Bar-Meir G. Fundamentals of Compressible Fluid
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Fundamentals of Compressible Fluid
Mechanics
Genick Bar–Meir, Ph. D.
1107 16
th
Ave S. E.
Minneapolis, MN 55414-2411
email: “barmeir@gmail.com”
Copyright 2009, 2008, 2007, 2006, 2005, and 2004 by Genick Bar-Meir
See the file copying.fdl or copyright.tex for copying conditions.
Version (0.4.8.6 October 23, 2009)
‘We are like dwarfs sitting on the shoulders of giants”
from The Metalogicon by John in 1159
CONTENTS
Nomenclature xv
Feb-21-2007 version . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
Jan-16-2007 version . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
Dec-04-2006 version . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
GNU Free Documentation License . . . . . . . . . . . . . . . . . . . . . . . xxv
1. APPLICABILITY AND DEFINITIONS . . . . . . . . . . . . . . . . xxvi
2. VERBATIM COPYING . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
3. COPYING IN QUANTITY . . . . . . . . . . . . . . . . . . . . . . . xxvii
4. MODIFICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii
5. COMBINING DOCUMENTS . . . . . . . . . . . . . . . . . . . . . xxx
6. COLLECTIONS OF DOCUMENTS . . . . . . . . . . . . . . . . . . xxx
7. AGGREGATION WITH INDEPENDENT WORKS . . . . . . . . . . xxxi
8. TRANSLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi
9. TERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi
10. FUTURE REVISIONS OF THIS LICENSE . . . . . . . . . . . . . . xxxi
ADDENDUM: How to use this License for your documents . . . . . . .
How to contribute to this book . . . . . . . . . . . . . . . . . . . . . . . . i
Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
John Martones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Grigory Toker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Ralph Menikoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Domitien Rataaforret . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Gary Settles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Your name here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Typo corrections and other ”minor” contributions . . . . . . . . . . . . ii
Version 0.4.8.5a . July 21, 2009 . . . . . . . . . . . . . . . . . . . . . . . . xiii
iii
iv CONTENTS
Version 0.4.8 Jan. 23, 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Version 0.4.3 Sep. 15, 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Version 0.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Version 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Version 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Version 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
Version 0.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii
Version 0.4.1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii
Speed of Sound [beta] . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi
Stagnation effects [advance] . . . . . . . . . . . . . . . . . . . . . . . xxvi
Nozzle [advance] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi
Normal Shock [advace] . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi
Minnor Loss [NSV] . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
Isothermal Flow [advace] . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
Fanno Flow [advace] . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
Rayleigh Flow [beta] . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
Evacuation and filling semi rigid Chambers [alpha] . . . . . . . . . . . . xxvii
Evacuating and filling chambers under external forces [alpha] . . . . . . xxvii
Oblique Shock [advace] . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii
Prandtl–Meyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii
Transient problem [NYP] . . . . . . . . . . . . . . . . . . . . . . . . . xxviii
General 1-D flow [NYP] . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii
1 Introduction 1
1.1 What is Compressible Flow ? . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Why Compressible Flow is Important? . . . . . . . . . . . . . . . . . . 2
1.3 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.1 Early Developments . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 The shock wave puzzle . . . . . . . . . . . . . . . . . . . . . . 5
1.3.3 Choking Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.4 External flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.5 Filling and Evacuating Gaseous Chambers . . . . . . . . . . . . 14
1.3.6 Biographies of Major Figures . . . . . . . . . . . . . . . . . . . 14
2 Review of Thermodynamics 25
2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 The Velocity–Temperature Diagram . . . . . . . . . . . . . . . . . . . 32
3 Fundamentals of Basic Fluid Mechanics 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Control Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Reynold’s Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . 37
CONTENTS v
4 Speed of Sound 39
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Speed of sound in ideal and perfect gases . . . . . . . . . . . . . . . . 41
4.4 Speed of Sound in Real Gas . . . . . . . . . . . . . . . . . . . . . . . 44
4.5 Speed of Sound in Almost Incompressible Liquid . . . . . . . . . . . . . 47
4.6 Speed of Sound in Solids . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.7 Sound Speed in Two Phase Medium . . . . . . . . . . . . . . . . . . . 49
5 Isentropic Flow 53
5.1 Stagnation State for Ideal Gas Model . . . . . . . . . . . . . . . . . . . 53
5.1.1 General Relationship . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1.2 Relationships for Small Mach Number . . . . . . . . . . . . . . 56
5.2 Isentropic Converging-Diverging Flow in Cross Section . . . . . . . . . 57
5.2.1 The Properties in the Adiabatic Nozzle . . . . . . . . . . . . . . 58
5.2.2 Isentropic Flow Examples . . . . . . . . . . . . . . . . . . . . . 62
5.2.3 Mass Flow Rate (Number) . . . . . . . . . . . . . . . . . . . . 65
5.3 Isentropic Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.1 Isentropic Isothermal Flow Nozzle . . . . . . . . . . . . . . . . 76
5.3.2 General Relationship . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 The Impulse Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4.1 Impulse in Isentropic Adiabatic Nozzle . . . . . . . . . . . . . . 83
5.4.2 The Impulse Function in Isothermal Nozzle . . . . . . . . . . . 85
5.5 Isothermal Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.6 The effects of Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Normal Shock 93
6.1 Solution of the Governing Equations . . . . . . . . . . . . . . . . . . . 95
6.1.1 Informal Model . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.1.2 Formal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.1.3 Prandtl’s Condition . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 Operating Equations and Analysis . . . . . . . . . . . . . . . . . . . . 100
6.2.1 The Limitations of the Shock Wave . . . . . . . . . . . . . . . 102
6.2.2 Small Perturbation Solution . . . . . . . . . . . . . . . . . . . . 102
6.2.3 Shock Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.4 Shock or Wave Drag . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 The Moving Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3.1 Shock or Wave Drag Result from a Moving Shock . . . . . . . . 106
6.3.2 Shock Result from a Sudden and Complete Stop . . . . . . . . 108
6.3.3 Moving Shock into Stationary Medium (Suddenly Open Valve) . 111
6.3.4 Partially Open Valve . . . . . . . . . . . . . . . . . . . . . . . 121
6.3.5 Partially Closed Valve . . . . . . . . . . . . . . . . . . . . . . . 122
6.3.6 Worked–out Examples for Shock Dynamics . . . . . . . . . . . 123
6.4 Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.5 Shock with Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . . 132
vi CONTENTS
6.6 Shock in Wet Steam . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.7 Normal Shock in Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.8 More Examples for Moving Shocks . . . . . . . . . . . . . . . . . . . . 133
6.9 Tables of Normal Shocks, k = 1.4 Ideal Gas . . . . . . . . . . . . . . . 134
7 Normal Shock in Variable Duct Areas 141
7.1 Nozzle efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.2 Diffuser Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8 Nozzle Flow With External Forces 153
8.1 Isentropic Nozzle (Q = 0) . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.2 Isothermal Nozzle (T = constant) . . . . . . . . . . . . . . . . . . . . 156
9 Isothermal Flow 157
9.1 The Control Volume Analysis/Governing equations . . . . . . . . . . . 158
9.2 Dimensionless Representation . . . . . . . . . . . . . . . . . . . . . . 158
9.3 The Entrance Limitation of Supersonic Branch . . . . . . . . . . . . . 163
9.4 Comparison with Incompressible Flow . . . . . . . . . . . . . . . . . . 164
9.5 Supersonic Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9.6 Figures and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.7 Isothermal Flow Examples . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.8 Unchoked situations in Fanno Flow . . . . . . . . . . . . . . . . . . . . 172
10 Fanno Flow 177
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.2 Fanno Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
10.3 Non–Dimensionalization of the Equations . . . . . . . . . . . . . . . . 179
10.4 The Mechanics and Why the Flow is Choked? . . . . . . . . . . . . . . 182
10.5 The Working Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 183
10.6 Examples of Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.7 Supersonic Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.8 Maximum Length for the Supersonic Flow . . . . . . . . . . . . . . . . 192
10.9 Working Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
10.9.1 Variations of The Tube Length (
4fL
D
) Effects . . . . . . . . . . 193
10.9.2 The Pressure Ratio,
P
2
P
1
, effects . . . . . . . . . . . . . . . . . . 197
10.9.3 Entrance Mach number, M
1
, effects . . . . . . . . . . . . . . . 201
10.10 Practical Examples for Subsonic Flow . . . . . . . . . . . . . . . . . . 207
10.10.1 Subsonic Fanno Flow for Given
4fL
D
and Pressure Ratio . . . . 208
10.10.2 Subsonic Fanno Flow for a Given M
1
and Pressure Ratio . . . . 210
10.11 The Approximation of the Fanno Flow by Isothermal Flow . . . . . . . 212
10.12 More Examples of Fanno Flow . . . . . . . . . . . . . . . . . . . . . . 213
10.13 The Table for Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . . 214
10.14 Appendix – Reynolds Number Effects . . . . . . . . . . . . . . . . . . 216
CONTENTS vii
11 Rayleigh Flow 219
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
11.2 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
11.3 Rayleigh Flow Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
11.4 Examples For Rayleigh Flow . . . . . . . . . . . . . . . . . . . . . . . 225
12 Evacuating SemiRigid Chambers 233
12.1 Governing Equations and Assumptions . . . . . . . . . . . . . . . . . . 234
12.2 General Model and Non-dimensioned . . . . . . . . . . . . . . . . . . 236
12.2.1 Isentropic Process . . . . . . . . . . . . . . . . . . . . . . . . . 237
12.2.2 Isothermal Process in The Chamber . . . . . . . . . . . . . . . 238
12.2.3 A Note on the Entrance Mach number . . . . . . . . . . . . . . 238
12.3 Rigid Tank with Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . 239
12.3.1 Adiabatic Isentropic Nozzle Attached . . . . . . . . . . . . . . . 239
12.3.2 Isothermal Nozzle Attached . . . . . . . . . . . . . . . . . . . . 241
12.4 Rapid evacuating of a rigid tank . . . . . . . . . . . . . . . . . . . . . 241
12.4.1 With Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . . . 241
12.4.2 Filling Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
12.4.3 The Isothermal Process . . . . . . . . . . . . . . . . . . . . . . 244
12.4.4 Simple Semi Rigid Chamb er . . . . . . . . . . . . . . . . . . . 244
12.4.5 The “Simple” General Case . . . . . . . . . . . . . . . . . . . . 245
12.5 Advance Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
13 Evacuating under External Volume Control 249
13.1 General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
13.1.1 Rapid Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
13.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
13.1.3 Direct Connection . . . . . . . . . . . . . . . . . . . . . . . . . 253
13.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
14 Oblique Shock 257
14.1 Preface to Oblique Shock . . . . . . . . . . . . . . . . . . . . . . . . . 257
14.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
14.2.1 Introduction to Oblique Shock . . . . . . . . . . . . . . . . . . 258
14.2.2 Introduction to Prandtl–Meyer Function . . . . . . . . . . . . . 258
14.2.3 Introduction to Zero Inclination . . . . . . . . . . . . . . . . . . 259
14.3 Oblique Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
14.4 Solution of Mach Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 262
14.4.1 Upstream Mach Number, M
1
, and Deflection Angle, δ . . . . . 262
14.4.2 When No Oblique Sho ck Exist or When D > 0 . . . . . . . . . 265
14.4.3 Upstream Mach Number, M
1
, and Shock Angle, θ . . . . . . . 273
14.4.4 Given Two Angles, δ and θ . . . . . . . . . . . . . . . . . . . 274
14.4.5 Flow in a Semi–2D Shape . . . . . . . . . . . . . . . . . . . . . 276
14.4.6 Small δ “Weak Oblique shock” . . . . . . . . . . . . . . . . . . 278
14.4.7 Close and Far Views of the Oblique Shock . . . . . . . . . . . . 278
viii CONTENTS
14.4.8 Maximum Value of Oblique shock . . . . . . . . . . . . . . . . 279
14.5 Detached Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
14.5.1 Issues Related to the Maximum Deflection Angle . . . . . . . . 281
14.5.2 Oblique Shock Examples . . . . . . . . . . . . . . . . . . . . . 282
14.5.3 Application of Oblique Shock . . . . . . . . . . . . . . . . . . . 285
14.5.4 Optimization of Suction Section Design . . . . . . . . . . . . . 296
14.5.5 Retouch of Shock or Wave Drag . . . . . . . . . . . . . . . . . 296
14.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
14.7 Appendix: Oblique Shock Stability Analysis . . . . . . . . . . . . . . . 298
15 Prandtl-Meyer Function 301
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
15.2 Geometrical Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . 302
15.2.1 Alternative Approach to Governing Equations . . . . . . . . . . 303
15.2.2 Comparison And Limitations between the Two Approaches . . . 307
15.3 The Maximum Turning Angle . . . . . . . . . . . . . . . . . . . . . . . 307
15.4 The Working Equations for the Prandtl-Meyer Function . . . . . . . . . 308
15.5 d’Alembert’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
15.6 Flat Body with an Angle of Attack . . . . . . . . . . . . . . . . . . . . 309
15.7 Examples For Prandtl–Meyer Function . . . . . . . . . . . . . . . . . . 309
15.8 Combination of the Oblique Sho ck and Isentropic Expansion . . . . . . 312
A Computer Program 317
A.1 About the Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
A.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
A.3 Program listings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Index 321
Subjects Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Authors Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
LIST OF FIGURES
1.1 The shock as a connection of Fanno and Rayleigh lines . . . . . . . . . 7
1.2 The schematic of deLavel’s turbine . . . . . . . . . . . . . . . . . . . . 9
1.3 The measured pressure in a nozzle . . . . . . . . . . . . . . . . . . . . 10
1.4 Flow rate as a function of the back pressure . . . . . . . . . . . . . . . 11
1.5 Portrait of Galileo Galilei . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Photo of Ernest Mach . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 The bullet photo of in a supersonic flow taken by Mach . . . . . . . . . 15
1.8 Lord Rayleigh portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.9 Portrait of Rankine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.10 The photo of Gino Fanno approximately in 1950 . . . . . . . . . . . . . 18
1.11 Photo of Prandtl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.12 The photo of Thedor Meyer . . . . . . . . . . . . . . . . . . . . . . . . 20
1.13 The diagrams taken from Meyer thesis. . . . . . . . . . . . . . . . . . . 21
1.14 The photo of Ernst Rudolf George Eckert with Bar-Meir’s family . . . . 22
2.1 The energy lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 The velocity temperature diagram . . . . . . . . . . . . . . . . . . . . 34
4.1 A very slow moving piston in a still gas . . . . . . . . . . . . . . . . . . 40
4.2 Stationary sound wave and gas moves relative to the pulse. . . . . . . . 40
4.3 The Compressibility Chart . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1 Flow through a converging diverging nozzle . . . . . . . . . . . . . . . 53
5.2 Perfect gas flows through a tub e . . . . . . . . . . . . . . . . . . . . . 55
5.3 The stagnation properties as a function of the Mach number, k = 1.4 . 56
5.4 Control volume inside a converging-diverging nozzle. . . . . . . . . . . 58
5.5 The relationship between the cross section and the Mach number . . . 62
ix
x LIST OF FIGURES
5.6 Various ratios as a function of Mach number for isothermal Nozzle . . . 79
5.7 The comparison of nozzle flow . . . . . . . . . . . . . . . . . . . . . . 80
5.8 Comparison of the pressure and temp erature drop (two scales) . . . . . 81
5.9 Schematic to explain the significances of the Impulse function . . . . . 83
5.10 Schematic of a flow through a nozzle example (5.8) . . . . . . . . . . . 84
6.1 A shock wave inside a tube . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 The intersection of Fanno flow and Rayleigh flow . . . . . . . . . . . . 95
6.3 The M
exit
and P
0
as a function M
upstream
. . . . . . . . . . . . . . . 99
6.4 The ratios of the static properties of the two sides of the shock. . . . . 101
6.5 The shock drag diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.6 Comparison between stationary shock and moving shock. . . . . . . . . 104
6.7 The shock drag diagram for moving shock. . . . . . . . . . . . . . . . . 106
6.8 The diagram for the common explanation for shock drag. . . . . . . . . 107
6.9 A stationary shock and a moving shock 1 . . . . . . . . . . . . . . . . 108
6.10 A stationary shock and a moving shock 2 . . . . . . . . . . . . . . . . 109
6.11 The moving shock a result of a sudden stop . . . . . . . . . . . . . . . 110
6.12 A shock as a result of a sudden Opening . . . . . . . . . . . . . . . . . 111
6.13 The number of iterations to achieve convergence. . . . . . . . . . . . . 112
6.14 Schematic of showing the piston pushing air. . . . . . . . . . . . . . . 114
6.15 Time the pressure at the nozzle for the French problem. . . . . . . . . . 116
6.16 Max Mach number as a function of k. . . . . . . . . . . . . . . . . . . 116
6.17 Time the pressure at the nozzle for the French problem. . . . . . . . . . 119
6.18 Moving shock as a result of valve opening . . . . . . . . . . . . . . . . 121
6.19 The results of the partial opening of the valve. . . . . . . . . . . . . . . 122
6.20 A shock as a result of partially a valve closing . . . . . . . . . . . . . . 123
6.21 Figure for Example (6.10) . . . . . . . . . . . . . . . . . . . . . . . . 128
6.22 The shock tube schematic with a pressure “diagram.” . . . . . . . . . . 129
6.23 Figure for Example (6.12) . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.24 The results for Example (6.12 ) . . . . . . . . . . . . . . . . . . . . . . 134
7.1 The flow in the nozzle with different back pressures. . . . . . . . . . . 141
7.2 A nozzle with normal shock . . . . . . . . . . . . . . . . . . . . . . . . 142
7.3 Description to clarify the definition of diffuser efficiency . . . . . . . . . 148
7.4 Schematic of a supersonic tunnel example(7.3) . . . . . . . . . . . . . 148
9.1 Control volume for isothermal flow . . . . . . . . . . . . . . . . . . . . 157
9.2 Working relationships for isothermal flow . . . . . . . . . . . . . . . . . 163
9.3 The entrance Mach for isothermal flow for
4fL
D
. . . . . . . . . . . . . 174
10.1 Control volume of the gas flow in a constant cross section . . . . . . . 177
10.2 Various parameters in Fanno flow as a function of Mach number . . . . 186
10.3 Schematic of Example (10.1) . . . . . . . . . . . . . . . . . . . . . . . 186
10.4 The schematic of Example (10.2) . . . . . . . . . . . . . . . . . . . . . 188
10.5 The maximum length as a function of specific heat, k . . . . . . . . . . 193