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.

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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

Contents xv

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