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Research Topics in Wind Energy
Volume 7
Series editors
Joachim Peinke, University of Oldenburg, Oldenburg, Germanye-mail: peinke@uni-oldenburg.deGerard van Bussel, Delft University of Technology, Delft, The Netherlandse-mail: g.j.w.vanbussel@tudelft.nl
About this Series
The series Research Topics in Wind Energy publishes new developments andadvances in the fields of Wind Energy Research and Technology, rapidly andinformally but with a high quality. Wind Energy is a new emerging research fieldcharacterized by a high degree of interdisciplinarity. The intent is to cover all thetechnical contents, applications, and multidisciplinary aspects of Wind Energy,embedded in the fields of Mechanical and Electrical Engineering, Physics,Turbulence, Energy Technology, Control, Meteorology and Long-Term WindForecasts, Wind Turbine Technology, System Integration and Energy Economics,as well as the methodologies behind them. Within the scope of the series aremonographs, lecture notes, selected contributions from specialized conferences andworkshops, as well as selected PhD theses. Of particular value to both thecontributors and the readership are the short publication timeframe and theworldwide distribution, which enable both wide and rapid dissemination of researchoutput. The series is promoted under the auspices of the European Academy ofWind Energy.
More information about this series at http://www.springer.com/series/11859
Emmanuel Branlard
Wind Turbine Aerodynamicsand Vorticity-Based MethodsFundamentals and Recent Applications
123
Emmanuel BranlardDepartment of Wind Energy, AeroelasticDesign
Technical University of DenmarkRoskildeDenmark
ISSN 2196-7806 ISSN 2196-7814 (electronic)Research Topics in Wind EnergyISBN 978-3-319-55163-0 ISBN 978-3-319-55164-7 (eBook)DOI 10.1007/978-3-319-55164-7
Library of Congress Control Number: 2017933865
© Springer International Publishing AG 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.
Printed on acid-free paper
This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To love, 2K � 2K
Preface
The standard approach in the study of wind turbine aerodynamics consists in usingmomentum analyses. The momentum theory of an actuator disk is an example ofmomentum analysis. Blade element momentum (BEM) and the conventionalcomputation fluid dynamics (CFD) are two numerical methods also based onmomentum analyses. Velocity and pressure are the main variables used inmomentum analysis. The equations can also be formulated using vorticity as mainvariable. This leads to an alternative approach referred to as vorticity-basedmethods. The great potential of vorticity-based methods comes from the multitudeof formulations they offer, ranging from simple analytical models to advancednumerical methods. The analytical model will be referred to as vortex theories andthe numerical methods as vortex methods.
The term vorticity often intimidates the newcomer, but this fear vanishes whenone realizes that velocity and vorticity offer two different, but often equivalent,points of view. For instance, the momentum theory of an actuator disk with con-stant loading can be equivalently studied by considering the tubular vorticity sheetthat is present at the surface of the streamtube. Vorticity plays an important role inwind turbine aerodynamics since strong vortices are present in the wakes inparticular. Vorticity and vorticity-based methods cannot be omitted in a book on thetopic. Most of the analytical models used in BEM methods are derived from ana-lytical vortex models. Further, numerical vortex methods are now competing withconventional CFD methods in terms of accuracy and computational time, and theyare becoming a common tool for the study of wind turbine aerodynamics.
The aim of this book is to show the relevance of vorticity-based methods for thestudy of wind turbine aerodynamics and to present historical and recent develop-ments in the field with a sufficient level of details for the book to be self-contained.
This book is intended for students and researchers curious about rotor aerody-namics and/or about vorticity-based methods. The book introduces the funda-mentals of fluid mechanics, momentum theories, vortex theories, and vortexmethods necessary for the study of rotors and wind turbines in particular. Rotortheories are presented in a great level of details at the beginning of the book. Thesetheories include the blade element theory, the Kutta–Joukowski theory, the
vii
momentum theory, and the BEM method. Different momentum theories are derivedfrom first principles using a critical approach. The remaining of the book focuses onvortex theory and vortex methods with application to wind turbine aerodynamics.Examples of vortex theory applications that are discussed in this book are optimalrotor design, tip-loss corrections, yaw models, and dynamic inflow models.Historical derivations and recent extensions of the models are presented. Thecylindrical vortex model is another example of a simple analytical vortex modelused in this book. In this model, a wind turbine and its wake are simplified using avortex system of cylindrical shape. Formulations equivalent to the ones used in aBEM algorithm are obtained. The model provides a wake-rotation correction whichgreatly improves the accuracy of BEM algorithms. The cylindrical model is alsoused to provide the analytical velocity field upstream of a turbine or a wind farm(i.e., the induction zone) under aligned or yawed conditions. Such results areobtained in a couple of seconds with an impressive accuracy compared to numericalresults from CFD methods which would require days of computation. Differentapplications of numerical vortex methods are presented in this book. Numericalmethods are used for instance to investigate the influence of a wind turbine on theincoming turbulence. Sheared inflows are also investigated. It is shown in particularthat most vortex methods omit a term resulting in excessive upward displacementof the wind turbine wake. Many analytical flows are derived in detail in this book:vortex rings, Hill’s vortex, vortex blobs, etc. They are used throughout the book todevise simple rotor models or to validate the implementation of numerical methods.Several MATLAB programs are provided to ease some of the most compleximplementations: BEM codes, vortex cylinder velocity functions, Goldstein’s cir-culation, lifting-line codes, Karman–Trefftz conformal map, projection functionsfor vortex particle methods, etc.
Part I introduces the fluid mechanics foundations relevant to this book. Part IIintroduces rotor aerodynamics, including momentum analyses, vortex models, andthe BEM method. Part III focuses on classical vortex theory results which origi-nated from the study of rotors with optimal circulation. Part IV presents the recentdevelopments in rotor aerodynamics based on analytical vortex flows. Part Vpresents recent applications of vortex methods. Part VI provides detailed analyticalsolutions that are relevant for rotor aerodynamics, either for the derivation of vortexmodels or for the implementation and validation of vortex methods. Part VII isdedicated to vortex methods. Part VIII provides mathematical complements to somechapters of the book.
Roskilde, Denmark Emmanuel BranlardJanuary 2017
viii Preface
Acknowledgements
The current work would not have been possible without the support and help of myPhD supervisor Mac Gaunaa and the contributions from Spyros Voutsinas, EwanMachefaux, Philippe Mercier, Gregoire Winckelmans, Niels Troldborg, GiorgiosPapadakis, and Henrik Brandenborg Sørensen. I would like to thank my colleaguesfor their inspiration and fruitful discussions: Jakob Mann, Niels Sørensen, CurranCrawford, Philippe Chatelain, Torben Larsen, Anders Hansen, Georg Pirrung,Frederik Zahle, Mads Hejlesen, Juan Pablo Murcia, Alexander Forsting, ChristianPavese, Michael McWilliams, Lucas Pascal, and Jacobus De Vaal.
I am grateful to the persons who accepted to review some chapters of this bookdespite a limited time: Damien Castaignet, Michael McWilliams, Mac Gaunaa, JensGengenbach, Gil-Arnaud Coche, Julien B., and Björn Schmidt.
Above all, I am glad for the moments of life and love I experienced thanks to myfamily and friends. I wish to share more of those with all of you: Ewan, François,Aghiad, Mika, Dim, Heidi, Mike, K, Ozi, Bertille, Julie, Kiki, Loïc, Milou, Romain,Sofie, Lucas P., Lucas M., Philipp, Jeanne, Alessandro, Julien, Sophie, Dad, andMom.
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Part I Fluid Mechanics Foundations
2 Theoretical Foundations for Flows Involving Vorticity . . . . . . . . . . 112.1 Fluid Mechanics Equations in Inertial and Non-inertial
Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.1 Physical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Fluid-Mechanic Equations
in a Non-inertial Frame . . . . . . . . . . . . . . . . . . . . . . . 172.1.4 Fluid Mechanics Assumptions. . . . . . . . . . . . . . . . . . 262.1.5 Usual Cases - Equations of Euler and Bernoulli . . . . 29
2.2 Flow Kinematics and Vorticity. . . . . . . . . . . . . . . . . . . . . . . . . 322.2.1 Flow Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.2 Vorticity and Related Definitions . . . . . . . . . . . . . . . 332.2.3 Helmholtz (First) Law. . . . . . . . . . . . . . . . . . . . . . . . 362.2.4 Helmholtz-(Hodge) Decomposition . . . . . . . . . . . . . . 362.2.5 Bounded and Unbounded Domain - Surface
Map - Generalized Helmholtz Decomposition . . . . . . 372.3 Main Dynamics Equations Involving Vorticity. . . . . . . . . . . . . 38
2.3.1 Circulation Equation . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.2 Vorticity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3.3 Stretching and Dilatation of Vorticity . . . . . . . . . . . . 402.3.4 Alternative Forms of the Vorticity Equation . . . . . . . 422.3.5 Vorticity Equation in Particular Cases. . . . . . . . . . . . 432.3.6 Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3.7 Vortex Force, Image/Generalized/Bound Vorticity,
Kutta–Joukowski Relation. . . . . . . . . . . . . . . . . . . . . 45
xi
2.4 Different Dimensions of Vorticity: Surface,Line and Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5 Vorticity Moments, Variables and Invariants - IncompressibleFlows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6 Main Theorems Involving Vorticity . . . . . . . . . . . . . . . . . . . . . 522.6.1 Kelvin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.6.2 Lagrange’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 522.6.3 Helmholtz Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 532.6.4 Biot–Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.7 Vortices in Viscous and Inviscid Fluid - Resultsand Classical Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.7.1 Vortex in Inviscid Fluid . . . . . . . . . . . . . . . . . . . . . . 572.7.2 Vortex in Viscous Fluid - Standard Solutions . . . . . . 572.7.3 Life of a Vortex - Vortex Decay, Collapse
and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.8 Surface Representations - Vortex Sheets . . . . . . . . . . . . . . . . . 60
2.8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.8.2 Vortex Sheets Kinematics . . . . . . . . . . . . . . . . . . . . . 602.8.3 Vortex Sheets Dynamics . . . . . . . . . . . . . . . . . . . . . . 612.8.4 Vortex Sheet Convection and Stability . . . . . . . . . . . 622.8.5 Vortex Surfaces in 2D. . . . . . . . . . . . . . . . . . . . . . . . 62
2.9 Incompressible Flow Equations in Polar Coordinates - 2Dand 3D Flows - Axisymmetric Flows. . . . . . . . . . . . . . . . . . . . 632.9.1 2D Arbitrary Flow (Cylindrical Coordinates) . . . . . . 642.9.2 3D Arbitrary Flow (Cylindrical Coordinates) . . . . . . 642.9.3 3D Axisymmetric Flows with Swirl (Cylindrical
Coordinates) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.9.4 3D Axisymmetric Flows Without Swirl
(Cylindrical Coordinates) . . . . . . . . . . . . . . . . . . . . . 672.9.5 3D Arbitrary Flow (Spherical Coordinates) . . . . . . . . 682.9.6 3D Axisymmetric Flows with Swirl
(Spherical Coordinates) . . . . . . . . . . . . . . . . . . . . . . . 692.9.7 3D Axisymmetric Flows Without Swirl
(Spherical Coordinates) . . . . . . . . . . . . . . . . . . . . . . . 692.10 2D Potential Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.11 Conformal Map Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.11.1 Conformal Mapping - Definitions and Properties . . . 732.11.2 Reference Airfoil Flow: Flow Around a Cylinder
and Kutta Condition . . . . . . . . . . . . . . . . . . . . . . . . . 742.11.3 Joukowski’s Conformal Map. . . . . . . . . . . . . . . . . . . 742.11.4 Karman-Trefftz Conformal Map . . . . . . . . . . . . . . . . 76
xii Contents
2.11.5 Van de Vooren Conformal Map . . . . . . . . . . . . . . . . 772.11.6 Matlab Source Code . . . . . . . . . . . . . . . . . . . . . . . . . 78
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3 Lifting Bodies and Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.1 Characteristics of Lifting Bodies . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.1 Fluid Force on a Body: Lift, Drag, Momentand Center of Pressure . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.2 Center of Pressure, Aerodynamic Centerand Quarter Chord Point of an Airfoil . . . . . . . . . . . 86
3.1.3 Vorticity Associated with Lifting Bodies . . . . . . . . . 893.1.4 Kutta Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.1.5 Kutta–Joukowski Relation. . . . . . . . . . . . . . . . . . . . . 91
3.2 Polar Data of an Airfoil and Related Engineering Models . . . . 933.2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.2.2 Models for Large Angle of Attacks. . . . . . . . . . . . . . 943.2.3 Dynamic Stall Models. . . . . . . . . . . . . . . . . . . . . . . . 953.2.4 Inviscid Performances . . . . . . . . . . . . . . . . . . . . . . . . 963.2.5 Model of Fully-Separated Polar
from Known Polar . . . . . . . . . . . . . . . . . . . . . . . . . . 973.3 Vorticity Based Theories of Two-Dimensional
Lifting Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.4 Vorticity Based Theories of Thick Three-Dimensional
Lifting Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.5 Inviscid Lifting-Surface Theory of a Wing. . . . . . . . . . . . . . . . 993.6 Inviscid Lifting-Line Theory of a Wing . . . . . . . . . . . . . . . . . . 100
3.6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.6.2 Lifting Line Theory - From Circulation Distribution
to Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.6.3 Prandtl’s Lifting Line
Equation - Integro-Differential Form . . . . . . . . . . . . . 1023.6.4 Elliptical Loading and Elliptical Wing
Under Lifting Line Assumptionsand Linear Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.6.5 Numerical Implementation of the Method - SampleCode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Part II Introduction to Rotors Aerodynamics
4 Rotor and Wind Turbine Formalism . . . . . . . . . . . . . . . . . . . . . . . . 1134.1 Main Assumptions and Conventions . . . . . . . . . . . . . . . . . . . . 1134.2 Wind Turbine Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.3 Loads and Dimensionless Coefficients . . . . . . . . . . . . . . . . . . . 116
Contents xiii
4.4 Velocity Induction Factors Under the Lifting LineApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.5 Solidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5 Vortex Systems and Models of a Rotor - Bound,Root and Wake Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.1 Main Components of Vorticity Involved About a Rotor. . . . . . 1215.2 Simplified Vorticity Models of Rotors . . . . . . . . . . . . . . . . . . . 123
5.2.1 Main Simplifications Used by the Models. . . . . . . . . 1235.2.2 Helical Vortex Models of a Rotor . . . . . . . . . . . . . . . 1255.2.3 Cylindrical and Tubular Vortex Model
of a Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.2.4 Vortex Ring Model of a Rotor . . . . . . . . . . . . . . . . . 130
5.3 Analytical Results for the Vortex Wake Models . . . . . . . . . . . 131References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6 Considerations and Challenges Specificto Rotor Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.1 Yaw and Tilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.2 Rotational Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.3 Airfoil Corrections for Rotating Blades . . . . . . . . . . . . . . . . . . 138References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7 Blade Element Theory (BET). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.2 Analysis of a Blade Element . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.3.1 Flow with Rotational Symmetry . . . . . . . . . . . . . . . . 1457.3.2 Particular Cases of Flows with Rotational
Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477.3.3 Introducing the Induction Factors on the Blade. . . . . 148
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8 Kutta–Joukowski (KJ) Theorem Applied to a Rotor . . . . . . . . . . . . 1518.1 Assumptions and Main Result . . . . . . . . . . . . . . . . . . . . . . . . . 1518.2 Rotor Performance Coefficients from the KJ Analyses . . . . . . . 152
8.2.1 Local Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.2.2 Global Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.3 Vortex Actuator Disk - KJ Analysis for an Infinite Numberof Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.4 Applications for Large Tip-Speed Ratios . . . . . . . . . . . . . . . . . 155
9 Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.2 Simplified Axial Momentum Theory (No Wake Rotation) . . . . 159
xiv Contents
9.2.1 Notations and Assumptions . . . . . . . . . . . . . . . . . . . . 1599.2.2 Determination of Power, Thrust
and Rotor Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 1619.2.3 Induction Factors and Rotor Performance . . . . . . . . . 1639.2.4 Discussion on the Assumptions. . . . . . . . . . . . . . . . . 165
9.3 General Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 1689.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1689.3.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.4 General Axial Momentum Theory (No Wake Rotation) . . . . . . 1749.4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1749.4.2 Results of the General Axial Momentum Theory . . . 175
9.5 Streamtube Theory (Simplified Momentum Theory). . . . . . . . . 1759.5.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1759.5.2 Derivation of the Main Streamtube
Theory Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1769.5.3 Loads from Streamtube Theory . . . . . . . . . . . . . . . . . 1779.5.4 Maximum Power Extraction
from STT - “Optimal Rotor” . . . . . . . . . . . . . . . . . . . 178References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
10 The Blade Element Momentum (BEM) Method . . . . . . . . . . . . . . . . 18110.1 The BEM Method for a Steady Uniform Inflow. . . . . . . . . . . . 182
10.1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18210.1.2 First Linkage: Velocity Triangle and Induction
Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18310.1.3 Second Linkage: Thrust and Torque from MT
and BET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18510.1.4 BEM Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18610.1.5 Summary of the BEM Algorithm . . . . . . . . . . . . . . . 188
10.2 Common Corrections to the Steady BEM Method . . . . . . . . . . 19010.2.1 Discrete Number of Blades, Tip-Losses
and Hub-Losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19010.2.2 Correction Due to Momentum
Theory Breakdown - a� Ct Relations . . . . . . . . . . . 19310.2.3 Wake Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
10.3 Unsteady BEM Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19710.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19710.3.2 Dynamic Wake/Inflow. . . . . . . . . . . . . . . . . . . . . . . . 19710.3.3 Yaw and Tilt Model . . . . . . . . . . . . . . . . . . . . . . . . . 19910.3.4 Dynamic Stall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20010.3.5 Tower and Nacelle Interference. . . . . . . . . . . . . . . . . 20110.3.6 Summary of the Unsteady BEM Algorithm . . . . . . . 202
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10.4 Typical Applications and Source Code. . . . . . . . . . . . . . . . . . . 20310.4.1 Examples of Applications . . . . . . . . . . . . . . . . . . . . . 20310.4.2 Source Code for Steady and Unsteady BEM
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Part III Classical Vortex Theory Results: Optimal Circulationand Tip-Losses
11 Far-Wake Analyses and the Rigid Helical Wake . . . . . . . . . . . . . . . 21511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21511.2 The Wake Screw Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21611.3 Relation with Rotor Parameters . . . . . . . . . . . . . . . . . . . . . . . . 21911.4 Dimensionless Circulation in Terms of Wake Parameters. . . . . 221References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
12 Betz Theory of Optimal Circulation . . . . . . . . . . . . . . . . . . . . . . . . . 22312.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22312.2 Betz Optimal Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22312.3 Inclusion of Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
13 Tip-Losses with Focus on Prandlt’s Tip Loss Factor . . . . . . . . . . . . 22713.1 Introduction to Tip-Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22713.2 Historical and Modern Tip-Loss Factors . . . . . . . . . . . . . . . . . 229
13.2.1 Historical Tip-Loss Factor. . . . . . . . . . . . . . . . . . . . . 22913.2.2 Modern Definitions of the Tip-Loss Factors . . . . . . . 230
13.3 Prandlt’s Tip-Loss Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23213.3.1 Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23213.3.2 Derivation of Prandtl’s Tip-Loss Factor . . . . . . . . . . 23313.3.3 General Expression . . . . . . . . . . . . . . . . . . . . . . . . . . 239
13.4 Different Expressions of Prandtl’s Tip-Loss Factor . . . . . . . . . 24013.5 Review of Tip-Loss Corrections. . . . . . . . . . . . . . . . . . . . . . . . 241
13.5.1 Theoretical Tip-Loss Corrections. . . . . . . . . . . . . . . . 24213.5.2 Semi-empirical Tip-Loss Corrections . . . . . . . . . . . . 24213.5.3 Semi-empirical Performance Tip-Loss
Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24213.5.4 The Historical Approach of Radius Reduction . . . . . 243
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
14 Goldstein’s Optimal Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24714.2 Goldstein’s Circulation, Factor and Tip-Loss Factor. . . . . . . . . 24814.3 Computation of Goldstein’s Factor. . . . . . . . . . . . . . . . . . . . . . 249
14.3.1 Main Methods of Evaluation. . . . . . . . . . . . . . . . . . . 249
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14.3.2 Computation Using Helical Vortex Solution:Algorithm and Source Code . . . . . . . . . . . . . . . . . . . 250
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
15 Wake Expansion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25515.1 Simple 1D Momentum Theory/Vortex Cylinder Model . . . . . . 25515.2 Cylinder Analog Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 25515.3 Theodorsen’s Wake Expansion. . . . . . . . . . . . . . . . . . . . . . . . . 25615.4 Far-Wake Expansion Models . . . . . . . . . . . . . . . . . . . . . . . . . . 25715.5 Comparison of Wake Expansions. . . . . . . . . . . . . . . . . . . . . . . 258References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
16 Relation Between Far-Wake and Near-Wake Parameters . . . . . . . . 25916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25916.2 Extension of the Work of Okulov and Sørensen
for Non-optimal Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26016.3 Extension of Theodorsen’s Theory . . . . . . . . . . . . . . . . . . . . . . 261References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Part IV Latest Developments in Vorticity-Based RotorAerodynamics
17 Cylindrical Vortex Model of a Rotor of Finite or InfiniteTip-Speed Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26517.1 Introduction and Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26517.2 Model and Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26717.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
18 Cylindrical Model of a Rotor with VaryingCirculation - Effect of Wake Rotation . . . . . . . . . . . . . . . . . . . . . . . . 27318.1 Context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27418.2 Model and Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27418.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
19 An Improved BEM Algorithm Accounting for Wake RotationEffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28319.1 Context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28319.2 Actuator Disk Models for the BEM-Like Method . . . . . . . . . . 284
19.2.1 Comparisons of Stream-Tube Theoryand Vortex Cylinder Results . . . . . . . . . . . . . . . . . . . 285
19.3 BEM Algorithm Including Wake Rotation . . . . . . . . . . . . . . . . 28619.3.1 General Structure of a Lifting-Line-Based
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28619.3.2 Step 6: Inductions for the Standard BEM
(STT-KJ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
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19.3.3 Step 6: Inductions for the Improved BEMof Madsen et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
19.3.4 Step 6: Inductions for the Actuator Disk Model(AD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
19.3.5 Step 6: Inductions for the Vortex Cylinder Model(VCT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
19.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28919.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
20 Helical Model for Tip-Losses: Development of a Novel Tip-LossFactor and Analysis of the Effect of Wake Expansion . . . . . . . . . . . 29320.1 Description of the Helical Wake Models . . . . . . . . . . . . . . . . . 29320.2 A Novel Tip-Loss Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29420.3 Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29520.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
21 Yaw-Modelling Using a Skewed Vortex Cylinder. . . . . . . . . . . . . . . 29921.1 Introduction and Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29921.2 Model and Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30121.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
22 Simple Implementation of a New Yaw-Model. . . . . . . . . . . . . . . . . . 30722.1 Context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30722.2 Model and Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30822.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
23 Advanced Implementation of the New Yaw-Model . . . . . . . . . . . . . 31523.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31523.2 Models for the Velocity Field Outside of the Skewed
Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31623.3 Helical Pitch for the Superposition of Skewed Cylinders . . . . . 31723.4 Yaw-Model Implementation Using a Superposition
of Skewed Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31823.5 Partial Approach - Focus on the Inboard Part of the Blade . . . 31923.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
24 Velocity Field Upstream of Aligned and Yawed Rotors:Wind Turbine and Wind Farm Induction Zone . . . . . . . . . . . . . . . . 32124.1 Context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32124.2 Model for the Velocity Field in the Induction Zone . . . . . . . . . 32224.3 Results for a Single Wind Turbine . . . . . . . . . . . . . . . . . . . . . . 323
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24.3.1 Aligned Case Without Swirl . . . . . . . . . . . . . . . . . . . 32424.3.2 Aligned Case with Swirl . . . . . . . . . . . . . . . . . . . . . . 32524.3.3 Yawed Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32624.3.4 Computational Time . . . . . . . . . . . . . . . . . . . . . . . . . 328
24.4 Results for a Wind Farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32824.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32824.4.2 Velocity Deficit Upstream of a Wind Farm. . . . . . . . 329
24.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
25 Analytical Model of a Wind Turbine in Sheared Inflow . . . . . . . . . 33325.1 Context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33325.2 Model and Key-Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33425.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
26 Model of a Wind Turbine with Unsteady Circulationor Unsteady Inflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33926.1 Context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33926.2 Model and Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34026.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Part V Latest Applications of Vortex Methods to RotorAerodynamics and Aeroelasticity
27 Examples of Applications of Vortex Methodsto Wind Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34727.1 Comparison with BEM and Actuator-Line Simulations . . . . . . 34727.2 Wakes and Flow Field for Uniform Inflows . . . . . . . . . . . . . . . 34927.3 Effect of Viscosity - Comparison with AD. . . . . . . . . . . . . . . . 34927.4 Effect of Turbulence - Comparison with Lidar and AD . . . . . . 35027.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
28 Representation of a (Turbulent) Velocity Field Using VortexParticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35528.1 Simple Velocity Reconstruction Using Vortex Particles . . . . . . 35528.2 Associated Errors and Discussions . . . . . . . . . . . . . . . . . . . . . . 35628.3 Example of Velocity Reconstruction for a Turbulent Field. . . . 35828.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
29 Effect of a Wind Turbine on the Turbulent Inflow . . . . . . . . . . . . . 36129.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36129.2 Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
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29.3 Model and Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36429.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
30 Aeroelastic Simulation of a Wind Turbine Under Turbulentand Sheared Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37130.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37130.2 Representation of Shear in Vortex Methods . . . . . . . . . . . . . . . 37230.3 Full Aeroelastic Simulation Including Shear
and Turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37330.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
Part VI Analytical Solutions for Vortex Methods and RotorAerodynamics
31 Elementary Three-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . . . 38131.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38131.2 Flow Induced by a Point-Wise Distribution . . . . . . . . . . . . . . . 382
31.2.1 Point Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38231.2.2 Vortex Point (Vortex Particle/Blobs). . . . . . . . . . . . . 384
31.3 Vortex Filaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38731.3.1 Vortex Segment and Line of Constant Strength . . . . 38731.3.2 Vortex Segment of Linearly Varying Strength . . . . . 390
31.4 Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39131.4.1 Dipole - Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39131.4.2 Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39231.4.3 Constant Panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39231.4.4 Equivalences Between Elements . . . . . . . . . . . . . . . . 392
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
32 Elementary Two-Dimensional Potential Flows . . . . . . . . . . . . . . . . . 39332.1 Uniform Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39332.2 Point Source, Point Vortex and Distributions of Points . . . . . . 393
32.2.1 Point Source/Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . 39332.2.2 Point Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39432.2.3 Periodic Point Vortices . . . . . . . . . . . . . . . . . . . . . . . 39532.2.4 Continuous Distribution of 2D Points . . . . . . . . . . . . 395
32.3 Doublet and Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39632.3.1 Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39632.3.2 Multi-poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
32.4 Cylinder/Ellipse Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39732.4.1 Cylinder Flow - Acyclic - No Lift . . . . . . . . . . . . . . 39732.4.2 Flow Around a 2D Ellipse - No Lift . . . . . . . . . . . . . 39832.4.3 Cylinder Flow - Cyclic - with Lift . . . . . . . . . . . . . . 398
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32.4.4 Flow About Quadrics . . . . . . . . . . . . . . . . . . . . . . . . 39932.5 Miscellaneous Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
32.5.1 Rigid Rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39932.5.2 Corner Flow, Flat Plate and Stagnation Point . . . . . . 40032.5.3 Cylinder and Vortex Point . . . . . . . . . . . . . . . . . . . . 400
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
33 Flows with a Spread Distribution of Vorticity . . . . . . . . . . . . . . . . . 40133.1 Axisymmetric Vorticity Patches . . . . . . . . . . . . . . . . . . . . . . . . 401
33.1.1 Examples of Vorticity Patches . . . . . . . . . . . . . . . . . 40133.1.2 Canonical Example: The Inviscid Vorticity Patch . . . 402
33.2 Rectangular Vorticity Patch (2D Brick) . . . . . . . . . . . . . . . . . . 405References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
34 Spherical Geometry Models: Flow About a Sphereand Hill’s Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40734.1 Sphere with Free Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40734.2 Hill’s Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41134.3 Ellipsoid and Spheroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
35 Vortex and Source Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41935.1 Vortex Rings - General Considerations . . . . . . . . . . . . . . . . . . 41935.2 Formulae for the Potential, Velocity and Gradient . . . . . . . . . . 42035.3 Flow at Particular Locations. . . . . . . . . . . . . . . . . . . . . . . . . . . 42135.4 Derivation of the Velocity and Vector Potential . . . . . . . . . . . . 42435.5 Further Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42835.6 Source Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
36 Flow Induced by a Right Vortex Cylinder . . . . . . . . . . . . . . . . . . . . 42936.1 Right Cylinder of Tangential Vorticity with Arbitrary Cross
Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43036.1.1 Finite Cylinder - General Velocity Field . . . . . . . . . . 43036.1.2 Finite Cylinder - Velocity in Terms
of Solid Angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43036.1.3 Infinite and Semi-infinite Cylinders of Arbitrary
Cross Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43236.1.4 Finite Cylinder of Tangential Vorticity
and Link to Source Surfaces . . . . . . . . . . . . . . . . . . . 43336.2 Right Vortex Cylinder of Tangential Vorticity - Circular
Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43536.2.1 Finite Vortex Cylinder of Tangential Vorticity . . . . . 43636.2.2 Semi-infinite Vortex Cylinder of Tangential
Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
Contents xxi
36.3 Vortex Cylinder of Longitudinal Vorticity . . . . . . . . . . . . . . . . 45036.3.1 Infinite Cylinder of Longitudinal Vorticity . . . . . . . . 45036.3.2 Finite Cylinder of Longitudinal Vorticity . . . . . . . . . 45136.3.3 Semi-infinite Cylinder of Longitudinal Vorticity . . . . 451
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
37 Flow Induced by a Vortex Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45537.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45537.2 Indefinite Form of the Biot–Savart Law . . . . . . . . . . . . . . . . . . 45637.3 Definite Form of the Biot–Savart Law . . . . . . . . . . . . . . . . . . . 45837.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
38 Flow Induced by a Skewed Vortex Cylinder . . . . . . . . . . . . . . . . . . 46138.1 Semi-infinite Skewed Cylinder of Tangential Vorticity . . . . . . 461
38.1.1 Preliminary Note on the Integrals Involved. . . . . . . . 46238.1.2 Extension of the Work of Castles and Durham . . . . . 46338.1.3 Longitudinal Axis - Work of Coleman et al. . . . . . . . 46438.1.4 Matlab Source Code . . . . . . . . . . . . . . . . . . . . . . . . . 466
38.2 Semi-infinite Skewed Cylinder with Longitudinal Vorticity . . . 46738.3 Infinite Skewed Cylinder with Longitudinal Vorticity
(Elliptic Cylinder). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
39 Flow Induced by Helical Vortex Filaments . . . . . . . . . . . . . . . . . . . . 47339.1 Preliminary Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
39.1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47339.1.2 Semi-infinite Helix and Rotor Terminology . . . . . . . 474
39.2 Exact Expressions for Infinite Helical Vortex Filaments. . . . . . 47539.3 Approximate Expressions for Infinite Helical Filaments . . . . . . 47539.4 Expressions for Semi-infinite Helices Evaluated
on the Lifting Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47639.5 Notations Introduced for Approximate Formulae . . . . . . . . . . . 47639.6 Summation of Several Helices - Link Between Okulov’s
Relation and Wrench’s Relation . . . . . . . . . . . . . . . . . . . . . . . . 478References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
Part VII Vortex Methods
40 A Brief Introduction to Vortex Methods . . . . . . . . . . . . . . . . . . . . . . 48340.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48340.2 Pros and Cons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48440.3 An Example of Vortex Method History . . . . . . . . . . . . . . . . . . 486
xxii Contents
40.4 Classification of Vortex Methods . . . . . . . . . . . . . . . . . . . . . . . 48740.5 Existing Vortex Codes and Application to Wind Energy . . . . . 489References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
41 The Different Aspects of Vortex Methods . . . . . . . . . . . . . . . . . . . . . 49341.1 Fundamental Equations and Concepts . . . . . . . . . . . . . . . . . . . 49341.2 Discretization and Initialization . . . . . . . . . . . . . . . . . . . . . . . . 495
41.2.1 Information Carried by the Vortex Elements . . . . . . . 49541.2.2 Initialization and Reinitialization . . . . . . . . . . . . . . . . 49741.2.3 Initialization - Inviscid Vortex Patch Example . . . . . 498
41.3 Viscous-Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49941.3.1 Viscous-Splitting Algorithm . . . . . . . . . . . . . . . . . . . 49941.3.2 Rate of Convergence of the Viscous-Splitting
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50041.3.3 Application to the Vorticity Transport Equation . . . . 501
41.4 Convection and Stretching of Vortex Elements . . . . . . . . . . . . 50141.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50141.4.2 Convection of Vortex Elements . . . . . . . . . . . . . . . . 50241.4.3 Stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50341.4.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
41.5 Grid-Free and Grid-Based Methods . . . . . . . . . . . . . . . . . . . . . 50441.5.1 Grid-Free Vortex Methods . . . . . . . . . . . . . . . . . . . . 50441.5.2 Grid-Based Vortex Methods
(Mixed Eulerian–Lagrangian Formulation) . . . . . . . . 50541.5.3 Coupled Lagrangian and Eulerian Solvers. . . . . . . . . 506
41.6 Viscous Diffusion - Solution of the Diffusion Equation . . . . . . 50641.6.1 Diffusion Equation and Vorticity Transport
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50641.6.2 Fundamental Solution and Lamb–Oseen Vortex . . . . 50741.6.3 Core-Spreading Method . . . . . . . . . . . . . . . . . . . . . . 50941.6.4 Random-Walk Method . . . . . . . . . . . . . . . . . . . . . . . 51041.6.5 Grid-Based Finite-Differences Method . . . . . . . . . . . 51141.6.6 Particle-Strength-Exchange (PSE) . . . . . . . . . . . . . . . 51141.6.7 Numerical Application: Lamb–Oseen Vortex . . . . . . 51341.6.8 Vorticity Redistribution Method . . . . . . . . . . . . . . . . 514
41.7 Boundaries, Boundary Conditions and Lifting-Bodies . . . . . . . 51441.7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51441.7.2 Fluid Boundary Conditions: Free-Flow
and Periodic Boundaries . . . . . . . . . . . . . . . . . . . . . . 51541.7.3 Solid Boundaries in Inviscid Flows. . . . . . . . . . . . . . 51541.7.4 Solid Boundaries in Viscous Flows - Vorticity
Generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
Contents xxiii
41.7.5 Viscous Boundaries Using Coupling(Viscous-Inviscid or Lagrangian–Eulerian) . . . . . . . . 517
41.7.6 Lifting-Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51741.8 Regularization - Kernel Smoothing - Mollification . . . . . . . . . . 517
41.8.1 Kernel Smoothing via Convolutionwith a Cut-Off Function . . . . . . . . . . . . . . . . . . . . . . 519
41.8.2 Requirements on the Cut-Off Function . . . . . . . . . . . 51941.8.3 Special Case of Spherical Symmetry. . . . . . . . . . . . . 52141.8.4 Examples Used in Particle Methods . . . . . . . . . . . . . 52441.8.5 Regularization Models for Vortex Filaments . . . . . . . 52641.8.6 Choice of Cut-Off/Smooth Parameter . . . . . . . . . . . . 52741.8.7 Application to the Inviscid Vortex Patch. . . . . . . . . . 529
41.9 Spatial Adaptation - Redistribution - Rezoning -Reinitialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53041.9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53041.9.2 Remeshing - Rezoning - Redistribution -
Reinitialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53041.9.3 Gain from Remeshing - Application
to Inviscid-Vortex Patch . . . . . . . . . . . . . . . . . . . . . . 53141.9.4 Problems Introduced by Remeshing . . . . . . . . . . . . . 531
41.10 Subgrid-Scale Models - LES - Turbulence . . . . . . . . . . . . . . . . 53241.11 Accuracy of Vortex Methods, Guidelines, Diagnostics
and Possible Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . 53341.11.1 Guidelines and Diagnostics for General Vortex
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53341.11.2 Boundary Elements - Guidelines and Diagnostics . . . 53541.11.3 Particle Methods - Convergence . . . . . . . . . . . . . . . . 53641.11.4 Application to the Inviscid Vortex Patch. . . . . . . . . . 537
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
42 Particularities of Vortex Particle Methods . . . . . . . . . . . . . . . . . . . . 54542.1 Particle Approximation and Lagrangian Methods . . . . . . . . . . . 545
42.1.1 Notion of Vortex Blob . . . . . . . . . . . . . . . . . . . . . . . 54542.1.2 Particle Approximation . . . . . . . . . . . . . . . . . . . . . . . 54542.1.3 Dynamics of Lagrangian Methods. . . . . . . . . . . . . . . 54642.1.4 Incompressible Vortex Particle Methods . . . . . . . . . . 547
42.2 Stretching Term - Different Schemes . . . . . . . . . . . . . . . . . . . . 54842.3 Divergence of the Vorticity Field . . . . . . . . . . . . . . . . . . . . . . . 549
42.3.1 Minimizing the Error Growth . . . . . . . . . . . . . . . . . . 54942.3.2 Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55042.3.3 Criteria for Correction . . . . . . . . . . . . . . . . . . . . . . . . 550
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
xxiv Contents
43 Numerical Implementation of Vortex Methods . . . . . . . . . . . . . . . . . 55343.1 Interpolation Method Required for Grid-Based Methods . . . . . 553
43.1.1 Interpolation in Vortex Methods . . . . . . . . . . . . . . . . 55343.1.2 Concept of Interpolation . . . . . . . . . . . . . . . . . . . . . . 55443.1.3 Interpolation to Grid (Projection, Griding,
Assignment, Particle-to-Mesh). . . . . . . . . . . . . . . . . . 55643.1.4 Interpolation from Grid (Mesh-to-Particle) . . . . . . . . 557
43.2 Tree-Codes and Fast Multipole Method . . . . . . . . . . . . . . . . . . 55843.2.1 Tree-Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . 55843.2.2 Tree-Based Method - Coefficients up to Order 2. . . . 560
43.3 Poisson Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56143.4 Numerical Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . . 562
43.4.1 Expression of the Different Schemes. . . . . . . . . . . . . 56243.4.2 Example of Application to the Inviscid Patch . . . . . . 56343.4.3 Work Presented by Leishman . . . . . . . . . . . . . . . . . . 564
43.5 Vorticity Splitting and Merging Schemes . . . . . . . . . . . . . . . . . 56443.6 Conversion from Segments to Particles . . . . . . . . . . . . . . . . . . 566
43.6.1 Canonical Examples for Validation . . . . . . . . . . . . . . 56643.6.2 Representation of One Segment by One Particle . . . . 56743.6.3 Representation Using Several Particles . . . . . . . . . . . 56743.6.4 Trailed and Shed Vorticity Behind a Wing . . . . . . . . 568
43.7 Distribution of Control Points . . . . . . . . . . . . . . . . . . . . . . . . . 56843.7.1 The Work of James - Chordwise Distribution . . . . . . 56843.7.2 Cosine Spacing and Other References
in the Topic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56943.8 The 3/4 Chord Collocation Point . . . . . . . . . . . . . . . . . . . . . . . 570References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
44 OmniVor: An Example of Vortex Code Implementation . . . . . . . . . 57544.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57544.2 Implementation and Features . . . . . . . . . . . . . . . . . . . . . . . . . . 57644.3 Specific Configurations Used in Publications . . . . . . . . . . . . . . 584References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
45 Vortex Code Validation and Illustration . . . . . . . . . . . . . . . . . . . . . . 58745.1 Simple Validation of the Vortex Particle Method. . . . . . . . . . . 58745.2 Lifting Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58845.3 Lifting Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58945.4 Thick Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59045.5 Unit-Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59145.6 Further Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
Contents xxv
Appendix A: Complements on the Right Cylindrical Modeland the Effect of Wake Rotation . . . . . . . . . . . . . . . . . . . . . 595
Appendix B: From Poisson’s Equation to the Biot–Savart Lawin an Unbounded Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 607
Appendix C: Useful Mathematical Relations . . . . . . . . . . . . . . . . . . . . . . 617
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
xxvi Contents
Acronyms
a Axial induction factoraB Axial induction factor local to the bladea Axial induction factor from 2D MTa0 Tangential induction factorc Chordcn Normal aerodynamic coefficientct Tangential aerodynamic coefficiente Internal energyet Total energyh Enthalpyht Total enthalpyh Typical grid spacing in vortex methodsh Helix pitchhB Apparent pitch h=Bh Normalized pitch h=Rk Dimensionless circulationk2 Elliptical parameter for elliptic integralskt Turbulent kinetic energyl Helix torsional parameterl Normalized torsional parameter l=Rm Elliptical parameter for elliptic integralsnrot Rotational speed in RPM: X=ð2pÞp Static pressurept Total pressure pþ 1
2 qu2
p Frequency associated with X, p ¼ X=2pq Heat fluxr Radial positionrc Viscous core radiust Timet0 Parameter in the core-spreading model
xxvii
r Dimensionless radial position r=R~r Dimensionless radial position r=Rs Signuh Tangential induced velocityuz Axial induced velocityu x-component of velocityv y-component of velocityw z-component of velocityw Wake relative longitudinal velocity (Betz)z0 Surface roughness lengthA Angular ImpulseA AreaAR See AbbreviationsB Number of bladesCC Dimensionless circulationCd Drag coefficientsCl Lift coefficientsCl;a Lift coefficient slope for small anglesCp Power coefficientCq Local torque coefficientCQ Total torque coefficientCt Tangential aerodynamic coefficientCt Local thrust coefficientCT Total thrust coefficientD Drag forceD Rotor diameterD Deformation matrixE Complete elliptic integral of the 2nd kindE EnergyE EnstrophyF Tip-loss factorFa Tip-loss factor based on axial inductionFC Tip-loss factor based on circulationFCl Performance tip-loss factorFGo Goldstein’s tip-loss factorFGl Glauert’s tip-loss factorFPr Prandtl’s tip-loss factorFSh Shen’s tip-loss factorF Complex velocity potential in 2DG� Green’s function associated with the operator �H Heaviside functionH Bernoulli constant, e.g., pþ 1
2 qu2
It Turbulence intensityI Linear Impulse
xxviii Acronyms
J HelicityK� Kernel (associated with a given operator �)K Complete elliptic integral of the 1st kindL Lift forceMa Mach numberP PowerP PalinstrophyQ Rotor torqueQ Vortical HelicityR Rotor radiusRe Reynolds numberS EntropyS SurfaceS Energy density spectrumSd Volume of the unit sphere in R
d
T Thrust forceT TemperatureU Longitudinal velocity at the rotor in 1DU Relative velocity at the rotorU0 Longitudinal velocity far upstreamUi Induced velocity in 1DUn Velocity normal to the rotorUref Reference velocity used, e.g., for the normalization of loadsUt Velocity tangent to the rotorV Velocity vectorVrel Relative velocityV VolumeW Induced velocity vector at the rotora Point/Blob vorticity intensitya Angle of attacka0 Angle of attack at zero liftb Twist anglec Surface vorticity - Distributed circulationct Vortex cylinder tangential vorticitycl Vortex cylinder longitudinal vorticitycb Bound vorticityd Dirac functione Pitch angle of the wake helix screwe Regularization parameterf Regularization/cutoff/smoothing functiong Efficiencyh Azimuthal coordinatej Goldstein’s factork Tip speed ratio ¼ XR=U0
Acronyms xxix
kr Local speed ratio ¼ kr=Rk First Lamé’s coefficient for Newtonian fluidl Second Lamé’s coefficient: dynamic viscositym Kinematic viscosity ¼ l=qq Air density � 1.225 kg/m3
r Local blade solidity ¼ Bc=2prr Cauchy stress tensors Shear stress, viscous stress tensor/ Flow anglev Wake skew angle, in yaw conditionsw Azimuthal coordinatew Vector potentialx Rotational speed of the wakex VorticityC CirculationD Laplacian operator r2
H DilatationP Gate functionP Complete elliptic integral of the 3rd kindU Velocity PotentialW Stream function (2D)W Stokes’ stream function (3D)X Rotational speed of the rotorX Rotation matrix (fluid kinematics)X Solid angleX Volume of the domainX Total vorticity@X Surface boundary of volume XtX TransposeXT Transposer Del operator, “nabla”div Divergence, divX ¼ r � X
divT ¼ @jðTijÞeigrad Gradient, grad X ¼ rXgrad Gradient of first-order tensorcurl Rotational, curl X ¼ r� Xe.g. exempli gratia: “for example”i.e. id est: “that is”viz. videlicet: “namely”w.r.t. “with respect to”1D One dimension2D Two dimensions3D Three dimensionsAC Aerodynamic center
xxx Acronyms
AD Actuator DiskAEP Annual Energy OutputAED Aeroelastic Design (section at DTU)AL Actuator LineAR Aspect ratio of a wing (b2=S)BEM Blade Element MomentumBET Blade Element TheoryBT Blade Element Theory (subscript)CFD Computational Fluid DynamicsCP Control PointCP Center of PressureCPU Central Processing UnitsCV Control volumeDOF Degree of FreedomDTU Technical University of DenmarkECN Energy Center of the NetherlandsGPU Graphical Processing UnitsHSS High-Speed ShaftIEC International Electrotechnical CommissionKJ Kutta–JoukowskiLE Leading edgeLES Large Eddy SimulationLHS Left-Hand SideLSS Low-Speed ShaftMT Momentum TheoryNTUA National Technical University of AthensPSE Particle Strength ExchangeVC Vortex CodeVC Vortex Cylinder (depending on context)VL Vortex LatticeRHS Right-Hand SideSGS Sub-grid scale modelST Streamtube Theory (also written STT)STT Streamtube Theory (also written ST)TE Trailing edgeTKE Turbulent Kinetic EnergyWD Wind DirectionWS Wind SpeedWT Wind Turbine
Acronyms xxxi
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