Branlard Wind Turbine Aerodynamics and Vorticity-Based Methods

Fundamentals and Recent Applications
1. Auflage 2017
ISBN: 978-3-319-55164-7
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 7, 632 Seiten

Reihe: Research Topics in Wind Energy

The book introduces the fundamentals of fluid-mechanics, momentum theories, vortex theories and vortex methods necessary for the study of rotors aerodynamics and wind-turbines aerodynamics 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 momentum theory and the blade element momentum method. A part of the book is dedicated to the description and implementation of vortex methods. The remaining of the book focuses on the study of wind turbine aerodynamics using vortex-theory analyses or vortex-methods. Examples of vortex-theory applications 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 presented in this book. This model leads to the development of different BEM models and it is also used to provide the analytical velocity field upstream of a turbine or a wind farm under aligned or yawed conditions. Different applications of numerical vortex methods are presented. Numerical methods are used for instance to investigate the influence of a wind turbine on the incoming turbulence. Sheared inflows and aero-elastic simulations are investigated using vortex methods for the first time. Many analytical flows are derived in details: vortex rings, vortex cylinders, 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.

Autoren/Hrsg.

Branlard, Emmanuel

Inhaltsverzeichnis

1;Preface;7
2;Acknowledgements;9
3;Contents;10
4;Acronyms;26
5;1 Introduction;31
5.1;References;36
6;Part I Fluid Mechanics Foundations;38
7;2 Theoretical Foundations for Flows Involving Vorticity;39
7.1;2.1 Fluid Mechanics Equations in Inertial and Non-inertial Frames;39
7.1.1;2.1.1 Physical Quantities;39
7.1.2;2.1.2 Conservation Laws;40
7.1.3;2.1.3 Fluid-Mechanic Equations in a Non-inertial Frame;45
7.1.4;2.1.4 Fluid Mechanics Assumptions;54
7.1.5;2.1.5 Usual Cases - Equations of Euler and Bernoulli;57
7.2;2.2 Flow Kinematics and Vorticity;60
7.2.1;2.2.1 Flow Kinematics;60
7.2.2;2.2.2 Vorticity and Related Definitions;61
7.2.3;2.2.3 Helmholtz (First) Law;64
7.2.4;2.2.4 Helmholtz-(Hodge) Decomposition;64
7.2.5;2.2.5 Bounded and Unbounded Domain - Surface Map - Generalized Helmholtz Decomposition;65
7.3;2.3 Main Dynamics Equations Involving Vorticity;66
7.3.1;2.3.1 Circulation Equation;66
7.3.2;2.3.2 Vorticity Equation;68
7.3.3;2.3.3 Stretching and Dilatation of Vorticity;68
7.3.4;2.3.4 Alternative Forms of the Vorticity Equation;70
7.3.5;2.3.5 Vorticity Equation in Particular Cases;71
7.3.6;2.3.6 Pressure;72
7.3.7;2.3.7 Vortex Force, Image/Generalized/Bound Vorticity, Kutta--Joukowski Relation;73
7.4;2.4 Different Dimensions of Vorticity: Surface, Line and Points;75
7.5;2.5 Vorticity Moments, Variables and Invariants - Incompressible Flows;77
7.6;2.6 Main Theorems Involving Vorticity;80
7.6.1;2.6.1 Kelvin's Theorem;80
7.6.2;2.6.2 Lagrange's Theorem;80
7.6.3;2.6.3 Helmholtz Theorem;81
7.6.4;2.6.4 Biot--Savart Law;82
7.7;2.7 Vortices in Viscous and Inviscid Fluid - Results and Classical Flows;85
7.7.1;2.7.1 Vortex in Inviscid Fluid;85
7.7.2;2.7.2 Vortex in Viscous Fluid - Standard Solutions;85
7.7.3;2.7.3 Life of a Vortex - Vortex Decay, Collapse and Stability;87
7.8;2.8 Surface Representations - Vortex Sheets;88
7.8.1;2.8.1 Introduction;88
7.8.2;2.8.2 Vortex Sheets Kinematics;88
7.8.3;2.8.3 Vortex Sheets Dynamics;89
7.8.4;2.8.4 Vortex Sheet Convection and Stability;90
7.8.5;2.8.5 Vortex Surfaces in 2D;90
7.9;2.9 Incompressible Flow Equations in Polar Coordinates - 2D and 3D Flows - Axisymmetric Flows;91
7.9.1;2.9.1 2D Arbitrary Flow (Cylindrical Coordinates);92
7.9.2;2.9.2 3D Arbitrary Flow (Cylindrical Coordinates);92
7.9.3;2.9.3 3D Axisymmetric Flows with Swirl (Cylindrical Coordinates);93
7.9.4;2.9.4 3D Axisymmetric Flows Without Swirl (Cylindrical Coordinates);95
7.9.5;2.9.5 3D Arbitrary Flow (Spherical Coordinates);96
7.9.6;2.9.6 3D Axisymmetric Flows with Swirl (Spherical Coordinates);97
7.9.7;2.9.7 3D Axisymmetric Flows Without Swirl (Spherical Coordinates);97
7.10;2.10 2D Potential Flows;99
7.11;2.11 Conformal Map Solutions;101
7.11.1;2.11.1 Conformal Mapping - Definitions and Properties;101
7.11.2;2.11.2 Reference Airfoil Flow: Flow Around a Cylinder and Kutta Condition;102
7.11.3;2.11.3 Joukowski's Conformal Map;102
7.11.4;2.11.4 Karman-Trefftz Conformal Map;104
7.11.5;2.11.5 Van de Vooren Conformal Map;105
7.11.6;2.11.6 Matlab Source Code;106
7.12;References;108
8;3 Lifting Bodies and Circulation;111
8.1;3.1 Characteristics of Lifting Bodies;111
8.1.1;3.1.1 Fluid Force on a Body: Lift, Drag, Moment and Center of Pressure;111
8.1.2;3.1.2 Center of Pressure, Aerodynamic Center and Quarter Chord Point of an Airfoil;114
8.1.3;3.1.3 Vorticity Associated with Lifting Bodies;117
8.1.4;3.1.4 Kutta Condition;118
8.1.5;3.1.5 Kutta--Joukowski Relation;119
8.2;3.2 Polar Data of an Airfoil and Related Engineering Models;121
8.2.1;3.2.1 Introduction;121
8.2.2;3.2.2 Models for Large Angle of Attacks;122
8.2.3;3.2.3 Dynamic Stall Models;123
8.2.4;3.2.4 Inviscid Performances;124
8.2.5;3.2.5 Model of Fully-Separated Polar from Known Polar;125
8.3;3.3 Vorticity Based Theories of Two-Dimensional Lifting Bodies;127
8.4;3.4 Vorticity Based Theories of Thick Three-Dimensional Lifting Bodies;127
8.5;3.5 Inviscid Lifting-Surface Theory of a Wing;127
8.6;3.6 Inviscid Lifting-Line Theory of a Wing;128
8.6.1;3.6.1 Introduction;128
8.6.2;3.6.2 Lifting Line Theory - From Circulation Distribution to Loads;129
8.6.3;3.6.3 Prandtl's Lifting Line Equation - Integro-Differential Form;130
8.6.4;3.6.4 Elliptical Loading and Elliptical Wing Under Lifting Line Assumptions and Linear Theory;131
8.6.5;3.6.5 Numerical Implementation of the Method - Sample Code;133
8.7;References;137
9;Part II Introduction to Rotors Aerodynamics;139
10;4 Rotor and Wind Turbine Formalism;140
10.1;4.1 Main Assumptions and Conventions;140
10.2;4.2 Wind Turbine Formalism;142
10.3;4.3 Loads and Dimensionless Coefficients;143
10.4;4.4 Velocity Induction Factors Under the Lifting Line Approximation;145
10.5;4.5 Solidity;146
10.6;References;146
11;5 Vortex Systems and Models of a Rotor - Bound, Root and Wake Vorticity;147
11.1;5.1 Main Components of Vorticity Involved About a Rotor;147
11.2;5.2 Simplified Vorticity Models of Rotors;149
11.2.1;5.2.1 Main Simplifications Used by the Models;149
11.2.2;5.2.2 Helical Vortex Models of a Rotor;151
11.2.3;5.2.3 Cylindrical and Tubular Vortex Model of a Rotor;153
11.2.4;5.2.4 Vortex Ring Model of a Rotor;156
11.3;5.3 Analytical Results for the Vortex Wake Models;157
11.4;References;159
12;6 Considerations and Challenges Specific to Rotor Aerodynamics;160
12.1;6.1 Yaw and Tilt;160
12.2;6.2 Rotational Effects;162
12.3;6.3 Airfoil Corrections for Rotating Blades;163
12.4;References;165
13;7 Blade Element Theory (BET);167
13.1;7.1 Introduction;167
13.2;7.2 Analysis of a Blade Element;168
13.3;7.3 Applications;169
13.3.1;7.3.1 Flow with Rotational Symmetry;169
13.3.2;7.3.2 Particular Cases of Flows with Rotational Symmetry;171
13.3.3;7.3.3 Introducing the Induction Factors on the Blade;172
13.4;References;173
14;8 Kutta--Joukowski (KJ) Theorem Applied to a Rotor;174
14.1;8.1 Assumptions and Main Result;174
14.2;8.2 Rotor Performance Coefficients from the KJ Analyses;175
14.2.1;8.2.1 Local Coefficients;175
14.2.2;8.2.2 Global Coefficients;176
14.3;8.3 Vortex Actuator Disk - KJ Analysis for an Infinite Number of Blades;177
14.4;8.4 Applications for Large Tip-Speed Ratios;178
15;9 Momentum Theory;180
15.1;9.1 Introduction;180
15.2;9.2 Simplified Axial Momentum Theory (No Wake Rotation);182
15.2.1;9.2.1 Notations and Assumptions;182
15.2.2;9.2.2 Determination of Power, Thrust and Rotor Velocity;184
15.2.3;9.2.3 Induction Factors and Rotor Performance;186
15.2.4;9.2.4 Discussion on the Assumptions;188
15.3;9.3 General Momentum Theory;191
15.3.1;9.3.1 Introduction;191
15.3.2;9.3.2 Derivation;192
15.4;9.4 General Axial Momentum Theory (No Wake Rotation);197
15.4.1;9.4.1 Assumptions;197
15.4.2;9.4.2 Results of the General Axial Momentum Theory;198
15.5;9.5 Streamtube Theory (Simplified Momentum Theory);198
15.5.1;9.5.1 Assumptions;198
15.5.2;9.5.2 Derivation of the Main Streamtube Theory Results;199
15.5.3;9.5.3 Loads from Streamtube Theory;200
15.5.4;9.5.4 Maximum Power Extraction from STT - ``Optimal Rotor'';201
15.6;References;203
16;10 The Blade Element Momentum (BEM) Method;204
16.1;10.1 The BEM Method for a Steady Uniform Inflow;205
16.1.1;10.1.1 Introduction;205
16.1.2;10.1.2 First Linkage: Velocity Triangle and Induction Factors;206
16.1.3;10.1.3 Second Linkage: Thrust and Torque from MT and BET;208
16.1.4;10.1.4 BEM Equations;209
16.1.5;10.1.5 Summary of the BEM Algorithm;211
16.2;10.2 Common Corrections to the Steady BEM Method;213
16.2.1;10.2.1 Discrete Number of Blades, Tip-Losses and Hub-Losses;213
16.2.2;10.2.2 Correction Due to Momentum Theory Breakdown - a-Ct Relations;216
16.2.3;10.2.3 Wake Rotation;218
16.3;10.3 Unsteady BEM Method;220
16.3.1;10.3.1 Introduction;220
16.3.2;10.3.2 Dynamic Wake/Inflow;220
16.3.3;10.3.3 Yaw and Tilt Model;222
16.3.4;10.3.4 Dynamic Stall;223
16.3.5;10.3.5 Tower and Nacelle Interference;224
16.3.6;10.3.6 Summary of the Unsteady BEM Algorithm;225
16.4;10.4 Typical Applications and Source Code;226
16.4.1;10.4.1 Examples of Applications;226
16.4.2;10.4.2 Source Code for Steady and Unsteady BEM Methods;229
16.5;References;233
17;Part III Classical Vortex Theory Results: Optimal Circulation and Tip-Losses;235
18;11 Far-Wake Analyses and the Rigid Helical Wake;236
18.1;11.1 Introduction;236
18.2;11.2 The Wake Screw Model;237
18.3;11.3 Relation with Rotor Parameters;240
18.4;11.4 Dimensionless Circulation in Terms of Wake Parameters;242
18.5;References;243
19;12 Betz Theory of Optimal Circulation;244
19.1;12.1 Introduction;244
19.2;12.2 Betz Optimal Circulation;244
19.3;12.3 Inclusion of Drag;245
19.4;References;246
20;13 Tip-Losses with Focus on Prandlt's Tip Loss Factor;247
20.1;13.1 Introduction to Tip-Losses;247
20.2;13.2 Historical and Modern Tip-Loss Factors;249
20.2.1;13.2.1 Historical Tip-Loss Factor;249
20.2.2;13.2.2 Modern Definitions of the Tip-Loss Factors;250
20.3;13.3 Prandlt's Tip-Loss Factor;252
20.3.1;13.3.1 Notations;252
20.3.2;13.3.2 Derivation of Prandtl's Tip-Loss Factor;253
20.3.3;13.3.3 General Expression;259
20.4;13.4 Different Expressions of Prandtl's Tip-Loss Factor;260
20.5;13.5 Review of Tip-Loss Corrections;261
20.5.1;13.5.1 Theoretical Tip-Loss Corrections;262
20.5.2;13.5.2 Semi-empirical Tip-Loss Corrections;262
20.5.3;13.5.3 Semi-empirical Performance Tip-Loss Corrections;262
20.5.4;13.5.4 The Historical Approach of Radius Reduction;263
20.6;References;264
21;14 Goldstein's Optimal Circulation;266
21.1;14.1 Introduction;266
21.2;14.2 Goldstein's Circulation, Factor and Tip-Loss Factor;267
21.3;14.3 Computation of Goldstein's Factor;268
21.3.1;14.3.1 Main Methods of Evaluation;268
21.3.2;14.3.2 Computation Using Helical Vortex Solution: Algorithm and Source Code;269
21.4;References;272
22;15 Wake Expansion Models;273
22.1;15.1 Simple 1D Momentum Theory/Vortex Cylinder Model;273
22.2;15.2 Cylinder Analog Expansion;273
22.3;15.3 Theodorsen's Wake Expansion;274
22.4;15.4 Far-Wake Expansion Models;275
22.5;15.5 Comparison of Wake Expansions;276
22.6;References;276
23;16 Relation Between Far-Wake and Near-Wake Parameters;277
23.1;16.1 Introduction;277
23.2;16.2 Extension of the Work of Okulov and Sørensen for Non-optimal Condition;278
23.3;16.3 Extension of Theodorsen's Theory;279
23.4;References;280
24;Part IV Latest Developments in Vorticity-Based Rotor Aerodynamics;281
25;17 Cylindrical Vortex Model of a Rotor of Finite or Infinite Tip-Speed Ratios;282
25.1;17.1 Introduction and Context;282
25.2;17.2 Model and Key Results;284
25.3;17.3 Conclusions;288
25.4;References;288
26;18 Cylindrical Model of a Rotor with Varying Circulation - Effect of Wake Rotation;290
26.1;18.1 Context;291
26.2;18.2 Model and Key Results;291
26.3;18.3 Conclusions;298
26.4;References;299
27;19 An Improved BEM Algorithm Accounting for Wake Rotation Effects;300
27.1;19.1 Context;300
27.2;19.2 Actuator Disk Models for the BEM-Like Method;301
27.2.1;19.2.1 Comparisons of Stream-Tube Theory and Vortex Cylinder Results;302
27.3;19.3 BEM Algorithm Including Wake Rotation;303
27.3.1;19.3.1 General Structure of a Lifting-Line-Based Algorithm;303
27.3.2;19.3.2 Step 6: Inductions for the Standard BEM (STT-KJ);304
27.3.3;19.3.3 Step 6: Inductions for the Improved BEM of Madsen et al.;304
27.3.4;19.3.4 Step 6: Inductions for the Actuator Disk Model (AD);305
27.3.5;19.3.5 Step 6: Inductions for the Vortex Cylinder Model (VCT);305
27.4;19.4 Results;306
27.5;19.5 Conclusions;308
27.6;References;308
28;20 Helical Model for Tip-Losses: Development of a Novel Tip-Loss Factor and Analysis of the Effect of Wake Expansion;309
28.1;20.1 Description of the Helical Wake Models;309
28.2;20.2 A Novel Tip-Loss Factor;310
28.3;20.3 Key Results;311
28.4;20.4 Conclusions;312
28.5;References;313
29;21 Yaw-Modelling Using a Skewed Vortex Cylinder;314
29.1;21.1 Introduction and Context;314
29.2;21.2 Model and Key Results;316
29.3;21.3 Conclusions;320
29.4;References;320
30;22 Simple Implementation of a New Yaw-Model;322
30.1;22.1 Context;322
30.2;22.2 Model and Key Results;323
30.3;22.3 Conclusions;327
30.4;References;327
31;23 Advanced Implementation of the New Yaw-Model;329
31.1;23.1 Introduction;329
31.2;23.2 Models for the Velocity Field Outside of the Skewed Cylinder;330
31.3;23.3 Helical Pitch for the Superposition of Skewed Cylinders;331
31.4;23.4 Yaw-Model Implementation Using a Superposition of Skewed Cylinders;332
31.5;23.5 Partial Approach - Focus on the Inboard Part of the Blade;333
31.6;23.6 Conclusions;334
31.7;References;334
32;24 Velocity Field Upstream of Aligned and Yawed Rotors: Wind Turbine and Wind Farm Induction Zone;335
32.1;24.1 Context;335
32.2;24.2 Model for the Velocity Field in the Induction Zone;336
32.3;24.3 Results for a Single Wind Turbine;337
32.3.1;24.3.1 Aligned Case Without Swirl;338
32.3.2;24.3.2 Aligned Case with Swirl;339
32.3.3;24.3.3 Yawed Case;340
32.3.4;24.3.4 Computational Time;342
32.4;24.4 Results for a Wind Farm;342
32.4.1;24.4.1 Introduction;342
32.4.2;24.4.2 Velocity Deficit Upstream of a Wind Farm;343
32.5;24.5 Conclusions;345
32.6;References;346
33;25 Analytical Model of a Wind Turbine in Sheared Inflow;347
33.1;25.1 Context;347
33.2;25.2 Model and Key-Results;348
33.3;25.3 Conclusions;351
33.4;References;351
34;26 Model of a Wind Turbine with Unsteady Circulation or Unsteady Inflow;352
34.1;26.1 Context;352
34.2;26.2 Model and Key Results;353
34.3;26.3 Conclusions;356
34.4;References;356
35;Part V Latest Applications of Vortex Methods to Rotor Aerodynamics and Aeroelasticity;358
36;27 Examples of Applications of Vortex Methods to Wind Energy;359
36.1;27.1 Comparison with BEM and Actuator-Line Simulations;359
36.2;27.2 Wakes and Flow Field for Uniform Inflows;361
36.3;27.3 Effect of Viscosity - Comparison with AD;361
36.4;27.4 Effect of Turbulence - Comparison with Lidar and AD;362
36.5;27.5 Conclusions;364
36.6;References;364
37;28 Representation of a (Turbulent) Velocity Field Using Vortex Particles;366
37.1;28.1 Simple Velocity Reconstruction Using Vortex Particles;366
37.2;28.2 Associated Errors and Discussions;367
37.3;28.3 Example of Velocity Reconstruction for a Turbulent Field;369
37.4;28.4 Conclusions;371
37.5;References;371
38;29 Effect of a Wind Turbine on the Turbulent Inflow;372
38.1;29.1 Introduction;372
38.2;29.2 Terminology;373
38.3;29.3 Model and Key Results;375
38.4;29.4 Conclusions;379
38.5;References;379
39;30 Aeroelastic Simulation of a Wind Turbine Under Turbulent and Sheared Conditions;381
39.1;30.1 Introduction;381
39.2;30.2 Representation of Shear in Vortex Methods;382
39.3;30.3 Full Aeroelastic Simulation Including Shear and Turbulence;383
39.4;30.4 Conclusions;387
39.5;References;387
40;Part VI Analytical Solutions for Vortex Methods and Rotor Aerodynamics;389
41;31 Elementary Three-Dimensional Flows;390
41.1;31.1 Introduction;390
41.2;31.2 Flow Induced by a Point-Wise Distribution;391
41.2.1;31.2.1 Point Source;391
41.2.2;31.2.2 Vortex Point (Vortex Particle/Blobs);393
41.3;31.3 Vortex Filaments;396
41.3.1;31.3.1 Vortex Segment and Line of Constant Strength;396
41.3.2;31.3.2 Vortex Segment of Linearly Varying Strength;399
41.4;31.4 Multipoles;400
41.4.1;31.4.1 Dipole - Doublet;400
41.4.2;31.4.2 Multipoles;401
41.4.3;31.4.3 Constant Panels;401
41.4.4;31.4.4 Equivalences Between Elements;401
41.5;References;401
42;32 Elementary Two-Dimensional Potential Flows;402
42.1;32.1 Uniform Flow;402
42.2;32.2 Point Source, Point Vortex and Distributions of Points;402
42.2.1;32.2.1 Point Source/Sink;402
42.2.2;32.2.2 Point Vortex;403
42.2.3;32.2.3 Periodic Point Vortices;404
42.2.4;32.2.4 Continuous Distribution of 2D Points;404
42.3;32.3 Doublet and Multipoles;405
42.3.1;32.3.1 Doublet;405
42.3.2;32.3.2 Multi-poles;406
42.4;32.4 Cylinder/Ellipse Flows;406
42.4.1;32.4.1 Cylinder Flow - Acyclic - No Lift;406
42.4.2;32.4.2 Flow Around a 2D Ellipse - No Lift;407
42.4.3;32.4.3 Cylinder Flow - Cyclic - with Lift;407
42.4.4;32.4.4 Flow About Quadrics;408
42.5;32.5 Miscellaneous Flows;408
42.5.1;32.5.1 Rigid Rotation;408
42.5.2;32.5.2 Corner Flow, Flat Plate and Stagnation Point;409
42.5.3;32.5.3 Cylinder and Vortex Point;409
42.6;References;409
43;33 Flows with a Spread Distribution of Vorticity;410
43.1;33.1 Axisymmetric Vorticity Patches;410
43.1.1;33.1.1 Examples of Vorticity Patches;410
43.1.2;33.1.2 Canonical Example: The Inviscid Vorticity Patch;411
43.2;33.2 Rectangular Vorticity Patch (2D Brick);414
43.3;References;415
44;34 Spherical Geometry Models: Flow About a Sphere and Hill's Vortex;416
44.1;34.1 Sphere with Free Stream;416
44.2;34.2 Hill's Vortex;420
44.3;34.3 Ellipsoid and Spheroid;426
44.4;References;426
45;35 Vortex and Source Rings;427
45.1;35.1 Vortex Rings - General Considerations;427
45.2;35.2 Formulae for the Potential, Velocity and Gradient;428
45.3;35.3 Flow at Particular Locations;429
45.4;35.4 Derivation of the Velocity and Vector Potential;432
45.5;35.5 Further Considerations;436
45.6;35.6 Source Rings;436
45.7;References;436
46;36 Flow Induced by a Right Vortex Cylinder;437
46.1;36.1 Right Cylinder of Tangential Vorticity with Arbitrary Cross Section;438
46.1.1;36.1.1 Finite Cylinder - General Velocity Field;438
46.1.2;36.1.2 Finite Cylinder - Velocity in Terms of Solid Angle;438
46.1.3;36.1.3 Infinite and Semi-infinite Cylinders of Arbitrary Cross Sections;440
46.1.4;36.1.4 Finite Cylinder of Tangential Vorticity and Link to Source Surfaces;441
46.2;36.2 Right Vortex Cylinder of Tangential Vorticity - Circular Cross Section;443
46.2.1;36.2.1 Finite Vortex Cylinder of Tangential Vorticity;444
46.2.2;36.2.2 Semi-infinite Vortex Cylinder of Tangential Vorticity;452
46.3;36.3 Vortex Cylinder of Longitudinal Vorticity;458
46.3.1;36.3.1 Infinite Cylinder of Longitudinal Vorticity;458
46.3.2;36.3.2 Finite Cylinder of Longitudinal Vorticity;459
46.3.3;36.3.3 Semi-infinite Cylinder of Longitudinal Vorticity;459
46.4;References;461
47;37 Flow Induced by a Vortex Disk;462
47.1;37.1 Introduction;462
47.2;37.2 Indefinite Form of the Biot--Savart Law;463
47.3;37.3 Definite Form of the Biot--Savart Law;465
47.4;37.4 Properties;466
47.5;Reference;467
48;38 Flow Induced by a Skewed Vortex Cylinder;468
48.1;38.1 Semi-infinite Skewed Cylinder of Tangential Vorticity;468
48.1.1;38.1.1 Preliminary Note on the Integrals Involved;469
48.1.2;38.1.2 Extension of the Work of Castles and Durham;470
48.1.3;38.1.3 Longitudinal Axis - Work of Coleman et al.;471
48.1.4;38.1.4 Matlab Source Code;473
48.2;38.2 Semi-infinite Skewed Cylinder with Longitudinal Vorticity;474
48.3;38.3 Infinite Skewed Cylinder with Longitudinal Vorticity (Elliptic Cylinder);475
48.4;References;478
49;39 Flow Induced by Helical Vortex Filaments;479
49.1;39.1 Preliminary Considerations;479
49.1.1;39.1.1 Introduction;479
49.1.2;39.1.2 Semi-infinite Helix and Rotor Terminology;480
49.2;39.2 Exact Expressions for Infinite Helical Vortex Filaments;481
49.3;39.3 Approximate Expressions for Infinite Helical Filaments;481
49.4;39.4 Expressions for Semi-infinite Helices Evaluated on the Lifting Line;482
49.5;39.5 Notations Introduced for Approximate Formulae;482
49.6;39.6 Summation of Several Helices - Link Between Okulov's Relation and Wrench's Relation;484
49.7;References;485
50;Part VII Vortex Methods;486
51;40 A Brief Introduction to Vortex Methods;487
51.1;40.1 Introduction;487
51.2;40.2 Pros and Cons;488
51.3;40.3 An Example of Vortex Method History;490
51.4;40.4 Classification of Vortex Methods;491
51.5;40.5 Existing Vortex Codes and Application to Wind Energy;493
51.6;References;494
52;41 The Different Aspects of Vortex Methods;497
52.1;41.1 Fundamental Equations and Concepts;497
52.2;41.2 Discretization and Initialization;499
52.2.1;41.2.1 Information Carried by the Vortex Elements;499
52.2.2;41.2.2 Initialization and Reinitialization;501
52.2.3;41.2.3 Initialization - Inviscid Vortex Patch Example;502
52.3;41.3 Viscous-Splitting;503
52.3.1;41.3.1 Viscous-Splitting Algorithm;503
52.3.2;41.3.2 Rate of Convergence of the Viscous-Splitting Algorithm;504
52.3.3;41.3.3 Application to the Vorticity Transport Equation;505
52.4;41.4 Convection and Stretching of Vortex Elements;505
52.4.1;41.4.1 Introduction;505
52.4.2;41.4.2 Convection of Vortex Elements;506
52.4.3;41.4.3 Stretching;507
52.4.4;41.4.4 Applications;507
52.5;41.5 Grid-Free and Grid-Based Methods;508
52.5.1;41.5.1 Grid-Free Vortex Methods;508
52.5.2;41.5.2 Grid-Based Vortex Methods (Mixed Eulerian--Lagrangian Formulation);509
52.5.3;41.5.3 Coupled Lagrangian and Eulerian Solvers;510
52.6;41.6 Viscous Diffusion - Solution of the Diffusion Equation;510
52.6.1;41.6.1 Diffusion Equation and Vorticity Transport Equation;510
52.6.2;41.6.2 Fundamental Solution and Lamb--Oseen Vortex;511
52.6.3;41.6.3 Core-Spreading Method;513
52.6.4;41.6.4 Random-Walk Method;514
52.6.5;41.6.5 Grid-Based Finite-Differences Method;515
52.6.6;41.6.6 Particle-Strength-Exchange (PSE);515
52.6.7;41.6.7 Numerical Application: Lamb--Oseen Vortex;517
52.6.8;41.6.8 Vorticity Redistribution Method;518
52.7;41.7 Boundaries, Boundary Conditions and Lifting-Bodies;518
52.7.1;41.7.1 Introduction;518
52.7.2;41.7.2 Fluid Boundary Conditions: Free-Flow and Periodic Boundaries;519
52.7.3;41.7.3 Solid Boundaries in Inviscid Flows;519
52.7.4;41.7.4 Solid Boundaries in Viscous Flows - Vorticity Generation;520
52.7.5;41.7.5 Viscous Boundaries Using Coupling (Viscous-Inviscid or Lagrangian--Eulerian);521
52.7.6;41.7.6 Lifting-Bodies;521
52.8;41.8 Regularization - Kernel Smoothing - Mollification;521
52.8.1;41.8.1 Kernel Smoothing via Convolution with a Cut-Off Function;523
52.8.2;41.8.2 Requirements on the Cut-Off Function;523
52.8.3;41.8.3 Special Case of Spherical Symmetry;525
52.8.4;41.8.4 Examples Used in Particle Methods;528
52.8.5;41.8.5 Regularization Models for Vortex Filaments;530
52.8.6;41.8.6 Choice of Cut-Off/Smooth Parameter;531
52.8.7;41.8.7 Application to the Inviscid Vortex Patch;533
52.9;41.9 Spatial Adaptation - Redistribution - Rezoning - Reinitialization;534
52.9.1;41.9.1 Introduction;534
52.9.2;41.9.2 Remeshing - Rezoning - Redistribution - Reinitialization;534
52.9.3;41.9.3 Gain from Remeshing - Application to Inviscid-Vortex Patch;535
52.9.4;41.9.4 Problems Introduced by Remeshing;535
52.10;41.10 Subgrid-Scale Models - LES - Turbulence;536
52.11;41.11 Accuracy of Vortex Methods, Guidelines, Diagnostics and Possible Improvements;537
52.11.1;41.11.1 Guidelines and Diagnostics for General Vortex Methods;537
52.11.2;41.11.2 Boundary Elements - Guidelines and Diagnostics;539
52.11.3;41.11.3 Particle Methods - Convergence;540
52.11.4;41.11.4 Application to the Inviscid Vortex Patch;541
52.12;References;543
53;42 Particularities of Vortex Particle Methods;548
53.1;42.1 Particle Approximation and Lagrangian Methods;548
53.1.1;42.1.1 Notion of Vortex Blob;548
53.1.2;42.1.2 Particle Approximation;548
53.1.3;42.1.3 Dynamics of Lagrangian Methods;549
53.1.4;42.1.4 Incompressible Vortex Particle Methods;550
53.2;42.2 Stretching Term - Different Schemes;551
53.3;42.3 Divergence of the Vorticity Field;552
53.3.1;42.3.1 Minimizing the Error Growth;552
53.3.2;42.3.2 Corrections;553
53.3.3;42.3.3 Criteria for Correction;553
53.4;References;554
54;43 Numerical Implementation of Vortex Methods;555
54.1;43.1 Interpolation Method Required for Grid-Based Methods;555
54.1.1;43.1.1 Interpolation in Vortex Methods;555
54.1.2;43.1.2 Concept of Interpolation;556
54.1.3;43.1.3 Interpolation to Grid (Projection, Griding, Assignment, Particle-to-Mesh);558
54.1.4;43.1.4 Interpolation from Grid (Mesh-to-Particle);559
54.2;43.2 Tree-Codes and Fast Multipole Method;560
54.2.1;43.2.1 Tree-Based Method;560
54.2.2;43.2.2 Tree-Based Method - Coefficients up to Order 2;562
54.3;43.3 Poisson Solvers;563
54.4;43.4 Numerical Integration Schemes;564
54.4.1;43.4.1 Expression of the Different Schemes;564
54.4.2;43.4.2 Example of Application to the Inviscid Patch;565
54.4.3;43.4.3 Work Presented by Leishman;566
54.5;43.5 Vorticity Splitting and Merging Schemes;566
54.6;43.6 Conversion from Segments to Particles;568
54.6.1;43.6.1 Canonical Examples for Validation;568
54.6.2;43.6.2 Representation of One Segment by One Particle;569
54.6.3;43.6.3 Representation Using Several Particles;569
54.6.4;43.6.4 Trailed and Shed Vorticity Behind a Wing;570
54.7;43.7 Distribution of Control Points;570
54.7.1;43.7.1 The Work of James - Chordwise Distribution;570
54.7.2;43.7.2 Cosine Spacing and Other References in the Topic;571
54.8;43.8 The 3/4 Chord Collocation Point;572
54.9;References;573
55;44 OmniVor: An Example of Vortex Code Implementation;576
55.1;44.1 Introduction;576
55.2;44.2 Implementation and Features;577
55.3;44.3 Specific Configurations Used in Publications;585
55.4;References;586
56;45 Vortex Code Validation and Illustration;588
56.1;45.1 Simple Validation of the Vortex Particle Method;588
56.2;45.2 Lifting Line;589
56.3;45.3 Lifting Surface;590
56.4;45.4 Thick Bodies;591
56.5;45.5 Unit-Tests;592
56.6;45.6 Further Validation;593
56.7;References;593
57;Appendix A Complements on the Right Cylindrical Model and the Effect of Wake Rotation;595
58;A.1 Elementary Cylindrical System;595
59;A.2 Superposition of Cylindrical Vortex Models for Rotor Modelling;598
60;A.3 System Closure Under Assumption of Large Tip-Speed Ratio;599
61;A.4 System Closure for Finite Tip-Speed Ratio;601
62;A.5 Superposition of Cylindrical Vortex Systems with Wrong Closure;603
63;A.6 Algorithm for System Closure;604
64;Appendix B From Poisson's Equation to the Biot--Savart Law in an Unbounded Domain;606
65;B.1 Poisson's Screened Equation and Green's Function;606
66;B.1.1 Poisson's Screened Equation;606
67;B.1.2 The Use of Green Function for Solving Differential Equations;606
68;B.1.3 Resolution of Poisson Screened Equation with the Use of Fourier Transform;608
69;B.1.4 Green's Function for Poisson's Screened Equation;609
70;B.1.5 Green's Function for Poisson's Equation;610
71;B.1.6 Resolution of Poisson's Equation with the Use of Green Function;610
72;B.2 Fluid Mechanics Application;611
73;B.2.1 Velocity Induced by a Vorticity Field in an Incompressible Flow;611
74;B.3 Biot--Savart Law in Terms of Solid Angle for a Closed Path;612
75;B.3.1 Solid Angle;612
76;B.3.2 Biot--Savart Law and Solid Angle;613
77;Appendix C Useful Mathematical Relations;616
78;C.1 Useful Formulae and Theorems;616
79;C.2 Relation Between Operators;622
80;C.3 Operators in Cartesian, Cylindrical and Spherical Coordinates;623
81;C.4 Elliptic Integrals;625
82;C.4.1 Definitions;625
83;C.4.2 Properties;626
84;Index;627