Wind Turbine Design

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Wind Turbine

Design

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Wind Turbine Design

www.polymtl.ca/pub ISBN : 978-2-553-00931-0

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With Emphasis on Darrieus Concept

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The depletion of global fossil fuel reserves combined with mount-ing environmental concerns has served to focus attention on the development of ecologically compatible and renewable «alterna-tive» sources of energy.

Wind energy, with its impressive growth rate of 50% over the last five years, is the fastest growing alternate source of energy in the world since its purely economic potential is complemented by its great positive environmental impact. The wind turbine, whether it may be a Horizontal-Axis Wind Turbine (HAWT) or a Vertical-Axis Wind Turbine (VAWT), offers a practical way to convert the wind energy into electrical or mechanical energy. Although this book focuses on the aerodynamic design and performance of VAWTs based on the Darrieus concept, it also discusses the compari-son between HAWTs and VAWTs, future trends in design and the inherent socio-economic and environmental friendly aspects of wind energy as an alternate source of energy.

This book will be of great interest to students in Mechanical and Aero nautical Engineering field, professional engineers, university professors and researchers in universities, government and industry. It will also be of interest to all researchers involved in theoretical, computational and experimental methods used in wind tur-bine design and wind energy development.

Dr. Ion Paraschivoiu is J.-A. Bombardier Aeronautical Chair Professor at École Polytechnique de Montréal where he is teaching undergraduate and graduate courses in Aerodynamics. He has made significant contributions to the theory of the aerodynamic performance of the Darrieus vertical axis wind turbine. His software programs for these calculations, described in the book, have been used successfully by others for design purposes and to assist in the evaluation of VAWT field tests. His other research interests include application of advanced aerodynamics methods in the study of aircraft icing, drag prediction and laminar-flow control.

Ion

Paraschivoiu

Excerpt of the full publication

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Wind Turbine

Design

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ith Emphasis on Darrieus Concept

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Presses internationales

P o l y t e c h n i q u e Excerpt of the full publication

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Wind Turbine Design – With Emphasis on Darrieus Concept Ion Paraschivoiu

Production team

Editorial management and production: Presses internationales Polytechnique Editing: Stephen Schettini

Illustrations: Farooq Saeed Cover Page: Cyclone Design

For information on distribution and points of sale, see our Website: www.polymtl.ca/pub E-mail of Presses internationales Polytechnique: [email protected]

E-mail of Ion Paraschivoiu: [email protected]

We acknowledge the financial support of the Government of Canada through the Book Pu-blishing Industry Development Program (BPIDP) for our puPu-blishing activities.

This book may not be duplicated in any way without the express written consent of the publisher. Legal deposit: 4th quarter 2002 ISBN 978-2-553-00931-0 (printed version) Bibliothèque et Archives nationales du Québec ISBN 978-2-553-01594-6 (pdf version) Library and Archives Canada Printed in Canada

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To my daughter Gloria and my wife Liliana

“When the wind is blowing

The wind turbine is turning The electricity is flowing

The gas emissions are ceasing The environment is refreshing

And people are cheering”

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Foreword v

Foreword

This book is intended to be a good reference for anyone interested in the design of Vertical-Axis Wind Turbine for electricity generation and other applications such as pumping water, irrigation, grinding and drying grain, and heating water to name a few.

The book is divided into ten chapters that are presented in a logical manner. The content is easy to follow and each chapter has its own conclusions. The innovative nature of this book is in its comprehensive review of state of the art in Vertical-Axis Wind Turbine (VAWT), correlation of existing knowledge base and the more recent developments in understanding the physics of flow associated with the Darrieus type vertical-axis wind turbine. The principal theories and aerodynamic models for performance calculations are presented with experimental data, not only from laboratory measurements but also from real prototypes.

The first chapter presents an introductory topic on the wind characteristics, a brief descrip-tion of the components of both major categories of wind machines: Horizontal-Axis Wind Turbine (HAWT) and Vertical-Axis Wind Turbine (VAWT) and an overview of the wind energy development in the world.

The state of the art of vertical-axis wind turbine including Savonius and Giromill rotors are described in Chapter 2.

The scope of Chapter 3 encompasses the mathematical formulation of the equations for the various Darrieus rotor configurations as well as geometries including: catenary, parabolic, troposkien and modified troposkien blade and also a practical Sandia type shape.

The aerodynamic performance prediction models are presented in Chapter 4 for: single streamtube, multiple streamtube, vortex and local-circulation models. The aerodynamic loads: normal and tangential components and performance, as well as, rotor torque and power coeffi-cient are calculated and the comparisons of different prediction models are shown.

The unsteady aerodynamics of Darrieus type VAWTs is dealt with in detail in Chapter 5. A CFD model based on the streamfunction-vorticity formulation of the Navier-Stokes equations is presented to study and highlight unsteady effects that may influence design and performance.

The real essence of the book is in Chapter 6 that provides a practical design model for the Darrieus type VAWTs based on the double-multiple streamtube model, originally developed by the author. Several variants of the software program CARDAAV, for use in performance calculations, are described. Other important aspects such as rotor geometries, conventional and natural laminar flow airfoils, dynamic-stall effects, secondary effects and stochastic wind model are also addressed here.

The subsequent chapters present aerodynamic load and performance data from water channel and wind tunnel experiments, the state of the art of innovative aerodynamic devices as applied to VAWTs and the future trends in the design of Darrieus type wind turbine.

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vi Foreword

A comparison between Horizontal-Axis and Vertical-Axis Wind Turbines is given in Chapter 9. The idea here is to keep in perspective the technical aspects and the global cost of the advanced designs for both kinds of machines.

Finally, Chapter 10 deals with the environmental and social aspects of wind energy since it is an emerging environmental technology of great impact and value.

The author is indebted to Research Institute of Hydro-Quebec (IREQ) and to his many graduate students and researchers: Drs. T. Brahimi, A. Allet, R. Martinuzzi, K. F. Tchon, C. Masson, S. Hallé and L. Surugiu formerly of the J.-A. Bombardier Aeronautical Chair, Department of Mechanical Engineering at École Polytechnique of Montreal, for their help in preparing this book. The author would like to extend his gratitude to the Department of Mechanical Engineering at École Polytechnique of Montreal, CANMET in Ottawa and Norbert Voutthi Dy, Ph.D. candidate (2009 edition) for all their assistance in preparing this book.

This book has been gracefully translated in Japanese with the help of a team: Professor Emeritus Tsutomu Hayashi (leader), and Dr. Yutaka Hara from Tottori University, and Professor Tetuya Kawamura from Ochanomizu University, Tokyo.

Special contributions in the preparation of this reference book were made by Mr. Jack R. Templin, formerly with the National Research Council of Canada, Dr. Claude Béguier, formerly with Institute of Research on Phenomena out of Equilibrium (IRPHE) − Marseilles, France, Prof. Raghu S. Raghunathan of Queen’s University of Belfast, Dr. Takao Maeda and Prof. Yukimaru Shimizu, Mie University, Japan, who provided useful comments and constructive suggestions as reviewers of the manuscript.

The author gratefully acknowledges the advice and valuable remarks of his many friends from Sandia National Laboratories during several meetings and conferences that spanned for two decades, as well as Drs. Paul C. Klimas, Jim H. Strickland, Dale E. Berg, Paul G. Migliore, Paul S. Veers, Herbert Sutherland, Williams N. Sullivan, Donald W. Lobitz, Tom Ashwill, etc.

The author would especially like to thank Dr. David Malcolm, Global Energy Concepts, LLC, and Dr. Lawrence Schienbein for providing important experimental data and extensive information on Darrieus wind turbine, Carl Brothers from Atlantic Wind Test Site at Prince Edward Island (Canada) for helpful discussion on the comparison between horizontal-axis and vertical-axis wind turbines, Prof. Kazuichi Seki of Tokai University, Japan, Prof. Gerald Gregorek, Ohio State University, Columbus, USA, for his interesting discussions, Dr. Ganesh Rajagopalan, Iowa State University, Ames, USA, and Dr. A. Jagadeesh of Nayudamma Center for Development of Alternatives, Andhra Pradesh, India, for his discussions specifically on the environmental aspects of wind energy. The author would like to acknowledge and thank, in general, the wind energy fraternity and, in particular, to Prof. Holt Ashley, Dr. Al Eggers, Prof. Robert E. Wilson, Mr. Raj Rangi and Dr. Robert Thresher.

The author would like to express his acknowledgments and special thanks to Dr. Farooq Saeed, formerly research associate of J.-A. Bombardier Aeronautical Chair, for his valuable assistance in the preparation of this manuscript. Last but not the least, the author would like to thank Mrs. Diane Ratel and Mrs. Martine Aubry for their skillful editing and typing of the book and also to Mr. Lucien Foisy and Mrs. Constance Forest (2009 edition) for their help in its publication by Presses internationales Polytechnique.

Ion Paraschivoiu

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Table of Contents vii

Table

of

List of

Figures

Chapter 1 Figure 1.1 Components - Upwind rotor and downwind HAWT rotor [Ref. 1.1] ... 2

Figure 1.2 VAWT of Darrieus type [Ref. 1.1] ... 3

Figure 1.3 Types of vertical-axis wind turbines - a) Fixed bladed Darrieus or articulating blade Giromill; b) Savonius rotor ... 4

Chapter 2 Figure 2.1 The Madaras concept for generating electricity using the Magnus effect [2.1] ... 15

Figure 2.2 Savonius rotor - Calculation scheme ... 17

Figure 2.3 Pressure distribution vs azimuthal angle ... 18

Figure 2.4 Starting torque for a rotation ... 19

Figure 2.5 Normalized power coefficient vs bucket tip-speed ratio ... 20

Figure 2.6 Two-bucket Savonius rotor ... 21

Figure 2.7 Three-bucket Savonius rotor ... 21

Figure 2.8 The static torque coefficient as a function of angular position for a two-bucket Savonius rotor, [2.17] ... 23

Figure 2.9 The static torque coefficient as a function of angular position for a three-bucket Savonius rotor, [2.17] ... 23

Figure 2.10 A comparison of the power coefficients for two- and three-bucket Savonius rotors with a gap width ratio of 0.15 at Re/m of 8.64 × 105 ...24

Figure 2.11 Normalized turbine power for 1-meter, two-bucket Savonius rotors as a function of normalized rotational speed for Re/m of 4.32 × 105 ...25

Figure 2.12 Translating drag device ... 26

Figure 2.13 Translating airfoil ... 27

Figure 2.14 Power from a translating airfoil vs lift-drag ratio ... 27

Figure 2.15 Translating airfoil with relative wind ... 28

Figure 2.16 Coordinate system and vortex sheet location for analysis of the Giromill ... 29

Figure 2.17 Streamlines and velocity profile at X = 3, a = 1/3. The velocity profile is given along the lines x/R = -0.05 and +2.0 ... 31

Figure 2.18 Vortex shedding of cross-wind axis actuator ... 33

Figure 2.19 Vortex system of single bladed cross-wind axis actuator ... 20

Figure 2.20 Relative velocity and aerodynamic forces for typical blade element ... 34

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xiv List of Figures

Chapter 3

Figure 3.1 Darrieus vertical-axis wind turbine (DOE/SANDIA 34-m) ... 38

Figure 3.2 Catenary shape ... 43

Figure 3.3 Troposkien shape ... 46

Figure 3.4 Length of Troposkien blade vs b and W ... 50

Figure 3.5 Tensions ratio vs blade length ... 52

Figure 3.6 Sandia shape ... 55

Figure 3.7 Darrieus rotor geometries ... 61

Chapter 4 Figure 4.1 Curved blade vertical-axis wind turbine with three blades ... 67

Figure 4.2 NACA 0012 Airfoil - Normal force and chordwise thrust coefficients ... 69

Figure 4.3 Comparison of theory and experiment - a) Power coefficient; b) Rotor drag coefficient ... 72

Figure 4.4 Effect of rotor solidity Nc/R ... 74

Figure 4.5 Effect of blade airfoil C_{do ...}75

Figure 4.6 Upstream and plan view of typical streamtube ... 77

Figure 4.7 Blade element forces ... 78

Figure 4.8 Relative velocity vector ... 79

Figure 4.9 Comparison of DART and single streamtube models with Sandia test data (2m diameter rotor) ... 81

Figure 4.10 Variation of streamtube velocities through the rotor (view looking upstream through the rotor) ... 82

Figure 4.11 The effect of solidity on C_P (Re = 3.0 × 106_{) ... 83}

Figure 4.12 Contribution of equatorial band to C_{P ...}84

Figure 4.13 Effect of wind shear on rotor performance ... 85

Figure 4.14 Vortex system for a single blade element ... 86

Figure 4.15 Velocity induced at a point by a vortex filament ... 86

Figure 4.16 Fixed-wake geometry ... 88

Figure 4.17 Rotor aerodynamic torque, Sandia 17-m-diameter research turbine, two blades, NACA 0015 section, 61-cm chord, 50.6 rpm, X = 2.18 ... 89

Figure 4.18 Fixed-wake theory and test results, Sandia 17-m-diameter research turbine, two blades, NACA 0015 section, 61-cm chord, 50.6 rpm ... 89

Figure 4.19 Schematic of a typical Darrieus turbine ... 90

Figure 4.20 Numerical representation of the Darrieus rotor ... 92

Figure 4.21 Vortex system for a single blade element [Ref. 4.14] ... 93

Figure 4.22 Normal force coefficient variation. - Two-dimensional VDART-TURBO, c/R = 0.135; VDART2, c/R = 0.15 [Ref. 4.14]; Experiment [Ref. 4.14] ... 94

Figure 4.23 Normal force coefficient variation, c/R = 0.135. ----- Three-dimensional VDART-TURBO; VDART3 [Ref. 4.14] ... 95

Figure 4.24 Tangential force coefficient variation. - Two-dimensional VDART-TURBO, c/R = 0.135; VDART2, c/R = 0.15 [Ref. 4.14] ... 95

Figure 4.25 Tangential force coefficient variation c/R = 0.135. - Three-dimensional VDART-TURBO; VDART3 [Ref. 4.14] ... 95

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List of Figures xv

Figure 4.26 Wake convection velocity as predicted by three-dimensional VDART-TURBO, c/R = 0.135 ... 96

Figure 4.27 Wake geometry as predicted by two-dimensional VDART-TURBO, c/R = 0.135 ... 96

Figure 4.28 Wake geometry as predicted by VDART3, c/R = 0.135 ... 96

Figure 4.29 Aerodynamic torque ... 98

Chapter 5 Figure 5.1 Airfoil in Darrieus motion ... 102

Figure 5.2 Dynamic-stall events on the Vertol VR-7 airfoil [5.1] ... 104

Figure 5.3 Non-inertial frame of reference ... 106

Figure 5.4 Computational domain ... 107

Figure 5.5 Algorithm ... 113

Figure 5.6 Wake definition ... 116

Figure 5.7 Computation of the eddy viscosity ... 117

Figure 5.8 Stations on the structured zone ... 119

Figure 5.9 Flat plate shape ... 121

Figure 5.10 Computational mesh for flat plate ... 121

Figure 5.11 Pressure distribution over flat plate ... 122

Figure 5.12 Boundary layer velocity profile – Cebeci-Simth ... 122

Figure 5.13 Boundary layer velocity profile – Johnson-King ... 122

Figure 5.14 Non-inertial frame - Pitching motion ... 123

Figure 5.15 Computational mesh – NACA 0015 pitching airfoil ... 124

Figure 5.16 Transitional function – Pitching motion ... 124

Figure 5.17 Lift coefficient – Cebeci-Smith model ... 125

Figure 5.18 Drag coefficient – Cebeci-Smith model ... 125

Figure 5.19 Lift coefficient – Johnson-King model ... 126

Figure 5.20 Drag coefficient – Johnson-King model ... 126

Figure 5.21 Computational mesh #2 – Darrieus motion ... 127

Figure 5.22 Evolution of the relative velocity and angle of attack for Darrieus motion ... 128

Figure 5.23 Darrieus motion simulation ... 128

Figure 5.24 Evolution of the effective Reynolds number ... 129

Figure 5.25 Computed streamlines – Cebeci-Smith model ... 131

Figure 5.26 Evolution of the vorticity field – Cebeci-Smith model ... 132

Figure 5.27 Computed streamlines – Johnson-King model ... 133

Figure 5.28 Evolution of the vorticity field – Johnson-King model ... 134

Figure 5.29 Dynamic-stall regions – Cebeci-Smith model ... 135

Figure 5.30 Dynamic-stall regions – Johnson-King model ... 135

Figure 5.31 Dynamic-stall regions – Laminar case ... 135

Figure 5.32 Evolution of the normal force – Laminar case ... 136

Figure 5.33 Evolution of the normal force – Cebeci-Smith model ... 136

Figure 5.34 Evolution of the normal force – Johnson-King model ... 137

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xvi List of Figures

Figure 5.36 Evolution of the tangential force – Cebeci-Smith model ... 138

Figure 5.37 Evolution of the tangential force – Johnson-King model ... 138

Figure 5.38 Evolution of the pitching moment ... 139

Figure 5.39 Wake convection ... 139

Chapter 6 Figure 6.1 A pair of actuator disks in tandem ... 147

Figure 6.2 Double actuator disks streamlines pattern ... 149

Figure 6.3 Control volumes 1 and 2 ... 149

Figure 6.4 Control volumes 3, 4 and 5 ... 150

Figure 6.5 Relative velocity and angle of attack ... 153

Figure 6.6 Force coefficients of a blade element airfoil ... 154

Figure 6.7 Elemental forces on a blade element ... 155

Figure 6.8 Elemental forces on a blade element airfoil (in a horizontal plane) ... 155

Figure 6.9 Definition of rotor geometry for a Darrieus wind turbine. Two actuator disks in tandem ... 159

Figure 6.10 Angles, forces and velocity vectors at the equator ... 160

Figure 6.11 Comparison between normal force coefficients calculated by the multiple streamtube theory, and the present model. Sandia 5-m, 162.5 rpm ... 165

Figure 6.12 Variation of the normal force coefficients with azimuthal angle q, for each blade, in the upwind and downwind zones ... 166

Figure 6.13 Variation of the normal force coefficients with azimuthal angle q, for two blades, at three tip-speed ratios ... 166

Figure 6.14 Comparison between tangential force coefficients calculated by the multiple streamtube theory and the present model ... 167

Figure 6.15 Variation of the tangential force coefficients with the azimuthal angle q, for each blade, in the upwind and downwind zones ... 167

Figure 6.16 Variation of the tangential force coefficients with the azimuthal angle q, for the two blades, at the three tip-speed ratios ... 168

Figure 6.17 Power coefficient as a function of the equatorial tip-speed ratio. Comparison between analytical model results and field test data [6.17] for the Sandia 5-m, two-blade rotor ... 169

Figure 6.18 Power coefficient as a function of the equatorial tip-speed ratio. Comparison between analytical model results and field test data [6.17] for the Sandia-5-m, three-blade rotor ... 169

Figure 6.19 Upwind and downwind velocity ratios as functions of tip-speed ratio ... 170

Figure 6.20 Variation of the angle of attack at the equator with the blade position ... 171

Figure 6.21 Blade element normal force coefficients at the equator as a function of the azimuthal angle q ... 171

Figure 6.22 Blade element tangential force coefficients at the equator as function of the azimuthal angle, q ... 172

Figure 6.23 Upwind and downwind normal force coefficients distribution on the rotor blades ... 172

Figure 6.24 Upwind and downwind tangential force coefficients distribution on the rotor blades ... 173

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List of Figures xvii

Figure 6.25 Rotor torque as a function of the azimuthal angle. Comparison between analytical results and experimental data ... 174

Figure 6.26 Upwind, downwind and total rotor power coefficients as functions of tip-speed ratio ... 175

Figure 6.27 Power coefficient vs tip-speed ratio. Comparison between present model results and field test data ... 176

Figure 6.28 Darrieus rotor power as a function of the wind velocity at the equator ... 176

Figure 6.29 A typical Darrieus rotor performance characteristic C_P as a function of the tip-speed ratio X_{EQ ...}177

Figure 6.30 Power coefficient vs tip-speed ratio ... 178

Figure 6.31 Performance coefficient vs advance ratio ... 179

Figure 6.32 Power coefficient vs tip-speed ratio for three types of airfoil ... 179

Figure 6.33 Tower wake-velocity deficit ... 181

Figure 6.34 Measurement of the distribution of mean velocities and relative turbulence intensities in the wake of a rotating cylinder ... 181

Figure 6.35 Power coefficient as a function of the tip-speed ratio. Comparison between experimental data and results predicted by CARDAA, CARDAAV, and VDART3 codes ... 185

Figure 6.36 Open spoiler effects on the performance of the Magdalen Islands rotor ... 186

Figure 6.37 Aerodynamic power as a function of wind speed at the equator. Comparison between experimental data and results predicted by CARDAAV code, including secondary effects ... 186

Figure 6.38 Induced velocity variation with blade position ... 187

Figure 6.39 Blade tangential force coefficient as a function of blade position ... 187

Figure 6.40 Average side-force coefficient as a function of tip-speed ratio ... 188

Figure 6.41 Simplified physical model of the flowfield in a horizontal slice of the rotor ... 189

Figure 6.42 Reduction of the streamtube in the undisturbed part of the rotor vs the tip-speed ratio ... 192

Figure 6.43 Curve streamlines through the rotor, calculation and experiments ... 194

Figure 6.44 Variation of the angle of attack at the equator with the blade position ... 195

Figure 6.45 Performance comparison between theoretical results and experimental data for the Sandia 17-m turbine ... 196

Figure 6.46 Contribution of vertical slices to the power coefficient versus tip-speed ratio ... 197

Figure 6.47 Performance comparison of theoretical results and experimental data for the Sandia 5-m turbine ... 197

Figure 6.48 Normal force coefficient as a function of the azimuthal angle ... 198

Figure 6.49 Tangential force coefficient as a function of the azimuthal angle ... 198

Figure 6.50 Schematic diagram of the vortex shedding for X = 2.14 ... 204

Figure 6.51 Gormont’s model adaptations: Magdalen Islands rotor at 29.4 rpm ... 205

Figure 6.52 Gormont’s model adaptations: Sandia 17-m at 42.2 rpm ... 206

Figure 6.53 Gormont’s model adaptations: Sandia 34-m at 28.0 rpm ... 206

Figure 6.54 VAWT: Angles, forces and velocities at the equator (MIT model) ... 208

Figure 6.55 Maximum lift and moment coefficients vs rate of change of angle of attack ... 211

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xviii List of Figures

Figure 6.56 Normal force coefficient vs angle of attack at the equator for Sandia 17-m,

38.7 rpm (experimental data and MIT model) ... 212

Figure 6.57 Normal force coefficient vs angle of attack at the equator for Sandia 17-m, 38.7 rpm (experimental data and Gormont’s model) ... 212

Figure 6.58 Rotor power vs wind speed at the equator for Sandia 17-m, 42.2 rpm. Dynamic-stall effects ... 213

Figure 6.59 Rotor power vs wind speed at the equator for Sandia 17-m, 46.6 rpm ... 214

Figure 6.60 Rotor power vs wind speed at the equator for Sandia 17-m, 50.6 rpm ... 214

Figure 6.61 The indicial functions as they vary with time ... 216

Figure 6.62 Typical curve of the position of the flow separation point function of a ... 218

Figure 6.63 Critical normal force coefficient C_NI for the onset of leading-edge separation function of the Mach number ... 219

Figure 6.64 Dynamic-stall vortex lift contribution ... 220

Figure 6.65 Normal force coefficient vs angle of attack ... 221

Figure 6.66 Aerodynamic torque vs azimuthal angle at low tip-speed ratio ... 221

Figure 6.67 Power output vs wind velocity ... 222

Figure 6.68 Blade shape geometry for 34-m wind turbine ... 223

Figure 6.69 Rotor power vs wind speed at equator ... 224

Figure 6.70 Power coefficient vs tip-speed ratio ... 224

Figure 6.71 Performance coefficient vs advance ratio ... 225

Figure 6.72 Rotor power vs wind speed at equator ... 225

Figure 6.73 Schematic of three-dimensional wind simulation for Darrieus rotor with 5 × 5 grids ... 231

Figure 6.74 Sectional normal force coefficient versus azimuthal angle at the rotor equator, X_EQ = 4.60 and turbulence intensity = (27 percent, 25 percent) ... 233

Figure 6.75 Sectional normal force coefficient versus azimuthal angle at the rotor equator, X_EQ = 2.49 and turbulence intensity = (27 percent, 25 percent). Comparison between CARDAAS-1D & 3D, CARDAAV (0 percent turbulence), and experimental data ... 234

Figure 6.76 Sectional tangential force coefficient versus azimuthal angle at the rotor equator, X_EQ = 2, and three turbulence intensity levels. Comparison between CARDAAS-1D & 3D, CARDAAV (0 percent turbulence) and experimental data ... 235

Figure 6.77 Rotor torque distribution, standard deviation, minimum and maximum values at X_EQ = 2.87 and turbulence intensity = (27 percent, 25 percent). Comparison between CARDAAS-D and experimental data ... 236

Figure 6.78 Performance comparison between theoretical results and experimental data for the Sandia 17-m wind turbine ... 237

Figure 6.79 Normal force coefficient F+_N as a function of the azimuthal angle q ... 238

Figure 6.80 RMS vibratory rotor tower stresses for the stiff cable configuration, CARDAA aerodynamic model [Ref. 6.80] ... 239

Figure 6.81 Structural capabilities using three aerodynamic models for studying Darrieus rotor ... 240

Figure 6.82 Velocity field near blade tip ... 242

Figure 6.83 Upwind and downwind interference factors vs rotor height for a 6-m

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List of Tables xxiii

List of

Tables

Chapter 1 Table 1.1 Average Power Output (kW) ... 5

Table 1.2 Europe’s Wind Power ... 9

Table 1.3 Cost of Wind Electricity Evolution ...11

Chapter 2 Table 2.1 Velocity Along the x-Axis for a = 1/3, X = 3 ... 32

Chapter 3 Table 3.1 Power Performance Data Available from Field Tests ... 40

Table 3.2 Power Output Performance Data Available From Wind Tunnel Tests ... 41

Table 3.3 Typical Relative Costs of VAWT Subsystems ... 41

Table 3.4 Geometrical Parameters for Two-Bladed Darrieus Rotors of Different Blade Shapes ... 57

Table 3.5 Dimensionless Coordinates and Meridian Angle d (Radians) ... 58

Table 3.6 Dimensionless Coordinates of the Magdalen Islands Darrieus Rotor ... 59

Table 3.7 Coordinates in Meters for an Ideal Troposkien and for the Magdalen-Islands Darrieus Rotor (M.I.D.R.) ... 60

Chapter 5 Table 5.1 Darrieus Motion Parameters ... 129

Chapter 6 Table 6.1 Predicted and measured performances ... 175

Chapter 7 Table 7.1 Darrieus Rotor Tests in the Vought Systems Division Low Speed Wind Tunnel ... 292

Table 7.2 Power Output Performance Data Available From Wind Tunnel Tests ... 295

Table 7.3 Sandia 17-m Turbine Rotor Configurations ... 309

Table 7.4 Aerodynamic Torques in Nm, 50.6 rpm ... 324

Table 7.5 Fourier Coefficients of Torque, 50.6 rpm (Coefficients normalized with mean torque) ... 325

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xxiv List of Tables

Chapter 8

Table 8.1 Ohio State University Wind Tunnel Tests ... 330

Table 8.2 34 Meter Wind Turbine Blade Data ... 334

Table 8.3 Performance Comparison Between Cam-bered and Symmetrical Blade Section

of the Sandia 5-Meter Research Turbine ... 349

Chapter 9

Table 9.1 Rotor Mass and Rotor Size ... 361

Table 9.2 Advantages of Two or Three Blades ... 364

Table 9.3 Darrieus Wind Turbine Design Alternatives ... 375

Table 9.4 Darrieus Wind Turbine Improvements ... 376

Table 9.5 Advantages and Disadvantages of HAWTs and VAWTs ... 378

Table 9.6 VAWT Aspect Ratios ... 379

Table 9.7 Area Required for Wind Plants ... 381

Chapter 10

Table 10.1 Survey on Energy Research Priority ... 388 Table 10.2 Environmental Aspects versus Type of Wind Turbine ... 389

Table 10.3 Carbon dioxide (CO₂). The Leading Greenhouse Gas ... 395

Table 10.4 Sulfur Dioxide (SO₂). The Leading Precursor of Acid Rain ... 395

Table 10.5 Nitrogen Oxides (NOx), Another Acid Rain Precursor and the Leading

Component of Smog ... 395

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Wind Energy 1

Wind Energy

1.1 WIND DEFINITION AND CHARACTERISTICS

WIND is the movement of the air between high pressure and low pressure regions in the atmosphere, caused by the uneven heating of the earth’s surface by the sun. When the air above hot surfaces is heated, it rises, creating a low pressure zone. The air surrounding higher pres-sure zones flows toward the low prespres-sure area, creating wind. For this reason, sometimes wind energy is called “indirect solar energy.”

Wind varies with time in intensity and direction, and the potential of a wind site is generally evaluated as a function of the annual average wind speed. Wind speeds can be calculated for other periods to determine hourly, daily or monthly averages. Winds vary with altitude and wind speed is also affected by ground features such as hills. The variation of wind speed with altitude is due to friction between air movement and the earth’s surface (the atmospheric boundary-layer). All weather offices report the wind speed at a standard height of 10 meters above ground. Wind near the ground gathers speed to climb a hill, then slows (and sometimes becomes very turbulent) on the far side of the hill. The wind speed strength and direction are measured by anemometers.

1.2 WIND TURBINES

The depletion of global fossil fuel reserves combined with mounting environmental concern has served to focus attention to the development of ecologically compatible and renewable alternative energy sources. The harnessing of wind energy is a promising technology able to provide a portion of the power requirements in many regions of the world. Wind generators are a practical way to capture and convert the kinetic energy of the atmosphere to either mechanical or, more significantly, electrical energy.

The term WINDMILL is applied to the wind-powered machine that grinds (or mills) grain. Modern machines are more correctly called WIND TURBINES because they can be used for a variety of applications, such as generating electricity and pumping water.

Windmills have a very simple design based on the drag-device that relies on different air resistance on the front and back of the rotor section to cause rotation.

An interesting and well documented survey concerning historical development of windmills is given in “Wind Turbine Technology” (ASME Press, 1994, D.A. Spera, editor), Ref. [1.1].

The most efficient way to convert wind energy into electrical or mechanical energy is offered by wind turbines that operate as a lifting-device. Wind turbines are classified into two categories, according to the direction of their rotational axis: Horizontal-Axis Wind Turbines

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2 Chapter 1

(HAWT) and Vertical-Axis Wind Turbines (VAWT). Horizontal-axis wind turbines capture kinetic wind energy with a propeller type rotor and their rotational axis is parallel to the direc-tion of the wind (Fig. 1.1). Vertical-axis wind turbines use straight or curved bladed (Darrieus type) rotors with rotating axes perpendicular to the wind stream. They can capture wind from any direction (Fig. 1.2). The most popular wind turbine systems are of the “propeller type,” but the VAWTs have not yet benefited from the years of development undergone by HAWTs. These two kinds of wind machine are compared in Chapter 9.

Figure 1.1 Components - Upwind rotor and downwind HAWT rotor [Ref. 1.1]

Both HAWTs and VAWTs have about the same ideal efficiency but the horizontal-axis wind tur-bine is more common. It has the entire rotor, gearbox and generator at the top of the tower, and must be turned to face the wind direction. The VAWT accepts wind from any direction, and its heavy machinery is at ground level. This is more convenient for maintenance, particularly on large units or when operating in potential icing conditions.

Both types of wind turbines have the same general components: - a rotor to convert wind energy into mechanical power, - a tower to support the rotor,

- a gearbox to adjust the rotational speed of the rotor shaft for the electric generator or pump,

- a control system to monitor operation of the wind turbine in automatic mode, including starting and stopping,

- a foundation (sometimes aided by guy wires) to prevent the turbine from blowing over in high winds.

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Wind Energy 3 Upper Bearing Upper Hub Central Column Cables Lower Hub Lower Bearing Support Stand Power Train Equipment Station Rotor Foundation Cable Foundation Ground Level Clearance Tensioner Rotor Height Rotor Diameter

Figure 1.2 VAWT of Darrieus type [Ref. 1.1]

The size of a wind turbine is measured in terms of swept area, or surface area swept by the rotating blades. The swept area of the rotor is calculated from the diameter of the rotor by:

S = 0.785 D2_{for HAWTs or by S = 1.000 D}2_{for typical VAWTs with an aspect ratio (height/}

diameter) of 1.5.

The control system of wind turbines is connected to an anemometer that continuously measures wind speed. When wind speed is high enough to overcome friction in the drive train, the control system allows the turbine to rotate, producing limited power. This is the “cut-in” wind speed, usually about 4 or 5 m/s. Wind turbines normally have a “rated wind speed,” corresponding to maximum output power. Typically, the rated wind speed is about 10-12 m/s. If wind speed exceeds rated wind speed, the control system prevents further power increases until “cut-out” wind speed is reached, at approximatively 25 m/s.

VAWTs are generally classified according to aerodynamic and mechanical characteristics, or the lifting surfaces, or the movement of the blades of the rotor, about a vertical-axis along a path in a horizontal plane. Today, there are four classes of VAWTs (Fig. 1.3):

a) the articulating straight-blade Giromill;

b) the Savonius rotor, a mostly drag-driven device;

c) the variable-geometry Musgrove, which permits reefing of the blades; and, d) the fixed-blade Darrieus rotor.

Vertical-axis wind turbines (VAWTs) have been studied by various researchers using modern analysis techniques. Common examples of these vertical-axis wind turbines are the Savonius and Darrieus turbines. In 1968, South and Rangi, from the National Research Council of Canada, reintroduced the Darrieus rotor concept. Since then, many analytical models predicting the aerodynamic performance of this type of wind turbine have been formulated.

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State of the Art of Vertical-Axis Wind Turbines 15

State of the Art of Vertical-Axis

Wind Turbines

The earliest practical wind machines were the “Panemones” (examples: Persian vertical-axis windmill in Sistan, A.D. 1300 and Chinese vertical-axis windmill, A.D. 1219). These ma-chines were of vertical-axis type driven by drag forces with a multi-bladed rotor operating at very low tip-speed ratios (much less than unity), which explains their poor efficiency. In spite of the simple design, the panemones need large amounts of material, are not able to withstand high wind loads and thus have not proven cost-effective.

2.1 THE MADARAS ROTOR CONCEPT

This concept was conceived as a “train” of vehicles, each vehicle supporting rotating cylinders mounted vertically on its flat-bed, moving to work on a circular track; each cylinder being driven by an electrical motor [2.1]. The Madaras rotor was designed on the principle of the Magnus effect known since the 1850s: the circulation induced around a rotating cylinder results in a lift force perpendicular to the flow direction as well as to the axis of the cylinder. On the side of the cylinder, where the flow and the cylinder are moving in the same direction, boundary layer separation is completely eliminated while on the opposite side a significant part undergoes separation. In 1933, Madaras conceived a plan for a large-scale test (for a 40 MW plant) that required building a full-scale rotating cylinders of 27.4 m hight and 8.5 m diameter mounted on a stationary platform in order to measure the forces due to the Magnus effect (see Fig. 2.1).

Figure 2.1 The Madaras concept for generating electricity using the Magnus effect [2.1]

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16 Chapter 2

The Magnus effect would propel the cars around the track and drive generators connected to the car axles. The Madaras concept for generating electricity using Magnus effect did not succeed because of mechanical complexity: the need to reverse direction of the cylinder at each end of the oval track, poor aerodynamic design (low “tip speed” with low aerodynamic effi-ciency), mechanical losses (high track loads and overturning moments), lower wind speeds near the ground and electrical losses.

2.2 SAVONIUS ROTOR

Nomenclature

As = Savonius turbine swept area, m2 C_P = wQ/(q•V•As), power coefficient C*

P = wQ/[q•V• (4rH)], normalized power coefficient

C_Q = Q/(q_•V_•A_s), torque coefficient

C*

Q = Q/[q• (4rH)(2r)], normalized torque coefficient

d = 2r, bucket diameter, m

H = rotor height, m

N = number of buckets

p_• = freestream static pressure, Pa

Q = turbine torque, N·m

Qf = friction (tare) torque, N·m (Eq. 2.12) q_• = 1

2 2

ρV_∞, freestream dynamic pressure, Pa

R = rotor radius of rotation (see Figs 2.6 and 2.7) (if s/d = 0, R = 2r, see Fig. 2.2)

Re_• = rV_•/m_•, Reynolds number per unit length, m-1

r = bucket radius (see Figs 2.6 and 2.7), m

s = bucket gap width (see Figs 2.6 and 2.7), m

s/d = gap width ratio

V_• = V_•(1 + x ), freestream velocity, m/s a = azimuthal angle (see Fig. 2.2), deg L = Rw/V_•, turbine tip-speed ratio

l = 2rw/V_•, bucket tip-speed ratio

x = wind tunnel blockage factor

q = bucket angular position (see Figs 2.6 and 2.7), deg

m_• = freestream viscosity, kg/(m·s)

r = freestream density, kg/m3

w = turbine rotational speed, rad/s

Subscripts

u = uncorrected for blockage • = freestream conditions

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State of the Art of Vertical-Axis Wind Turbines 17

Another vertical-axis machine based on the low lift-to-drag ratio is the Savonius rotor named after its Finnish inventor [2.1-2.3]. The Savonius rotor has an “S-shaped” cross-section and appears as a vertical cylinder sliced in half from top to bottom. It operates as a cup anemometer with the addition that wind is allowed to pass between the bent sheets (or buckets). The Savonius rotor has been studied using wind tunnel tests by several researchers since the 1920s [2.4-2.12]. Generally speaking, Savonius rotors can reach maximum power coefficient of 30%. Moreover, it is not efficient with respect to weight/unit power output since it would require as much as 30 times the surface to output the same power as a conventional wind turbine. For this reason, the Savonius machine is only useful and economical for small power requirements such as water pumping, driving a small electrical generator, providing ventilation, and providing water agitation to keep stock ponds ice-free during winter. It is also commonly used as an ocean current meter. The technology required to design and manufacture a Savonius rotor is very simple and is recommended for applications in developing countries or in isolated areas without electrical power. A simple Savonius rotor can be manufactured by cutting an oil barrel in half, inverting one of the halves, and welding the two pieces together in a S-shaped cross-section.

Figure 2.2 Savonius rotor - Calculation scheme

2.2.1 Mathematical Model

A mathematical model based on the pressure drop on each side of the blades was proposed by Chauvin et al. [2.13] to evaluate the power of a two-bucket Savonius rotor with a gap spac-ing s/d = 0. From Fig. 2.2, if w = ka is the instantaneous rotation vector and, due to the sym-metry of the Savonius rotor, _α˙ ₌ _ω ₌constant, then the torque is given by:

Q OM Fi k i

=

∑

e

×

j

⋅ _(2.1)

This sum has two components:

a) the first is associated with the retreating blade, a driven component, QM

b) the second is associated with the advancing blade, a resistant component, Q_D

Q = Q_M + Q_D (2.2)

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!

The Darrieus Wind-Turbine

Concept

3.1 INTRODUCTION

The great majority of wind turbines in the world are aerodynamically improved versions of the traditional horizontal-axis propeller-type device. Over the past two decades, the Darrieus type vertical-axis wind turbine (VAWT) has undergone considerable research and significant engineering development. However, it did not benefit from R&D as much as propeller-type machines.

The Darrieus wind turbine was patented by the U.S. Patent Office in the name of G.J.M. Darrieus in 1931 [3.1]. The Darrieus patent states that each blade should “have a streamline outline curved in the form of skipping rope.” In other words, the Darrieus rotor has curved blades that approximate the shape of a perfectly flexible cable, of uniform density and cross-section, hanging freely from two fixed points; under the action of centripetal forces such a shape minimizes inherent bending stresses. This blade shape is called Troposkien (from the Greek roots: trots, turning and sXOLuLOu, rope; or “turning rope”) pure Troposkien shape (gravity

neglected) does not depend on angular velocity. The first known wind tunnel measurements of Darrieus wind-turbine performance were carried out by R.S. Rangi and P. South of the National Research Council of Canada, [3.2, 3.3]. Later measurements included fundamental investigations of the number of blades, the rotor’s solidity, and the effects of spoilers and aerobrakes. In the early 1970’s, engineers at the National Research Council of Canada (NRC) independently developed a similar concept of VAWT by assuming an approximate shape of a catenary for the curved blades.

In Great Britain, the H-type or Musgrove rotor VAWT was introduced by Vertical-Axis Wind Turbines Limited [3.4]. The Musgrove rotor is straight bladed and can be reefed to provide speed control. Two prototypes of H-type machine were built in 1986: a 25-m rotor sponsored by the U.K. Department of Energy, and a 14-m machine funded by Tema SpA of Italy. The HM-Rotor-300, another straight-bladed Darrieus rotor, was manufactured by the Heidelberg Motor Company. An interesting H-Type prototype was tested in 1994 at Kaiser-Wilhelm-Koog Wind Test site; this rotor has no gearbox and its low rotor speed reduces noise [IEA 1992].

The Darrieus curved blade rotor has been developed and commercialized mainly in North America at institutions such as the National Research Council of Canada and by companies such as FloWind Corp. and Vawtpower in the U.S. and Indal Technologies Inc., Lavalin Inc. and Adecon Inc. in Canada. A detailed survey and bibliography on the vertical-axis wind turbines is presented in Ref. [3.5]. Sandia National Laboratories (SNL) deployed considerable effort for the research and development of the curve-bladed Darrieus rotor. Thus, in 1974 SNL built a 5-m diameter research VAWT, followed by a 17-m diameter rated at 60 kW in 1977 [3.6-3.18].

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38 Chapter 3

A significant step in the development of larger and more efficient commercial Darrieus VAWT’s was the installation and operation of 34-m Sandia-DOE VAWT in 1987, rated at 625 kW. The Sandia 34-m turbine (Fig. 3.1) was the first curved-blade Darrieus turbine rotor originally designed to incorporate step tapered blades using varying blade-section airfoils and a blade airfoil section specifically designed for VAWTs. The equator and transition sections of that ro-tor use the SAND 0018/50 airfoil section while the root sections are NACA 0021, [3.19-3.20]. The test beds are designed so that configurations can be quickly and easily changed to investigate the basic physics of wind turbines. For example, the Sandia 34-m test bed is equipped with a variable speed drive system to permit, among other things, performance tests of new blade airfoils and blade shapes over a wide range of Reynolds numbers. Test beds are normally operated on a limited basis and only for specific tests.

Figure 3.1 Darrieus vertical-axis wind turbine (DOE/SANDIA 34-m)

(Courtesy of Sandia National Laboratories)

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The Darrieus Wind-Turbine Concept 39

The Canadians manufactured the first large-scale Darrieus turbine rated at 230 kW with an estimated average output of 100 kW on Magdalen-Islands in May 1977. An unexpected self-start with no brakes destroyed this prototype, and a similar VAWT was installed in 1978, [3.21]. Performance test data for this turbine operating at 29.4 rpm [3.22], are believed to be the first field data gathered on large scale Darrieus turbines that clearly show the performance in the post stall regime (at low tip-speed ratios). A complete data set for operation at 36.6 rpm could not be obtained because high wind operation was limited to about 15 m/s. The performance data obtained from this turbine were an important element in the design of the Indal 6400-500 kW turbine since the effects of dynamic stall were not included in performance prediction models, and peak power output was seriously underestimated by the models.

Under Sandia technical guidance and DOE support funding, Alcoa constructed four 17-m, 100-kW units, two of which were grid-connected. One of these was tested successfully for over 10,000 h in storm winds exceeding 120 mph, [3.23-3.25]. The performance testing of the Sandia National Laboratories 2, 5, 17 and 34-m research turbines resulted in the most rigorous and exhaustive set of performance data and comparisons to theoretical predictions. SNL routinely presented test and predicted data in non-dimensional form, to facilitate comparison with other data, including those for HAWTs.

The greatest power output measured for any Darrieus wind turbine constructed to date has been from the Lavalin Eole (64 m) Research Turbine [3.26]. Built in 1986 in Cap Chat, Quebec, Canada, Eole is a two-bladed NACA 0018 rotor at fixed rotational speeds of 10 and 11.35 rpm respectively. The maximum power output is in excess of 1.3 MW at 14.7 m/s and corresponds to 11.35 rpm. The Eole wind turbine was designed to operate in a variable speed mode up to a rotor speed of 16.3 rpm with the maximum power reaching about 3.6 MW at 17 m/s and then being held constant by decreasing rotor speed at higher wind speeds [3.27]. However, fatigue life predictions showed that the turbine should be limited to 13.25 rpm with a nominal cut-out of 15 m/s (about 2 MW maximum power output) in order to operate successfully for the five year duration of the energy purchase agreement.

FloWind was a leader in delivering wind generated electricity to U.S. utilities, and designed, manufactured and operated wind turbines from 1982 to 1997. They developed a VAWT FloWind 19-m using a two-bladed NACA 0015 operating at 51.8 rpm and producing 250 kW at a wind speed of about 20 m/s, [3.28-3.29]. Drawing upon this experience, FloWind developed a new generation advanced vertical-axis wind turbine, with an extended height-to-diameter (EHD) ratio. This class of advanced VAWT maximizes production from any given wind area. In this case, an optimal balance between aerodynamic efficiency, wake loss and swept area is achieved by varying rotor height and diameter. For example, the three bladed FloWind EHD 17-m wind turbine, using a laminar airfoil SNLA 0021/50, can produce 175 kW at 51.8 rpm operating in a wind of 16 m/s, [3.30].

The power performance data available for Darrieus wind turbines from field tests in several countries is summarised in Table 3.1. Table 3.2 shows a few Darrieus wind turbines for which power output data are available from wind tunnel tests. In both cases, both the predicted power and the aerodynamic model used for calculation are indicated.

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Aerodynamic Performance Prediction Models 65

"

Aerodynamic Performance

Prediction Models

Nomenclature

a = velocity interference factor (Eq. 4.37)

c = chord length of blade, m

CDD = disk drag coefficient C_N = normal force coefficient

CP = average coefficient of power

CPe = elemental coefficient of power (Eq. 4.72) C_T = tangential force coefficient

c/R = chord-to-radius ratio

D = wind turbine drag, N

FN = normal force on turbine blade, N

FN

*

= dimensionless normal force on turbine blade

FT = tangential force on turbine blade, N

F_T* = dimensionless tangential force on turbine blade

h = height of streamtube, m 2H = rotor height, m

L = lift force, N

N = number of blades

Nc/R = rotor solidity (Eq. 4.15)

NLEV = number of vertically spaced blade divisions (see Fig. 4.20) NSTA = number of angular blade positions (Eq. 4.58 and Fig. 4.20) q = local relative dynamic pressure, N/m2

r = local turbine radius, m

R = radius of turbine at equator, m

S = frontal area of turbine (or disk area), m2

t = time, s

T_B = total torque, N◊◊◊◊◊m (Eq. 4.20)

Te = elemental blade torque, N◊◊◊◊◊m (Eq. 4.70)

T_e* = dimensionless blade torque (Eq. 4.71)

TS = single blade torque, N◊◊◊◊◊m (Eq. 4.19)

V = fluid velocity, m/s

d

v = velocity through wind turbine disk, m/s

d

V = disturbance velocity, m/s

r

V = relative fluid velocity, m/s

t

V = tip-speed, m/s

T

V = tangential blade velocity at equator, m/s

w

V = wake convection velocity, m/s

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66 Chapter 4 V_∞ = freestream velocity, m/s W = relative velocity, m/s

( )

w y = downwash velocity, m/s X = tip-speed ratio

z = height with respect to equator, m

a = angle of attack, deg

d = blade slope angle (or meridian angle), deg

g = vorticity, m2_/s gS = shed vorticity, m2/s gt = trailing vorticity, m2/s gw = wake vorticity, m2/s G = circulation, m2_/s h = r/R

q = azimuthal angle of turbine blade, deg

r = fluid density, kg/m3

w = angular velocity, rad-1

z = z/H Subscripts EQ = equator • = freestream value Superscripts (-₎ _{= mean value} (*) = dimensionless value

4.1 SINGLE STREAMTUBE MODEL

The single streamtube model was first developed by Templin [4.1] to calculate the aerodynamic performance of a curved-blade vertical-axis wind turbine. This model is based on the approach of the propeller or windmill actuator disk theories that assume induced velocity to be constant through the disk and related directly to wind turbine drag. The induced velocity is thus assumed to be the same through upwind and downwind faces of the rotor.

According to Glauert’s theory [4.2], the velocity through a windmill disk VD is the

arithmetic mean of the undisturbed velocity V• and the velocity in the wake. The wind turbine

drag is given by

D = 2ρSVD

(

V∞ − VD

)

(4.1)

where r represents the fluid density and S the disk area.

A disk drag coefficient CDD based on the dynamic pressure and the disk area is defined as:

C D V S DD D = ₁ 2 2 ρ (4.2)

and from equation (4.1),

C V V DD D =  −    ∞ 4 1 _(4.3)

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Aerodynamic Performance Prediction Models 67 Hence V V_D CDD ∞ _{= +}₁ 1 4 (4.4)

For structural design purposes, a more convenient drag coefficient CD is based on the ambient

dynamic pressure, where

C D V S C V V C C D DD D DD DD = =     = _ ₊   ∞ ∞ 1 2 1 1 4 2 2 2 ρ (4.5)

For a given wind turbine geometry and rotational speed w, the aerodynamic performance,

turbine power and rotor drag are calculated using the blade element theory. In general, the curved shape of the vertical-axis wind rotor is that of a skipping rope, spinning about a vertical-axis and assuming the gravity forces to be negligible. For a ratio of rotor height to rotor diameter of unity, the shape can be approximated by a parabola and the blade shape is given by the expression:

r R z H = − 1  2 (4.6) which in nondimensional form is h = 1 - z2_{, with}_{h = r/R and z = z/H, where r is the local}

rotor radius and z is the height above the equatorial plane. By differentiating the relation (4.6) we can obtain the local blade slope given by angle d (Fig. 4.1).

δ =  _ζ    − tan 1 1 2 (4.7)

Figure 4.1 Curved blade vertical-axis wind turbine with three blades

Chap_04.p65 67 12/11/2009, 08:59

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Unsteady Aerodynamics − CFD Models 101

#

Unsteady Aerodynamics

CFD Models

5.1 INTRODUCTION

The environment-friendly nature of wind energy and recent advances in wind turbine technology have made this renewable energy source a promising alternative for the future. Although the horizontal-axis wind turbine is the most common device of its type, the Darrieus vertical-axis model has proven one of the most efficient systems of wind energy conversion. Its many advantages include its independence of wind direction and its simplicity. Some of the most complex and least understood phenomena in the field of Computational Fluid Dynamics (CFD) are associated with the description of the flow past rotating blades (Fig. 5.1). A major aspect of the unsteady aerodynamics of the Darrieus rotor is dynamic stall, which occurs at low tip-speed ratios. Its effects have a significant influence on the overall system design. According to many experimental tests, the feature of dynamic stall that distinguishes it from static stall is the shedding of significant concentrated vorticity from the leading-edge region. This vortex disturbance subsequently sweeps over the airfoil surface causing pressure changes and resulting in significant increases in airfoil lift and large nose-down pitching that exceeds static values.

This chapter describes a two-dimensional unsteady flow analysis around an airfoil in Darrieus motion under dynamic-stall conditions (Fig. 5.2). A numerical solver based on the solution of the Reynolds-averaged Navier-Stokes equations expressed in a streamfunction-vorticity formulation in a non-inertial frame of reference is developed. The governing equations are solved by the streamline upwind Petrov-Galerkin finite element method (FEM). Temporal discretization is achieved by second-order-accurate finite differences. The resulting global matrix system is linearized by the Newton method and solved by the generalized minimum residual method (GMRES) with an incomplete triangular factorization preconditioning (ILU). Turbulence effects are introduced in the solver by eddy viscosity models, namely the algebraic Cebeci-Smith model and the nonequilibrium Johnson-King model. To validate the turbulent solver, a flat plate in pure translation and a pitching NACA 0015 airfoil are used as test cases. The Johnson-King model shows better performance than the Cebeci-Smith or the k-e turbulence

models for the pitching NACA 0015 airfoil test case. The solver is then used to simulate the flow around a NACA 0015 airfoil in a Darrieus motion (Fig. 5.1). The computed results show clearly some distinctive features of the dynamic stall on an airfoil in Darrieus motion despite the fact that the generation of the leading-edge vortex typical for dynamic stall is not observed.

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102 Chapter 5

Figure 5.1 Airfoil in Darrieus motion

Nomenclature

A = cross-section of the body surrounded by Bs, nondimensionalized by c2, (Fig. 5.4)

A+ = constant in the law of the wall coordinate (A+ = 26 (CSM), A+ = 17 (JKM), (Eq. 5.48)

B• = external boundary of B, (Fig. 5.4)

Bs = internal boundary of B, (Fig. 5.4)

B = computational domain

CM = pitching moment coefficient

CN = normal force coefficient

Cp = pressure coefficient

CT = tangential force coefficient

c = airfoil chord, m e = finite element domain (e₁, e₂, e₃) =

(

e e e1, 2, 3

)

, unit vectors along x, y and z directions

FKleb = Klebanoff intermittence function

g = function defined as τm−1 2, (Eq. 5.59)

k = turbulent kinetic energy

k* =wc/(2u_∞), reduced frequency

nP n n

e e e

, _ψ, _ω = number of nodes associated to finite element

P = perturbation pressure, nondimensionalized by u_∞2, (Eq. 5.59)

p = pressure, nondimensionalized by

R = equatorial radius, nondimensionalized by c

Re = Reynolds number, Re = u•c/n

s, n = unit vectors tangent and normal to boundaries

t = time, nondimensionalized by c/u•

D t = time step, nondimensionalized by c/u•

u = velocity vector, nondimensionalized by u• 2

u

ρ _∞

Wind Turbine Design

Wind Turbine

Design

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P

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iv

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Wind Turbine Design

With Emphasis on Darrieus Concept

W

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ph

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p

t

Ion

Paraschivoiu

Wind Turbine

Design

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I

P

Foreword

Table

of

Contents

List of

Figures

List of

Tables



Wind Energy

State of the Art of Vertical-Axis

Wind Turbines

∑

e

j

!

The Darrieus Wind-Turbine

Concept

"

Aerodynamic Performance

Prediction Models