Research Topics in Wind Energy

Transcription

1 Research Topics in Wind Energy Volume 7 Series editors Joachim Peinke, University of Oldenburg, Oldenburg, Germany peinke@uni-oldenburg.de Gerard van Bussel, Delft University of Technology, Delft, The Netherlands g.j.w.vanbussel@tudelft.nl

2 About this Series The series Research Topics in Wind Energy publishes new developments and advances in the fields of Wind Energy Research and Technology, rapidly and informally but with a high quality. Wind Energy is a new emerging research field characterized by a high degree of interdisciplinarity. The intent is to cover all the technical 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 Wind Forecasts, Wind Turbine Technology, System Integration and Energy Economics, as well as the methodologies behind them. Within the scope of the series are monographs, lecture notes, selected contributions from specialized conferences and workshops, as well as selected PhD theses. Of particular value to both the contributors and the readership are the short publication timeframe and the worldwide distribution, which enable both wide and rapid dissemination of research output. The series is promoted under the auspices of the European Academy of Wind Energy. More information about this series at

3 Emmanuel Branlard Wind Turbine Aerodynamics and Vorticity-Based Methods Fundamentals and Recent Applications 123

4 Emmanuel Branlard Department of Wind Energy, Aeroelastic Design Technical University of Denmark Roskilde Denmark ISSN ISSN (electronic) Research Topics in Wind Energy ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of 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 transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the 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 this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

5 To love, 2K 2K

6 Preface The standard approach in the study of wind turbine aerodynamics consists in using momentum analyses. The momentum theory of an actuator disk is an example of momentum analysis. Blade element momentum (BEM) and the conventional computation fluid dynamics (CFD) are two numerical methods also based on momentum analyses. Velocity and pressure are the main variables used in momentum analysis. The equations can also be formulated using vorticity as main variable. This leads to an alternative approach referred to as vorticity-based methods. The great potential of vorticity-based methods comes from the multitude of formulations they offer, ranging from simple analytical models to advanced numerical methods. The analytical model will be referred to as vortex theories and the numerical methods as vortex methods. The term vorticity often intimidates the newcomer, but this fear vanishes when one realizes that velocity and vorticity offer two different, but often equivalent, points of view. For instance, the momentum theory of an actuator disk with constant loading can be equivalently studied by considering the tubular vorticity sheet that is present at the surface of the streamtube. Vorticity plays an important role in wind turbine aerodynamics since strong vortices are present in the wakes in particular. Vorticity and vorticity-based methods cannot be omitted in a book on the topic. Most of the analytical models used in BEM methods are derived from analytical vortex models. Further, numerical vortex methods are now competing with conventional CFD methods in terms of accuracy and computational time, and they are 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 the study of wind turbine aerodynamics and to present historical and recent developments 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 aerodynamics and/or about vorticity-based methods. The book introduces the fundamentals of fluid mechanics, momentum theories, vortex theories, and vortex methods necessary for the study of rotors and wind turbines in particular. Rotor theories are presented in a great level of details at the beginning of the book. These theories include the blade element theory, the Kutta Joukowski theory, the vii

7 viii Preface momentum theory, and the BEM method. Different momentum theories are derived from first principles using a critical approach. The remaining of the book focuses on vortex theory and vortex methods with application to wind turbine aerodynamics. Examples of vortex theory applications that are discussed in this book are optimal rotor design, tip-loss corrections, yaw models, and dynamic inflow models. Historical derivations and recent extensions of the models are presented. The cylindrical vortex model is another example of a simple analytical vortex model used in this book. In this model, a wind turbine and its wake are simplified using a vortex system of cylindrical shape. Formulations equivalent to the ones used in a BEM algorithm are obtained. The model provides a wake-rotation correction which greatly improves the accuracy of BEM algorithms. The cylindrical model is also used 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 are obtained in a couple of seconds with an impressive accuracy compared to numerical results from CFD methods which would require days of computation. Different applications of numerical vortex methods are presented in this book. Numerical methods are used for instance to investigate the influence of a wind turbine on the incoming turbulence. Sheared inflows are also investigated. It is shown in particular that most vortex methods omit a term resulting in excessive upward displacement of 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 to devise simple rotor models or to validate the implementation of numerical methods. Several MATLAB programs are provided to ease some of the most complex implementations: BEM codes, vortex cylinder velocity functions, Goldstein s circulation, lifting-line codes, Karman Trefftz conformal map, projection functions for vortex particle methods, etc. Part I introduces the fluid mechanics foundations relevant to this book. Part II introduces rotor aerodynamics, including momentum analyses, vortex models, and the BEM method. Part III focuses on classical vortex theory results which originated from the study of rotors with optimal circulation. Part IV presents the recent developments in rotor aerodynamics based on analytical vortex flows. Part V presents recent applications of vortex methods. Part VI provides detailed analytical solutions that are relevant for rotor aerodynamics, either for the derivation of vortex models or for the implementation and validation of vortex methods. Part VII is dedicated to vortex methods. Part VIII provides mathematical complements to some chapters of the book. Roskilde, Denmark January 2017 Emmanuel Branlard

8 Acknowledgements The current work would not have been possible without the support and help of my PhD supervisor Mac Gaunaa and the contributions from Spyros Voutsinas, Ewan Machefaux, Philippe Mercier, Gregoire Winckelmans, Niels Troldborg, Giorgios Papadakis, and Henrik Brandenborg Sørensen. I would like to thank my colleagues for their inspiration and fruitful discussions: Jakob Mann, Niels Sørensen, Curran Crawford, Philippe Chatelain, Torben Larsen, Anders Hansen, Georg Pirrung, Frederik Zahle, Mads Hejlesen, Juan Pablo Murcia, Alexander Forsting, Christian Pavese, Michael McWilliams, Lucas Pascal, and Jacobus De Vaal. I am grateful to the persons who accepted to review some chapters of this book despite a limited time: Damien Castaignet, Michael McWilliams, Mac Gaunaa, Jens Gengenbach, Gil-Arnaud Coche, Julien B., and Björn Schmidt. Above all, I am glad for the moments of life and love I experienced thanks to my family 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, and Mom. ix

9 Contents 1 Introduction... 1 References... 6 Part I Fluid Mechanics Foundations 2 Theoretical Foundations for Flows Involving Vorticity Fluid Mechanics Equations in Inertial and Non-inertial Frames Physical Quantities Conservation Laws Fluid-Mechanic Equations in a Non-inertial Frame Fluid Mechanics Assumptions Usual Cases - Equations of Euler and Bernoulli Flow Kinematics and Vorticity Flow Kinematics Vorticity and Related Definitions Helmholtz (First) Law Helmholtz-(Hodge) Decomposition Bounded and Unbounded Domain - Surface Map - Generalized Helmholtz Decomposition Main Dynamics Equations Involving Vorticity Circulation Equation Vorticity Equation Stretching and Dilatation of Vorticity Alternative Forms of the Vorticity Equation Vorticity Equation in Particular Cases Pressure Vortex Force, Image/Generalized/Bound Vorticity, Kutta Joukowski Relation xi

10 xii Contents 2.4 Different Dimensions of Vorticity: Surface, Line and Points Vorticity Moments, Variables and Invariants - Incompressible Flows Main Theorems Involving Vorticity Kelvin s Theorem Lagrange s Theorem Helmholtz Theorem Biot Savart Law Vortices in Viscous and Inviscid Fluid - Results and Classical Flows Vortex in Inviscid Fluid Vortex in Viscous Fluid - Standard Solutions Life of a Vortex - Vortex Decay, Collapse and Stability Surface Representations - Vortex Sheets Introduction Vortex Sheets Kinematics Vortex Sheets Dynamics Vortex Sheet Convection and Stability Vortex Surfaces in 2D Incompressible Flow Equations in Polar Coordinates - 2D and 3D Flows - Axisymmetric Flows D Arbitrary Flow (Cylindrical Coordinates) D Arbitrary Flow (Cylindrical Coordinates) D Axisymmetric Flows with Swirl (Cylindrical Coordinates) D Axisymmetric Flows Without Swirl (Cylindrical Coordinates) D Arbitrary Flow (Spherical Coordinates) D Axisymmetric Flows with Swirl (Spherical Coordinates) D Axisymmetric Flows Without Swirl (Spherical Coordinates) D Potential Flows Conformal Map Solutions Conformal Mapping - Definitions and Properties Reference Airfoil Flow: Flow Around a Cylinder and Kutta Condition Joukowski s Conformal Map Karman-Trefftz Conformal Map... 76

11 Contents xiii Van de Vooren Conformal Map Matlab Source Code References Lifting Bodies and Circulation Characteristics of Lifting Bodies Fluid Force on a Body: Lift, Drag, Moment and Center of Pressure Center of Pressure, Aerodynamic Center and Quarter Chord Point of an Airfoil Vorticity Associated with Lifting Bodies Kutta Condition Kutta Joukowski Relation Polar Data of an Airfoil and Related Engineering Models Introduction Models for Large Angle of Attacks Dynamic Stall Models Inviscid Performances Model of Fully-Separated Polar from Known Polar Vorticity Based Theories of Two-Dimensional Lifting Bodies Vorticity Based Theories of Thick Three-Dimensional Lifting Bodies Inviscid Lifting-Surface Theory of a Wing Inviscid Lifting-Line Theory of a Wing Introduction Lifting Line Theory - From Circulation Distribution to Loads Prandtl s Lifting Line Equation - Integro-Differential Form Elliptical Loading and Elliptical Wing Under Lifting Line Assumptions and Linear Theory Numerical Implementation of the Method - Sample Code References Part II Introduction to Rotors Aerodynamics 4 Rotor and Wind Turbine Formalism Main Assumptions and Conventions Wind Turbine Formalism Loads and Dimensionless Coefficients

12 xiv Contents 4.4 Velocity Induction Factors Under the Lifting Line Approximation Solidity References Vortex Systems and Models of a Rotor - Bound, Root and Wake Vorticity Main Components of Vorticity Involved About a Rotor Simplified Vorticity Models of Rotors Main Simplifications Used by the Models Helical Vortex Models of a Rotor Cylindrical and Tubular Vortex Model of a Rotor Vortex Ring Model of a Rotor Analytical Results for the Vortex Wake Models References Considerations and Challenges Specific to Rotor Aerodynamics Yaw and Tilt Rotational Effects Airfoil Corrections for Rotating Blades References Blade Element Theory (BET) Introduction Analysis of a Blade Element Applications Flow with Rotational Symmetry Particular Cases of Flows with Rotational Symmetry Introducing the Induction Factors on the Blade References Kutta Joukowski (KJ) Theorem Applied to a Rotor Assumptions and Main Result Rotor Performance Coefficients from the KJ Analyses Local Coefficients Global Coefficients Vortex Actuator Disk - KJ Analysis for an Infinite Number of Blades Applications for Large Tip-Speed Ratios Momentum Theory Introduction Simplified Axial Momentum Theory (No Wake Rotation)

13 Contents xv Notations and Assumptions Determination of Power, Thrust and Rotor Velocity Induction Factors and Rotor Performance Discussion on the Assumptions General Momentum Theory Introduction Derivation General Axial Momentum Theory (No Wake Rotation) Assumptions Results of the General Axial Momentum Theory Streamtube Theory (Simplified Momentum Theory) Assumptions Derivation of the Main Streamtube Theory Results Loads from Streamtube Theory Maximum Power Extraction from STT - Optimal Rotor References The Blade Element Momentum (BEM) Method The BEM Method for a Steady Uniform Inflow Introduction First Linkage: Velocity Triangle and Induction Factors Second Linkage: Thrust and Torque from MT and BET BEM Equations Summary of the BEM Algorithm Common Corrections to the Steady BEM Method Discrete Number of Blades, Tip-Losses and Hub-Losses Correction Due to Momentum Theory Breakdown - a C t Relations Wake Rotation Unsteady BEM Method Introduction Dynamic Wake/Inflow Yaw and Tilt Model Dynamic Stall Tower and Nacelle Interference Summary of the Unsteady BEM Algorithm

14 xvi Contents 10.4 Typical Applications and Source Code Examples of Applications Source Code for Steady and Unsteady BEM Methods References Part III Classical Vortex Theory Results: Optimal Circulation and Tip-Losses 11 Far-Wake Analyses and the Rigid Helical Wake Introduction The Wake Screw Model Relation with Rotor Parameters Dimensionless Circulation in Terms of Wake Parameters References Betz Theory of Optimal Circulation Introduction Betz Optimal Circulation Inclusion of Drag References Tip-Losses with Focus on Prandlt s Tip Loss Factor Introduction to Tip-Losses Historical and Modern Tip-Loss Factors Historical Tip-Loss Factor Modern Definitions of the Tip-Loss Factors Prandlt s Tip-Loss Factor Notations Derivation of Prandtl s Tip-Loss Factor General Expression Different Expressions of Prandtl s Tip-Loss Factor Review of Tip-Loss Corrections Theoretical Tip-Loss Corrections Semi-empirical Tip-Loss Corrections Semi-empirical Performance Tip-Loss Corrections The Historical Approach of Radius Reduction References Goldstein s Optimal Circulation Introduction Goldstein s Circulation, Factor and Tip-Loss Factor Computation of Goldstein s Factor Main Methods of Evaluation

15 Contents xvii Computation Using Helical Vortex Solution: Algorithm and Source Code References Wake Expansion Models Simple 1D Momentum Theory/Vortex Cylinder Model Cylinder Analog Expansion Theodorsen s Wake Expansion Far-Wake Expansion Models Comparison of Wake Expansions References Relation Between Far-Wake and Near-Wake Parameters Introduction Extension of the Work of Okulov and Sørensen for Non-optimal Condition Extension of Theodorsen s Theory References Part IV Latest Developments in Vorticity-Based Rotor Aerodynamics 17 Cylindrical Vortex Model of a Rotor of Finite or Infinite Tip-Speed Ratios Introduction and Context Model and Key Results Conclusions References Cylindrical Model of a Rotor with Varying Circulation - Effect of Wake Rotation Context Model and Key Results Conclusions References An Improved BEM Algorithm Accounting for Wake Rotation Effects Context Actuator Disk Models for the BEM-Like Method Comparisons of Stream-Tube Theory and Vortex Cylinder Results BEM Algorithm Including Wake Rotation General Structure of a Lifting-Line-Based Algorithm Step 6: Inductions for the Standard BEM (STT-KJ)

16 xviii Contents Step 6: Inductions for the Improved BEM of Madsen et al Step 6: Inductions for the Actuator Disk Model (AD) Step 6: Inductions for the Vortex Cylinder Model (VCT) Results Conclusions References Helical Model for Tip-Losses: Development of a Novel Tip-Loss Factor and Analysis of the Effect of Wake Expansion Description of the Helical Wake Models A Novel Tip-Loss Factor Key Results Conclusions References Yaw-Modelling Using a Skewed Vortex Cylinder Introduction and Context Model and Key Results Conclusions References Simple Implementation of a New Yaw-Model Context Model and Key Results Conclusions References Advanced Implementation of the New Yaw-Model Introduction Models for the Velocity Field Outside of the Skewed Cylinder Helical Pitch for the Superposition of Skewed Cylinders Yaw-Model Implementation Using a Superposition of Skewed Cylinders Partial Approach - Focus on the Inboard Part of the Blade Conclusions References Velocity Field Upstream of Aligned and Yawed Rotors: Wind Turbine and Wind Farm Induction Zone Context Model for the Velocity Field in the Induction Zone Results for a Single Wind Turbine

17 Contents xix Aligned Case Without Swirl Aligned Case with Swirl Yawed Case Computational Time Results for a Wind Farm Introduction Velocity Deficit Upstream of a Wind Farm Conclusions References Analytical Model of a Wind Turbine in Sheared Inflow Context Model and Key-Results Conclusions References Model of a Wind Turbine with Unsteady Circulation or Unsteady Inflow Context Model and Key Results Conclusions References Part V Latest Applications of Vortex Methods to Rotor Aerodynamics and Aeroelasticity 27 Examples of Applications of Vortex Methods to Wind Energy Comparison with BEM and Actuator-Line Simulations Wakes and Flow Field for Uniform Inflows Effect of Viscosity - Comparison with AD Effect of Turbulence - Comparison with Lidar and AD Conclusions References Representation of a (Turbulent) Velocity Field Using Vortex Particles Simple Velocity Reconstruction Using Vortex Particles Associated Errors and Discussions Example of Velocity Reconstruction for a Turbulent Field Conclusions References Effect of a Wind Turbine on the Turbulent Inflow Introduction Terminology

18 xx Contents 29.3 Model and Key Results Conclusions References Aeroelastic Simulation of a Wind Turbine Under Turbulent and Sheared Conditions Introduction Representation of Shear in Vortex Methods Full Aeroelastic Simulation Including Shear and Turbulence Conclusions References Part VI Analytical Solutions for Vortex Methods and Rotor Aerodynamics 31 Elementary Three-Dimensional Flows Introduction Flow Induced by a Point-Wise Distribution Point Source Vortex Point (Vortex Particle/Blobs) Vortex Filaments Vortex Segment and Line of Constant Strength Vortex Segment of Linearly Varying Strength Multipoles Dipole - Doublet Multipoles Constant Panels Equivalences Between Elements References Elementary Two-Dimensional Potential Flows Uniform Flow Point Source, Point Vortex and Distributions of Points Point Source/Sink Point Vortex Periodic Point Vortices Continuous Distribution of 2D Points Doublet and Multipoles Doublet Multi-poles Cylinder/Ellipse Flows Cylinder Flow - Acyclic - No Lift Flow Around a 2D Ellipse - No Lift Cylinder Flow - Cyclic - with Lift

19 Contents xxi Flow About Quadrics Miscellaneous Flows Rigid Rotation Corner Flow, Flat Plate and Stagnation Point Cylinder and Vortex Point References Flows with a Spread Distribution of Vorticity Axisymmetric Vorticity Patches Examples of Vorticity Patches Canonical Example: The Inviscid Vorticity Patch Rectangular Vorticity Patch (2D Brick) References Spherical Geometry Models: Flow About a Sphere and Hill s Vortex Sphere with Free Stream Hill s Vortex Ellipsoid and Spheroid References Vortex and Source Rings Vortex Rings - General Considerations Formulae for the Potential, Velocity and Gradient Flow at Particular Locations Derivation of the Velocity and Vector Potential Further Considerations Source Rings References Flow Induced by a Right Vortex Cylinder Right Cylinder of Tangential Vorticity with Arbitrary Cross Section Finite Cylinder - General Velocity Field Finite Cylinder - Velocity in Terms of Solid Angle Infinite and Semi-infinite Cylinders of Arbitrary Cross Sections Finite Cylinder of Tangential Vorticity and Link to Source Surfaces Right Vortex Cylinder of Tangential Vorticity - Circular Cross Section Finite Vortex Cylinder of Tangential Vorticity Semi-infinite Vortex Cylinder of Tangential Vorticity

20 xxii Contents 36.3 Vortex Cylinder of Longitudinal Vorticity Infinite Cylinder of Longitudinal Vorticity Finite Cylinder of Longitudinal Vorticity Semi-infinite Cylinder of Longitudinal Vorticity References Flow Induced by a Vortex Disk Introduction Indefinite Form of the Biot Savart Law Definite Form of the Biot Savart Law Properties Reference Flow Induced by a Skewed Vortex Cylinder Semi-infinite Skewed Cylinder of Tangential Vorticity Preliminary Note on the Integrals Involved Extension of the Work of Castles and Durham Longitudinal Axis - Work of Coleman et al Matlab Source Code Semi-infinite Skewed Cylinder with Longitudinal Vorticity Infinite Skewed Cylinder with Longitudinal Vorticity (Elliptic Cylinder) References Flow Induced by Helical Vortex Filaments Preliminary Considerations Introduction Semi-infinite Helix and Rotor Terminology Exact Expressions for Infinite Helical Vortex Filaments Approximate Expressions for Infinite Helical Filaments Expressions for Semi-infinite Helices Evaluated on the Lifting Line Notations Introduced for Approximate Formulae Summation of Several Helices - Link Between Okulov s Relation and Wrench s Relation References Part VII Vortex Methods 40 A Brief Introduction to Vortex Methods Introduction Pros and Cons An Example of Vortex Method History

21 Contents xxiii 40.4 Classification of Vortex Methods Existing Vortex Codes and Application to Wind Energy References The Different Aspects of Vortex Methods Fundamental Equations and Concepts Discretization and Initialization Information Carried by the Vortex Elements Initialization and Reinitialization Initialization - Inviscid Vortex Patch Example Viscous-Splitting Viscous-Splitting Algorithm Rate of Convergence of the Viscous-Splitting Algorithm Application to the Vorticity Transport Equation Convection and Stretching of Vortex Elements Introduction Convection of Vortex Elements Stretching Applications Grid-Free and Grid-Based Methods Grid-Free Vortex Methods Grid-Based Vortex Methods (Mixed Eulerian Lagrangian Formulation) Coupled Lagrangian and Eulerian Solvers Viscous Diffusion - Solution of the Diffusion Equation Diffusion Equation and Vorticity Transport Equation Fundamental Solution and Lamb Oseen Vortex Core-Spreading Method Random-Walk Method Grid-Based Finite-Differences Method Particle-Strength-Exchange (PSE) Numerical Application: Lamb Oseen Vortex Vorticity Redistribution Method Boundaries, Boundary Conditions and Lifting-Bodies Introduction Fluid Boundary Conditions: Free-Flow and Periodic Boundaries Solid Boundaries in Inviscid Flows Solid Boundaries in Viscous Flows - Vorticity Generation

22 xxiv Contents Viscous Boundaries Using Coupling (Viscous-Inviscid or Lagrangian Eulerian) Lifting-Bodies Regularization - Kernel Smoothing - Mollification Kernel Smoothing via Convolution with a Cut-Off Function Requirements on the Cut-Off Function Special Case of Spherical Symmetry Examples Used in Particle Methods Regularization Models for Vortex Filaments Choice of Cut-Off/Smooth Parameter Application to the Inviscid Vortex Patch Spatial Adaptation - Redistribution - Rezoning - Reinitialization Introduction Remeshing - Rezoning - Redistribution - Reinitialization Gain from Remeshing - Application to Inviscid-Vortex Patch Problems Introduced by Remeshing Subgrid-Scale Models - LES - Turbulence Accuracy of Vortex Methods, Guidelines, Diagnostics and Possible Improvements Guidelines and Diagnostics for General Vortex Methods Boundary Elements - Guidelines and Diagnostics Particle Methods - Convergence Application to the Inviscid Vortex Patch References Particularities of Vortex Particle Methods Particle Approximation and Lagrangian Methods Notion of Vortex Blob Particle Approximation Dynamics of Lagrangian Methods Incompressible Vortex Particle Methods Stretching Term - Different Schemes Divergence of the Vorticity Field Minimizing the Error Growth Corrections Criteria for Correction References

23 Contents xxv 43 Numerical Implementation of Vortex Methods Interpolation Method Required for Grid-Based Methods Interpolation in Vortex Methods Concept of Interpolation Interpolation to Grid (Projection, Griding, Assignment, Particle-to-Mesh) Interpolation from Grid (Mesh-to-Particle) Tree-Codes and Fast Multipole Method Tree-Based Method Tree-Based Method - Coefficients up to Order Poisson Solvers Numerical Integration Schemes Expression of the Different Schemes Example of Application to the Inviscid Patch Work Presented by Leishman Vorticity Splitting and Merging Schemes Conversion from Segments to Particles Canonical Examples for Validation Representation of One Segment by One Particle Representation Using Several Particles Trailed and Shed Vorticity Behind a Wing Distribution of Control Points The Work of James - Chordwise Distribution Cosine Spacing and Other References in the Topic The 3/4 Chord Collocation Point References OmniVor: An Example of Vortex Code Implementation Introduction Implementation and Features Specific Configurations Used in Publications References Vortex Code Validation and Illustration Simple Validation of the Vortex Particle Method Lifting Line Lifting Surface Thick Bodies Unit-Tests Further Validation References

24 xxvi Contents Appendix A: Complements on the Right Cylindrical Model and the Effect of Wake Rotation Appendix B: From Poisson s Equation to the Biot Savart Law in an Unbounded Domain Appendix C: Useful Mathematical Relations Index

25 Acronyms a a B ^a a 0 c c n c t e e t h h t h h h B h k k 2 k t l l m n rot p p t p q r r c t t 0 Axial induction factor Axial induction factor local to the blade Axial induction factor from 2D MT Tangential induction factor Chord Normal aerodynamic coefficient Tangential aerodynamic coefficient Internal energy Total energy Enthalpy Total enthalpy Typical grid spacing in vortex methods Helix pitch Apparent pitch h=b Normalized pitch h=r Dimensionless circulation Elliptical parameter for elliptic integrals Turbulent kinetic energy Helix torsional parameter Normalized torsional parameter l=r Elliptical parameter for elliptic integrals Rotational speed in RPM: X=ð2pÞ Static pressure Total pressure p þ 1 2 qu2 Frequency associated with X, p ¼ X=2p Heat flux Radial position Viscous core radius Time Parameter in the core-spreading model xxvii

26 xxviii Acronyms r Dimensionless radial position r=r ~r Dimensionless radial position r=r s Sign u h Tangential induced velocity u z Axial induced velocity u x-component of velocity v y-component of velocity w z-component of velocity w Wake relative longitudinal velocity (Betz) z 0 Surface roughness length A Angular Impulse A Area AR See Abbreviations B Number of blades C C Dimensionless circulation C d Drag coefficients C l Lift coefficients C l;a Lift coefficient slope for small angles C p Power coefficient C q Local torque coefficient C Q Total torque coefficient C t Tangential aerodynamic coefficient C t Local thrust coefficient C T Total thrust coefficient D Drag force D Rotor diameter D Deformation matrix E Complete elliptic integral of the 2nd kind E Energy E Enstrophy F Tip-loss factor F a Tip-loss factor based on axial induction F C Tip-loss factor based on circulation F Cl Performance tip-loss factor F Go Goldstein s tip-loss factor F Gl Glauert s tip-loss factor F Pr Prandtl s tip-loss factor F Sh Shen s tip-loss factor F Complex velocity potential in 2D G Green s function associated with the operator H Heaviside function H Bernoulli constant, e.g., p þ 1 2 qu2 I t Turbulence intensity I Linear Impulse

27 Acronyms xxix J Helicity K Kernel (associated with a given operator ) K Complete elliptic integral of the 1st kind L Lift force Ma Mach number P Power P Palinstrophy Q Rotor torque Q Vortical Helicity R Rotor radius Re Reynolds number S Entropy S Surface S Energy density spectrum S d Volume of the unit sphere in R d T Thrust force T Temperature U Longitudinal velocity at the rotor in 1D U Relative velocity at the rotor U 0 Longitudinal velocity far upstream U i Induced velocity in 1D U n Velocity normal to the rotor U ref Reference velocity used, e.g., for the normalization of loads U t Velocity tangent to the rotor V Velocity vector V rel Relative velocity V Volume W Induced velocity vector at the rotor a Point/Blob vorticity intensity a Angle of attack a 0 Angle of attack at zero lift b Twist angle c Surface vorticity - Distributed circulation c t Vortex cylinder tangential vorticity c l Vortex cylinder longitudinal vorticity c b Bound vorticity d Dirac function e Pitch angle of the wake helix screw e Regularization parameter f Regularization/cutoff/smoothing function g Efficiency h Azimuthal coordinate j Goldstein s factor k Tip speed ratio ¼ XR=U 0

28 xxx Acronyms k r Local speed ratio ¼ kr=r k First Lamé s coefficient for Newtonian fluid l Second Lamé s coefficient: dynamic viscosity m Kinematic viscosity ¼ l=q q Air density kg/m 3 r Local blade solidity ¼ Bc=2pr r Cauchy stress tensor s Shear stress, viscous stress tensor / Flow angle v Wake skew angle, in yaw conditions w Azimuthal coordinate w Vector potential x Rotational speed of the wake x Vorticity C Circulation D Laplacian operator r 2 H Dilatation P Gate function P Complete elliptic integral of the 3rd kind U Velocity Potential W Stream function (2D) W Stokes stream function (3D) X Rotational speed of the rotor X Rotation matrix (fluid kinematics) X Solid angle X Volume of the domain X Total Surface boundary of volume X t X Transpose X T Transpose r Del operator, nabla div Divergence, divx ¼r X grad grad curl e.g. i.e. viz. w.r.t. 1D 2D 3D AC divt j ðt ij Þe i Gradient, grad X ¼rX Gradient of first-order tensor Rotational, curl X ¼r X exempli gratia: for example id est: that is videlicet: namely with respect to One dimension Two dimensions Three dimensions Aerodynamic center

29 Acronyms xxxi AD AEP AED AL AR BEM BET BT CFD CP CP CPU CV DOF DTU ECN GPU HSS IEC KJ LE LES LHS LSS MT NTUA PSE VC VC VL RHS SGS ST STT TE TKE WD WS WT Actuator Disk Annual Energy Output Aeroelastic Design (section at DTU) Actuator Line Aspect ratio of a wing (b 2 =S) Blade Element Momentum Blade Element Theory Blade Element Theory (subscript) Computational Fluid Dynamics Control Point Center of Pressure Central Processing Units Control volume Degree of Freedom Technical University of Denmark Energy Center of the Netherlands Graphical Processing Units High-Speed Shaft International Electrotechnical Commission Kutta Joukowski Leading edge Large Eddy Simulation Left-Hand Side Low-Speed Shaft Momentum Theory National Technical University of Athens Particle Strength Exchange Vortex Code Vortex Cylinder (depending on context) Vortex Lattice Right-Hand Side Sub-grid scale model Streamtube Theory (also written STT) Streamtube Theory (also written ST) Trailing edge Turbulent Kinetic Energy Wind Direction Wind Speed Wind Turbine