Applied Gas Dynamics

A revised edition to applied gas dynamics with exclusive coverage on jets and additional sets of problems and examples

The revised and updated second edition of Applied Gas Dynamics offers an authoritative guide to the science of gas dynamics. Written by a noted expert on the topic, the text contains a comprehensive review of the topic; from a definition of the subject, to the three essential processes of this science: the isentropic process, shock and expansion process, and Fanno and Rayleigh flows.

In this revised edition, there are additional worked examples that highlight many concepts, including moving shocks, and a section on critical Mach number is included that helps to illuminate the concept. The second edition also contains new exercise problems with the answers added. In addition, the information on ram jets is expanded with helpful worked examples. It explores the entire spectrum of the ram jet theory and includes a set of exercise problems to aid in the understanding of the theory presented. This important text:

• Includes a wealth of new solved examples that describe the features involved in the design of gas dynamic devices 

• Contains a chapter on jets; this is the first textbook material available on high-speed jets
• Offers comprehensive and simultaneous coverage of both the theory and application
• Includes additional information designed to help with an understanding of the material covered

Written for graduate students and advanced undergraduates in aerospace engineering and mechanical engineering, Applied Gas Dynamics, Second Edition expands on the original edition to include not only the basic information on the science of gas dynamics but also contains information on high-speed jets.

ETHIRAJAN RATHAKRISHNAN is professor of Aerospace Engineering at the Indian Institute of Technology Kanpur, India. He is well known internationally for his research in the area of high-speed jets.

Preface xv

Author Biography xvii

About the Companion Website xix

1 Basic Facts 1

1.1 Definition of Gas Dynamics 1

1.2 Introduction 1

1.3 Compressibility 2

1.3.1 Limiting Conditions for Compressibility 3

1.4 Supersonic Flow – What is it? 4

1.5 Speed of Sound 5

1.6 Temperature Rise 7

1.7 Mach Angle 8

1.7.1 Small Disturbance 10

1.7.2 Finite Disturbance 10

1.8 Thermodynamics of Fluid Flow 11

1.9 First Law of Thermodynamics (Energy Equation) 11

1.9.1 Energy Equation for an Open System 12

1.9.2 Adiabatic Flow Process 14

1.10 The Second Law of Thermodynamics (Entropy Equation) 15

1.11 Thermal and Calorical Properties 16

1.11.1 Thermally Perfect Gas 16

1.12 The Perfect Gas 17

1.12.1 Entropy Calculation 18

1.12.2 Isentropic Relations 20

1.12.3 Limitations on Air as a Perfect Gas 25

1.13 Wave Propagation 26

1.14 Velocity of Sound 26

1.15 Subsonic and Supersonic Flows 27

1.16 Similarity Parameters 28

1.17 Continuum Hypothesis 28

1.18 Compressible Flow Regimes 30

1.19 Summary 31

Exercise Problems 34

2 Steady One-Dimensional Flow 43

2.1 Introduction 43

2.2 Fundamental Equations 43

2.3 Discharge from a Reservoir 45

2.3.1 Mass Flow Rate per Unit Area 47

2.3.2 Critical Values 51

2.4 Streamtube Area–Velocity Relation 54

2.5 de Laval Nozzle 57

2.5.1 Mass Flow Relation in Terms of Mach Number 65

2.5.2 Maximum Mass Flow Rate per Unit Area 65

2.6 Supersonic Flow Generation 66

2.6.1 Nozzles 68

2.6.2 Physics of the Nozzle Flow Process 69

2.7 Performance of Actual Nozzles 71

2.7.1 Nozzle Efficiency 71

2.7.2 Nozzle Discharge Coefficient 73

2.8 Diffusers 75

2.8.1 Special Features of Supersonic Diffusers 77

2.8.2 Supersonic Wind Tunnel Diffusers 78

2.8.3 Supersonic Inlets 81

2.8.4 Fixed-Geometry Inlet 82

2.8.5 Variable-Geometry Inlet 83

2.8.6 Diffuser Efficiency 84

2.9 Dynamic Head Measurement in Compressible Flow 88

2.9.1 Compressibility Correction to Dynamic Pressure 91

2.10 Pressure Coefficient 95

2.11 Summary 97

Exercise Problems 99

3 Normal Shock Waves 113

3.1 Introduction 113

3.2 Equations of Motion for a Normal Shock Wave 113

3.3 The Normal Shock Relations for a Perfect Gas 115

3.4 Change of Stagnation or Total Pressure Across a Shock 118

3.5 Hugoniot Equation 121

3.5.1 Moving Shocks 123

3.6 The Propagating Shock Wave 123

3.6.1 Weak Shock 128

3.6.2 Strong Shock 130

3.7 Reflected Shock Wave 133

3.8 Centered Expansion Wave 138

3.9 Shock Tube 139

3.9.1 Shock Tube Applications 142

3.10 Summary 145

Exercise Problems 148

4 Oblique Shock and Expansion Waves 155

4.1 Introduction 155

4.2 Oblique Shock Relations 156

4.3 Relation Between 𝛽 and 𝜃 158

4.4 Shock Polar 160

4.5 Supersonic Flow Over a Wedge 162

4.6 Weak Oblique Shocks 165

4.7 Supersonic Compression 167

4.8 Supersonic Expansion by Turning 169

4.9 The Prandtl–Meyer Expansion 170

4.9.1 Velocity Components V_r and V_𝜙 172

4.9.2 The Prandtl–Meyer Function 175

4.9.3 Compression 177

4.10 Simple and Nonsimple Regions 178

4.11 Reflection and Intersection of Shocks and Expansion Waves 178

4.11.1 Intersection of Shocks of the Same Family 181

4.11.2 Wave Reflection from a Free Boundary 183

4.12 Detached Shocks 189

4.13 Mach Reflection 191

4.14 Shock-Expansion Theory 197

4.15 Thin Airfoil Theory 202

4.15.1 Application of Thin Aerofoil Theory 203

4.16 Summary 210

Exercise Problems 212

5 Compressible Flow Equations 221

5.1 Introduction 221

5.2 Crocco’s Theorem 221

5.2.1 Basic Solutions of Laplace’s Equation 224

5.3 General Potential Equation for Three-Dimensional Flow 225

5.4 Linearization of the Potential Equation 226

5.4.1 Small Perturbation Theory 227

5.5 Potential Equation for Bodies of Revolution 229

5.5.1 Conclusions 230

5.5.2 Solution of Nonlinear Potential Equation 231

5.6 Boundary Conditions 231

5.6.1 Bodies of Revolution 232

5.7 Pressure Coefficient 233

5.7.1 Bodies of Revolution 234

5.8 Summary 234

Exercise Problems 237

6 Similarity Rule 239

6.1 Introduction 239

6.2 Two-Dimensional Flow: The Prandtl–Glauert Rule for Subsonic Flow 239

6.2.1 Prandtl–Glauert Transformations 239

6.2.2 The Direct Problem (Version I) 241

6.2.3 The Indirect Problem (Case of Equal Potentials): P–G Transformation (Version II) 243

6.2.4 Streamline Analogy (Version III): Gothert’s Rule 244

6.3 Prandtl–Glauert Rule for Supersonic Flow: Versions I and II 245

6.3.1 Subsonic Flow 246

6.3.2 Supersonic Flow 246

6.4 The von Karman Rule for Transonic Flow 248

6.4.1 Use of the von Karman Rule 249

6.5 Hypersonic Similarity 250

6.6 Three-Dimensional Flow: Gothert’s Rule 252

6.6.1 General Similarity Rule 252

6.6.2 Gothert’s Rule 254

6.6.3 Application toWings of Finite Span 255

6.6.4 Application to Bodies of Revolution and Fuselages 255

6.6.5 The Prandtl–Glauert Rule 257

6.6.6 The von Karman Rule for Transonic Flow 261

6.7 Critical Mach Number 261

6.7.1 Calculation of M^∗_∞ 264

6.8 Summary 266

Exercise Problems 269

7 Two-Dimensional Compressible Flows 271

7.1 Introduction 271

7.2 General Linear Solution for Supersonic Flow 271

7.2.1 Existence of Characteristics in a Physical Problem 273

7.2.2 Equation for the Streamlines from Kinematic Flow Condition 274

7.3 Flow over a Wave-Shaped Wall 276

7.3.1 Incompressible Flow 276

7.3.2 Compressible Subsonic Flow 277

7.3.3 Supersonic Flow 278

7.3.4 Pressure Coefficient 278

7.4 Summary 280

Exercise Problems 280

8 Flow with Friction and Heat Transfer 283

8.1 Introduction 283

8.2 Flow in Constant Area Duct with Friction 283

8.2.1 The Fanno Line 284

8.3 Adiabatic, Constant-Area Flow of a Perfect Gas 285

8.3.1 Definition of Friction Coefficient 286

8.3.2 Effects of Wall Friction on Fluid Properties 287

8.3.3 Second Law of Thermodynamics 288

8.3.4 Working Relations 289

8.4 Flow with Heating or Cooling in Ducts 294

8.4.1 Governing Equations 294

8.4.2 Simple-Heating Relations for a Perfect Gas 295

8.5 Summary 300

Exercise Problems 303

9 Method of Characteristics 309

9.1 Introduction 309

9.2 The Concepts of Characteristics 309

9.3 The Compatibility Relation 310

9.4 The Numerical Computational Method 312

9.4.1 Solid and Free Boundary Points 313

9.4.2 Sources of Error 316

9.4.3 Axisymmetric Flow 316

9.4.4 Nonisentropic Flow 317

9.5 Theorems for Two-Dimensional Flow 318

9.6 Numerical Computation with Weak Finite Waves 320

9.6.1 Reflection of Waves 320

9.7 Design of Supersonic Nozzle 323

9.7.1 Contour Design Details 324

9.8 Summary 328

10 Measurements in Compressible Flow 329

10.1 Introduction 329

10.2 Pressure Measurements 329

10.2.1 Liquid Manometers 329

10.2.2 Measuring Principle of Manometers 330

10.2.3 Dial-Type Pressure Gauges 332

10.2.4 Pressure Transducers 333

10.3 Temperature Measurements 335

10.4 Velocity and Direction 338

10.5 Density Problems 339

10.6 Compressible Flow Visualization 339

10.6.1 Supersonic Flows 340

10.7 Interferometer 341

10.7.1 Formation of Interference Patterns 341

10.7.2 Quantitative Evaluation 342

10.7.3 Fringe-Displacement Method 344

10.8 Schlieren System 344

10.8.1 Range and Sensitivity of the Schlieren System 347

10.8.2 Optical Components Quality Requirements 347

10.8.3 Sensitivity of the Schlieren Method for Shock and Expansion Studies 350

10.9 Shadowgraph 352

10.9.1 Comparison of the Schlieren and Shadowgraph Methods 353

10.10 Wind Tunnels 354

10.10.1 High-SpeedWind Tunnels 354

10.10.2 Blowdown TypeWind Tunnels 354

10.10.3 Induction Type Tunnels 355

10.10.4 Continuous Supersonic Wind Tunnels 356

10.10.5 Losses in Supersonic Tunnels 357

10.10.6 Supersonic Wind Tunnel Diffusers 358

10.10.7 Effects of Second Throat 360

10.10.8 Compressor Tunnel Matching 362

10.10.9 The Mass Flow Rate 365

10.10.10 Blowdown Tunnel Operation 369

10.10.11 Optimum Conditions 372

10.10.12 Running Time of Blowdown Wind Tunnels 373

10.11 Hypersonic Tunnels 375

10.11.1 Hypersonic Nozzle 377

10.12 Instrumentation and Calibration ofWind Tunnels 380

10.12.1 Calibration of SupersonicWind Tunnels 380

10.12.2 Calibration 381

10.12.3 Mach Number Determination 381

10.12.4 Pitot Pressure Measurement 382

10.12.5 Static Pressure Measurement 382

10.12.6 Determination of Flow Angularity 383

10.12.7 Determination of Turbulence Level 383

10.12.8 Determination of Test-Section Noise 384

10.12.9 Use of Calibration Results 384

10.12.10 Starting of Supersonic Tunnels 384

10.12.11 Starting Loads 385

10.12.12 Reynolds Number Effects 385

10.12.13 Model Mounting-Sting Effects 385

10.13 Calibration and Use of Hypersonic Tunnels 386

10.13.1 Calibration of Hypersonic Tunnels 386

10.13.2 Mach Number Determination 386

10.13.3 Determination of Flow Angularity 388

10.13.4 Determination of Turbulence Level 388

10.13.5 Reynolds Number Effects 389

10.13.6 Force Measurements 389

10.14 Flow Visualization 390

10.15 Summary 390

Exercise Problems 393

11 Ramjet 395

11.1 Introduction 395

11.2 The Ideal Ramjet 396

11.3 Aerodynamic Losses 401

11.4 Aerothermodynamics of Engine Components 404

11.4.1 Engine Inlets 404

11.5 Flow Through Inlets 405

11.5.1 Inlet Flow Process 406

11.5.2 Boundary Layer Separation 406

11.5.3 Flow Over the Inlet 406

11.6 Performance of Actual Intakes 410

11.6.1 Isentropic Efficiency 410

11.6.2 Stagnation Pressure Ratio 411

11.6.3 Supersonic Inlets 411

11.6.4 Supersonic Diffusers 412

11.6.5 Starting Problem 413

11.7 Shock–Boundary Layer Interaction 418

11.8 Oblique Shock Wave Incident on Flat Plate 419

11.9 Normal Shocks in Ducts 420

11.10 External Supersonic Compression 422

11.11 Two-Shock Intakes 423

11.12 Multi-Shock Intakes 427

11.13 Isentropic Compression 429

11.14 Limits of External Compression 431

11.15 External Shock Attachment 433

11.16 Internal Shock Attachment 433

11.17 Pressure Loss 434

11.18 Supersonic Combustion 442

11.19 Summary 444

Exercise Problems 447

12 Jets 451

12.1 Introduction 451

12.1.1 Subsonic Jets 453

12.2 Mathematical Treatment of Jet Profiles 454

12.3 Theory of Turbulent Jets 455

12.3.1 Mean Velocity and Mean Temperature 456

12.3.2 Turbulence Characteristics of Free Jets 457

12.3.3 Mixing Length 458

12.4 Experimental Methods for Studying Jets and the Techniques Used for Analysis 461

12.4.1 Pressure Measurement 462

12.5 Expansion Levels of Jets 464

12.5.1 Overexpanded Jets 464

12.5.2 Correctly Expanded Jets 467

12.5.3 Underexpanded Jets 469

12.6 Control of Jets 471

12.6.1 Classification of Control Methods 473

12.6.2 Role of Shear Layer in Flow Control 474

12.6.3 Supersonic Shear Layers 475

12.6.4 Use of Tabs for Jet Control 477

12.6.5 Evaluation of the Effectiveness of Some Specific Passive Controls 481

12.6.6 Grooves and Cutouts 519

12.7 Noncircular Jets and Shifted Tabs 519

12.7.1 Jet Control with Tabs 523

12.7.2 Shifted Tabs 527

12.7.3 Ventilated Triangular Tabs 532

12.7.4 Tab Edge Effect 535

12.8 Summary 541

Appendix A 547

References 619

Index 625

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