Cover
About the Author
Preface
Acknowledgments
Abbreviations
List of Symbols
About the Companion Website
1 Introduction: Wind Turbines and the Wind Resource
1. 1.1 A Brief History of Wind Turbine Development
2. 1.2 Wind Resource Characterization
3. References
4. Further Reading
2 Momentum Theory
1. 2.1 Actuator Disk Model
2. 2.2 Rotor Disk Model
3. References
4. Further Reading
3 Blade Element Momentum Theory (BEMT)
1. 3.1 The Blade Element – Incremental Torque and Thrust
2. 3.2 Combining Momentum Theory and Blade Element Theory through a, a', and Φ
3. 3.3 Aerodynamic Design and Performance of an Ideal Rotor
4. 3.4 Tip and Root Loss Factors
5. 3.5 BEM Solution Method
6. 3.6 Simplified BEMT (Wilson and Lissaman 1974)
7. 3.7 Effect of Design Parameters on Power Coefficient
8. 3.8 Validity of BEMT
9. References
10. Further Reading
4 Wind Turbine Airfoils
1. 4.1 Fundamentals of Airfoil Theory
2. 4.2 Design Characteristics of Wind Turbine Airfoils
3. 4.3 Development of Wind Turbine Airfoils
4. References
5. Further Reading
5 Unsteady Aerodynamics and 3‐D Correction Models for Airfoil Characteristics
1. 5.1 Unsteady Aerodynamics on Wind Turbine Blades
2. 5.2 Rotational Augmentation and Stall Delay
3. 5.3 Airfoil Characteristics at High Angles of Attack
4. References
5. Further Reading
6 Vortex Wake Methods
1. 6.1 Fundamentals of Prandtl Lifting‐Line Theory
2. 6.2 Prescribed‐Wake Methods
3. 6.3 Free‐Wake Methods
4. References
5. Further Reading
7 Advanced Computational Methods
1. 7.1 High‐Fidelity Blade‐Resolved CFD Solutions
2. 7.2 Numerical Modeling of Wind Turbine Wakes
3. 7.3 Wake Modeling – Effect of Atmospheric Stability State
4. References
5. Further Reading
8 Design Principles, Scaled Design, and Optimization
1. 8.1 Design Principles for Horizontal‐Axis Wind Turbines
2. 8.2 Scaled Design of Wind Turbine Blades
3. 8.3 Aerodynamic Optimization of Wind Turbine Blades
4. 8.4 Summary – Scaled Design and Optimization
5. References
6. Further Reading
Index
End User License Agreement

List of Tables

Chapter 1
1. Table 1.1 Wind power density,P/A, for steady uniform wind (ρ = 1.225...
2. Table 1.2 Surface roughness length,z ₀, for various natural surfaces.
Chapter 2
1. Table 2.1 Example values of maximum power coefficient,C _{P, max}, as a functio...
Chapter 3
1. Table 3.1 Airfoil distribution (PSU 1.5‐MW turbine).
Chapter 4
1. Table 4.1 Benchmark airfoils for utility‐scale wind turbines with available meas...
2. Table 4.2 S‐series airfoils.
3. Table 4.3 Risø‐A airfoil family.
4. Table 4.4 Risø‐P airfoil family.
5. Table 4.5 Risø‐B airfoil family.
6. Table 4.6 DU airfoil family.
Chapter 5
1. Table 5.1 “Napkin” estimate of periodic angle‐of‐attack variations (PSU 1.5‐MW T...
2. Table 5.2 Dimensional parameters relevant to stall delay.
3. Table 5.3 Stall‐delay dimensionless groupings.
Chapter 6
1. Table 6.1 Relative pitch angles due to blade twist (NREL phase VI rotor).
Chapter 8
1. Table 8.1 IEC standards for wind turbines.
2. Table 8.2 Scaling wind turbine loads and resonances with blade radius,R, at co...
3. Table 8.3 Examples of wind turbine analysis and design codes.
4. Table 8.4 Dimensionless groupings relevant to wind turbine aerodynamics and stru...
5. Table 8.5 Example – scaled NREL 5‐MW turbine (V ₀ = 8 m s^‐1

List of Illustrations

Chapter 1
1. Figure 1.1 European smock mill near Waldfeucht (Germany).
2. Figure 1.2 Smeaton's windmill apparatus.
3. Figure 1.3 American water‐pumping fan mill (U.S. Department of Agriculture...
4. Figure 1.4 Wind in transition: Brush postmill (a) and Poul La Cour in fron...
5. Figure 1.5 Wind across scales: Gedser turbine, 200 kW (a) and Smith‐P...
6. Figure 1.6 Examples of modern wind turbines: (a) a Siemens 2.3 MW tur...
7. Figure 1.7 Design evolution of modern utility‐scale HAWTs.
8. Figure 1.8 Key components of a modern utility‐scale wind turbine.
9. Figure 1.9 Atmospheric motion – time and space scales.
10. Figure 1.10 Example of simulated short‐term wind variations, OpenFOAM ABL‐...
11. Figure 1.11 Flow at wind speed, V ₀, through area, A.
12. Figure 1.12 Power curve of a pitch‐controlled HAWT.
13. Figure 1.13 Basic characteristics of the ABL.
14. Figure 1.14 Vertical wind profiles – effect of stability state.
15. Figure 1.15 Rayleigh probability density function for variable mean wind s...
16. Figure 1.16 Weibull probability density function for variable parameter, ...
Chapter 2
1. Figure 2.1 Simplified axi‐symmetric streamtube.
2. Figure 2.2 Wind speed and pressure distribution inside axi‐symmetric strea...
3. Figure 2.3 Thrust and power coefficients C _T and C _P versu...
4. Figure 2.4 Streamtube expansion for increasing thrust coefficient, C...
5. Figure 2.5 Streamtube area ratios A / A ₀ and A ₁/ A versus a...
6. Figure 2.6 Validity of actuator disk theory.
7. Figure 2.7 Simplified streamtube including wake rotation.
8. Figure 2.8 Illustration of a rotor disk annulus.
9. Figure 2.9 Universal solutions for optimum induction factors a and a...
10. Figure 2.10 Spanwise distributions of optimum induction factors a and
11. Figure 2.11 Maximum power coefficient, C _P,max, versus tip speed r...
Chapter 3
1. Figure 3.1 Principle of blade element theory.
2. Figure 3.2 Airfoil nomenclature.
3. Figure 3.3 Velocity and force/torque triangles at blade element.
4. Figure 3.4 Example – ideal rotor blade planforms with/without wake rotatio...
5. Figure 3.5 Radial distributions of optimum induction factors a and a...
6. Figure 3.6 Prandtl's approximation of helical vortex sheets using semi‐inf...
7. Figure 3.7 Radial comparisons of Glauert tip correction (or Prandtl tip‐lo...
8. Figure 3.8 Radial variation of total tip‐/root loss correction for a three...
9. Figure 3.9 General g function parameters (Schmitz and Maniaci 2016, 2017)....
10. Figure 3.10 Flowchart of BEM solution algorithms: (a) two‐equation method ...
11. Figure 3.11 BEM solution in a turbulent wake state, F = 0.85...
12. Figure 3.12 BEM solution in a turbulent wake state, F = 0.85...
13. Figure 3.13 Maximum power coefficient, C _P,max, versus tip speed r...
14. Figure 3.14 Maximum power coefficient, C _P,max, versus tip speed r...
15. Figure 3.15 Maximum power coefficient, C _P,max, versus tip speed r...
16. Figure 3.16 Rotor power, P , versus wind speed, V ₀ – effec...
17. Figure 3.17 Rotor power, P , versus wind speed, V ₀ – effec...
18. Figure 3.18 Power coefficient, C _P, versus tip speed ratio, λ...
19. Figure 3.19 Radial distribution of non‐dimensional blade chord, c / R ...
20. Figure 3.20 Radial distribution of blade twist, β _Twist (PSU ...
21. Figure 3.21 Rotor power, P , and thrust, T , versus wind speed, V...
22. Figure 3.22 Tip speed ratio, λ , and blade tip pitch, βTip...
23. Figure 3.23 Rotor power coefficient, C _P, versus tip speed ratio,
24. Figure 3.24 Rotor thrust coefficient, C _T, versus tip speed ratio,...
25. Figure 3.25 Radial distribution of lift coefficient, c _l (PSU 1.5‐...
26. Figure 3.26 Radial distribution of axial induction factor, a (PSU 1.5‐...
27. Figure 3.27 Blade operating polar, c _l versus c _d (PSU 1.5...
28. Figure 3.28 Blade operating polar, c _l versus cl/cd...
29. Figure 3.29 Optimum blade parameter, σ ′ c _l, at C...
Chapter 4
1. Figure 4.1 Re‐dependence of viscous flow region (gray) around an airfoil...
2. Figure 4.2 Flight (or wind) speed versus chord‐based Reynolds number, Re...
3. Figure 4.3 Vortex‐sheet representation of an airfoil.
4. Figure 4.4 Inviscid flow – Kutta–Joukowski lift theorem and d'Alembert's p...
5. Figure 4.5 Pressure integration around airfoil (inviscid).
6. Figure 4.6 General viscous flow around an airfoil.
7. Figure 4.7 Displacement body model in viscous flow.
8. Figure 4.8 Control‐volume analysis of viscous airfoil flow.
9. Figure 4.9 Lift curve in inviscid and viscous flow.
10. Figure 4.10 Effect of pressure gradient on airfoil boundary‐layer dynamics...
11. Figure 4.11 Integrating surface pressure and viscous stress (NACA 4412, R...
12. Figure 4.12 Effect of free versus forced transition on airfoil drag coeffi...
13. Figure 4.13 Force/torque and velocity triangles along blade radius.
14. Figure 4.14 Examples of S‐series airfoils (S809, S814, S826).
15. Figure 4.15 DU series airfoils.
Chapter 5
1. Figure 5.1 Force/torque and velocity triangles in time‐dependent flow.
2. Figure 5.2 Unsteady thin‐airfoil theory – mathematical model (freestream a...
3. Figure 5.3 Theodorsen function and effect on circulatory part of the unste...
4. Figure 5.4 Illustration of Øye dynamic stall model – FFA‐W3‐360 Airfoil (
5. Figure 5.5 Effect of yaw, tower interference, and wind shear on rotor infl...
6. Figure 5.6 Himmelskamp's measurements of rotational augmentation (or stall...
7. Figure 5.7 Direction of Coriolis forces acting at blade root and tip.
8. Figure 5.8 Effect of Coriolis force on boundary‐layer lift dynamics at inb...
9. Figure 5.9 Effect of centrifugal pumping on a rotating wind turbine blade....
10. Figure 5.10 Methodology of a solution‐based stall‐delay model.
11. Figure 5.11 Example of 3‐D stall‐delay corrected airfoil polar for the NRE...
12. Figure 5.12 Radial distributions of local chord ratio, c/r, and modified...
13. Figure 5.13 NREL Phase VI rotor – Equivalent radial circulation distributi...
14. Figure 5.14 NREL Phase VI rotor – Radial distribution of normal force (a:
15. Figure 5.15 NREL Phase VI rotor – Radial distribution of tangential force ...
16. Figure 5.16 Example of high AoA corrected airfoil polar for the NREL Phase...
Chapter 6
1. Figure 6.1 Lifting‐line concept and horseshoe vortices.
2. Figure 6.2 Trailing vortex sheet behind lifting line.
3. Figure 6.3 Lift curves and drag polars for elliptic‐planform wings of vary...
4. Figure 6.4 Spanwise circulation and downwash distribution (AR = 10...
5. Figure 6.5 Parked (sequences L and O) NREL phase VI rotor in the NASA Ames...
6. Figure 6.6 NREL phase VI rotor (parked) – normal and tangential force coef...
7. Figure 6.7 Kinematic description of helicoidal vortex filament.
8. Figure 6.8 Example of helix advance ratio vs. axial distance traveled into...
9. Figure 6.9 Example of helicoidal vortex structure downstream of a two‐blad...
10. Figure 6.10 Bound vortex element (a) and trailing vortex filament (b).
11. Figure 6.11 Flowchart of prescribed‐wake solution algorithm.
12. Figure 6.12 Rotor power coefficient, C _P, and thrust coefficient, C...
13. Figure 6.13 Radial distributions of induction factors a and a′ ob...
14. Figure 6.14 Blade trailing versus shed wake vorticity.
15. Figure 6.15 Lagrangian wake markers (a) and Weissinger blade model (b).
16. Figure 6.16 Flowchart of free‐wake solution algorithm.
17. Figure 6.17 Schematic of stretched vortex filament.
18. Figure 6.18 Example of DVE‐computed wake (NREL phase VI rotor S‐sequence,
19. Figure 6.19 NREL phase VI rotor – normal force coefficient (S – sequence,
Chapter 7
1. Figure 7.1 Limit surface streaklines over surface pressure contours (NREL ...
2. Figure 7.2 Coupled Navier–Stokes/Vortex‐Panel Solver (Schmitz and Chattot ...
3. Figure 7.3 Distribution of bound circulation (NREL Phase VI rotor S‐Sequen...
4. Figure 7.4 Concept of actuator‐type methods.
5. Figure 7.5 Volumetric body‐force projection (ALM) – Jha and Schmitz (2018)...
6. Figure 7.6 Volumetric body‐force projection (ALM*, ACE) – Jha and Schmitz ...
7. Figure 7.7 Wake structure and strength for the NREL 5‐MW turbine (V0 = 8...
8. Figure 7.8 Actuator curve embedding (ACE) computation of p _n and ps...
9. Figure 7.9 Sectional forces on rotating NREL 5‐MW rotor, V0 = 8...
10. Figure 7.10 Comparison of models and measurements for Horns Rev. (directio...
11. Figure 7.11 Wind turbine wake computed with LES (Jha et al. 2015).
12. Figure 7.12 Turbine‐turbine interaction (Jha et al. 2015) in neutral bound...
13. Figure 7.13 Turbine‐turbine interaction – Mean and standard deviation of t...
14. Figure 7.14 Turbine‐turbine interaction – ALM/ALM* flowfield in horizontal...
15. Figure 7.15 Turbine‐turbine interaction – ALM/ALM* mean and standard devia...
Chapter 8
1. Figure 8.1 Dependence of rotor diameter, D , with power, P (ρ = 1...
2. Figure 8.2 Scaling example – NACA 64618 (FS tip airfoil) and NACA 4412 (MS...
3. Figure 8.3 Re‐dependence of airfoil characteristics (NACA 64618) – circle ...
4. Figure 8.4 Re‐dependence of airfoil characteristics (NACA 4412) – circle d...
5. Figure 8.5 Scaled NREL 5‐MW turbines – radial distribution of lift coeffic...
6. Figure 8.6 Scaled NREL 5‐MW turbines – blade planforms and pitch.
7. Figure 8.7 Scaled NREL 5‐MW turbines – radial distribution of normal and t...
8. Figure 8.8 Examples of MS wind turbines..
9. Figure 8.9 NREL phase VI rotor – normal and tangential force coefficients,...
10. Figure 8.10 New MEXICO rotor – normal and tangential force, F _n an...
11. Figure 8.11 Krogstad turbine – rotor power and thrust coefficients, C...
12. Figure 8.12 Blade geometry parameter, Ξ – Wake Rotation (λ = 6...
13. Figure 8.13 Blade geometry parameter, Ξ – Wake Rotation (λ = 6...
14. Figure 8.14 Blade geometry parameter, Ξ – Wake Rotation (λ = 6...
15. Figure 8.15 Ideal rotor with wake rotation – optimum circulation distribut...
16. Figure 8.16 (a) Rotor power, P , and (b) thrust, T , versus wind spe...
17. Figure 8.17 Radial distribution of local power/thrust coefficient, dC...
18. Figure 8.18 Blade geometry parameter, Ξ, and radial circulation dis...
19. Figure 8.19 Radial distribution of non‐dimensional blade chord, c / R ,...
20. Figure 8.20 Simplified objective function, C _P/ C _T ^2/3, and ...
21. Figure 8.21 Blade geometry parameter, Ξ, and radial circulation distr...
22. Figure 8.22 Radial distribution of blade chord, c, and blade pitch, β...
23. Figure 8.23 Rotor power, P, and root‐flap bending moment, Be, versus w...

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Library of Congress Cataloging‐in‐Publication Data

Names: Schmitz, Sven, 1976‐ author.

Title: Aerodynamics of wind turbines : a physical basis for analysis and

design / Sven Schmitz, Department of Aerospace Engineering, Pennsylvania

State University.

Description: First edition. | Hoboken : Wiley, 2020. | Includes

bibliographical references and index.

Identifiers: LCCN 2019024050 (print) | LCCN 2019024051 (ebook) | ISBN

9781119405610 (paperback) | ISBN 9781119405597 (adobe pdf) | ISBN

9781119405641 (epub)

Subjects: LCSH: Wind turbines–Aerodynamics.

Classification: LCC TJ828 .S37 2019 (print) | LCC TJ828 (ebook) | DDC

621.4/5–dc23

LC record available at https://lccn.loc.gov/2019024050

LC ebook record available at https://lccn.loc.gov/2019024051

Cover Design: Wiley

Cover Images: © YaiSirichai/Shutterstock, © nito/Shutterstock,

Preface

The vision for the book was to develop a self‐contained and affordable unique text with focus on the aerodynamics, scaled design and analysis, and aerodynamic optimization of horizontal‐axis wind turbines. It is not a systems‐engineering text on wind energy, which distinguishes the book from other available texts. On the contrary, the author was encouraged by many colleagues over the past several years to develop a well‐integrated and focused account on the blade aerodynamics of horizontal‐axis wind turbines. The technical content is based on lecture notes developed by the author at the senior‐level undergraduate and graduate level. A further unique aspect of the book is the close integration of the text with the wind turbine design and analysis software, XTurb, developed and maintained by the author at The Pennsylvania State University. The XTurb code is essentially a teaching and research tool used since 2011 by the author in his graduate course and by Penn State students participating in the U.S. DOE Collegiate Wind Competition. The XTurb examples in the book add a “hands‐on” component, thus enhancing the learning experience to readers and resulting in a deeper and more complete understanding of the subject matter. This gives readers from interdisciplinary backgrounds in the area of wind energy the opportunity to fully absorb and understand the design principles and governing concepts in blade aerodynamics through the text and independent analyses using XTurb.

Chapter 1 concerns a brief description of horizontal‐axis wind turbine development, with particular considerations for the history of aerodynamics and its impact on the design evolution of wind turbines. In addition, the reader is introduced to the atmospheric boundary layer and the wind resource. Chapter 2 is a classical account on momentum theory for horizontal‐axis wind turbines, with some differences and additions compared to most texts. Chapter 3 covers classical Blade Element Momentum (BEM) theory. A special section includes a detailed description of root and tip loss factors used in today's BEM‐type methods and a discussion of their respective limitations. As far as BEM solution techniques are concerned, the presentation of classical work is accompanied by a description of various numerical techniques of solving the BEM equations (new in this book), followed by a complete description of models for the turbulent wake state (also not in any other text). A simplified BEM theory is described as a classical means to introduce simplified dependencies of the effect of design parameters on power coefficient, with the later subsection containing multiple examples of XTurb analyses and associated input decks. Chapter 4 begins with an introduction to thin‐airfoil theory and the foundations of viscous airfoil flow. This is followed by a brief historical review on wind turbine airfoil design and airfoil design criteria along the blade radius. The chapter concludes with a catalog of wind turbine airfoils. Chapter 5 focuses on introducing unsteady aerodynamics occurring on horizontal‐axis wind turbines, with subsections describing yaw effects, tower interaction, and dynamic stall. A big portion of this chapter is devoted to a comprehensive review on rotational augmentation and available stall‐delay models (also new in this text). Chapter 6 concerns vortex‐wake methods and starts off by a comprehensive introduction to lifting‐line theory and describes the basics of computing induced velocities from planar and vortical wake sheets. A unique subsection on prescribed wake methods (not in any other text) includes additional XTurb examples. A subsection on free‐wake methods includes classical descriptions and introduces the reader to the numerical problem of vortex cores to avoid wake singularities, including recent advances on singularity‐free‐wake methods. Chapter 7 gives an introduction to advanced computational methods including Computational Fluid Dynamics (CFD), hybrid CFD methods, and an introduction to recent advances in actuator‐type methods for wake modeling (i.e. actuator disk, actuator line, actuator surface). Chapter 8 introduces the reader to principles of (scaled) wind turbine design and optimization. The technical content draws from the author's experience in designing scaled experiments in both wind‐/water tunnels for wind turbine and rotorcraft applications. This is a very unique chapter of the book and one that is vitally important to the wind energy community at large. The XTurb software is used extensively in this chapter and brings the material of previous chapters into an overarching context to the reader, thus allowing a deeper understanding of the book material by considering the difficult and non‐unique aspects of scaled blade design and optimization. Supplementary files including XTurb input decks are available on a respective Wiley website.

The primary intended audience consists of senior undergraduate students and graduate students in MSc. and Ph.D. programs at universities, focusing on coursework and research in wind energy in engineering, atmospheric science, and meteorology. Further audiences are instructors and university professors, as well as practicing engineers and scientists in industry and national laboratories. Readers are expected to have basic knowledge of incompressible flows through coursework or work experience. The book is written at an intermediate level using college‐level algebra and analysis. The book can be used as a primary text for courses on wind turbine aerodynamics and/or as an affordable instructional aid for a multitude of courses on wind energy systems and power engineering in the international wind energy community.

In essence, this is a book “for Students, written by a student,” as the author sees himself as a continuous learner. I hope that you will find the book helpful to your careers. Wind energy has the potential to playing a major role in battling climate change by powering the world with a clean and renewable source of energy. Never give up, not as long as you have strength left.

January 2019

Sven Schmitz
State College, PA

Acknowledgments

A number of individuals have assisted the author and contributed to the book in different ways. I want to start by thanking my beloved wife Cristina for her patience and encouragement while writing the manuscript. I am especially grateful to my parents Helmut and Irmgard Schmitz who have always supported me in the path I have taken far away from home. I am also thankful to my Ph.D. advisor Professor Jean‐Jacques Chattot at the University of California Davis who educated me in wind turbine aerodynamics and numerical methods.

Many thanks are directed at The Pennsylvania State University for allowing me to spend my sabbatical leave at home. Furthermore, I am grateful to Professor Carlo L. Bottasso of the Technical University of Munich (TUM) and Professor Jens. N. Sørensen of the Technical University of Denmark (DTU) for hosting me a week each and their many valuable suggestions. The discussions with many colleagues at TUM and DTU were inspiring and helpful in organizing some of the book content.

I also acknowledge very much the help of my dear colleague Professor Mark D. Maughmer for numerous discussions on aerodynamics during my time at Penn State and for checking many subtle details thoroughly in the book. Furthermore, I am thankful to the former Aerospace Department Head Professor George A. Lesieutre for his encouragement in developing a graduate course focused on wind turbine aerodynamics, as well as colleagues Dr. Susan W. Stewart and Professor Dennis K. McLaughlin without whose dedication there would not be a continuing graduate certificate in wind energy in the department.

Special thanks to Dr. Scott Larwood of the University of the Pacific and Dr. Mike P. Kinzel of the University of Central Florida for reading the entire manuscript. Scott and Mike represent a portion of a diverse readership with different background and experience; their many questions and comments helped clarifying derivations, figures, and the narrative. I would also like to thank Mr. Dan Somers of Airfoils Inc. in Port Matilda, PA for his review and suggestions on the history of airfoil design for wind turbines over the past 30 years. In addition, comments and suggestions by Dr. Niels Troldborg of Risø National Laboratories on advanced computational methods and actuator line modeling are very much appreciated.

Last but not least, I want to thank my students in wind energy for their commitment to hard work, inspiration, and passion for wind energy as a viable source of renewable energy. The future lies with our students who will advance wind energy to the next generation. I would like to close by thanking the U.S. Department of Energy (DOE), the National Science Foundation (NSF), the University Corporation for Atmospheric Research (UCAR), and industry for their generously supporting my students at Penn State.

List of Symbols

Note: This list includes the most relevant symbols used in the book, and omits some symbols that are unique to particular subchapters.

English

a: axial (flow) induction factor
a′: angular (flow) induction factor
adv _i: (vortex) sheet advance ratio (Chapter 6)
A: actuator/rotor disk area
A ₀: entrance area of streamtube (Chapter 2); Fourier coefficient in airfoil theory (Chapter 4)
A ₁: exit area of streamtube
A _n: Fourier coefficients in airfoil/wing theory (Chapters 4, 6)
A _t: airfoil cross‐sectional area (Chapter 4)
b: wing span
B: blade number
c: blade chord; Weibull scale parameter (Chapter 1)
: mean aerodynamic chord
c ₀: wing mid‐span chord
c _d: section (profile) drag coefficient
c _d,min: minimum section (profile) drag coefficient
c _l: section lift coefficient
c _l,max: maximum section (airfoil) lift coefficient
c _l,α: airfoil lift‐curve slope
c _m: section moment coefficient about quarter‐chord location
c _m,0: section moment coefficient about leading edge
c _p: pressure coefficient; specific heat constant (at constant pressure)
C(k): complex Theodorsen function
c _D: wing drag coefficient
: wing induced drag coefficient
c _L: wing lift coefficient
C _P: power coefficient
C _P,max: maximum power coefficient
C _Q: torque coefficient
C _T: thrust coefficient
d: spacing between semi‐infinite planar vortex sheets; airfoil camber
dr: incremental width of blade element (or rotor annulus)
dA: incremental disk‐annulus area (2πr · dr )
dA _c: wetted area of infinitesimal blade element ( c · dr )
dD: incremental/sectional profile drag force
dF _N: incremental/sectional (or blade‐element) normal force – i.e. force normal to local airfoil/section chord line towards upper airfoil/section surface
dF _T: incremental/sectional (or blade‐element) tangential force – i.e. force parallel to local airfoil/section chord line towards leading edge of airfoil/section
dL: incremental/sectional lift force
: incremental mass‐flow rate
dQ: incremental torque
dQ _B: blade‐element torque, incremental torque (one blade)
dP: incremental power
dT _B: blade‐element thrust, incremental thrust (one blade)
D: rotor diameter; (profile) drag force
D _i: induced drag
e: Oswald efficiency factor
f( ): functional relationship
f _N,m: elemental rotor/blade force vector (Chapter 7)
F: total loss factor – i.e. product of local root‐ and tip‐loss factor ( F _R · F _T )
F( ): objective function (Chapter 8)
F(k): real part of Theodorsen function
: airfoil force vector
Fr: Froude number
F _C: centrifugal force ( mΩ² R )
F _Co: Coriolis force (−2mΩv _rel )
F ₁: tip correction factor applied to section normal‐/tangential force coefficients (Section 3.4)
F _B: Prandtl's approximate loss factor to account for a finite number of blades (Section 3.4)
F _P: volumetric body‐force vector (Chapter 7)
F _R: root‐loss factor
F _T: tip‐loss factor
g: gravitational acceleration; coefficient or function used in tip corrections (Section 3.4)
G(k): imaginary part of Theodorsen function
H: shape factor of airfoil boundary layer ( δ ^*/Θ)
: unit vector in the x‐direction
I: turbulence intensity; mass moment of inertia
: unit vector in the y‐direction
k: integer; shape parameter for Weibull distribution (Chapter 1); reduced frequency (Chapter 5); specific turbulent kinetic energy (Chapter 7)
: unit vector in the normal/vertical z‐direction
l: turbulent mixing length
L: lift force; Monin–Obukhov length scale (Chapter 1)
Lo: Lock number ( c _l,α ρcR ⁴/I )
m: mass
: mass flow rate
M ₀: pitching moment about leading edge
Ma: (tip) Mach number ( λV ₀/a ₀ )
n: integer; coordinate normal to surface (element)
: outer unit normal vector (along contour or control‐volume surface)
p: fluid static pressure
p ₀: ambient static pressure
p _∞: freestream static pressure
p(U): probability density function
P: actuator/rotor/turbine power
: average wind/turbine power
P _Rated: rated wind/turbine power
q _s: surface heat flux
Q: rotor torque
r: radial blade coordinate (radial location)
: marker position vector (Chapter 6)
r′: normalized radial blade coordinate ( r/R )
r _c: viscous core radius (Chapter 6)
r _j: radius of vortex filament j emanating from blade trailing edge in HVM (Chapter 6)
: initial position vector (or vector to starting point) of vortical trailer in HVM (Chapter 6)
r _Root: (blade) root cut‐out radial location
R: blade radius; specific gas constant (air)
Re: Reynolds number
Re _c: chord‐based (blade section) Reynolds number ( ρV _rel c/μ )
Re _D: Reynolds number based on rotor diameter (2ρV ₀ R/μ )
s: coordinate tangential to surface (element)
S: wing area
t: time; airfoil thickness (Chapter 4); blade thickness distribution (Chapter 8)
t*: eddy turnover time
T, T(z): actuator/rotor thrust force; absolute temperature
u: uniform axial (x‐direction) velocity across actuator/rotor disk
u ₁: uniform axial (x‐direction) velocity across streamtube exit plane
u _i: axial induced velocity (Chapter 6)
u, u(t): instantaneous wind speed along the x‐direction
u′, u′(t): fluctuating wind speed along the x‐direction; perturbation velocity in the x‐direction
u*: horizontal velocity scale (friction velocity) in boundary layer
U, U(z): mean wind speed in the x‐direction (mean value of u(t))
: average wind speed in the x‐direction (typically taken over a period of 10 minutes or 1 hour)
U _e: edge velocity of airfoil boundary layer
U _∞: freestream speed (referring to airfoil flows)
: fluid‐parcel velocity vector relative to rotating system
: wind speed (average) at hub height
V _In: cut‐in wind speed
V _Out: cut‐out wind speed
V _Rated: rated wind speed
V _rel: (local) inflow/relative velocity
: marker advection velocity (Chapter 6)
w′: fluctuating wind speed along the z‐direction; perturbation velocity in the z‐direction
w*: vertical velocity scale in boundary layer
w _i: angular induced velocity (Chapter 6)
w _T: downwash in Trefftz plane
w _W: downwash at lifting line (or wing)
x: downwind/streamwise coordinate (fixed and rotating axis systems)
x _a.c.: chordwise location of airfoil aerodynamic center (measured from leading edge)
x _c.p.: chordwise location of airfoil center of pressure (measured from leading edge)
y: lateral (across‐wind) coordinate with respect to vertical axis in fixed axis system; wing‐span coordinate; radial (along blade length) coordinate in rotating axis system
z: vertical coordinate (upwards positive) in fixed axis system
z ₀: surface roughness
z _i: height of capping inversion; boundary‐layer height/depth

Greek

α: angle of attack – i.e. angle between (local) flow incident on the airfoil/blade and the blade chord line; (wind‐shear) power‐law exponent
: mean angle of attack
: time rate‐of‐change of angle of attack
α ₀: airfoil zero‐lift angle of attack
α _g: wing setting angle (or geometric angle of attack)
α _e: effective angle of attack ( α _g + α _i )
α _i: induced angle of attack
α _Stall: deep‐stall angle (of attack)
α _W: wing setting angle at mid‐span
β: blade pitch angle – i.e. angle between local chord line and rotor plane ( β _Twist + β ₀ )
β ₀: collective blade pitch angle ( β _Tip − β _Twist(R)) – i.e. the collective blade pitch angle is chosen such that when applied to a built‐in twist, results in the desired blade tip pitch angle.
β _Tip: blade tip pitch angle, β(R) – i.e. angle used to describe “pitch setting” of blade
β _Twist: built‐in (manufactured) twist
γ: vorticity distribution along airfoil camber line (Chapters 4, 5); rotor/turbine yaw angle (Chapter 5)
γ _w: wake vorticity (Chapter 5)
Γ: (bound) circulation on airfoil (and along wing or blade)
Γ_B: circulation of rotor with B blades
Γ_max: maximum circulation
Γ_∞: circulation of rotor with infinite number of blades
Γ( ): gamma function (Chapter 1)
Γ_d: adiabatic lapse rate
δ: boundary‐layer thickness (Chapter 4); relative drag increase (Chapter 6)
δ ^*: displacement thickness
Δc _l: viscous lift‐coefficient correction (Chapter 4)
Δc _p: local change/difference in pressure coefficient from upper to lower airfoil surface
Δ_grid: grid spacing
Δp: pressure discontinuity/jump at actuator disk
Δt: time step
Δα: amplitude of angle‐of‐attack oscillation; range of angle of attack
δΓ: trailing vorticity
Δδ _TE: difference between upper and lower boundary‐layer thickness at airfoil trailing edge ( δ _u − δ _l )
ΔΓ: viscous circulation correction (Chapter 4)
ε: specific turbulent energy dissipation rate; Gaussian projection radius (Chapter 7)
η _N,m: Gaussian projection function (Chapter 7)
Θ: transformed chordwise coordinate in airfoil frame (Chapter 4); momentum thickness (Chapter 4)
Θ₀: potential temperature on the ground (z = 0)
φ: local blade flow angle ( α + β ) – i.e. angle between local flow incident and rotor plane
κ: von Kármán constant; thrust constraint parameter (Chapter 8)
λ: tip speed ratio
λ _r: local speed ratio ( r/R · λ )
Λ: Lagrange multiplier (Chapter 8)
μ: air (or fluid) dynamic viscosity
μ _t: turbulent eddy viscosity
ν: air (or fluid) kinematic viscosity
Ξ: blade geometry parameter ( λσ′r′c _l )
ρ: air (or fluid) density
ρ _∞: freestream air (or fluid) density
σ: rotor solidity (integral of local solidity along blade radius)
σ′: local solidity ( Bc/2πr )
σ _u: standard deviation of wind speed in the x‐direction
σ _v: standard deviation of wind speed in the (across‐wind) y‐direction
σ _z: standard deviation of wind speed in the (vertical) z‐direction
τ: shear stress; time constant; characteristic time
τ ₀: wall shear stress (on boundary‐layer surface)
: skin friction (or wall shear stress) on airfoil
: turbulent shear stress in the x‐direction (normal vector in the z‐direction)
ϕ(k): phase angle of Theodorsen function
ϕ _j: phase angle of initial position vector (Chapter 6)
ψ: azimuthal (or circumferential) angle – i.e. blade angle in rotor plane with respect to tower axis
ω: wake angular speed (or wake swirl, added wake rotation, Chapter 2); vorticity (Chapter 4); angular frequency (Chapter 5); turbulent energy frequency (Chapter 7)
ω _N: natural frequency
: vorticity vector ()
Ω: disk/rotor angular speed (or rotor speed); simply‐connected domain (Chapter 4)
: angular velocity vector
∂Ω: boundary of simply connected domain
: nabla operator ()

Subscripts

0: reference condition (upstream or wall/surface); reference angle; reference to airfoil leading edge
1: reference condition (downstream); reference in tip loss correction
∞: farfield/freestream condition; reference to infinite number of blades
–: normalized quantity in airfoil boundary layer
A: aerodynamic
a. c.: aerodynamic center
B: blade; reference to finite number of blades
Bo: bound
c: chord; core (Chapter 6)
crit: critical
c. p.: center of pressure
C: reference to arbitrary point/location; centrifugal
CO: Coriolis
d: drag; adiabatic
e: effective
D: reference to rotor diameter; drag
fs: fully separated
hub: hub height
He: helix
i: integer; inversion; induced
inv: inviscid
iter: iteration
j: integer
k: integer
l: lift; lower
m: integer; moment
n: integer index used for fourier coefficients
N: integer; normal; natural
opt: optimum
p: pressure; pitching
P: power; reference to arbitrary point/location
r: radial coordinate along blade; reference quantity
rel: relative (at blade section)
rot: rotor entrainment
s: surface
S: small scale
t: turbulent; thickness
T: thrust; tangential; Trefftz
TE: trailing edge
u: reference to wind speed in the (streamwise) x‐direction; upper
U: utility scale
v: reference to wind speed in the (lateral) y‐direction
w: reference to wind speed in the (vertical) z‐direction; wake
W: wing; wake
x: vector component in the x‐direction; streamwise variable
Xtr: chordwise location of laminar‐turbulent transition (measured from leading edge)
y: vector component in the y‐direction
z: vector component in the z‐direction

Superscripts

*: reference to velocity scale in boundary layer
′: normalized/dimensionless; fluctuating component; per unit length/width; slope (first derivative) of function; perturbation
: time average
: normalized
rem: remainder
st: static

Aerodynamics of Wind Turbines

A Physical Basis for Analysis and Design