Wind Turbine Aerodynamics and Vorticity-Based Methods (eBook)

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