Introduction to Compressible Fluid Flow, Second Edition, 2nd Edition

vii

Contents

Nomenclature .................................................................................................................................xi

Preface .............................................................................................................................................xv

Acknowledgments ...................................................................................................................... xix

Authors ......................................................................................................................................... xxi

1. Introduction ............................................................................................................................. 1

Compressibility ........................................................................................................................1

Fundamental Assumptions .................................................................................................... 5

Units ...........................................................................................................................................8

SI Units ..................................................................................................................................8

English System ....................................................................................................................8

Conservation Laws ................................................................................................................ 10

Concluding Remarks .............................................................................................................18

2. Equations for Steady One-Dimensional Compressible Fluid Flow ...........................21

Introduction ............................................................................................................................21

Control Volume ......................................................................................................................23

Continuity Equation .............................................................................................................. 24

Momentum Equation (Euler’s Equation) ............................................................................26

Steady Flow Energy Equation .............................................................................................. 28

Equation of State ....................................................................................................................29

Entropy Considerations ........................................................................................................32

Use of the One-Dimensional Flow Equations ...................................................................35

Concluding Remarks ............................................................................................................. 36

3. Some Fundamental Aspects of Compressible Flow ...................................................... 39

Introduction ............................................................................................................................ 39

Isentropic Flow in a Streamtube ..........................................................................................39

Speed of Sound.......................................................................................................................43

Mach Waves ............................................................................................................................ 51

Concluding Remarks ............................................................................................................. 55

4. One-Dimensional Isentropic Flow ...................................................................................59

Introduction ............................................................................................................................ 59

Governing Equations ............................................................................................................59

Stagnation Conditions ...........................................................................................................68

Critical Conditions ................................................................................................................. 76

Maximum Discharge Velocity .............................................................................................78

Isentropic Relations in Tabular and Graphical Form and from Software .....................79

Concluding Remarks .............................................................................................................81

5. Normal Shock Waves ........................................................................................................... 87

Shock Waves ...........................................................................................................................87

Stationary Normal Shock Waves ......................................................................................... 91

viii Contents

Normal Shock Wave Relations in Terms of Mach Number ...........................................100

Limiting Cases of Normal Shock Wave Relations ..........................................................108

Normal Shock Wave Tables and Software ....................................................................... 110

The Pitot Tube in Supersonic Flow .................................................................................... 112

Moving Normal Shock Waves ........................................................................................... 117

Concluding Remarks ...........................................................................................................129

6. Oblique Shock Waves ........................................................................................................ 135

Introduction ..........................................................................................................................135

Oblique Shock Wave Relations ..........................................................................................135

Reection of Oblique Shock Waves ................................................................................... 146

Interaction of Oblique Shock Waves .................................................................................156

Conical Shock Waves ........................................................................................................... 162

Concluding Remarks ...........................................................................................................165

7. Expansion Waves: Prandtl–Meyer Flow ........................................................................ 169

Introduction .......................................................................................................................... 169

Prandtl–Meyer Flow ............................................................................................................ 169

Reection and Interaction of Expansion Waves ..............................................................180

Boundary Layer Effects on Expansion Waves .................................................................184

Flow over Bodies Involving Shock and Expansion Waves ............................................ 185

Unsteady Expansion Waves ............................................................................................... 191

Concluding Remarks ........................................................................................................... 204

8. Variable Area Flow .............................................................................................................209

Introduction .......................................................................................................................... 209

Effects of Area Changes on Flow ......................................................................................209

Equations for Variable Area Flow ..................................................................................... 215

Operating Characteristics of Nozzles ............................................................................... 224

Convergent Nozzle ..............................................................................................................225

Convergent–Divergent Nozzle ...........................................................................................229

Convergent–Divergent Supersonic Diffusers .................................................................. 241

Transonic Flow over a Body ............................................................................................... 251

Concluding Remarks ........................................................................................................... 255

9. Adiabatic Flow in a Duct with Friction ..........................................................................263

Introduction .......................................................................................................................... 263

Flow in a Constant Area Duct ............................................................................................263

Friction Factor Variations ....................................................................................................278

The Fanno Line ....................................................................................................................284

Frictional Flow in a Duct Preceded by an Isentropic Nozzle ........................................287

The Effects of Friction on Variable Area Flow .................................................................295

Concluding Remarks ...........................................................................................................298

10. Flow with Heat Transfer....................................................................................................305

Introduction .......................................................................................................................... 305

Aerodynamic Heating .........................................................................................................305

The Adiabatic Surface Temperature ..................................................................................306

ixContents

Heat Transfer in High-Speed Flow ................................................................................... 310

Internal Flows with Heat Addition or Removal .............................................................. 318

One-Dimensional Flow in a Constant Area Duct Neglecting Viscosity ..................... 319

Entropy–Temperature Relations ........................................................................................ 336

Variable Area Flow with Heat Addition ...........................................................................338

One-Dimensional Constant Area Flow with Both Heat Exchange and Friction ........343

Isothermal Flow with Friction in a Constant Area Duct ...............................................348

Combustion Waves .............................................................................................................. 353

Condensation Shocks ..........................................................................................................362

Concluding Remarks ........................................................................................................... 364

External Flows .................................................................................................................364

Internal Flows .................................................................................................................. 365

Combustion and Condensation Waves ........................................................................365

11. Hypersonic Flow .................................................................................................................373

Introduction ..........................................................................................................................373

Characteristics of Hypersonic Flow .................................................................................. 373

Newtonian Theory ..............................................................................................................380

Modied Newtonian Theory .............................................................................................384

Forces on a Body ..................................................................................................................389

Concluding Remarks ...........................................................................................................396

12. Hig h-Temper atu re F lows ..................................................................................................399

Introduction .......................................................................................................................... 399

Effect of Temperature on Specic Heats ...........................................................................401

Perfect Gas Law .................................................................................................................... 410

Dissociation and Ionization ............................................................................................... 413

Nonequilibrium Effects ......................................................................................................424

Concluding Remarks ........................................................................................................... 425

13. Low-Density Flows ............................................................................................................427

Introduction .......................................................................................................................... 427

Knudsen Number ................................................................................................................427

Low-Density Flow Regimes ...............................................................................................430

Slip Flow ................................................................................................................................434

Free Molecular Flow ............................................................................................................436

Concluding Remarks ........................................................................................................... 440

14. An Introduction to Two-Dimensional Compressible Flow .......................................441

Introduction .......................................................................................................................... 441

Governing Equations ..........................................................................................................441

Continuity Equation .......................................................................................................441

Momentum Equation......................................................................................................443

Energy Equation ..............................................................................................................445

Vorticity Considerations .....................................................................................................448

The Velocity Potential..........................................................................................................451

Linearized Solutions ............................................................................................................454

Linearized Subsonic Flow ..................................................................................................459

x Contents

Linearized Supersonic Flow ...............................................................................................464

Numerical Solutions ............................................................................................................473

Concluding Remarks ...........................................................................................................473

Appendix A: COMPROP2 Code for Compressible Flow Properties ..............................475

Appendix B: Numerical Examples .........................................................................................483

Appendix C: Mini-Project–Type Assignments ....................................................................497

Appendix D: Optical Methods in Compressible Flows .....................................................499

Appendix E: Isentropic Flow Table for γ = 1.4 ..................................................................... 507

Appendix F: Normal Shock Table for γ = 1.4 ........................................................................ 513

Appendix G: Table for One-Dimensional Adiabatic Flow with Friction for γ = 1.4 .... 517

Appendix H: Table for One-Dimensional Flow with Heat Exchange for γ = 1.4 .........523

Appendix I: Table for One-Dimensional Isothermal Flow with Friction for γ = 1.4 ... 529

Appendix J: Oblique Shock Wave Charts for γ = 1.4 .......................................................... 535

Appendix K: Approximate Properties of the Standard Atmosphere ..............................541

Appendix L: Properties of Dry Air at Atmospheric Pressure ...........................................543

Appendix M: Constants, Conversion Factors, and Units ...................................................545

Bibliography ................................................................................................................................547