## Contents

Foreword xi

Series Preface xiii

Preface xv

List of Symbols xix

1 Introduction to the Conceptual Landscape 1

2 From Elementary Particles to Aerodynamic Flows 5

3 Continuum Fluid Mechanics and the Navier-Stokes Equations 13

3.1 The Continuum Formulation and Its Range of Validity 13

3.2 Mathematical Formalism 16

3.3 Kinematics: Streamlines, Streaklines, Timelines, and Vorticity 18

3.3.1 Streamlines and Streaklines 18

3.3.2 Streamtubes, Stream Surfaces, and the Stream Function 19

3.3.3 Timelines 22

3.3.4 The Divergence of the Velocity and Green’s Theorem 23

3.3.5 Vorticity and Circulation 24

3.3.6 The Velocity Potential in Irrotational Flow 26

3.3.7 Concepts that Arise in Describing the Vorticity Field 26

3.3.8 Velocity Fields Associated with Concentrations of Vorticity 29

3.3.9 The Biot-Savart Law and the “Induction” Fallacy 31

3.4 The Equations of Motion and their Physical Meaning 33

3.4.1 Continuity of the Flow and Conservation of Mass 34

3.4.2 Forces on Fluid Parcels and Conservation of Momentum 35

3.4.3 Conservation of Energy 36

3.4.4 Constitutive Relations and Boundary Conditions 37

3.4.5 Mathematical Nature of the Equations 37

3.4.6 The Physics as Viewed in the Eulerian Frame 38

3.4.7 The Pseudo-Lagrangian Viewpoint 40

3.5 Cause and Effect, and the Problem of Prediction 40

3.6 The Effects of Viscosity 43

3.7 Turbulence, Reynolds Averaging, and Turbulence Modeling 48

3.8 Important Dynamical Relationships 55

3.8.1 Galilean Invariance, or Independence of Reference Frame 55

3.8.2 Circulation Preservation and the Persistence of Irrotationality 56

3.8.3 Behavior of Vortex Tubes in Inviscid and Viscous Flows 57

3.8.4 Bernoulli Equations and Stagnation Conditions 58

3.8.5 Crocco’s Theorem 60

3.9 Dynamic Similarity 60

3.9.1 Compressibility Effects and the Mach Number 63

3.9.2 Viscous Effects and the Reynolds Number 63

3.9.3 Scaling of Pressure Forces: the Dynamic Pressure 64

3.9.4 Consequences of Failing to Match All of the Requirements

for Similarity 65

3.10 “Incompressible” Flow and Potential Flow 66

3.11 Compressible Flow and Shocks 70

3.11.1 Steady 1D Isentropic Flow Theory 71

3.11.2 Relations for Normal and Oblique Shock Waves 74

4 Boundary Layers 79

4.1 Physical Aspects of Boundary-Layer Flows 80

4.1.1 The Basic Sequence: Attachment, Transition, Separation 80

4.1.2 General Development of the Boundary-Layer Flowfield 82

4.1.3 Boundary-Layer Displacement Effect 90

4.1.4 Separation from a Smooth Wall 93

4.2 Boundary-Layer Theory 99

4.2.1 The Boundary-Layer Equations 100

4.2.2 Integrated Momentum Balance in a Boundary Layer 108

4.2.3 The Displacement Effect and Matching with the Outer Flow 110

4.2.4 The Vorticity “Budget” in a 2D Incompressible Boundary Layer 113

4.2.5 Situations That Violate the Assumptions of Boundary-Layer

Theory 114

4.2.6 Summary of Lessons from Boundary-Layer Theory 117

4.3 Flat-Plate Boundary Layers and Other Simplified Cases 117

4.3.1 Flat-Plate Flow 117

4.3.2 2D Boundary-Layer Flows with Similarity 121

4.3.3 Axisymmetric Flow 123

4.3.4 Plane-of-Symmetry and Attachment-Line Boundary Layers 125

4.3.5 Simplifying the Effects of Sweep and Taper in 3D 128

4.4 Transition and Turbulence 130

4.4.1 Boundary-Layer Transition 131

4.4.2 Turbulent Boundary Layers 138

4.5 Control and Prevention of Flow Separation 150

4.5.1 Body Shaping and Pressure Distribution 150

4.5.2 Vortex Generators 150

4.5.3 Steady Tangential Blowing through a Slot 155

4.5.4 Active Unsteady Blowing 157

4.5.5 Suction 157

4.6 Heat Transfer and Compressibility 158

4.6.1 Heat Transfer, Compressibility, and the Boundary-Layer

Temperature Field 158

4.6.2 The Thermal Energy Equation and the Prandtl Number 159

4.6.3 The Wall Temperature and Other Relations for an Adiabatic Wall 159

4.7 Effects of Surface Roughness 162

5 General Features of Flows around Bodies 163

5.1 The Obstacle Effect 164

5.2 Basic Topology of Flow Attachment and Separation 168

5.2.1 Attachment and Separation in 2D 169

5.2.2 Attachment and Separation in 3D 171

5.2.3 Streamline Topology on Surfaces and in Cross Sections 176

5.3 Wakes 186

5.4 Integrated Forces: Lift and Drag 189

6 Drag and Propulsion 191

6.1 Basic Physics and Flowfield Manifestations of Drag and Thrust 192

6.1.1 Basic Physical Effects of Viscosity 193

6.1.2 The Role of Turbulence 193

6.1.3 Direct and Indirect Contributions to the Drag Force

on the Body 194

6.1.4 Determining Drag from the Flowfield: Application

of Conservation Laws 196

6.1.5 Examples of Flowfield Manifestations of Drag in Simple

2D Flows 204

6.1.6 Pressure Drag of Streamlined and Bluff Bodies 207

6.1.7 Questionable Drag Categories: Parasite Drag, Base Drag,

and Slot Drag 210

6.1.8 Effects of Distributed Surface Roughness on Turbulent

Skin Friction 212

6.1.9 Interference Drag 222

6.1.10 Some Basic Physics of Propulsion 225

6.2 Drag Estimation 241

6.2.1 Empirical Correlations 242

6.2.2 Effects of Surface Roughness on Turbulent Skin Friction 243

6.2.3 CFD Prediction of Drag 250

6.3 Drag Reduction 250

6.3.1 Reducing Drag by Maintaining a Run of Laminar Flow 251

6.3.2 Reduction of Turbulent Skin Friction 251

7 Lift and Airfoils in 2D at Subsonic Speeds 259

7.1 Mathematical Prediction of Lift in 2D 260

7.2 Lift in Terms of Circulation and Bound Vorticity 265

7.2.1 The Classical Argument for the Origin of the Bound Vorticity 267

7.3 Physical Explanations of Lift in 2D 269

7.3.1 Past Explanations and their Strengths and Weaknesses 269

7.3.2 Desired Attributes of a More Satisfactory Explanation 284

7.3.3 A Basic Explanation of Lift on an Airfoil, Accessible

to a Nontechnical Audience 286

7.3.4 More Physical Details on Lift in 2D, for the Technically Inclined 302

7.4 Airfoils 307

7.4.1 Pressure Distributions and Integrated Forces

at Low Mach Numbers 307

7.4.2 Profile Drag and the Drag Polar 316

7.4.3 Maximum Lift and Boundary-Layer Separation

on Single-Element Airfoils 319

7.4.4 Multielement Airfoils and the Slot Effect 329

7.4.5 Cascades 335

7.4.6 Low-Drag Airfoils with Laminar Flow 338

7.4.7 Low-Reynolds-Number Airfoils 341

7.4.8 Airfoils in Transonic Flow 342

7.4.9 Airfoils in Ground Effect 350

7.4.10 Airfoil Design 352

7.4.11 Issues that Arise in Defining Airfoil Shapes 354

8 Lift and Wings in 3D at Subsonic Speeds 359

8.1 The Flowfield around a 3D Wing 359

8.1.1 General Characteristics of the Velocity Field 359

8.1.2 The Vortex Wake 362

8.1.3 The Pressure Field around a 3D Wing 371

8.1.4 Explanations for the Flowfield 371

8.1.5 Vortex Shedding from Edges Other Than the Trailing Edge 375

8.2 Distribution of Lift on a 3D Wing 376

8.2.1 Basic and Additional Spanloads 376

8.2.2 Linearized Lifting-Surface Theory 379

8.2.3 Lifting-Line Theory 380

8.2.4 3D Lift in Ground Effect 382

8.2.5 Maximum Lift, as Limited by 3D Effects 384

8.3 Induced Drag 385

8.3.1 Basic Scaling of Induced Drag 385

8.3.2 Induced Drag from a Farfield Momentum Balance 386

8.3.3 Induced Drag in Terms of Kinetic Energy and an Idealized

Rolled-Up Vortex Wake 389

8.3.4 Induced Drag from the Loading on the Wing Itself:

Trefftz-Plane Theory 391

8.3.5 Ideal (Minimum) Induced-Drag Theory 394

8.3.6 Span-Efficiency Factors 396

8.3.7 The Induced-Drag Polar 397

8.3.8 The Sin-Series Spanloads 398

8.3.9 The Reduction of Induced Drag in Ground Effect 401

8.3.10 The Effect of a Fuselage on Induced Drag 402

8.3.11 Effects of a Canard or Aft Tail on Induced Drag 404

8.3.12 Biplane Drag 409

8.4 Wingtip Devices 411

8.4.1 Myths Regarding the Vortex Wake, and Some Questionable

Ideas for Wingtip Devices 411

8.4.2 The Facts of Life Regarding Induced Drag and Induced-Drag

Reduction 414

8.4.3 Milestones in the Development of Theory and Practice 420

8.4.4 Wingtip Device Concepts 422

8.4.5 Effectiveness of Various Device Configurations 423

8.5 Manifestations of Lift in the Atmosphere at Large 427

8.5.1 The Net Vertical Momentum Imparted to the Atmosphere 427

8.5.2 The Pressure Far above and below the Airplane 429

8.5.3 Downwash in the Trefftz Plane and Other

Momentum-Conservation Issues 431

8.5.4 Sears’s Incorrect Analysis of the Integrated Pressure

Far Downstream 435

8.5.5 The Real Flowfield Far Downstream of the Airplane 436

8.6 Effects of Wing Sweep 444

8.6.1 Simple Sweep Theory 444

8.6.2 Boundary Layers on Swept Wings 449

8.6.3 Shock/Boundary-Layer Interaction on Swept Wings 464

8.6.4 Laminar-to-Turbulent Transition on Swept Wings 465

8.6.5 Relating a Swept, Tapered Wing to a 2D Airfoil 468

8.6.6 Tailoring of the Inboard Part of a Swept Wing 469

9 Theoretical Idealizations Revisited 471

9.1 Approximations Grouped According to how the Equations

were Modified 471

9.1.1 Reduced Temporal and/or Spatial Resolution 472

9.1.2 Simplified Theories Based on Neglecting Something Small 472

9.1.3 Reductions in Dimensions 472

9.1.4 Simplified Theories Based on Ad hoc Flow Models 472

9.1.5 Qualitative Anomalies and Other Consequences

of Approximations 481

9.2 Some Tools of MFD (Mental Fluid Dynamics) 482

9.2.1 Simple Conceptual Models for Thinking about Velocity Fields 482

9.2.2 Thinking about Viscous and Shock Drag 485

9.2.3 Thinking about Induced Drag 486

9.2.4 A Catalog of Fallacies 487

10 Modeling Aerodynamic Flows in Computational Fluid Dynamics 491

10.1 Basic Definitions 493

10.2 The Major Classes of CFD Codes and Their Applications 493

10.2.1 Navier-Stokes Methods 493

10.2.2 Coupled Viscous/Inviscid Methods 497

10.2.3 Inviscid Methods 498

10.2.4 Standalone Boundary-Layer Codes 501

10.3 Basic Characteristics of Numerical Solution Schemes 501

10.3.1 Discretization 501

10.3.2 Spatial Field Grids 502

10.3.3 Grid Resolution and Grid Convergence 506

10.3.4 Solving the Equations, and Iterative Convergence 507

10.4 Physical Modeling in CFD 508

10.4.1 Compressibility and Shocks 508

10.4.2 Viscous Effects and Turbulence 510

10.4.3 Separated Shear Layers and Vortex Wakes 511

10.4.4 The Farfield 513

10.4.5 Predicting Drag 514

10.4.6 Propulsion Effects 515

10.5 CFD Validation? 515

10.6 Integrated Forces and the Components of Drag 516

10.7 Solution Visualization 517

10.8 Things a User Should Know about a CFD Code before Running it 524

References 527

Index 539

## Preface

The field of aerospace is wide ranging and multi-disciplinary, covering a large variety of

products, disciplines and domains, not merely in engineering but in many related supporting

activities. These combine to enable the aerospace industry to produce exciting and technologically

advanced vehicles. The wealth of knowledge and experience that has been gained

by expert practitioners in the various aerospace fields needs to be passed onto others working

in the industry, including those just entering from University.

The Aerospace Series aims to be a practical and topical series of books aimed at engineering

professionals, operators, users and allied professions such as commercial and legal

executives in the aerospace industry, and also engineers in academia. The range of topics is

intended to be wide ranging, covering design and development, manufacture, operation and

support of aircraft as well as topics such as infrastructure operations and developments in

research and technology. The intention is to provide a source of relevant information that

will be of interest and benefit to all those people working in aerospace.

Aerodynamics is the fundamental enabling science that underpins the world-wide aerospace

industry – without the ability to generate lift from airflow passing over wings, helicopter

rotors and other lifting surfaces, it would not be possible to fly heavier-than-air vehicles as

efficiently as is taken for granted nowadays. Much of the development of today’s highly

efficient aircraft is due to the ability to accurately model aerodynamic flows using sophisticated

computational codes and thus design high-performance wings; however, a thorough

understanding and insight of the aerodynamic flows is vital for engineers to comprehend

these designs.

This book, Understanding Aerodynamics, has the objective of providing a physical understanding

of aerodynamics, with an emphasis on how and why particular flow patterns around

bodies occur, and what relation these flows have to the underlying physical laws. It is a welcome

addition to the Wiley Aerospace Series. Unlike most aerodynamics textbooks, there

is a refreshing lack of detailed mathematical analysis, and the reader is encouraged instead

to consider the overall picture. As well as consideration of classical topics – continuum

fluid mechanics, boundary layers, lift, drag and the flow around wings, etc. – there is also a

very useful coverage of modelling aerodynamic flows using Computational Fluid Dynamics

(CFD).