' �I*[0�=I=?n corresponds to "3# J[L^J[NPO 'k�+*[-\� and "F#/7'k�+* =I�approximately from interpolation. The effective wake generated from :È;²' �I*[0�=I=In ,

:Zq�' �I*[� and :Zd¥' �I* � is considered to be the inflow for this design case, as shown

in Figure 5.17.

145

x1

KT

tota

l,K

TP,

KT

D

1 1.1 1.2 1.3 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 KTtotal

KTP

KTD

KTtotal =

KTP =

x1* = 1.2448 0.5

x1* = 1.2448

0.4

Figure 5.16: Interpolation for "3#&J[L^J[NPOZ'h�/*1-\� .

UX1.821.751.681.621.551.481.411.341.281.211.14

Figure 5.17: Effective wake geometry for ducted fully wetted propeller blade design.

146

Thus the new optimization problem of Problem 5.17 with respect to "�#/ is:

Minimize "�]subject to "3#/ß'.�+*�=

�!�� Ê =�*1-�I* � Ê :X; Ê �I*[-�+* n Ê :Zq Ê 0/* ��+* � Ê :Zd Ê 2+* �

(5.20)

Ten computational points are assigned to each design variable. The � �� g�� database is interpolated by LINTP. The optimization solutions solved by CAVOPT-

BASE are shown in Table 5.10 while the design results are shown in Table 5.11.

In Table 5.10, :eq reaches the lower bound of the limits. This means that if the

chord length is further decreased, the efficiency may increase. Table 5.11 compares

the interpolated results from LINTP and the feedback results from MPUF3A. They

are reasonably close to each other.

Figures 5.19 and 5.20 show the geometric parameters distribution and the cir-

culation distribution. Figure 5.21 shows the blade geometry from LINTP. In Fig-

ure 5.22, the pressure coefficients satisfy the inequality constraint �� Ê =�*[- .

The designed propeller is then used to generate the corresponding effective wake

by performing MPUF3A and GBFLOW. The effective wake is shown in Figure 5.18

and is close to the approximated effective wake as shown in Figure 5.17 which is

used to start the design procedure.

The later effective wake is used to compute the performance characteristics of

the designed propeller by MPUF3A, which are shown in Table 5.12 together with

the required values from Figure 5.16. As expected, the design results satisfy duct

147

Table 5.10: Optimal solutions of ducted propeller subject to non-axisymmetric in-flow solved by LINTP (N3745-based).

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# � �I"$]LINTP 1.2043 0.8000 2.0098 1.1250

Table 5.11: Design results of ducted propeller subject to non-axisymmetric inflowsolved by LINTP (N3745-based).

CAVOPT-BASE Recheck by MPUF3A"5#/ ��I"�] �!�p�+� � "F#/ � �I"$] �!�� ÁLINTP 0.4000 1.1250 4.5000 0.4004 1.1250 4.5042 56.6%

Table 5.12: Performance characteristics generated from the designed geometry bycoupling MPUF3A and GBFLOW, together with the required characteristics frominterpolation.

"F#/ "F#+� "5# J[L^J[NPOMPUF3A 0.4000 0.0992 0.4992

Required values 0.4000 0.1000 0.5000

and the total thrust requirements very closely.

148

UX1.821.751.681.621.551.481.411.341.281.211.14

Figure 5.18: Effective wake from the designed propeller (N3745-based).

149

r / R

P/D

,c/D

f/c

0.2 0.4 0.6 0.8 10.00

0.50

1.00

1.50

2.00

2.50

0.00

0.05

0.10

P/Dc/Df/c

Figure 5.19: Geometry parameters distribution solved by LINTP (N3745-based).

r / R

Γ/(

2πR

VS)

0.2 0.4 0.6 0.8 10.00

0.01

0.02

0.03

0.04

0.05

Figure 5.20: Mean circulation distribution solved by LINTP (N3745-based).

150

X

Y

Z

Figure 5.21: Propeller blade geometry solved by LINTP (N3745-based).

x / c

-CP

0 0.25 0.5 0.75 1-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5 CPMIN = 4.5

Figure 5.22: Pressure coefficient distribution solved by LINTP (N3745-based).

151


Three geometric parameters�¥�\�

, u �\�and

�� u are considered in the design proce-

dure. Linearly distributed multipliers are used for�¥�\�

. The optimization problem

is as follows:

Minimize "�]subject to "3#/ß'.�+*�=?�

�!�� Ê =�*1-�I* � Ê :X; Ê 0/* ��I* � Ê :Zq Ê 0/* ��+* n Ê :Zd Ê 0/* ��+* � Ê : � Ê 2+* �

(5.21)


�¥�\�. The design variable :Û; is the multiplier cor-

responding to the pitch ratio at the root while :jq is the multiplier corresponding to

the pitch ratio at the tip. The design variables :rd and : � are the multipliers assigned

to u ��and

�E� � u , respectively. There are 5 equally distributed computational points

for :V; and :Zq while 10 equally distributed computational points for :rd and : � , re-

spectively. Thus the size of the generated database is -3�7-F�4� �$�H� � . The LINTP


Tables 5.13 and 5.14 show the optimization solutions and the design results ob-

tained by LINTP. The optimal efficiency, Á s¥'f-\½+* 5 6 , with four design variables is

almost the same as that (56.6%) with three design variables.

152

Table 5.13: Optimal solutions for the design problem with four variables.

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# : � �/¸ +# Õ3ÜÞÝ � �\"�]LINTP 1.4862 1.1250 0.8000 1.6667 1.1230

Table 5.14: Design results for the design problem with four variables.


Figure 5.23 compares the three geometric parameters distribution for three de-

sign variables and four design variables. The design with four variables results in

smaller camber and the same chord length. The result has bigger�¥�\�

for § � � Ê �+* nwhile smaller

�¥�\�for § � � Ô �+* n . The distribution of

�¥�\�, u �\�

and�� u leads

to the circulation distribution shown in Figure 5.24. For the design problem with

four variables, the circulation strength is bigger for § � � Ê �/* 5 - while smaller for§ � � Ô �+* 5 - . The minimum pressure coefficient �� is reduced to 4.0748, which

does not reach the limit. Thus the designed propeller blade with four variables

reduces the possibility of cavity occurrence. This is because that the lower pitch

reduces the loading at the tip, where cavity is most likely to occur. The pressure co-

efficients distribution from three variables and four variables solved by LINTP are

shown in Figures 4.37 and 4.38. As also noted that in the case of open propellers,

the four design variables allow for tip unloading without any sacrifice (in fact some

it gain) in efficiency, due to the fact that unloading is driven by optimization process.

153

r / R

P/D

,c/

D

f/c

0.2 0.4 0.6 0.8 10.0

1.0

2.0

3.0

0.00

0.05

0.10

0.15


P/D

f/c

c/D

Figure 5.23: Comparison of geometric parameters from three and four design vari-ables by LINTP (N3745-based).

r / R

Γ/(

2πR

VS)

0.2 0.4 0.6 0.8 1.00.00

0.01

0.02

0.03

0.04

0.05


Figure 5.24: Comparison of mean circulation distribution from three and four designvariables by LINTP (N3745-based).

154

x / c

-CP

0 0.25 0.5 0.75 1-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5 CPMIN = 4.5

Figure 5.25: Pressure coefficients distribution from three design variables by LINTP(N3745-based).

x / c

-CP

0 0.25 0.5 0.75 1-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5 CPMIN = 4.5

Figure 5.26: Pressure coefficients distribution from four design variables by LINTP(N3745-based).

155

5.3 Unsteady cavitating propeller inside tunnel subject to non-axisymmetric inflow

This section applies CAVOPT-BASE to the cavitating propeller design inside cylin-

drical tunnel. The inflow is non-axisymmetric as shown in Figure 5.27. A specified

percentage of cavitation is allowed. In the optimization scheme, only the thrust co-

efficient "F#+ from the propeller is considered.


The operating conditions are specified as follows:

� � ' �+* � (5.22)Ã _ ' 0/*[- (5.23)� _ ' -/* � (5.24)¾ ' - (5.25)





and (3.4)):

�I*[0 Ê v �¥�\�gwyx Y{z^| Ê �\*[n�+*[0 Ê v u �\�gw¦x Y¢zM| Ê �/*1-�+* � Ê v � � � u w¦x Y¢zM| Ê �/*[�In (5.26)

156

Five-bladed propeller is used in the design procedure. The optimization problem is:

Minimize "�]subject to "F#/�')�+*�=

�B� Ê 2I� 6: ³ ª; Ê :X; Ê : ©�ª;:Z³ ªq Ê :Zq Ê : ©�ªq:Z³ ªd Ê :Zd Ê : ©�ªd

(5.27)

The requirement for the thrust coefficient due to the propeller is "�#/8'f�+*�= , which

serves as an equality constraint. The inequality constraint �B� Ê 2I� 6 allows partial

cavitation no greater than 30% of the blade surface area. The geometric parameters

considered in the design include��

, u �\�and

�� u , with their ranges specified

in Equation 5.26. The corresponding design variables are :È; , :Zq and :Zd , respec-

tively. The ranges of the three design variables are determined by the selected base

geometry. The skew constraint is not included in this case.



The bounds of the design variables are specified as follows, satisfying the ranges of

the parameters:

�/*[nI� Ê :X; Ê �I*10��/*[½I� Ê :Zq Ê �I*10��/*[�I� Ê :Zd Ê 2+*[�\� (5.28)

157

UX1.10

1.06

1.02

0.98

0.94

0.90

0.86

0.82

0.78

0.74

0.70

Figure 5.27: Inflow for cavitating propeller design inside tunnel [43].

158

Table 5.15: Optimal solutions for cavitating blade design inside tunnel (N3745-based).

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# Õ3ÜÞÝ � �\"�]LINTP 0.8728 0.9567 1.3333 0.7961

Table 5.16: Design results for cavitating blade design inside tunnel (N3745-based).

CAVOPT-BASE Recheck by MPUF3A"F#/ � �I"$] �� "F#/ � �I"$] Á ��LINTP 0.4000 0.7961 30.0% 0.3971 0.7892 76.1% 30.2%

The size of the database is � �� . In MPUF3A, the blade is discretized by 20

panels in chordwise direction while 9 panels in spanwise direction.

The optimization solutions from LINTP are shown in Table 5.15. The design

results are shown in Table 5.16 , together with the feedback results from MPUF3A.

The comparison validates that the interpolation works well.

The geometric parameters distributions and mean circulation distributions from

LINTP are shown in Figures 5.28 and 5.29, respectively. The designed propeller

geometry and the resulting cavity shape from LINTP are shown in Figures 5.30 and

5.31.

Effect of cavity constraints

Table 5.17 and Figure 5.32 show the optimal efficiency for different "�#/s and cavity

constraints. LINTP is used to approximate the � �ß�)� �ß�)� � database generated

from N3745 propeller geometry. As the cavity constraint is relaxed, the optimal

efficiency increases. This is because that the feasible region of the optimization

159

r / R

P/D

,c/

D

f/c

0.2 0.4 0.6 0.8 1.00.00

0.50

1.00

1.50

2.00

0.00

0.05

0.10P/Dc/Df/c

Figure 5.28: Geometry Parameters solved by LINTP and =b>A@ order LSM (N3745-based).

r / R

Γ/(2

πRV

S)

0.2 0.4 0.6 0.8 1.00.00

0.01

0.02

0.03

0.04

0.05

Figure 5.29: Mean circulation distribution solved by LINTP and =b>A@ order LSM(N3745-based).

160

X

Y

Z

Figure 5.30: Designed Propeller geometry solved by LINTP (N3745-based).

CA = 30.2 %

η = 76.1%

Figure 5.31: Cavity shape of the designed geometry by LINTP (N3745-based).

161

Table 5.17: Optimal efficiency for different �� constraints (N3745-based).

Cavity constraints-76 � �76 0\� 6 Ê 2I�76 Ê =?� 6"5#+�'h�/*[2?- 72.0% 74.6% 77.4% 77.7% 77.7%"5#+�'h�/* =?� 69.5% 73.3% 75.3% 76.1% 76.1%"5#+�'h�/* =b- 66.8% 70.3% 74.2% 74.2% 74.2%

problem is enlarged when larger cavity area ratio is allowed.

Optimization with two geometry parameters

The two geometric parameters considered in the optimization problem are�¥�\�

and�E� � u . The optimization problem is:

Minimize "�]subject to "F#�'h�/* =?�

�� ÊCAMAX

�/*[n Ê :X; Ê �I* ��/*[� Ê :Zd Ê 2+* �

(5.29)

The chord length distribution u �\�of the designed geometry is the same as the

base geometry. The contour plots of constant efficiency, constant "�#/ and constant

�� are given in Figure 5.35. The optimal efficiency and the solutions points for

different �� constraints are shown in Table 5.18. In Table 5.18, the solution differ-

ences between CAVOPT-BASE and the contour values are due to the LSM errors in

generating the contour plots. Note that for this case, the solutions remain the same

even though the cavity constraint changes.

162

CA

η

0 0.1 0.2 0.3 0.40.65

0.70

0.75

0.80

KTP = 0.35KTP = 0.40KTP = 0.45

Figure 5.32: Effect of the �� constraints (N3745-based).

Table 5.18: Optimal efficiency for different �� constraints (N3745-based).

"F#/�'h�+*�=?� Cavity Constraints15% 20% 30% 40%

CAVOPT-BASE 74.2% 75.3% 76.0% 76.0%Optimal Á

Contour Figure 74.1% 75.2% 75.9% 75.9%:X; 0.8395 0.8542 0.8701 0.8701:Zd 1.8992 1.6183 1.3333 1.3333

163

CA = 10%

η = 73.3%

CA = 19.3 %

η = 75.4%

CA = 30.2 %

η = 76.1%

CA = 32.1 %

η = 76.1%

Figure 5.33: Cavity shape for different �B� constraints.

164

CA = 14.4%

η = 74.2%

CA = 19.3 %

η = 75.4%

CA = 24.8%

η = 76.0%

CA = 24.8%

η = 76.0%

Figure 5.34: Cavity shape for different �B� constraints with :rq�'f�I*[� .

165

X1

X3

0.80 0.85 0.90 0.95 1.000.00

1.00

2.00

3.00

KT = 0.40

X1

X3

0.80 0.85 0.90 0.95 1.000.00

1.00

2.00

3.00

η=

74%

η=

78%

η=

77%

η=

76%

η=

75%η = 66%

η=

73%η

=72%

η = 71%

η = 70%

η = 69%

η = 68%

η = 67%

X1

X3

0.80 0.85 0.90 0.95 1.000.00

1.00

2.00

3.00

CA = 25%

CA=

65%

CA

=20%

CA = 25%

CA

=15%

CA

=30%

CA

=35

%

CA

=40

%

CA = 45%

CA = 50%

CA = 55%

CA = 60%

Figure 5.35: Contour plot generated by LINTP with :rq²'É�I* � (N3745-based).

166


Geometric parameters,�¥�\�

, u �\�and

�� u are considered in this case. The design

variables :V; and :Zq are assigned to��

. The multipliers for u �\�and

�� u are :Zdand : � , respectively. The cavitating open propeller design problem is as follows:

Minimize "�]subject to "3#�'h�+*�=

�� Ê 2\� 6�+* n Ê :X; Ê �I*[0�+* n Ê :Zq Ê �I*[0�+* ½ Ê :Zd Ê 0/* ��+* � Ê : � Ê 2+* �

(5.30)

N3745 propeller is selected as the base geometry. Ten equally distributed computa-

tional points are assigned to each design variable. Thus the size of the database is

� �F�o� �F�4� �5�4�� .

Table 5.19 shows the optimal solutions for the design problem, while Table 5.20

compares the corresponding designed propeller performance with the designed re-

sults from three variables problem. The optimal efficiency (76.7%) from four design

variables is slightly higher than the efficiency (76.1%) from three design variables.

The increment is due to larger feasible region which allows linearly distributed mul-

tipliers for�¥�\�

. Again, as mentioned before, tip unloading was achieved with some

actually small gain in efficiency.

The geometric parameters distributions from three and four design variables are

compared in Figure 5.36 while the mean circulation distributions are compared in

167

Figure 5.37.

Table 5.19: Optimal solutions for cavitating propeller inside tunnel with four vari-ables.

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# : � �/¸ +# Õ3ÜÞÝ � �\"�]LINTP 1.0210 0.8000 1.0386 1.1439 0.7886

Table 5.20: Design results for cavitating propeller inside tunnel with four and threevariables.

CAVOPT-BASE Recheck by MPUF3A"F# ��I"�] �� "F# � �\"�] �� Á4-variable 0.4000 0.7886 30.0% 0.3934 0.7758 26.0% 76.7%3-variable 0.4000 0.7961 30.0% 0.3971 0.7892 30.2% 76.1%

168

r / R

P/D

,c/D

f/c

0.2 0.4 0.6 0.8 10.0

0.5

1.0

1.5

2.0

0

0.05

0.1


P/D

f/c

c/D

Figure 5.36: Geometric parameters distribution from three and four design variablessolved by LINTP (N3745-based).

r / R

Γ/(2

πRV

S)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05


Figure 5.37: Mean circulation distribution from three and four design variablessolved by LINTP (N3745-based).

169

Chapter 6

Applications of Blade Design Methods to OtherPropulsion Systems

6.1 Unsteady fully wetted pushing type podded propeller subjectto non-axisymmetric inflow

The nominal inflow wake is non-axisymmetric, as shown in Figure 6.1. Similar to

the case of ducted propeller, the effective wake has to be obtained iteratively by

MPUF3A and GBFLOW.

The pod and strut geometries are shown in Figures 6.2, 6.3 and 6.4. The

length of the pod is from :D'çÄü0/*[0\½?- to :D' 0/*[0\½?- while the length of the strut is

from :H'µÄt�/*[½ to :4' �/*[½ . The propeller to be designed is a push type located at

:�' �I*i� n �7� . The effective wake is evaluated at :ß'ì�+* 5 �E-/� . The length dimensions

are made nondimensional by propeller radius.

Both the thrust due to the propeller and the drag due to the pod and strut should

be considered in computing the total thrust of the propulsion system.

"5#KJ LMJ NPOZ'h"5#+�ÄH"5#RQSLUT (6.1)

where "F# J[L^J[NPO is the total thrust coefficient of the podded propulsion system; "�#/is the thrust coefficient of the propeller; "$# Q�LUT is the drag coefficient including the

pressure drag force and the frictional drag force. A uniform friction coefficient of

170

0.004 was applied in this case.



� � ' �\*[� (6.2)

Ã _ ' ��I�I� (6.3)� _ ' ��I�I� (6.4)¾ ' 2 (6.5)





and (3.4)):

�I* � Ê v �¥�\�gwyx Y{z^| Ê 0í*[��+*[0 Ê v u �\�gw¦x Y¢zM| Ê �/*1-�+* � Ê v � � � u w¦x Y¢zM| Ê �/*[�In (6.6)

171

UX1.10

1.06

1.02

0.98

0.94

0.90

0.86

0.82

0.78

0.74

0.70

Figure 6.1: Nominal wake geometry for the podded propulsor design with one com-ponent.

X

Y

Z

-0.12 0.12

-0.4049 0.4049

Figure 6.2: Pod and strut 2D geometries of the podded propulsor design with onecomponent.

172

X

Y

Z

-2.265 2.265

-0.6 0.6 1.1899


X

Y

Z


173


Minimize "�]subject to "F# J[L^J[NPO 'h�/*,�E-

�!�p�+� � Ê =�*[-:Z³ ª; Ê :X; Ê : ©�ª;: ³ ªq Ê :Zq Ê : ©�ªq:Z³ ªd Ê :Zd Ê : ©�ªd

(6.7)

The total thrust of the propulsion system is required to be 0.15. �� Ê =+*1- is

required for fully wetted run. The geometric parameters considered are�¥�\�

, u �\�and

� � � u , with the corresponding design variables :Û; , :Zq and :ed .



The N4148 propeller is selected as the base geometry. The bounds on the design

variables are determined as follows:

�I*[� Ê :X; Ê 0/*[��+*[n Ê :Zq Ê 0/*[��+*[� Ê :Zd Ê 2+*[� (6.8)

The effective wake

One geometric parameter�¥�\�

is considered in generating the effective wake. The

coefficients "3# J[L^J[NSO , "F#/ and "F# Q�L^T are obtained for different :Û; , as shown in Ta-

174

ble 6.1.

Table 6.1: "F#KJ LMJ NPO , "F#/ and "F#RQ�LUT from different :V; by coupling MPUF3A andGBFLOW (N4148-based).

Selected :X; values:X;�'k�I*�= :X;c'É�I*[- :X;c'É�I* ½ :X; 'f�I* 5"5# J LMJ NPO 0.1188 0.1424 0.1641 0.1828"F#/ 0.1595 0.1854 0.2100 0.2318"5#RQSLUT 0.0406 0.0430 0.0459 0.0490

Figure 6.5 shows the interpolation scheme for "$# J[L^J[NPO . The interpolated multipli-

ers :X;D' �I*[-\2?-\� , and the propeller thrust coefficient and drag corresponding to

"F#&J[L^J[NSOX'f�+*i�E- are "5#+á'f�/*,� � =?� and "F#RQ�LUTü'ì�+* �\=I=I� . The wake corresponding to

:X;B' �\*1-\2I-\� is considered to be the effective wake, as shown in Figure 6.6. Thus

the design problem can be transformed to one with respect to "�#+ instead of "F#&J[L^J[NPOas follows:

Minimize "�]subject to "3#/ß'.�+*i� � =I�

�!�� Ê =�*1-�I* � Ê :X; Ê 0/* ��+* n Ê :Zq Ê 0/* ��+* � Ê :Zd Ê 2+* �

(6.9)

175

x1

KT

tota

l,K

TP,

KT

pod

1.4 1.5 1.6 1.70.00

0.10

0.20

0.30 KTtotal

KTP

KTpod

x1* = 1.5350

KTtotal = 0.1500

KTpod =0.0440

KTP =0.1940

Figure 6.5: "F#&J[L^J[NPO , "5#+ and "F#RQSLUT from different :V; by coupling MPUF3A andGBFLOW (N4148-based).

UX1.051.031.010.990.970.950.930.910.890.870.850.830.810.790.770.750.730.710.690.670.65

Figure 6.6: Effective wake of the podded propeller design (N4148-based).

176

Optimal solutions and design results

Ten computational points are assigned to each design variable. Thus the size of the

database is � �t�g� �� . The optimal solutions are shown in Table 6.2 while the de-

sign results are shown in Table 6.3. The design results are also shown in Figures 6.8,

6.9, 6.10 and 6.11.

The effective wake generated from the designed propeller by performing MPUF3A

and GBFLOW iteratively is shown in Figure 6.7, which is very closed the wake con-

tour generated by interpolation as shown in Figure 6.6. The thrust coefficient of

the design propeller solved by coupling MPUF3A and GBFLOW is 0.1963 which

is close to the required value 0.1940 while the non-dimensional drag forces due to

the pod and strut is 0.0447 which is close to the required value 0.0440, as shown in

Table 6.4.

177

Table 6.2: Optimal solutions of podded propeller design with respect to "�#/ solvedby LINTP (N4148-based).

:X; �/¸ +# :Zq �/¸ +# :Zd �/¸ +# min � �I"$]LINTP 1.2516 2.0000 2.7185 0.4316

Table 6.3: Design results of the podded propeller design with respect to "�#/ solvedby LINTP (N4148-based).

CAVOPT-BASE Recheck by MPUF3A"5#/ ��I"�] �!�p�+� � "F#/ � �I"$] Á �!�p�+� �LINTP 0.1940 0.4316 4.5000 0.1945 0.4310 71.8% 4.5071

Table 6.4: Propeller performance solved by coupling MPUF3A and GBFLOW withthe designed effective wake.

"F#/ "5#RQSLUT "5#&J[L^J[NPOMPUF3A 0.1963 0.0447 0.1516

Required values 0.1940 0.0440 0.1500

178

UX1.051.031.010.990.970.950.930.910.890.870.850.830.810.790.770.750.730.710.690.670.65

Figure 6.7: Effective wake generated from the designed propeller (N4148-based).

179

X

Y

Z

Figure 6.8: Designed podded propeller geometry solved by LINTP (N4148-based).

x / c

-CP

0 0.2 0.4 0.6 0.8 1-5

-4

-3

-2

-1

0

1

2

3

4

5 CPMIN = 4.5

Figure 6.9: Pressure coefficients distribution from the designed podded propeller(N4148-based).

180

r / R

P/D

,c/

D

f/c

0.2 0.4 0.6 0.8 10

1

2

0.00

0.05

0.10P/Dc/Df/c

Figure 6.10: Designed podded propeller geometric parameters distribution solvedby LINTP (N4148-based).

r / R

Γ/(

2πR

VS)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

Figure 6.11: Mean circulation distribution from the designed podded propeller(N4148-based). 181

6.2 Steady fully wetted contra-rotating propellers subject to uni-form inflow

Two propellers, denoted as forward propeller (For) and after propeller (Aft) , are

installed onto a dual shaft rotating in opposite directions. The inflow to one of the

propellers will be affected by the induced velocity from the other propeller. Thus

MPUF3A and GBFLOW are coupled to solve for the effective wakes to each pro-

peller.

All of the lengths are nondimensionalized by the radius of For propeller. The

radius of Aft propeller is 0.9799. The distance between the two propellers is 0.6,

with For propeller located at :ß'mÄ��+* 2 while Aft propeller located at :�'f�+* 2 . The

hub is considered to be infinite long with hub ratio § @ 'ì�+*[0 . The effective wake to

For propeller is evaluated at :�' Ä��+*1-�2 while the effective wake to Aft propeller is

evaluated at :�')�+*i� � . The friction coefficient is taken equal to 0.004.

Both the thrust and torque from the two propellers should be considered in com-

puting the total thrust and torque.

"5#KJ LMJ NPOe'h"F#�F Ñ "F# Â"�]·J[L^J[NPO·'."�]GF Ñ "$] Â

where "F#&J[L^J[NSO is the total thrust coefficient of the propulsion system; "�]·J[L^J[NPO is the

total torque coefficient of the propulsion system; "�#�F and "�]GF are the thrust and

torque coefficients of For propeller while "$# Âand "�] Â

are the thrust and torque

coefficients of Aft propeller.

Since the two propellers are rotating in opposite directions, the torque coef-

ficients "�]GF and "�] Âwill have opposite sign. To increase the total propulsion

efficiency, the total torque coefficient "�]·J[L^J[NPO is required to be zero so that (mean)

182

swirl cancellation can be achieved downstream of the Aft propeller.



Ã _ ' ��I�I�� _ ' ��I�I�¾ ' 2Advance ratio:

For propeller:� � 'É�I* �I�I�\�

Aft propeller:� � 'É�I* �?0\�I-





and (3.4)):

�I*[0 Ê v �¥�\�gwyx Y{z^| Ê 0í*[��+*[0 Ê v u �\�gw¦x Y¢zM| Ê �/*1-�+* � Ê v � � � u w¦x Y¢zM| Ê �/*[�In

183


Minimize "�]·J[L^J[NPOsubject to "F# J[L^J[NPO 'h�/*1-\�

�!�p�+� � Ê �I*[-:Z³ ª; Ê :X; Ê : ©�ª;: ³ ªq Ê :Zq Ê : ©�ªq:Z³ ªd Ê :Zd Ê : ©�ªd

(6.10)

where :X; , :Zq and :Zd are the design variables assigned to�¥�\�

, u �\�and

� � u , respec-

tively.

Since the total torque coefficient "�]·J[L^J[NPO can be negative or positive, the equiv-

alent objective is to require "�]·J[L^J[NPOñ' � . Additionally, the total thrust "3#&J[L^J[NPO is

required to be 0.50. Thus the designed contra-rotating propellers have to satisfy the

following equivalent requirements:

"5#�F Ñ "F# Â ')�+*1-��"$]GF Ñ "�] Â ')�+*[�\�

The additional inequality constraint �� Ê �\*1- ensures that the propellers are

operating under non-cavitating conditions.


Base geometry

The base geometry has same geometric parameters as N4148 except that the pitch

is 1.2 times larger. The distribution of base geometric parameters is shown in Fig-

184

r / R

P/D

,c/

D

f/c

0.2 0.4 0.6 0.8 10.0

1.0

2.0

0.00

0.05

0.10

P/Dc/Df/c

Figure 6.12: Base geometric parameters for the design of contra-rotating propellers.

ure 6.12. The bounds of the design variables are determined as follows:

�I*[� Ê :X; Ê �I*[½�+*[n Ê :Zq Ê 0/*[��+*[� Ê :Zd Ê 2+*[�

The effective wakes

Only geometric parameter�¥�\�

is considered to approximate the effective wakes.

The multiplier :eW�; is assigned to�¥�\�

of For propeller while :jYR; is assigned to�¥�\�

of Aft propeller. For different combinations of :rWK; and :ZY�; , the thrust and torque

185

coefficients are evaluated, as shown in Table 6.5 and Figures 6.13, 6.14 and 6.15.

Table 6.5: Thrust and torque coefficients for different combinations of :VWK; and :ZYR; .:eWK; :ZY�; "F#�F "F# Â "F#&J[L^J[NPO "�]IH "$]GJ "�]·J[L^J[NPO1.2 1.2 0.1825 0.1790 0.3616 0.4021 -0.3973 0.00481.2 1.4 0.1802 0.2564 0.4366 0.3974 -0.6509 -0.25351.2 1.6 0.1783 0.3157 0.4940 0.3934 -0.9081 -0.51471.4 1.2 0.2484 0.1880 0.4364 0.6290 -0.4171 0.21191.4 1.4 0.2465 0.2702 0.5168 0.6246 -0.6856 0.06091.4 1.6 0.2451 0.3327 0.5779 0.6212 -0.9566 -0.33541.6 1.2 0.3007 0.2001 0.5007 0.8643 -0.4432 0.42111.6 1.4 0.2992 0.2865 0.5858 0.8604 -0.7262 0.13421.6 1.6 0.2981 0.3578 0.6499 0.8573 -1.0105 -0.1532

Least squares method is used for interpolation based on the following polynomial

function:

� ' � ;y: q WK; Ñ � q : qYR; Ñ � d :eWK;y:ZYR; Ñ � � :eW�; Ñ � � :ZYR; Ñ � where

�denotes "3#�F , "5# Â

, "F# J[L^J[NSO , "�]GF , "�] Âand "�] J[L^J[NPO . As required, the inter-

polation results for "F# J[L^J[NPO 'h�/*1-\� and "�] J[L^J[NSO 'h� are shown in Table 6.6.

186

0.20

0.25

0.30

1.2

1.3

1.4

1.5

1.6

xf1

1.2

1.3

1.4

1.5

1.6

x a1

Y

X

Z

KTF

KTF vs. (xf1, xa1)

0.40

0.50

0.60

0.70

0.80

1.21.3

1.41.5

1.6xf1

1.21.3

1.41.5

1.6

x a1

Y

X

Z

KQF

KQF vs. (xf1, xa1)

Figure 6.13: Thrust and torque coefficients from For propeller.

0.20

0.30

0.40

1.21.3

1.41.5

1.61.7

xf1

1.21.3

1.41.5

1.61.7

x a1

Y

X

Z

KTA

KTA vs. (xf1, xa1)

-1.00

-0.80

-0.60

-0.40

1.2

1.4

1.6x f1

1.2

1.4

1.6xa1

X Y

Z

KQA

KQA vs. (xf1, xa1)

Figure 6.14: Thrust and torque coefficients from Aft propeller.

0.50

1.2

1.3

1.4

1.5

1.6xf1 1.2

1.3

1.4

1.5

1.6

x a1

Y

X

Z

KTtotal

KTtotal vs. (xf1, xa1)

-0.50

0.00

0.50

1.2

1.4

1.6xf1 1.2

1.4

1.6

x a1

Y

X

Z

KQtotal

KQtotal vs. (xf1, xa1)

Figure 6.15: Total thrust and torque coefficients from the propulsion system.

187

Table 6.6: Interpolation results for "3#&J[L^J[NPOe'h�+*[-\� and "�]·J[L^J[NPO·'.� .

For Aft:eWK; 'f�I*[2 �7� � :ZYR;�'k�I* 2?-�=In"F#�F�'h�+*[0�=?½ 5 "F# Â '.�/*10I-�2I2"�]GF�'h�/*[½?0�=b- "�] Â 'kÄt�+* ½?0�=?-"F#&J[L^J[NPOe')�+*1-��I�I�"�]·J[L^J[NSOZ'.�+* �I�I�\�

The effective wake to For propeller corresponding to :XWK; 'f�I* 2 �7� � and the effective

wake to Aft propeller corresponding to :jYR;ß' �I* 2?-�=?n are computed by coupling

MPUF3A and GBFLOW, as shown in Figures 6.16 and 6.17.

Design problems for each component

Thus the design problem is transformed to the following problems: The optimal

For propeller Aft propeller

Minimize "�]GF Minimize "�] Âsubject to subject to

"F#�F�'.�/*10�=I½ 5 "F# Â 'h�+*[0I-\2\2�!�� Ê �I*[- �!�p�+� � Ê �\*1-�I* � Ê :X; Ê �I* ½ �I*[� Ê :X; Ê �I*[½�+* n Ê :Zq Ê 0/* � �+*[n Ê :Zq Ê 0/*[��+* � Ê :Zd Ê 2+* � �+*[� Ê :Zd Ê 2+*[�

solutions of the design problems are shown in Table 6.7 while the designed results

188

UX1.0141.0111.0091.0061.0031.0010.9980.9950.9930.9900.9870.9850.9820.9790.976

Figure 6.16: Approximate effective wake for For propeller.

UX1.2701.2491.2281.2071.1871.1661.1451.1241.1041.0831.0621.0411.0211.0000.979

Figure 6.17: Approximate effective wake for Aft propeller.

189

Table 6.7: Optimal solutions for the CRP propeller design.

For Aft:X; �/¸ +# 1.1864 1.2022:Zq �/¸ +# 1.1587 1.0571:Zd �/¸ +# 3.0000 2.5542

Table 6.8: Designed results from CAVOPT-BASE and rechecked by MPUF3A.

CAVOPT-BASE Recheck by MPUF3AFor Aft For Aft"F# 0.2467 0.2533 0.2468 0.2533� �I"$] 0.5538 -0.5726 0.5538 -0.5722Á 70.9% 71.9% 70.9% 71.9%�!�p�+� � 1.5000 1.5000 1.4448 1.2987"5# J LMJ NPO 0.5000 0.5001"�] J[L^J[NPO -0.0187 -0.0184

are shown in Table 6.8. The geometric parameters distributions are shown in Fig-

ure 6.18. The pressure coefficients distribution and circulation distribution of For

and Aft propellers are shown in Figures 6.20 6.22, 6.21 and 6.23. The designed

propeller geometry is shown in Figure 6.19.

Recheck the effective wakes from the design propeller geometries

The designed For and Aft propeller geometries are used by MPUF3A and GBFLOW

to recheck the effective wakes. The effective wakes from the designed geometries

are shown in Figures 6.24 and 6.25, which are very close to the approximate effec-

tive wakes shown in Figures 6.16 and 6.17.

190

r / R

P/D

,c/

D

f/c

0.2 0.4 0.6 0.8 10.0

0.5

1.0

1.5

2.0

0

0.05

0.1For propellerAft propeller

P/D

f/c

c/D

Figure 6.18: Geometric parameters distributions of For and Aft propellers.

X

Y

Z

Figure 6.19: Designed CRP propeller geometry.

191

x / c

-CP

0.2 0.4 0.6 0.8 1-3

-2

-1

0

1

2

CPMIN = 1.5

Figure 6.20: Pressure coefficients distribution of For propeller.

x / c

-CP

0.2 0.4 0.6 0.8 1-3

-2

-1

0

1

2

CPMIN = 1.5

Figure 6.21: Pressure coefficients distribution of Aft propeller.192

r / R

Γ/(

2πR

VS)

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

Figure 6.22: Mean circulation distribution of For propeller.

r / R

Γ/(

2πR

VS)

0.2 0.4 0.6 0.8 1

-0.04

-0.03

-0.02

-0.01

0

Figure 6.23: Mean circulation distribution of Aft propeller.193

UX1.0141.0111.0091.0061.0031.0000.9980.9950.9920.9900.9870.9840.9810.9790.976

Figure 6.24: Effective wake to For propeller generated from designed geometries.

UX1.2701.2491.2281.2081.1871.1661.1451.1251.1041.0831.0621.0411.0211.0000.979

Figure 6.25: Effective wake to Aft propeller generated from design geometries.194

Chapter 7

Conclusions and Recommendations

This chapter discusses the conclusions of the work in this thesis and presents recom-

mendations for improvements in future research.

7.1 Conclusions

A general blade design method is presented in this thesis for non-cavitating or cav-

itating open, ducted, podded or contra-rotating propellers design subject to uniform

or non-axisymmetric inflow. The objective of the design method is to achieve the

highest efficiency while satisfying the thrust requirements and constraints.

The current design method (named CAVOPT-BASE) searches for the optimal

blade geometry within a performance database by using constrained nonlinear op-

timization. The vortex/lattice method (named MPUF3A) is used to construct the

database for a propeller family generated from a base propeller by changing the

selected geometric parameters. In the case of non-axisymmetric inflow, the finite

volume method (named GBFLOW) is coupled with MPUF3A in order to determine

the effective inflow in which the optimum blade is to be designed.

The constrained nonlinear optimization problem is solved by augmented La-

195

grangian penalty function. The algorithm, originally developed by Mishima [44],

consists of outer iterations and inner iterations. In the outer iterations, the La-

grangian multipliers and penalty coefficients are updated based on the Karush-Kuhn-

Tucker conditions, and the constrained nonlinear optimization problem is converted

to an unconstrained nonlinear optimization problem. The inner iterations solve the

unconstrained nonlinear optimization by using quasi-Newton method with the de-

scent direction determined by Broyden-Fletcher-Goldfarb-Shanno method, and the

step size determined by the line search method. The algorithm solves constrained

nonlinear optimization problems efficiently and allows for unlimited number of

equality and inequality constraints.

At the beginning of the design procedure, a base geometry is selected to gener-

ate a propeller family by multiplying the geometric parameters such as pitch, chord

and camber with factors, which are regarded as the design variables. A database

corresponding to the propeller family is constructed by performing MPUF3A runs

to analyze the performance characteristics for each geometry of the family. The

database is then interpolated by either the least squares method or a piecewise liner

interpolation method to construct the objective and constraint functions in terms of

the design variables. The constrained nonlinear optimization incorporates the func-

tions and searches for the optimal solution until all the constraints are satisfied. In

the case of piecewise linear interpolation method the optimum solution is searched

for through all cells of the domain, and that actually leads to the optimum global

design over the specified parameters.

If the inflow is non-axisymmetric, one extra step for evaluating the effective

wake should be performed before the design procedure. The situation is simplified

by only considering pitch as the geometric parameter and generating a small pro-

196

peller family from the base propeller. For each geometry in the family, MPUF3A

and GBFLOW are coupled to evaluate the loading and effective wake until forces

converge. The results are interpolated to evaluate the desired pitch providing the

required loading. The effective wake is then determined from the geometry corre-

sponding to the evaluated pitch.

The method is first applied to open propeller design subject to uniform or non-

axisymmetric inflow for both non-cavitating and cavitating conditions. The design

results compare well with other design methods such as PVL based on Betz/Lerb’s

condition, and CAVOPT-3D.

The method is then applied to the design of ducted, podded, and contra-rotating

propulsors subject to uniform or non-axisymmetric inflow with given duct/pod ge-

ometry. Both non-cavitating and cavitating cases are designed successfully with the

results satisfying equality constraint for the thrust coefficient, and inequality con-

straints on the minimum pressure coefficient and on the cavity area to blade area

ratio.

7.2 Recommendations

The current design method can be improved and extended to design more compli-

cated propulsion systems. Suggestions for future work are given as follows.

Construct larger database

The accuracy of interpolation is directly related to the size of the database, especially

for linear interpolation. More computational points can be placed equally inside the

197

ranges of the design variables. Furthermore, the ranges of the design variables can

be increased to enlarge the feasible domain such that the optimal solutions are not

restricted onto the boundaries.

Include more geometric parameters

Currently the three geometric parameters, pitch, chord and camber, are considered

in the design procedure. More geometric parameters such as rake and skew can be

included into the design so that the design results are more general and practical. The

present method can be used to provide a very good initiate guess for CAVOPT-3D,

which can then perform a less restrictive design in terms of geometry characteristics.

Improve the interpolation method

The database interpolation in the current method is performed by either the least

squares method or piecewise linear interpolation method. The piecewise linear in-

terpolation method has been proven to be more accurate in approximating pressure

coefficients and cavity to blade area ratio since their distributions are not as smooth

as the distributions of thrust and torque coefficients. However, the linear interpola-

tion method requires the database structure to be regular such that other data infor-

mation can not be included. In addition to these two interpolation methods, a general

interpolation method can be developed to capture the local structure of the database,

and the function value can be predicted locally. In that case, other data information

can be easily incorporated into the existing database to allow more precise design.

198

Include more physical constraints

More physical constraints, such as on the maximum values of the amplitudes of the

unsteady forces harmonics, or on the amount of face cavitation, can be incorporated

into the design method to make it more practical.

Extend to complicated propulsion systems

The current method can be extended to the design of multi-component propellers and

hybrid propulsion systems, as shown in the podded propeller design with two com-

ponents. It should be noted that the presented applications on podded and contra-

rotating propellers were somewhat limited, and more are required to render the cur-

rent method into a general design tool.

199

Appendix A

Geometric Parameters for N4148 Propeller

The geometry parameters for N4148 [39] propeller is shown in Table 1 and in Fig-

ure 1.

Table 1: Geometry Parameters for N4148 propeller [39].

r/R0.2000 0.3000 0.4000 0.5000 0.6000

P/D 0.9921 0.9967 0.9987 0.9975 0.9944RAKE/D 0.0000 0.0000 0.0000 0.0000 0.0000SKEW 0.0000 0.0000 0.0000 0.0000 0.0000

c/D 0.1600 0.1818 0.2024 0.2196 0.2305f/c 0.0174 0.0195 0.0192 0.0175 0.0158t/D 0.0329 0.0282 0.0239 0.0198 0.0160

r/R0.7000 0.8000 0.9000 0.9500 1.000

P/D 0.9907 0.9850 0.9788 0.9740 0.9680RAKE/D 0.0000 0.0000 0.0000 0.0000 0.0000SKEW 0.0000 0.0000 0.0000 0.0000 0.0000

c/D 0.2311 0.2173 0.1806 0.1387 0.0010f/c 0.0143 0.0133 0.0125 0.0115 0.0000t/D 0.0125 0.0091 0.0060 0.0045 0.0000

200

r / R

P/D

,c/D

f/c,t

/D

0.2 0.4 0.6 0.8 10.00

0.50

1.00

1.50

0.00

0.01

0.02

0.03

0.04

0.05P/Dc/Df/ct/D

Figure 1: Geometry parameters distribution of N4148 propeller.

201



ure 2.


r/R0.2000 0.3000 0.4000 0.5000 0.6000

P/D 1.1050 1.1020 1.0980 1.0930 1.0880RAKE/D 0.0000 0.0000 0.0000 0.0000 0.0000SKEW 0.0000 0.0000 0.0000 0.0000 0.0000

c/D 0.3200 0.3625 0.4048 0.4392 0.4610f/c 0.0143 0.0232 0.0230 0.0218 0.0207t/D 0.0658 0.0563 0.0478 0.0396 0.0321

r/R0.7000 0.8000 0.9000 0.9500 1.000

P/D 1.0840 1.0810 1.0790 1.0770 1.0750RAKE/D 0.0000 0.0000 0.0000 0.0000 0.0000SKEW 0.0000 0.0000 0.0000 0.0000 0.0000

c/D 0.4622 0.4347 0.3613 0.2775 0.0925f/c 0.0200 0.0197 0.0182 0.0163 0.0118t/D 0.0250 0.0183 0.0120 0.0090 0.0000

202

r / R

P/D

,c/D

f/c,t

/D

0.2 0.4 0.6 0.8 10.00

0.50

1.00

1.50

2.00

0.00

0.05

0.10P/Dc/Df/ct/D


203



ure 3.


r/R0.2000 0.3000 0.4000 0.5000 0.6000

P/D 1.432 1.402 1.389 1.380 1.379RAKE/D 0.0000 0.0000 0.0000 0.0000 0.0000SKEW 0.0000 0.0000 0.0000 0.0000 0.0000

c/D 0.1790 0.2080 0.2320 0.2540 0.2730f/c 0.0175 0.0199 0.0215 0.0226 0.0230t/D 0.0390 0.0310 0.0230 0.0160 0.0150

r/R0.7000 0.8000 0.9000 0.9500 1.000

P/D 1.386 1.408 1.446 1.472 1.502RAKE/D 0.0000 0.0000 0.0000 0.0000 0.0000SKEW 0.0000 0.0000 0.0000 0.0000 0.0000

c/D 0.2880 0.2990 0.3060 0.3080 0.3090f/c 0.0225 0.0198 0.0135 0.0092 0.0048t/D 0.0150 0.0150 0.0150 0.0150 0.0150

204

r / R

P/D

,c/D

f/c,t

/D

0.2 0.4 0.6 0.8 10.00

0.50

1.00

1.50

2.00

0.00

0.05

0.10P/Dc/Df/ct/D


205



ure 4.


r/R0.2000 0.3500 0.4000 0.4500 0.5000

P/D 0.8620 1.3600 1.5160 1.6420 1.7310RAKE/D 0.0000 -0.0297 -0.0414 -0.0515 -0.0591SKEW 0.0000 -18.9567 -21.9617 -23.6477 -24.1847

c/D 0.1184 0.2099 0.2412 0.2714 0.3020f/c 0.0000 0.0053 0.0106 0.0166 0.0230t/D 0.0610 0.0382 0.0338 0.0312 0.0304

r/R0.6000 0.7000 0.8000 0.8500 0.9000

P/D 1.7950 1.7190 1.5470 1.4442 1.3410RAKE/D -0.0635 -0.0572 -0.0455 -0.0387 -0.0317SKEW -22.3917 -17.5507 -10.3138 -5.9162 -1.0197

c/D 0.3620 0.4200 0.4690 0.4798 0.4650f/c 0.0298 0.0283 0.0204 0.0151 0.0093t/D 0.0293 0.0278 0.0253 0.0236 0.0210

206

r/R0.9300 0.9500 0.9700 0.9900 1.0000

P/D 1.2821 1.2450 1.2101 1.1780 1.1630RAKE/D -0.0276 -0.0249 -0.0221 -0.0192 -0.0177SKEW 2.1584 4.3803 6.6871 9.0803 10.3103

c/D 0.4308 0.3900 0.3249 0.2044 0.0000f/c 0.0057 0.0033 0.0009 -0.0015 -0.0027t/D 0.1862 0.0164 0.0134 0.0082 0.0000

r / R

P/D

,c/D

f/c,t

/D

0.2 0.4 0.6 0.8 10.00

0.50

1.00

1.50

2.00

2.50

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10P/Dc/Df/ct/D

Figure 4: Pith, chord and camber distribution of N4990 propeller.

207

r / R

RA

KE

/D

0.2 0.4 0.6 0.8 1

-0.08

-0.06

-0.04

-0.02

0.00

Figure 5: Rake distribution of N4990 propeller.

r / R

Skew

0.2 0.4 0.6 0.8 1-25.0

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

Figure 6: Skew distribution of N4990 propeller.

208

Geometric Parameters for CB-1 Propellers

The geometry parameters for CB-1 propeller is shown in Figures 7, 8 and 9.

r / R

P/D

,c/D

f m/c

,tm/D

0.2 0.4 0.6 0.8 10

0.5

1

1.5

0

0.05

0.1P/Dc/Dfm/ctm/D

CB-1 Propeller

Figure 7: Pitch, chord camber distribution of CB-1 propeller.

209

r / R

RA

KE

0.2 0.4 0.6 0.8 1

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02CB-1 Propeller

Figure 8: Rake distribution of CB-1 propeller.

r / R

SKE

W

0.2 0.4 0.6 0.8 1-15

-10

-5

0

5

10

15

20

25

30 CB-1 Propeller

Figure 9: Skew distribution of CB-1 propeller.

210

Geometric Parameters for CB-2 Propellers

The geometry parameters for CB-2 propeller is shown in Figures 10, 11 and 12.

r / R

P/D

,c/D

f m/c

,tm/D

0.2 0.4 0.6 0.8 10

0.5

1

1.5

0

0.05

0.1P/Dc/Dfm/ctm/D

CB-2 Propeller

Figure 10: Pitch, chord camber distribution of CB-2 propeller.

211

r / R

RA

KE

0.2 0.4 0.6 0.8 1

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02CB-2 Propeller

Figure 11: Rake distribution of CB-2 propeller.

r / R

SKE

W

0.2 0.4 0.6 0.8 1-15

-10

-5

0

5

10

15

20

25

30 CB-2 Propeller

Figure 12: Skew distribution of CB-2 propeller.

212

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Vita

Yumin Deng was born on August 19th, 1980 in Guangdong Province, P.R.China.

He studied at the No.1 High School of Huizhou from 1996 to 1999. After finishing

the high school study, he was admitted to the prestigious Shanghai Jiao Tong Uni-

versity in Shanghai, P.R.China. He got his Bachelor of Engineering degree in Naval

Architecture and Ocean Engineering Department in 2003. He joined the University

of Texas at Austin to pursue his Master degree in Civil Engineering in 2003, with a

focus on Ocean Engineering.

Permanent address: APT403 No.15 Lane1, Jiangbian Rd, XiajiaoDisctrict, Huizhou, Guangdong, P.R.China.

This thesis was typeset with LATEX K by the author.

LLATEX is a document preparation system developed by Leslie Lamport as a special version of

Donald Knuth’s TEX Program.

221

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