Fundamentals of compressible fluid mechanics

Fundamentals of Compressible
Fluid Mechanics

Genick Bar–Meir, Ph. D.
1107 16th Ave S. E.
Minneapolis, MN 55414-2411
email: “”

Copyright © 2007, 2006, 2005, and 2004 by Genick Bar-Meir
See the file copying.fdl or copyright.tex for copying conditions.
Version (0.4.8.5

January 13, 2009)


‘We are like dwarfs sitting on the shoulders of giants”

from The Metalogicon by John in 1159


CONTENTS

Nomenclature
Feb-21-2007 version . . . . . . . . . . . . . . . . . . . .
Jan-16-2007 version . . . . . . . . . . . . . . . . . . . . .
Dec-04-2006 version . . . . . . . . . . . . . . . . . . . .
GNU Free Documentation License . . . . . . . . . . . . . . . .
1. APPLICABILITY AND DEFINITIONS . . . . . . . . . .
2. VERBATIM COPYING . . . . . . . . . . . . . . . . . .
3. COPYING IN QUANTITY . . . . . . . . . . . . . . . . .
4. MODIFICATIONS . . . . . . . . . . . . . . . . . . . . .

5. COMBINING DOCUMENTS . . . . . . . . . . . . . . .
6. COLLECTIONS OF DOCUMENTS . . . . . . . . . . .
7. AGGREGATION WITH INDEPENDENT WORKS . . .
8. TRANSLATION . . . . . . . . . . . . . . . . . . . . . .
9. TERMINATION . . . . . . . . . . . . . . . . . . . . . .
10. FUTURE REVISIONS OF THIS LICENSE . . . . . . .
ADDENDUM: How to use this License for your documents
How to contribute to this book . . . . . . . . . . . . . . . . . .
Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
John Martones . . . . . . . . . . . . . . . . . . . . . . . .
Grigory Toker . . . . . . . . . . . . . . . . . . . . . . . . .
Ralph Menikoff . . . . . . . . . . . . . . . . . . . . . . . .
Domitien Rataaforret . . . . . . . . . . . . . . . . . . . .
Gary Settles . . . . . . . . . . . . . . . . . . . . . . . . . .
Your name here . . . . . . . . . . . . . . . . . . . . . . .
Typo corrections and other ”minor” contributions . . . . .
Version 0.4.8 Jan. 23, 2008 . . . . . . . . . . . . . . . . . . . .

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iv

CONTENTS
Version 0.4.3 Sep. 15, 2006 . . . . . . . . . . . . . . . . . .
Version 0.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . .
Version 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . .

Version 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . .
Version 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . .
Version 0.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . .
Version 0.4.1.7 . . . . . . . . . . . . . . . . . . . . . . . . .
Speed of Sound . . . . . . . . . . . . . . . . . . . . .
Stagnation effects . . . . . . . . . . . . . . . . . . . .
Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . .
Normal Shock . . . . . . . . . . . . . . . . . . . . . . .
Isothermal Flow . . . . . . . . . . . . . . . . . . . . . .
Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . .
Rayleigh Flow . . . . . . . . . . . . . . . . . . . . . . .
Evacuation and filling semi rigid Chambers . . . . . .
Evacuating and filling chambers under external forces
Oblique Shock . . . . . . . . . . . . . . . . . . . . . .
Prandtl–Meyer . . . . . . . . . . . . . . . . . . . . . .
Transient problem . . . . . . . . . . . . . . . . . . . .

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xliii
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1
1
2
2
4
5
9
13
15
15

2 Review of Thermodynamics
2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25
25

3 Fundamentals of Basic Fluid Mechanics
3.1 Introduction . . . . . . . . . . . . . . .
3.2 Fluid Properties . . . . . . . . . . . . .
3.3 Control Volume . . . . . . . . . . . . .
3.4 Reynold’s Transport Theorem . . . . .

1 Introduction
1.1 What is Compressible Flow ? . . . . . . . . . . . .
1.2 Why Compressible Flow is Important? . . . . . . .
1.3 Historical Background . . . . . . . . . . . . . . . .
1.3.1 Early Developments . . . . . . . . . . . . .
1.3.2 The shock wave puzzle . . . . . . . . . . .
1.3.3 Choking Flow . . . . . . . . . . . . . . . . .
1.3.4 External flow . . . . . . . . . . . . . . . . .
1.3.5 Filling and Evacuating Gaseous Chambers
1.3.6 Biographies of Major Figures . . . . . . . .

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33
33
33
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4 Speed of Sound
4.1 Motivation . . . . . . . . . . . . . . . . . .
4.2 Introduction . . . . . . . . . . . . . . . . .

4.3 Speed of sound in ideal and perfect gases
4.4 Speed of Sound in Real Gas . . . . . . .

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35
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37
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CONTENTS

v

4.5 Speed of Sound in Almost Incompressible Liquid . . . . . . . . . . .
4.6 Speed of Sound in Solids . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Sound Speed in Two Phase Medium . . . . . . . . . . . . . . . . . .
5 Isentropic Flow
5.1 Stagnation State for Ideal Gas Model . . . . . . . . . .
5.1.1 General Relationship . . . . . . . . . . . . . . .
5.1.2 Relationships for Small Mach Number . . . . .
5.2 Isentropic Converging-Diverging Flow in Cross Section
5.2.1 The Properties in the Adiabatic Nozzle . . . . .
5.2.2 Isentropic Flow Examples . . . . . . . . . . . .
5.2.3 Mass Flow Rate (Number) . . . . . . . . . . .
5.3 Isentropic Tables . . . . . . . . . . . . . . . . . . . . .
5.3.1 Isentropic Isothermal Flow Nozzle . . . . . . .
5.3.2 General Relationship . . . . . . . . . . . . . . .
5.4 The Impulse Function . . . . . . . . . . . . . . . . . .
5.4.1 Impulse in Isentropic Adiabatic Nozzle . . . .
5.4.2 The Impulse Function in Isothermal Nozzle . .
5.5 Isothermal Table . . . . . . . . . . . . . . . . . . . . .
5.6 The effects of Real Gases . . . . . . . . . . . . . . . .

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84

6 Normal Shock
89
6.1 Solution of the Governing Equations . . . . . . . . . . . . . . . . . . 92
6.1.1 Informal Model . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.1.2 Formal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.1.3 Prandtl’s Condition . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2 Operating Equations and Analysis . . . . . . . . . . . . . . . . . . . 97
6.2.1 The Limitations of the Shock Wave . . . . . . . . . . . . . . . 98

6.2.2 Small Perturbation Solution . . . . . . . . . . . . . . . . . . . 98
6.2.3 Shock Thickness . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2.4 Shock or Wave Drag . . . . . . . . . . . . . . . . . . . . . . . 99
6.3 The Moving Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3.1 Shock or Wave Drag Result from a Moving Shock . . . . . . 103
6.3.2 Shock Result from a Sudden and Complete Stop . . . . . . . 105
6.3.3 Moving Shock into Stationary Medium (Suddenly Open Valve) 108
6.3.4 Partially Open Valve . . . . . . . . . . . . . . . . . . . . . . . 117
6.3.5 Partially Closed Valve . . . . . . . . . . . . . . . . . . . . . . 118
6.3.6 Worked–out Examples for Shock Dynamics . . . . . . . . . . 119
6.4 Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.5 Shock with Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.6 Shock in Wet Steam . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.7 Normal Shock in Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.8 More Examples for Moving Shocks . . . . . . . . . . . . . . . . . . . 129
6.9 Tables of Normal Shocks, k = 1.4 Ideal Gas . . . . . . . . . . . . . . 132


vi

CONTENTS

7 Normal Shock in Variable Duct Areas
139
7.1 Nozzle efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2 Diffuser Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8 Nozzle Flow With External Forces
151
8.1 Isentropic Nozzle (Q = 0) . . . . . . . . . . . . . . . . . . . . . . . . 152
8.2 Isothermal Nozzle (T = constant) . . . . . . . . . . . . . . . . . . . 154

9 Isothermal Flow
9.1 The Control Volume Analysis/Governing equations
9.2 Dimensionless Representation . . . . . . . . . . .
9.3 The Entrance Limitation of Supersonic Branch . .
9.4 Comparison with Incompressible Flow . . . . . . .
9.5 Supersonic Branch . . . . . . . . . . . . . . . . . .
9.6 Figures and Tables . . . . . . . . . . . . . . . . . .
9.7 Isothermal Flow Examples . . . . . . . . . . . . . .
9.8 Unchoked situations in Fanno Flow . . . . . . . . .

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155
156
156
161
162
164
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165
170

10 Fanno Flow
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Fanno Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Non–Dimensionalization of the Equations . . . . . . . . . . . . .
10.4 The Mechanics and Why the Flow is Choked? . . . . . . . . . . .
10.5 The Working Equations . . . . . . . . . . . . . . . . . . . . . . .
10.6 Examples of Fanno Flow . . . . . . . . . . . . . . . . . . . . . . .
10.7 Supersonic Branch . . . . . . . . . . . . . . . . . . . . . . . . . .
10.8 Maximum Length for the Supersonic Flow . . . . . . . . . . . . .
10.9 Working Conditions . . . . . . . . . . . . . . . . . . . . . . . . .
10.9.1 Variations of The Tube Length ( 4fDL ) Effects . . . . . . . .
2
10.9.2 The Pressure Ratio, P

P1 , effects . . . . . . . . . . . . . . .
10.9.3 Entrance Mach number, M1 , effects . . . . . . . . . . . .
10.10Practical Examples for Subsonic Flow . . . . . . . . . . . . . . .
10.10.1Subsonic Fanno Flow for Given 4fDL and Pressure Ratio .
10.10.2Subsonic Fanno Flow for a Given M1 and Pressure Ratio
10.11The Approximation of the Fanno Flow by Isothermal Flow . . . .
10.12More Examples of Fanno Flow . . . . . . . . . . . . . . . . . . .
10.13The Table for Fanno Flow . . . . . . . . . . . . . . . . . . . . . .
10.14Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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175
175
176
177
180
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185
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190
191

192
197
199
206
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211
211
213
214

11 Rayleigh Flow
11.1 Introduction . . . . . . . . . .
11.2 Governing Equation . . . . .
11.3 Rayleigh Flow Tables . . . . .
11.4 Examples For Rayleigh Flow

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221
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CONTENTS

vii

12 Evacuating SemiRigid Chambers
12.1 Governing Equations and Assumptions . . .
12.2 General Model and Non-dimensioned . . . .
12.2.1 Isentropic Process . . . . . . . . . . .
12.2.2 Isothermal Process in The Chamber .
12.2.3 A Note on the Entrance Mach number
12.3 Rigid Tank with Nozzle . . . . . . . . . . . . .
12.3.1 Adiabatic Isentropic Nozzle Attached .
12.3.2 Isothermal Nozzle Attached . . . . . .

12.4 Rapid evacuating of a rigid tank . . . . . . .
12.4.1 With Fanno Flow . . . . . . . . . . . .
12.4.2 Filling Process . . . . . . . . . . . . .
12.4.3 The Isothermal Process . . . . . . . .
12.4.4 Simple Semi Rigid Chamber . . . . .
12.4.5 The “Simple” General Case . . . . . .
12.5 Advance Topics . . . . . . . . . . . . . . . . .

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231
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236
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237
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239
239
239
241
242
243
243
245

13 Evacuating under External Volume Control
13.1 General Model . . . . . . . . . . . . . . .
13.1.1 Rapid Process . . . . . . . . . . .
13.1.2 Examples . . . . . . . . . . . . . .
13.1.3 Direct Connection . . . . . . . . .
13.2 Summary . . . . . . . . . . . . . . . . . .

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247
247
248
251
251
252

14 Oblique Shock
14.1 Preface to Oblique Shock . . . . . . . . . . . . . . . . . . . .
14.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2.1 Introduction to Oblique Shock . . . . . . . . . . . . . .
14.2.2 Introduction to Prandtl–Meyer Function . . . . . . . .

14.2.3 Introduction to Zero Inclination . . . . . . . . . . . . .
14.3 Oblique Shock . . . . . . . . . . . . . . . . . . . . . . . . . .
14.4 Solution of Mach Angle . . . . . . . . . . . . . . . . . . . . .
14.4.1 Upstream Mach Number, M1 , and Deflection Angle, δ
14.4.2 When No Oblique Shock Exist or When D > 0 . . . .
14.4.3 Upstream Mach Number, M1 , and Shock Angle, θ . .
14.4.4 Given Two Angles, δ and θ . . . . . . . . . . . . . . .
14.4.5 Flow in a Semi–2D Shape . . . . . . . . . . . . . . . .
14.4.6 Small δ “Weak Oblique shock” . . . . . . . . . . . . .
14.4.7 Close and Far Views of the Oblique Shock . . . . . .
14.4.8 Maximum Value of Oblique shock . . . . . . . . . . . .
14.5 Detached Shock . . . . . . . . . . . . . . . . . . . . . . . . .
14.5.1 Issues Related to the Maximum Deflection Angle . . .
14.5.2 Oblique Shock Examples . . . . . . . . . . . . . . . .
14.5.3 Application of Oblique Shock . . . . . . . . . . . . . .
14.5.4 Optimization of Suction Section Design . . . . . . . .

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255
255
256
256
256
257
257
260
260
263
271
273
274
276
277
277
278
279
281

283
294

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viii

CONTENTS
14.5.5 Retouch of Shock or Wave Drag . . . . . . . . . . . . . . . . 294
14.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
14.7 Appendix: Oblique Shock Stability Analysis . . . . . . . . . . . . . . 296

15 Prandtl-Meyer Function
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.2 Geometrical Explanation . . . . . . . . . . . . . . . . . . . . . . . .
15.2.1 Alternative Approach to Governing Equations . . . . . . . .
15.2.2 Comparison And Limitations between the Two Approaches
15.3 The Maximum Turning Angle . . . . . . . . . . . . . . . . . . . . .
15.4 The Working Equations for the Prandtl-Meyer Function . . . . . . .

15.5 d’Alembert’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . .
15.6 Flat Body with an Angle of Attack . . . . . . . . . . . . . . . . . . .
15.7 Examples For Prandtl–Meyer Function . . . . . . . . . . . . . . .
15.8 Combination of the Oblique Shock and Isentropic Expansion . . .

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299
299
300
301
305
305
306
306
308
308
311

A Computer Program
315

A.1 About the Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
A.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
A.3 Program listings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Index
319
Subjects Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Authors Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322


LIST OF FIGURES

1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12

The shock as a connection of Fanno and Rayleigh lines . . . . . . .
The schematic of deLavel’s turbine . . . . . . . . . . . . . . . . . . .
The measured pressure in a nozzle . . . . . . . . . . . . . . . . . .
Flow rate as a function of the back pressure . . . . . . . . . . . . . .
Portrait of Galileo Galilei . . . . . . . . . . . . . . . . . . . . . . . . .
Photo of Ernest Mach . . . . . . . . . . . . . . . . . . . . . . . . . .

The photo of thebullet in a supersonic flow not taken in a wind tunnel
Photo of Lord Rayleigh . . . . . . . . . . . . . . . . . . . . . . . . . .
Portrait of Rankine . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The photo of Gino Fanno approximately in 1950 . . . . . . . . . . .
Photo of Prandtl . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The photo of Ernst Rudolf George Eckert with the author’s family . .

7
9
11
12
16
17
17
18
19
20
21
22

4.1 A very slow moving piston in a still gas . . . . . . . . . . . . . . . . .
4.2 Stationary sound wave and gas moves relative to the pulse. . . . . .
4.3 The Compressibility Chart . . . . . . . . . . . . . . . . . . . . . . . .

36
36
40

5.1
5.2

5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10

49
51
52
54
58
75
76
77
80
81

Flow thorough a converging diverging nozzle . . . . . . . . . . . . .
Perfect gas flows through a tube . . . . . . . . . . . . . . . . . . . .
The stagnation properties as a function of the Mach number, k = 1.4
Control volume inside a converging-diverging nozzle. . . . . . . . . .
The relationship between the cross section and the Mach number .
Various ratios as a function of Mach number for isothermal Nozzle .
The comparison of nozzle flow . . . . . . . . . . . . . . . . . . . . .
Comparison of the pressure and temperature drop (two scales) . . .
Schematic to explain the significances of the Impulse function . . . .
Schematic of a flow thorough a nozzle example (5.8) . . . . . . . . .

ix


x

LIST OF FIGURES
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20
6.21
6.22
6.23

6.24
6.25
6.26
6.27

A shock wave inside a tube . . . . . . . . . . . . . . . . . . . . . . . 89
The intersection of Fanno flow and Rayleigh flow . . . . . . . . . . . 91
The Mexit and P0 as a function Mupstream . . . . . . . . . . . . . . . 95
The ratios of the static properties of the two sides of the shock. . . . 97
The shock drag diagram . . . . . . . . . . . . . . . . . . . . . . . . . 99
Comparison between stationary shock and moving shock . . . . . . 101
The shock drag diagram for moving shock. . . . . . . . . . . . . . . 103
The diagram for the common explanation for shock drag. . . . . . . . 104
Comparison between a stationary shock and a moving shock in a stationary medium
Comparison between a stationary shock and a moving shock in a stationary medium
The moving shock a result of a sudden stop . . . . . . . . . . . . . . 107
A shock as a result of a sudden Opening . . . . . . . . . . . . . . . 108
The number of iterations to achieve convergence. . . . . . . . . . . . 109
Schematic of showing the piston pushing air. . . . . . . . . . . . . . 111
Time the pressure at the nozzle for the French problem. . . . . . . . 113
Max Mach number as a function of k. . . . . . . . . . . . . . . . . . 113
Time the pressure at the nozzle for the French problem. . . . . . . . 117
Moving shock as a result of valve opening . . . . . . . . . . . . . . . 117
The results of the partial opening of the valve. . . . . . . . . . . . . . 118
A shock as a result of partially a valve closing . . . . . . . . . . . . . 119
Schematic of a piston pushing air in a tube. . . . . . . . . . . . . . . 122
Figure for Example (6.10) . . . . . . . . . . . . . . . . . . . . . . . . 124
The shock tube schematic with a pressure ”diagram.” . . . . . . . . . 125
Figure for Example (6.13) . . . . . . . . . . . . . . . . . . . . . . . . 129
The results for Example (6.13) . . . . . . . . . . . . . . . . . . . . . 130

Figure for example (6.13) . . . . . . . . . . . . . . . . . . . . . . . . 130
The results for Example (6.13) . . . . . . . . . . . . . . . . . . . . . 131

7.1
7.2
7.3
7.4

The flow in the nozzle with different back pressures. .
A nozzle with normal shock . . . . . . . . . . . . . . .
Description to clarify the definition of diffuser efficiency
Schematic of a supersonic tunnel example(7.3) . . . .

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139
140
146
146

9.1 Control volume for isothermal flow . . . . . . . . . . . . . . . . . . . 155
9.2 Working relationships for isothermal flow . . . . . . . . . . . . . . . . 161

9.3 The entrance Mach for isothermal flow for 4fDL . . . . . . . . . . . . 172
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8

Control volume of the gas flow in a constant cross section . . . .
Various parameters in Fanno flow as a function of Mach number
Schematic of Example (10.1) . . . . . . . . . . . . . . . . . . . .
The schematic of Example (10.2) . . . . . . . . . . . . . . . . . .
The maximum length as a function of specific heat, k . . . . . . .
The effects of increase of 4fDL on the Fanno line . . . . . . . . .
The development properties in of converging nozzle . . . . . . .
Min and m
˙ as a function of the 4fDL . . . . . . . . . . . . . . . . .

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175
184
185
186
191
192
193
194


LIST OF FIGURES

xi

10.9 M1 as a function M2 for various 4fDL . . . . . . . . . . . . . . . . . . 195
10.10M1 as a function M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10.11The pressure distribution as a function of 4fDL for a short 4fDL . . . . 198
10.12The pressure distribution as a function of 4fDL for a long 4fDL . . . . 199
10.13The effects of pressure variations on Mach number profile . . . . . . 200
10.14Mach number as a function of 4fDL when the total 4fDL = 0.3 . . . . . 201
10.15Schematic of a “long” tube in supersonic branch . . . . . . . . . . . 202
10.16The extra tube length as a function of the shock location . . . . . . . 202

10.17The maximum entrance Mach number as a function of 4fDL . . . . . 203
10.18Unchoked flow calculations showing the hypothetical “full” tub when choked206
10.19The results of the algorithm showing the conversion rate. . . . . . . 208
10.20Solution to a missing diameter . . . . . . . . . . . . . . . . . . . . . 210
10.21M1 as a function of 4fDL comparison with Isothermal Flow . . . . . . 212
10.22“Moody” diagram on the name Moody who netscape H. Rouse work to claim as his own. In this section the turbulent are
11.1
11.2
11.3
11.4

The control volume of Rayleigh Flow . . . . . . . .
The temperature entropy diagram for Rayleigh line
The basic functions of Rayleigh Flow (k=1.4) . . .
Schematic of the combustion chamber. . . . . . .

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217
219
224
228

12.1
12.2
12.3
12.4
12.5
12.6

The two different classifications of models . . . . . . . . . . . . .
A schematic of two possible . . . . . . . . . . . . . . . . . . . . .
A schematic of the control volumes used in this model . . . . . .
The pressure assumptions in the chamber and tube entrance . .
The reduced time as a function of the modified reduced pressure
The reduced time as a function of the modified reduced pressure

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231
232
232
233
241
242

13.1
13.2
13.3
13.4

The control volume of the “Cylinder”. . . . . . . . . . . . . . . . .
The pressure ratio as a function of the dimensionless time . . . .
P¯ as a function of t¯ for choked condition . . . . . . . . . . . . . .
The pressure ratio as a function of the dimensionless time . . .

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248
253
254
254

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255
256
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263
264
265
269
270
271
273
274
275
277

14.1 A view of a normal shock as a limited case for oblique shock
14.2 The oblique shock or Prandtl–Meyer function regions . . . . .
14.3 A typical oblique shock schematic . . . . . . . . . . . . . . .
14.4 Flow around spherically blunted 30◦ cone-cylinder . . . . . .
14.5 The different views of a large inclination angle . . . . . . . . .
14.6 The three different Mach numbers . . . . . . . . . . . . . . .
14.7 The various coefficients of three different Mach numbers . . .
14.8 The “imaginary” Mach waves at zero inclination. . . . . . . .
14.9 The D, shock angle, and My for M1 = 3 . . . . . . . . . . . .
14.10The possible range of solutions . . . . . . . . . . . . . . . . .
14.11Two Dimensional Wedge . . . . . . . . . . . . . . . . . . . . .
14.12Schematic of finite wedge with zero angle of attack. . . . . .

14.13/; A local and a far view of the oblique shock. . . . . . . . . .

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xii

LIST OF FIGURES
14.14The schematic for a round–tip bullet in a supersonic flow. . . . . . . 278
14.15The schematic for a symmetrical suction section with Mach reflection. 279
14.16 The “detached” shock in a complicated configuration . . . . . . . . 280
14.17 Oblique shock around a cone . . . . . . . . . . . . . . . . . . . . . 281
14.18 Maximum values of the properties in an oblique shock . . . . . . . 282
14.19 Two variations of inlet suction for supersonic flow. . . . . . . . . . . 283
14.20 Schematic for Example (14.5). . . . . . . . . . . . . . . . . . . . . . 283
14.21 Schematic for Example (14.6). . . . . . . . . . . . . . . . . . . . . . 285
14.22 Schematic of two angles turn with two weak shocks. . . . . . . . . 285
14.23Revisiting of shock drag diagram for the oblique shock. . . . . . . . . 294
14.24 Typical examples of unstable and stable situations. . . . . . . . . . 296
14.25The schematic of stability analysis for oblique shock. . . . . . . . . . 297
15.1 The definition of the angle for the Prandtl–Meyer function. .
15.2 The angles of the Mach line triangle . . . . . . . . . . . . .
15.3 The schematic of the turning flow. . . . . . . . . . . . . . .
15.4 The mathematical coordinate description . . . . . . . . . .
15.5 Prandtl-Meyer function after the maximum angle . . . . . .
15.7 Diamond shape for supersonic d’Alembert’s Paradox . . . .
15.6 The angle as a function of the Mach number . . . . . . . .
15.8 The definition of the angle for the Prandtl–Meyer function. .
15.9 The schematic of Example 15.1 . . . . . . . . . . . . . . . .

15.10 The schematic for the reversed question of example (15.2)
15.11Schematic of the nozzle and Prandtle–Meyer expansion. . .

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299
299
300
301
306
306
307
308
308
310
312

A.1 Schematic diagram that explains the structure of the program . . . . 318


LIST OF TABLES

1
1

Books Under Potto Project . . . . . . . . . . . . . . . . . . . . . . . . xxxix
continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl

2.1 Properties of Various Ideal Gases [300K] . . . . . . . . . . . . . . .

30

4.1 Water speed of sound from different sources . . . . . . . . . . . . .
4.2 Liquids speed of sound . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 Solids speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . .

43
44
45

5.1
5.1
5.1
5.2
5.3
5.3

Fliegner’s number a function of Mach number
continue . . . . . . . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . .
Isentropic Table k = 1.4 . . . . . . . . . . . .
Isothermal Table . . . . . . . . . . . . . . .
Isothermal Table (continue) . . . . . . . . . .

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66
67
68
71
82
83

6.1
6.1
6.2
6.2
6.3
6.3
6.3
6.4
6.4

The shock wave table for k = 1.4 . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . . . . . .
Table for a Reflective Shock suddenly closed valve . .
continue . . . . . . . . . . . . . . . . . . . . . . . . . .
Table for shock suddenly opened valve (k=1.4) . . . .
continue . . . . . . . . . . . . . . . . . . . . . . . . . .
continue . . . . . . . . . . . . . . . . . . . . . . . . . .
Table for shock from a suddenly opened valve (k=1.3)
continue . . . . . . . . . . . . . . . . . . . . . . . . . .

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132
133
133
134
134
135
136
136
137

9.1
9.2

The Isothermal Flow basic parameters . . . . . . . . . . . . . . . . 165
The flow parameters for unchoked flow . . . . . . . . . . . . . . . . 170

xiii

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xiv

LIST OF TABLES

9.2

The flow parameters for unchoked flow (continue) . . . . . . . . . . 171

10.1 Fanno Flow Standard basic Table . . . . . . . . . . . . . . . . . . . 213
10.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
11.1 Rayleigh Flow k=1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
11.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
14.1 Table of maximum values of the oblique Shock k=1.4 . . . . . . . . 277
14.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278


NOMENCLATURE

¯
R

Universal gas constant, see equation (2.26), page 29
Units length., see equation (2.1), page 25

ρ

Density of the fluid, see equation (4.1), page 36

B

bulk modulus, see equation (4.35), page 43

Bf

Body force, see equation (2.9), page 27

c

Speed of sound, see equation (4.1), page 36

Cp

Specific pressure heat, see equation (2.23), page 29

Cv

Specific volume heat, see equation (2.22), page 29

E

Young’s modulus, see equation (4.37), page 44

EU

Internal energy, see equation (2.3), page 26

Eu

Internal Energy per unit mass, see equation (2.6), page 26

Ei

System energy at state i, see equation (2.2), page 26

H

Enthalpy, see equation (2.18), page 28

h

Specific enthalpy, see equation (2.18), page 28

k

the ratio of the specific heats, see equation (2.24), page 29

M

Mach number, see equation (5.8), page 50

n

The poletropic coefficient, see equation (4.32), page 42

xv


xvi

LIST OF TABLES

P

Pressure, see equation (4.3), page 36

q

Energy per unit mass, see equation (2.6), page 26

Q12

The energy transfered to the system between state 1 and state 2, see equation (2.2), page 26

R

Specific gas constant, see equation (2.27), page 30

Rmix

The universal gas constant for mixture, see equation (4.48), page 46

S

Entropy of the system, see equation (2.13), page 28

t

Time, see equation (4.15), page 39

U

velocity , see equation (2.4), page 26

w

Work per unit mass, see equation (2.6), page 26

W12

The work done by the system between state 1 and state 2, see equation (2.2), page 26

z

The compressibility factor, see equation (4.19), page 39


The Book Change Log

Version 0.4.8.5rc
On 31st December 2008 (3.3M pp. 380)
• Add Gary Settles’s color image in wedge shock and an example.
• Improve the wrap figure issue to oblique shock.
• Add Moody diagram to Fanno flow.
• English corrections to the oblique shock chapter.

Version 0.4.8.4
On 7th October 2008 (3.2M pp. 376)
• More work on the nomenclature issue.
• Important equations and useful equations issues inserted.
• Expand the discussion on the friction factor in isothermal and fanno flow.

Version 0.4.8.3
On 17th September 2008 (3.1M pp. 369)
• Started the nomenclature issue so far only the thermodynamics chapter.

• Started the important equations and useful equations issue.
• Add the introduction to thermodynamics chapter.

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• Add the discussion on the friction factor in isothermal and fanno flow.

Version 0.4.8.2
On 25th January 2008 (3.1M pp. 353)
• Add several additions to the isentropic flow, normal shock,
• Rayleigh Flow.
• Improve some examples.
• More changes to the script to generate separate chapters sections.
• Add new macros to work better so that php and pdf version will be similar.
• More English revisions.

Version 0.4.8
November-05-2007
• Add the new unchoked subsonic Fanno Flow section which include the “unknown” diameter question.
• Shock (Wave) drag explanation with example.
• Some examples were add and fixing other examples (small perturbations of
oblique shock).
• Minor English revisions.

Version 0.4.4.3pr1

July-10-2007
• Improvement of the pdf version provide links.

Version 0.4.4.2a
July-4-2007 version
• Major English revisions in Rayleigh Flow Chapter.
• Continue the improvement of the HTML version (imageonly issues).
• Minor content changes and addition of an example.


LIST OF TABLES

xix

Version 0.4.4.2
May-22-2007 version
• Major English revisions.
• Continue the improvement of the HTML version.
• Minor content change and addition of an example.

Version 0.4.4.1
Feb-21-2007 version
• Include the indexes subjects and authors.
• Continue the improve the HTML version.
• solve problems with some of the figures location (float problems)
• Improve some spelling and grammar.
• Minor content change and addition of an example.
• The main change is the inclusion of the indexes (subject and authors). There
were some additions to the content which include an example. The ”naughty
professor’s questions” section isn’t completed and is waiting for interface

of Potto-GDC to be finished (engine is finished, hopefully next two weeks).
Some grammar and misspelling corrections were added.
Now include a script that append a title page to every pdf fraction of the book
(it was fun to solve this one). Continue to insert the changes (log) to every
source file (latex) of the book when applicable. This change allows to follow
the progression of the book. Most the tables now have the double formatting
one for the html and one for the hard copies.

Version 0.4.4pr1
Jan-16-2007 version
• Major modifications of the source to improve the HTML version.
• Add the naughty professor’s questions in the isentropic chapter.
• Some grammar and miss spelling corrections.


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Version 0.4.3.2rc1
Dec-04-2006 version
• Add new algorithm for Fanno Flow calculation of the shock location in the
supersonic flow for given fld (exceeding Max) and M1 (see the example).
• Minor addition in the Sound and History chapters.
• Add analytical expression for Mach number results of piston movement.

Version 0.4.3.1rc4 aka 0.4.3.1
Nov-10-2006 aka Roy Tate’s version
For this release (the vast majority) of the grammatical corrections are due to Roy
Tate

• Grammatical corrections through the history chapter and part of the sound
chapter.
• Very minor addition in the Isothermal chapter about supersonic branch.

Version 0.4.3.1rc3
Oct-30-2006
• Add the solutions to last three examples in Chapter Normal Shock in variable
area.
• Improve the discussion about partial open and close moving shock dynamics
i.e. high speed running into slower velocity
• Clean other tables and figure and layout.

Version 0.4.3rc2
Oct-20-2006
• Clean up of the isentropic and sound chapters
• Add discussion about partial open and close moving shock dynamics i.e.
high speed running into slower velocity.
• Add the partial moving shock figures (never published before)


LIST OF TABLES

xxi

Version 0.4.3rc1
Sep-20-2006
• Change the book’s format to 6x9 from letter paper
• Clean up of the isentropic chapter.
• Add the shock tube section
• Generalize the discussion of the the moving shock (not including the change

in the specific heat (material))
• Add the Impulse Function for Isothermal Nozzle section
• Improve the discussion of the Fliegner’s equation
• Add the moving shock table (never published before)

Version 0.4.1.9 (aka 0.4.1.9rc2)
May-22-2006
• Added the Impulse Function
• Add two examples.
• Clean some discussions issues .

Version 0.4.1.9rc1
May-17-2006
• Added mathematical description of Prandtl-Meyer’s Function
• Fixed several examples in oblique shock chapter
• Add three examples.
• Clean some discussions issues .

Version 0.4.1.8 aka Version 0.4.1.8rc3
May-03-2006
• Added Chapman’s function
• Fixed several examples in oblique shock chapter
• Add two examples.
• Clean some discussions issues .


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Version 0.4.1.8rc2
Apr-11-2006
• Added the Maximum Deflection Mach number’s equation
• Added several examples to oblique shock


Notice of Copyright For This
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This document published Modified FDL. The change of the license is to prevent
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GNU Free Documentation License
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Version 1.2, November 2002
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LIST OF TABLES

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