Bar-Meir G. Fundamentals of Compressible Fluid
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LIST OF FIGURES xi
10.6 The effects of increase of
4fL
D
on the Fanno line . . . . . . . . . . . . 194
10.7 The development properties in of converging nozzle . . . . . . . . . . . 195
10.8 M
in
and ˙m as a function of the
4fL
D
. . . . . . . . . . . . . . . . . . . 195
10.9 M
1
as a function M
2
for various
4fL
D
. . . . . . . . . . . . . . . . . . 197
10.10 M
1
as a function M
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.11 The pressure distribution as a function of
4fL
D
for a short
4fL
D
. . . . 199
10.12 The pressure distribution as a function of
4fL
D
for a long
4fL
D
. . . . . 200
10.13 The effects of pressure variations on Mach number profile . . . . . . . 201
10.14 Mach number as a function of
4fL
D
when the total
4fL
D
= 0.3 . . . . . 202
10.15 Schematic of a “long” tub e in supersonic branch . . . . . . . . . . . . 203
10.16 The extra tube length as a function of the shock location . . . . . . . 204
10.17 The maximum entrance Mach numb er as a function of
4fL
D
. . . . . . 205
10.18 Unchoked flow showing the hypothetical “full” tube . . . . . . . . . . 208
10.19 The results of the algorithm showing the conversion rate. . . . . . . . 209
10.20 Solution to a missing diameter . . . . . . . . . . . . . . . . . . . . . . 212
10.21 M
1
as a function of
4fL
D
comparison with Isothermal Flow . . . . . . . 213
10.22 “Moody” diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
11.1 The control volume of Rayleigh Flow . . . . . . . . . . . . . . . . . . . 219
11.2 The temperature entropy diagram for Rayleigh line . . . . . . . . . . . 221
11.3 The basic functions of Rayleigh Flow (k=1.4) . . . . . . . . . . . . . . 226
12.1 The two different classifications of models . . . . . . . . . . . . . . . . 233
12.2 A schematic of two possible . . . . . . . . . . . . . . . . . . . . . . . . 234
12.3 A schematic of the control volumes used in this model . . . . . . . . . 234
12.4 The pressure assumptions in the chamber and tube entrance . . . . . . 235
12.5 The reduced time as a function of the modified reduced pressure . . . . 242
12.6 The reduced time as a function of the modified reduced pressure . . . . 244
13.1 The control volume of the “Cylinder”. . . . . . . . . . . . . . . . . . . 250
13.2 The pressure ratio as a function of the dimensionless time . . . . . . . 255
13.3
¯
P as a function of
¯
t for choked condition . . . . . . . . . . . . . . . . . 256
13.4 The pressure ratio as a function of the dimensionless time . . . . . . . 256
14.1 A view of a normal sho ck as a limited case for oblique shock . . . . . . 257
14.2 The oblique shock or Prandtl–Meyer function regions . . . . . . . . . . 258
14.3 A typical oblique shock schematic . . . . . . . . . . . . . . . . . . . . 259
14.4 Flow around spherically blunted 30
◦
cone-cylinder . . . . . . . . . . . . 265
14.5 The different views of a large inclination angle . . . . . . . . . . . . . . 266
14.6 The three different Mach numb ers . . . . . . . . . . . . . . . . . . . . 267
14.7 The various coefficients of three different Mach numbers . . . . . . . . 271
14.8 The “imaginary” Mach waves at zero inclination. . . . . . . . . . . . . 272
14.9 The D, shock angle, and M
y
for M
1
= 3 . . . . . . . . . . . . . . . . . 273
14.10 The possible range of solutions . . . . . . . . . . . . . . . . . . . . . 275
14.11 Two dimensional wedge . . . . . . . . . . . . . . . . . . . . . . . . . . 276
xii LIST OF FIGURES
14.12 Schematic of finite wedge with zero angle of attack. . . . . . . . . . . 277
14.13 A local and a far view of the oblique shock. . . . . . . . . . . . . . . 278
14.14 The schematic for a round–tip bullet in a supersonic flow. . . . . . . . 280
14.15 The schematic for a symmetrical suction section with reflection. . . . . 281
14.16The “detached” shock in a complicated configuration . . . . . . . . . . 282
14.17 Oblique shock around a cone . . . . . . . . . . . . . . . . . . . . . . 283
14.18 Maximum values of the properties in an oblique shock . . . . . . . . . 284
14.19 Two variations of inlet suction for supersonic flow. . . . . . . . . . . . 285
14.20 Schematic for Example (14.5). . . . . . . . . . . . . . . . . . . . . . 285
14.21 Schematic for Example (14.6). . . . . . . . . . . . . . . . . . . . . . 287
14.22 Schematic of two angles turn with two weak shocks. . . . . . . . . . . 287
14.23 Revisiting of shock drag diagram for the oblique shock. . . . . . . . . 296
14.24 Typical examples of unstable and stable situations. . . . . . . . . . . . 298
14.25 The schematic of stability analysis for oblique shock. . . . . . . . . . . 299
15.1 The definition of the angle for the Prandtl–Meyer function. . . . . . . . 301
15.2 The angles of the Mach line triangle . . . . . . . . . . . . . . . . . . . 301
15.3 The schematic of the turning flow. . . . . . . . . . . . . . . . . . . . . 302
15.4 The mathematical coordinate description . . . . . . . . . . . . . . . . . 303
15.5 Prandtl-Meyer function after the maximum angle . . . . . . . . . . . . 308
15.7 Diamond shape for supersonic d’Alembert’s Paradox . . . . . . . . . . 308
15.6 The angle as a function of the Mach number . . . . . . . . . . . . . . 309
15.8 The definition of attack angle for the Prandtl–Meyer function . . . . . 309
15.9 The schematic of Example 15.1 . . . . . . . . . . . . . . . . . . . . . . 310
15.10 The schematic for the reversed question of example (15.2) . . . . . . 311
15.11 Schematic of the nozzle and Prandtl–Meyer expansion. . . . . . . . . 314
A.1 Schematic diagram that explains the structure of the program . . . . . 320
LIST OF TABLES
1 Books Under Potto Project . . . . . . . . . . . . . . . . . . . . . . . . ix
1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
2.1 Properties of Various Ideal Gases [300K] . . . . . . . . . . . . . . . . . 30
4.1 Water speed of sound from different sources . . . . . . . . . . . . . . . 48
4.2 Liquids speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Solids speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.1 Fliegner’s number a function of Mach number . . . . . . . . . . . . . . 70
5.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Isentropic Table k = 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Isothermal Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Isothermal Table (continue) . . . . . . . . . . . . . . . . . . . . . . . 87
6.1 The shock wave table for k = 1.4 . . . . . . . . . . . . . . . . . . . . 134
6.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.2 Table for a Reflective Shock suddenly closed valve . . . . . . . . . . . . 136
6.2 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3 Table for shock suddenly opened valve (k=1.4) . . . . . . . . . . . . . 137
6.3 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.4 Table for shock from a suddenly opened valve (k=1.3) . . . . . . . . . 138
6.4 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.1 The Isothermal Flow basic parameters . . . . . . . . . . . . . . . . . 167
xiii
xiv LIST OF TABLES
9.2 The flow parameters for unchoked flow . . . . . . . . . . . . . . . . . 172
9.2 The flow parameters for unchoked flow (continue) . . . . . . . . . . . 173
10.1 Fanno Flow Standard basic Table . . . . . . . . . . . . . . . . . . . . 214
10.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
10.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
11.1 Rayleigh Flow k=1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
11.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
11.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
14.1 Table of maximum values of the oblique Shock k=1.4 . . . . . . . . . 279
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 40
B bulk modulus, see equation (4.38), page 48
B
f
Body force, see equation (2.9), page 27
c Speed of sound, see equation (4.1), page 40
C
p
Specific pressure heat, see equation (2.23), page 29
C
v
Specific volume heat, see equation (2.22), page 29
E Young’s modulus, see equation (4.40), page 49
E
U
Internal energy, see equation (2.3), page 26
E
u
Internal Energy per unit mass, see equation (2.6), page 26
E
i
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 sp ecific heats, see equation (2.24), page 29
M Mach number, see equation (5.8), page 54
n The polytropic coefficient, see equation (4.35), page 46
xv
xvi LIST OF TABLES
P Pressure, see equation (4.3), page 40
q Energy per unit mass, see equation (2.6), page 26
Q
12
The energy transferred to the system between state 1 and state 2, see equa-
tion (2.2), page 26
R Specific gas constant, see equation (2.27), page 30
R
mix
The universal gas constant for mixture, see equation (4.51), page 51
S Entropy of the system, see equation (2.13), page 28
t Time, see equation (4.18), page 43
U velocity , see equation (2.4), page 26
w Work per unit mass, see equation (2.6), page 26
W
12
The work done by the system between state 1 and state 2, see equation (2.2),
page 26
z The compressibility factor, see equation (4.22), page 44
The Book Change Log
Version 0.4.8.6
On 23rd Oct 2009 (3.6M pp. 384)
Add the section about Theodor Meyer’s biography
Addition of Temperature Velocity diagram. (The addition to the other chapters
was not added yet).
Version 0.4.8.5b
On 07th Sep 2009 (3.5M pp. 376)
Corrections in the Fanno chapter in Trends section.
English corrections.
Version 0.4.8.5a
On 04th July 2009 (3.5M pp. 376)
Corrections in the thermodynamics chapter to the gases properties table.
English corrections.
Improve the multilayer sound traveling example (Heru’s suggestion)
xvii
xviii LIST OF TABLES
Version 0.4.8.5a
On 04th July 2009 (3.3M pp. 380)
Correction to the gases properties table (Michael Madden and Heru Reksoprodjo)
English corrections.
Improving the multilayer sound wave traveling
Version 0.4.8.5
On 14th January 2009 (3.3M pp. 380)
Improve images macro (two captions issue).
English corrections.
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.
Add the discussion on the friction factor in isothermal and fanno flow.
LIST OF TABLES xix
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.
xx LIST OF TABLES
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.