Unsteady Aerodynamics Potential and Vortex Methods

A comprehensive overview of unsteady aerodynamics and its applications

The study of unsteady aerodynamics goes back a century and has only become more significant as aircraft become increasingly sophisticated, fly faster, and their structures are lighter and more flexible. Progress in the understanding of flow physics, computing power and techniques, and modelling technologies has led to corresponding progress in unsteady aerodynamics, with a wide range of methods currently used to predict the performance of engineering structures under unsteady conditions.

Unsteady Aerodynamics offers a comprehensive and systematic overview of the application of potential and vortex methods to the subject. Beginning with an introduction to the fundamentals of unsteady flow, it then discusses the modelling of attached and separated, incompressible and compressible flows around two-dimensional and three-dimensional bodies. The result is an essential resource for design and simulation in aerospace engineering.

Unsteady Aerodynamics readers will also find:

• MATLAB examples and exercises throughout, with codes and solutions on an accompanying website
• Detailed discussion of most classes of unsteady phenomena, including flapping flight, transonic flow, dynamic stall, flow around bluff bodies and more
• Validation of theoretical and numerical predictions using comparisons to experimental data from the literature

Unsteady Aerodynamics is ideal for researchers, engineers, and advanced students in aerospace engineering.

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 vii

1 Introduction 1

1.1 Why potential and vortex methods? 2

1.2 Outline of this book 3

References 4

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 15

2.3.1 Green’s third identity 22

2.3.2 Solutions in two dimensions 43

2.4 From Navier-Stokes to unsteady compressible potential flow 44

2.4.1 The compressible Bernoulli equation 45

2.4.2 The full potential equation 46

2.4.3 The transonic small disturbance equation 48

2.4.4 The linearized small disturbance equation 49

2.4.5 The compressible unsteady pressure coefficient 51

2.4.6 Motion in a compressible, inviscid, irrotational fluid 54

2.5 Subsonic linearised potential flow solutions 56

2.6 Supersonic linearised potential flow solutions 63

2.7 Vorticity and circulation 69

2.7.1 Solutions of the vorticity transport equations 73

2.7.2 Vorticity-Moment and Kutta-Joukowski Theorems 78

2.7.3 The wake and the Kutta condition 80

2.8 Concluding Remarks 82

References 83

3 Analytical incompressible 2D models 85

3.1 Introduction 85

3.2 Steady thin airfoil theory 85

3.3 Fundamentals of Wagner and Theodorsen theory 96

3.3.1 Flow induced by the source distribution 100

3.3.2 Flow induced by the vortex distribution 105

3.3.3 Imposing the impermeability boundary condition 108

3.3.4 Calculating the loads due to the source distribution 112

3.3.5 Imposing the Kutta condition 114

3.4 Wagner Theory 117

3.4.1 TheWagner function 123

3.4.2 Drag and thrust 127

3.4.3 General motion 134

3.4.4 Total loads 136

3.4.5 Quasi-steady aerodynamics 143

3.5 Theodorsen Theory 145

3.5.1 Theodorsen’s function 148

3.5.2 Total loads for sinusoidal motion 152

3.5.3 General motion 159

3.6 Finite state theory 164

3.6.1 Glauert expansions 168

3.6.2 Solution of the impermeability equation 178

3.6.3 Completing the equations 180

3.6.4 Kutta condition and aerodynamic loads 183

3.7 Concluding Remarks 192

3.8 Exercises 193

References 193

4 Numerical incompressible 2D models 195

4.1 Introduction 195

4.2 Lumped vortex method 195

4.2.1 Unsteady flows 205

4.2.2 Free wakes 215

4.3 Gust encounters 221

4.3.1 Pitching and plunging wings 225

4.4 Frequency domain formulation of the lumped vortex method 237

4.5 Source and vortex panel method 243

4.5.1 Impulsively started flow 257

4.5.2 Thrust and propulsive efficiency 266

4.6 Theodorsen’s function and wake shape 271

4.7 Steady and unsteady Kutta conditions 273

4.7.1 The unsteady Kutta condition 280

4.8 Concluding Remarks 288

4.9 Exercises 288

References 288

5 Finite wings 291

5.1 Introduction 291

5.1.1 Rigid wings and flexible wings 292

5.2 Finite wings in steady flow 293

5.3 The impulsively started elliptical wing 302

5.3.1 The solution by Jones 302

5.3.2 Unsteady lifting line solution 314

5.4 The unsteady vortex lattice method 319

5.4.1 Impulsive start of an elliptical wing 333

5.4.2 Other planforms 340

5.5 Rigid harmonic motion 343

5.5.1 Longitudinal harmonic motion 344

5.5.2 Frequency domain load calculations 350

5.5.3 Lateral harmonic motion 356

5.5.4 Aerodynamic stability derivatives 360

5.6 The 3D source and doublet panel method 366

5.7 Flexible motion 381

5.7.1 Source and doublet panel method in the frequency domain 390

5.8 Concluding Remarks 396

5.9 Exercises 397

References 397

6 Unsteady compressible flow 399

6.1 Introduction 399

6.2 Steady subsonic potential flow 399

6.3 Unsteady subsonic potential flow 406

6.3.1 The Doublet Lattice Method 407

6.3.2 Unsteady 3D subsonic source and doublet panel method 419

6.3.3 Steady correction of the Doublet Lattice Method 432

6.3.4 Unsteady 2D subsonic source and doublet panel method 435

6.4 Unsteady supersonic potential flow 437

6.4.1 The Mach box method 438

6.4.2 The Mach panel method 447

6.5 Transonic flow 453

6.5.1 Steady transonic flow 454

6.5.2 Time linearized transonic small perturbation equation 460

6.5.3 Unsteady transonic correction methods 463

6.6 Concluding Remarks 474

6.7 Exercises 475

References 475

7 Viscous flow 477

7.1 Introduction 477

7.1.1 Steady flow separation mechanisms 479

7.1.2 Dynamic stall 484

7.2 Impulsively started flow around a 2D flat plate at high angles of attack 490

7.2.1 Flow separation criteria 498

7.3 Flow around a 2D circular cylinder 504

7.3.1 The Discrete Vortex Method for bluff bodies 507

7.3.2 Modelling the flow past a circular cylinder using the DVM 510

7.4 Flow past 2D rectangular cylinders 521

7.4.1 Modelling the flow past rectangular cylinders using the DVM 522

7.5 Concluding Remarks 527

7.6 Exercises 527

References 527

A Fundamental solutions of Laplace’s equation 529

A.1 The 2D point source 529

A.2 The 2D point vortex 531

A.3 The source line panel 533

A.4 The vortex line panel 537

A.5 The horseshoe vortex 539

A.6 The vortex line segment 542

A.7 The vortex ring 543

A.8 The 3D point source 545

A.9 The 3D point doublet 547

A.10 The source surface panel 548

A.11 The doublet surface panel 554

References 558

B Fundamental solutions of the linearized small disturbance equation 559

B.1 The subsonic doublet surface panel 559

B.2 The acoustic source surface panel 561

B.3 The acoustic doublet surface panel 562

B.4 The supersonic source surface panel 563

References 568

C Wagner’s derivation of the Kutta condition 569

References 570

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