Grigorios Dimitriadis, PhD, is Professor of Fluid Structure Interaction and Experimental Aerodynamics in the Aerospace and Mechanical Engineering Department, University of Liège, Belgium and Adjunct Professor in Aeroelasticity at the von Karman Institute for Fluid Dynamics, Belgium. He has published extensively on unsteady aerodynamics and related fields.

Preface xi

About the Companion Website xiii

1 Introduction 1

1.1 Why Potential and Vortex Methods? 2

1.2 Outline of This Book 3

2 Unsteady Flow Fundamentals 5

2.1 Introduction 5

2.2 From Navier-Stokes to Unsteady Incompressible Potential Flow 5

2.2.1 Irrotational Flow 6

2.2.2 Laplace's and Bernoulli's Equations 7

2.2.3 Motion in an Incompressible, Inviscid, Irrotational Fluid 9

2.3 Incompressible Potential Flow Solutions 14

2.3.1 Green's Third Identity 21

2.3.2 Solutions in Two Dimensions 40

2.4 From Navier-Stokes to Unsteady Compressible Potential Flow 42

2.4.1 The Compressible Bernoulli Equation 42

2.4.2 The Full Potential Equation 44

2.4.3 The Transonic Small Disturbance Equation 46

2.4.4 The Linearised Small Disturbance Equation 47

2.4.5 The Compressible Unsteady Pressure Coefficient 49

2.4.6 Motion in a Compressible, Inviscid, Irrotational Fluid 52

2.5 Subsonic Linearised Potential Flow Solutions 53

2.6 Supersonic Linearised Potential Flow Solutions 61

2.7 Vorticity and Circulation 66

2.7.1 Solutions of the Vorticity Transport Equations 71

2.7.2 Vorticity-Moment and Kutta-Joukowski Theorems 76

2.7.3 TheWake and the Kutta Condition 77

2.8 Concluding Remarks 79

3 Analytical Incompressible 2D Models 83

3.1 Introduction 83

3.2 Steady Thin Airfoil Theory 83

3.3 Fundamentals ofWagner and Theodorsen Theory 93

3.3.1 Flow Induced by the Source Distribution 97

3.3.2 Flow Induced by the Vortex Distribution 101

3.3.3 Imposing the Impermeability Boundary Condition 104

3.3.4 Calculating the Loads Due to the Source Distribution 108

3.3.5 Imposing the Kutta Condition 111

3.4 Wagner Theory 113

3.4.1 TheWagner Function 120

3.4.2 Drag and Thrust 123

3.4.3 General Motion 129

3.4.4 Total Loads 131

3.4.5 Quasi-Steady Aerodynamics 138

3.5 Theodorsen Theory 139

3.5.1 Theodorsen's Function 143

3.5.2 Total Loads for Sinusoidal Motion 146

3.5.3 General Motion 153

3.6 Finite State Theory 157

3.6.1 Glauert Expansions 161

3.6.2 Solution of the Impermeability Equation 170

3.6.3 Completing the Equations 172

3.6.4 Kutta Condition and Aerodynamic Loads 175

3.7 Concluding Remarks 183

3.8 Exercises 184

4 Numerical Incompressible 2D Models 187

4.1 Introduction 187

4.2 Lumped Vortex Method 187

4.2.1 Unsteady Flows 197

4.2.2 FreeWakes 206

4.3 Gust Encounters 212

4.3.1 Pitching and Plunging Wings 216

4.4 Frequency Domain Formulation of the Lumped Vortex Method 227

4.5 Source and Vortex Panel Method 233

4.5.1 Impulsively Started Flow 245

4.5.2 Thrust and Propulsive Efficiency 254

4.6 Theodorsen's Function andWake Shape 259

4.7 Steady and Unsteady Kutta Conditions 261

4.7.1 The Unsteady Kutta Condition 267

4.8 Concluding Remarks 275

4.9 Exercises 275

5 Finite Wings 279

5.1 Introduction 279

5.1.1 Rigid Wings and Flexible Wings 280

5.2 Finite Wings in Steady Flow 281

5.3 The Impulsively Started Elliptical Wing 290

5.3.1 The Solution by Jones 290

5.3.2 Unsteady Lifting Line Solution 302

5.4 The Unsteady Vortex Lattice Method 306

5.4.1 Impulsive Start of an Elliptical Wing 320

5.4.2 Other Planforms 326

5.5 Rigid Harmonic Motion 329

5.5.1 Longitudinal Harmonic Motion 329

5.5.2 Frequency Domain Load Calculations 335

5.5.3 Lateral Harmonic Motion 341

5.5.4 Aerodynamic Stability Derivatives 345

5.6 The 3D Source and Doublet Panel Method 351

5.7 Flexible Motion 364

5.7.1 Source and Doublet Panel Method in the Frequency Domain 372

5.8 Concluding Remarks 378

5.9 Exercises 379

6 Unsteady Compressible Flow 383

6.1 Introduction 383

6.2 Steady Subsonic Potential Flow 383

6.3 Unsteady Subsonic Potential Flow 390

6.3.1 The Doublet Lattice Method 391

6.3.2 Unsteady 3D Subsonic Source and Doublet Panel Method 402

6.3.3 Steady Correction of the Doublet Lattice Method 414

6.3.4 Unsteady 2D Subsonic Source and Doublet Panel Method 416

6.4 Unsteady Supersonic Potential Flow 419

6.4.1 The Mach Box Method 420

6.4.2 The Mach Panel Method 428

6.5 Transonic Flow 434

6.5.1 Steady Transonic Flow 435

6.5.2 Time Linearised Transonic Small Perturbation Equation 440

6.5.3 Unsteady Transonic Correction Methods 443

6.6 Concluding Remarks 453

6.7 Exercises 454

7 Viscous Flow 459

7.1 Introduction 459

7.1.1 Steady Flow Separation Mechanisms 461

7.1.2 Dynamic Stall 466

7.2 Impulsively Started Flow around a 2D Flat Plate at High Angles of Attack 472

7.2.1 Flow Separation Criteria 480

7.3 Flow Around a 2D Circular Cylinder 485

7.3.1 The Discrete Vortex Method for Bluff Bodies 488

7.3.2 Modelling the Flow Past a Circular Cylinder Using the DVM 491

7.4 Flow Past 2D Rectangular Cylinders 501

7.4.1 Modelling the Flow Past Rectangular Cylinders Using the DVM 502

7.5 Concluding Remarks 507

7.6 Exercises 507

A Fundamental Solutions of Laplace's Equation 511

A.1 The 2D Point Source 511

A.2 The 2D Point Vortex 513

A.3 The Source Line Panel 515

A.4 The Vortex Line Panel 518

A.5 The Horseshoe Vortex 521

A.6 The Vortex Line Segment 523

A.7 The Vortex Ring 525

A.8 The 3D Point Source 526

A.9 The 3D Point Doublet 528

A.10 The Source Surface Panel 528

A.11 The Doublet Surface Panel 534

B Fundamental Solutions of the Linearized Small Disturbance Equation 539

B.1 The Subsonic Doublet Surface Panel 539

B.2 The Acoustic Source Surface Panel 541

B.3 The Acoustic Doublet Surface Panel 542

B.4 The Supersonic Source Surface Panel 543

C Wagner's Derivation of the Kutta Condition 549

Reference 550

Index 551