Fundamentals of Compressible Flow Mechanics

Fundamentals of Compressible Fluid
Mechanics
Genick Bar–Meir, Ph. D.
1107 16th 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)

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‘We are like dwarfs sitting on the shoulders of giants”
from The Metalogicon by John in 1159

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CONTENTS
Nomenclature
xv
Feb-21-2007 version . . . . . . . . . . . . . . . . . . . . . . . . . . .
xx
Jan-16-2007 version
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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 . . . . . . . . . . . . . . . . . . . . . . . .
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Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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John Martones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Grigory Toker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Ralph Menikoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Domitien Rataaforret . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Gary Settles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Your name here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Typo corrections and other ”minor” contributions . . . . . . . . . . . .
ii
Version 0.4.8.5a . July 21, 2009 . . . . . . . . . . . . . . . . . . . . . . . . xiii
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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

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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

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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 ) Effects . . . . . . . . . . 193
D
10.9.2 The Pressure Ratio, P2 , effects . . . . . . . . . . . . . . . . . . 197
P1
10.9.3 Entrance Mach number, M1, effects . . . . . . . . . . . . . . . 201
10.10 Practical Examples for Subsonic Flow . . . . . . . . . . . . . . . . . . 207
10.10.1 Subsonic Fanno Flow for Given 4fL and Pressure Ratio . . . . 208
D
10.10.2 Subsonic Fanno Flow for a Given M1 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

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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 Chamber . . . . . . . . . . . . . . . . . . . 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, M1, and Deflection Angle, δ . . . . . 262
14.4.2 When No Oblique Shock Exist or When D > 0 . . . . . . . . . 265
14.4.3 Upstream Mach Number, M1, 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

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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 Shock 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

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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 tube . . . . . . . . . . . . . . . . . . . . .
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

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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 temperature 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 Mexit and P0 as a function Mupstream . . . . . . . . . . . . . . .
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 . . . . . . . . . . . . . 174
D
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

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