1 Introduction 1

1.1 Fluids as a State of Matter 1

1.2 The Fundamental Equations for Flow of a Dissipationless Fluid 3

1.3 Lagrangian Frame 4

1.4 Eulerian Frame 8

1.5 Hydrostatics 12

1.6 Streamlines 16

1.7 Bernoulli’s Equation: Weak Form 16

1.8 Polytropic Gases 17

2 Flow of Ideal Fluids 25

2.1 Introduction 25

2.2 Kelvin’s Theorem 26

2.3 Irrotational Flow 31

2.4 Irrotational Flow–Velocity Potential and the Strong Form of Bernoulli’s Equation 32

2.5 Incompressible Flow–Streamfunction 33

2.6 Irrotational Incompressible Flow 35

2.7 Induced Velocity 38

2.8 Sources and Sinks 42

2.9 Two-Dimensional Flow 51

2.10 Applications of Analytic Functions in Fluid Mechanics 52

2.11 Force on a Body in Steady Two-Dimensional Incompressible Ideal Flow 66

2.12 Conformal Transforms 69

3 Viscous Fluids 75

3.1 Basic Concept of Viscosity 75

3.2 Differential Motion of a Fluid Element 76

3.3 Strain Rate 76

3.4 Stress 77

3.5 Viscous Stress 78

3.6 Incompressible Flow–Navier–Stokes Equation 80

3.7 Stokes’ or Creeping Flow 82

3.8 Dimensionless Analysis and Similarity 86

4 Waves and Instabilities in Fluids 93

4.1 Introduction 93

4.2 Small-Amplitude Surface Waves 94

4.3 Surface Waves in Infinite fluids 102

4.4 Surface Waves with Velocity Shear Across a Contact Discontinuity 104

4.5 Shallow Water Waves 106

4.6 Waves in a Stratified Fluid 108

4.7 Stability of Laminar Shear Flow 112

4.8 Nonlinear Instability 115

5 Turbulent Flow 117

5.1 Introduction 117

5.2 Fully Developed Turbulence 121

5.3 Turbulent Stress–Reynolds Stresses 126

5.4 Similarity Model of Shear in a Turbulent Flow–von Karman’s Hypothesis 127

5.5 Velocity Profile near a Wall in Fully Developed Turbulence–Law of the Wall 127

5.6 Turbulent Flow Through a Duct 129

6 Boundary Layer Flow 139

6.1 Introduction 139

6.2 The Laminar Boundary Layer in Steady Incompressible Two-Dimensional Flow–Prandtl’s Approximation 141

6.3 Laminar Boundary Layer over an Infinite Flat Plate–Blasius’s Solution 144

6.4 Laminar Boundary Layer–von Karman’s Momentum Integral Method 146

6.4.1 Application to Boundary Layers with an Applied Pressure Gradient 149

6.5 Boundary Layer Instability and the Onset of Turbulence–Tollmein–Schlichting Instability 151

6.6 Turbulent Boundary Layer on a Flat Smooth Plate 152

6.7 Boundary Layer Separation 156

6.8 Drag 161

6.9 Laminar Wake 163

6.10 Separation in the Turbulent Boundary Layer 166

7 Convective Heat Transfer 175

7.1 Introduction 175

7.2 Forced Convection 176

7.3 Heat Transfer in a Laminar Boundary Layer 189

7.4 Heat Transfer in a Turbulent Boundary Layer on a Smooth Flat Plate 193

7.5 Free or Natural Convection 194

8 Compressible Flow and Sound Waves 209

8.1 Introduction 209

8.2 Propagation of Small Disturbances 211

8.3 Reflection and Transmission of a Sound Wave at an Interface 214

8.4 Spherical Sound Waves 215

8.5 Cylindrical Sound Waves 217

9 Characteristics and Rarefactions 219

9.1 Mach Lines and Characteristics 219

9.2 Characteristics 221

9.3 One-Dimensional Time-Dependent Expansion 224

9.4 Steady Two-Dimensional Irrotational Expansion 231

10 Shock Waves 241

10.1 Introduction 241

10.2 The Shock Transition and the Rankine–Hugoniot Equations 242

10.3 The Shock Adiabat 245

10.4 Shocks in Real Gases 250

10.5 The Hydrodynamic Structure of the Shock Front 254

10.6 The Shock Front in Real Gases 264

10.7 Shock Tubes 267

10.8 Shock Interaction 271

10.9 Oblique Shocks 277

10.10 Adiabatic Compression 287

11 Aerofoils in Low-Speed Incompressible Flow 295

11.1 Introduction 295

11.2 Two-Dimensional Aerofoils 298

11.3 Generation of Lift on an Aerofoil 301

11.4 Pitching Moment about the Wing 302

11.5 Lift from a Thin Wing 304

11.6 Application of Conformal Transforms to the Properties of Aerofoils 309

11.7 The Two-Dimensional Panel Method 314

11.8 Three-Dimensional Wings 315

11.9 Three-Dimensional Panel Method 330

12 Aerofoils in High-Speed Compressible Fluid Flow 341

12.1 Introduction 341

12.2 Linearised Theory for Two-Dimensional Flows: Subsonic Compressible Flow around a Long Thin Aerofoil – Prandtl–Glauert Correction 344

12.3 Linearised Theory for Two-Dimensional Flows: Supersonic Flow about an Aerofoil – Ackeret’s Formula 347

12.4 Drag in High-Speed Compressible Flow 350

12.5 Linearised Theory of Three-Dimensional Supersonic Flow – von Karman Ogives and Sears–Haack Bodies 354

13 Deflagrations and Detonations 363

13.1 Introduction 363

13.2 Detonations, Deflagrations and the Hugoniot Plot 368

14 Self-similar Methods in Compressible Gas Flow and Intermediate Asymptotics 383

14.1 Introduction 383

14.2 Homogeneous Self-similar Flow of a Compressible Fluid 386

14.3 Centred Self-similar Flows 395

14.4 Flow Resulting from a Point Explosion in Gas – Blast Waves 397

14.5 Adiabatic Collapse of a Sphere 402

14.6 Convergent Shock Waves – Guderley’s Solution 407

Problems 417

Solutions 427

Bibliography 455

Index 463