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Fundamentals
of
Compressible
Fluid Mechanics
by
Genick Bar–Meir
Fundamentals of Compressible
Fluid Mechanics
Genick Bar–Meir, Ph. D.

8118 Kolmar Ave
Skokie, IL 60076
barmeir at gmail
Copyright c 2022, 2021, 2020, 2019, 2017,

2013, 2012, 2009, 2008, 2007, 2006,
2005, and 2004 by Genick Bar-Meir
See the file copying.fdl or copyright.tex for copying conditions.
doi: 0.5281/zenodo.5523349
Version (0.5.2 April 18, 2022)

How to cite this book:
Bar-Meir, Genick, “Fundamentals of Compressible Fluid Mechanics”, Version 0.5.2{last
modified or Accessed}: insert the date and version you are using, www.potto.org/
downloads.php
ver 0.5.2 doi:10.5281/zenodo.6467559
Example:
If you are using the latest version

Bar-Meir, Genick, “Fundamentals of Compressible Fluid Mechanics”,Last modified: Ver-
sion 0.5.3.0 April 15, 2022, www.potto.org/downloads.php
ver 0.5 doi:10.5281/zenodo.5523349
ver 0.5.2 doi:10.5281/zenodo.6467559
If you are using older version

Bar-Meir, Genick, “Fundamentals of Compressible Fluid Mechanics”, Accessed: Version
0.5.0 July 10, 2021, www.potto.org/downloads.php
doi:10.5281/zenodo.5523349
iii
‘We are like dwarfs sitting on the shoulders of giants”
from The Metalogicon by John in 1159

iv
Please Update
This book became victim of its own successes. More than 60% of downloads of this book
are for the old versions because the search engines keep track the previous downloads.
That is, when the old version with 80,000 downloads from one web site for example, like
researchgate.net, the new version cannot surface up. The book is released on a rolling
fashion. It means that it released several times during the year. In other words, if you
have a copy of the book and it is older than a month, the chances are that you have
an old version. Please do yourself a favor and download a new version. You can get
the last version from zenodo https://zenodo.org/record/5521908#.YhxIaVRMFhF.
While you are there you can download several items:
• “Stability of Ships and Other bodies”.
https://zenodo.org/record/5784893#.Yd1uuYpME-0.
• “Fundamentals of Compressible Flow”,
https://zenodo.org/record/5523349#.YhxNZ1RMFhE.
• the world largest gad dynamics tables (over 600 pages).
https://zenodo.org/record/5523532#.YhxOD1RMFhE
• “Basics of Die Casting Design”
https://zenodo.org/record/5523594#.YhxNxFRMFhE , and
• other material like “15 Years Experience Creating Open Content Engineering Ma-
terial” describing the depth of the great depth analytically.
Like the largest gas table in world published by Potto Project NFP. All these materials
are authored by the undersigned. If you would like to learn more about this author
you can grab the article “15 Years Experience Creating Open Content Engineering Ma-
terial.” https://zenodo.org/record/5791182#.Yd1v3YpME-0 In the near future the
article “20 years of producing open content engineering.”
Thank you for using this book, Genick
v
vi PLEASE UPDATE
Abstract
Why Abstract
The doi registration of the book forced the examination of what is written in this book.
This abstract is the result of this examination.
Short Abstract
As many topics in the thromo-fluid area the presentation of the material has two main
approaches: first is the presentation and emphasis of the big equations that represented
by the book of Liepmann and Roshko “Elements of Gas Dynamics” which was the
mandated textbook when this author took compressible class. On the other hand, more
practice approach is where various pipe flows regimes. The second approach seems more
logical because more engineers deal with other than the flow around airplane’s wings.
Large part of the problems are focused around the pipe flow, shocks, chocking. The
fact that large part of communication in the book was on these issues and not about
flow around wings.
The book starts with brief history of the developments in this area. Then the book
cover basic topics like thermodynamics, give wide review of the fluid mechanics. Not
commonly presented, the speed of sound is discussed and several features are exhibited.
Later the fundamentals of the compressible flow isentropic flow is presented including the
real gases. The unique features of the compressible flow when boundary conditions are
forced a jump is created and it is call shock. This chapter is very extensive. The chapter
includes shock dynamics (moving shocks). The following chapter deals with shock in
variable area. The next three chapter allocates for various specially pipe flows. One
of the interesting problem is the evacuating chamber which appear in many industrial
application. It close with with two dimensional flow compressible flow.
Long Abstract
under construction.
vii
viii ABSTRACT
Prologue For This Book
Version 0.5.2 491 pages, 5.6M bytes April 18, 2022

over 500,000 downloads
The number of pages shrinked because more efficient techniques were developed utilizing
the tcolorbox, exBox and prllTextImgage macros. Two more examples were added other
were cleaned (hopefully no other issues were damaged).
Version 0.5.0 525 pages, 5.5M bytes July 13, 2021

After 9 years of work on other topics, the book has a major revisiting. The book has
large present on the net and there are discussion about this book and how to download
it (since the potto web site was not work temporally).
Version 0.4.9 pp. ? Feb ?, 2012

In the last three years the focus was on building the fluid mechanics book. In the
construction of the fluid book the potto style file significantly changed to the point
that render the old files of book as un–compilable. This work was to bring these file
up to date. Several chapters from that the fluid book were summarized into single
introduction chapter on Fluid Mechanics. There are several additions which include
better description of the shock tube, and sound in variable liquid density etc.
Version 0.4.8.5a . July 21, 2009

The spread of the book was the biggest change that can be observed during the last
year (more than a year). Number of download reached to over 160,000 copies. The
book became the main textbook in many universities. This time, the main work focused
on corrections and minor additions. The fluid mechanics book is under construction
and reached to 0.17x version. Hopefully when finished, with good help in the coming
ix
x PROLOGUE
months will be used in this book to make better introduction. Other material in this
book like the gas dynamics table and equation found their life and very popular today.
This additions also include GDC which become the standard calculator for the gas
dynamics class.
Version 0.4.8 Jan. 23, 2008

It is more than a year ago, when the previous this section was modified. Many things
have changed, and more people got involved. It nice to know that over 70,000 copies
have been download from over 130 countries. It is more pleasant to find that this
book is used in many universities around the world, also in many institutes like NASA
(a tip from Dr. Farassat, NASA ”to educate their “young scientist, and engineers”)
and others. Looking back, it must be realized that while, this book is the best in many
areas, like oblique shock, moving shock, fanno flow, etc there are missing some sections,
like methods of characteristics, and the introductory sections (fluid mechanics, and
thermodynamics). Potto–GDC is much more mature and it is changing from “advance
look up” to a real gas dynamics calculator (for example, calculation of unchoked Fanno
Flow). Today Potto–GDC has the only capability to produce the oblique shock figure.
Potto-GDC is becoming the major educational educational tool in gas dynamics. To
kill two birds in one stone, one, continuous requests from many and, two, fill the
introductory section on fluid mechanics in this book this area is major efforts in the
next few months for creating the version 0.2 of the “Basic of Fluid Mechanics” are
underway.
Version 0.4.3 Sep. 15, 2006

The title of this section is change to reflect that it moved to beginning of the book.
While it moves earlier but the name was not changed. Dr. Menikoff pointed to this
inconsistency, and the author is apologizing for this omission.
Several sections were add to this book with many new ideas for example on the
moving shock tables. However, this author cannot add all the things that he was asked
and want to the book in instant fashion. For example, one of the reader ask why not
one of the example of oblique shock was not turn into the explanation of von Neumann
paradox. The author was asked by a former client why he didn’t insert his improved tank
filling and evacuating models (the addition of the energy equation instead of isentropic
model). While all these requests are important, the time is limited and they will be
inserted as time permitted.
The moving shock issues are not completed and more work is needed also in the
shock tube. Nevertheless, the ideas of moving shock will reduced the work for many
student of compressible flow. For example solving homework problem from other text
books became either just two mouse clicks away or just looking at that the tables in
this book. I also got request from a India to write the interface for Microsoft. I am
sorry will not be entertaining work for non Linux/Unix systems, especially for Microsoft.
If one want to use the software engine it is okay and permitted by the license of this
VERSION 0.4.2 xi
work.
The download to this mount is over 25,000.
Version 0.4.2
It was surprising to find that over 14,000 downloaded and is encouraging to receive over
200 thank you eMail (only one from U.S.A./Arizona) and some other reactions. This
textbook has sections which are cutting edge research1 .
The additions of this version focus mainly on the oblique shock and related issues
as results of questions and reactions on this topic. However, most readers reached to
www.potto.org by searching for either terms “Rayleigh flow” (107) and “Fanno flow”
((93). If the total combined variation search of terms “Fanno” and “Rayleigh” (mostly
through google) is accounted, it reaches to about 30% (2011). This indicates that these
topics are highly is demanded and not many concerned with the shock phenomena as
this author believed and expected. Thus, most additions of the next version will be
concentrated on Fanno flow and Rayleigh flow. The only exception is the addition to
Taylor–Maccoll flow (axisymmetricale conical flow) in Prandtl–Meyer function (currently
in a note form).
Furthermore, the questions that appear on the net will guide this author on what
is really need to be in a compressible flow book. At this time, several questions were
about compressibility factor and two phase flow in Fanno flow and other kind of flow
models. The other questions that appeared related two phase and connecting several
chambers to each other. Also, an individual asked whether this author intended to write
about the unsteady section, and hopefully it will be near future.
Version 0.4
Since the last version (0.3) several individuals sent me remarks and suggestions. In the
introductory chapter, extensive description of the compressible flow history was written.
In the chapter on speed of sound, the two phase aspects were added. The isothermal
nozzle was combined with the isentropic chapter. Some examples were added to the
normal shock chapter. The fifth chapter deals now with normal shock in variable area
ducts. The sixth chapter deals with external forces fields. The chapter about oblique
shock was added and it contains the analytical solution. At this stage, the connection
between Prandtl–Meyer flow and oblique is an note form. The a brief chapter on
Prandtl–Meyer flow was added.
Version 0.3
In the traditional class of compressible flow it is assumed that the students will be
aerospace engineers or dealing mostly with construction of airplanes and turbomachin-
1 A reader asked this author to examine a paper on Triple Shock Entropy Theorem and Its Conse-
quences by Le Roy F. Henderson and Ralph Menikoff. This led to comparison between maximum to
ideal gas model to more general model.
xii PROLOGUE
ery. This premise should not be assumed. This assumption drives students from other
fields away from this knowledge. This knowledge should be spread to other fields be-
cause it needed there as well. This “rejection” is especially true when students feel that
they have to go through a “shock wave” in their understanding.
This book is the second book in the series of POTTO project books. POTTO
project books are open content textbooks. The reason the topic of Compressible Flow
was chosen, while relatively simple topics like fundamentals of strength of material were
delayed, is because of the realization that manufacture engineering simply lacks funda-
mental knowledge in this area and thus produces faulty designs and understanding of
major processes. Unfortunately, the undersigned observed that many researchers who
are dealing with manufacturing processes are lack of understanding about fluid mechan-
ics in general but particularly in relationship to compressible flow. In fact one of the
reasons that many manufacturing jobs are moving to other countries is because of the
lack of understanding of fluid mechanics in general and compressible in particular. For
example, the lack of competitive advantage moves many of the die casting operations
to off shore2 . It is clear that an understanding of Compressible Flow is very important
for areas that traditionally have ignored the knowledge of this topic3 .
As many instructors can recall from their time as undergraduates, there were
classes during which most students had a period of confusion, and then later, when
the dust settled, almost suddenly things became clear. This situation is typical also for
Compressible Flow classes, especially for external compressible flow (e.g. flow around a
wing, etc.). This book offers a more balanced emphasis which focuses more on internal
compressible flow than the traditional classes. The internal flow topics seem to be com-
mon for the “traditional” students and students from other fields, e.g., manufacturing
engineering.
This book is written in the spirit of my adviser and mentor E.R.G. Eckert. Who,
aside from his research activity, wrote the book that brought a revolution in the heat
transfer field of education. Up to Eckert’s book, the study of heat transfer was without
any dimensional analysis. He wrote his book because he realized that the dimensional
analysis utilized by him and his adviser (for the post doc), Ernst Schmidt, and their
colleagues, must be taught in engineering classes. His book met strong criticism in
which some called to burn his book. Today, however, there is no known place in world
that does not teach according to Eckert’s doctrine. It is assumed that the same kind of
individuals who criticized Eckert’s work will criticize this work. This criticism will not
change the future or the success of the ideas in this work. As a wise person says “don’t
tell me that it is wrong, show me what is wrong”; this is the only reply. With all the
above, it must be emphasized that this book will not revolutionize the field even though
considerable new materials that have never been published are included. Instead, it will
provide a new emphasis and new angle to Gas Dynamics.
Compressible flow is essentially different from incompressible flow in mainly two
2 Please read the undersigned’s book “Fundamentals of Die Casting Design,” which demonstrates
how ridiculous design and research can be.

3 The fundamental misunderstanding of choking results in poor models (research) in the area of
die casting, which in turn results in many bankrupt companies and the movement of the die casting
industry to offshore.
VERSION 0.3 xiii
respects: discontinuity (shock wave) and choked flow. The other issues, while impor-
tant, are not that crucial to the understanding of the unique phenomena of compressible
flow. These unique issues of compressible flow are to be emphasized and shown. Their
applicability to real world processes is to be demonstrated4 .
The book is organized into several chapters which, as a traditional textbook,
deals with a basic introduction of thermodynamics concepts (under construction). The
second chapter deals with speed of sound. The third chapter provides the first example
of choked flow (isentropic flow in a variable area). The fourth chapter deals with a simple
case of discontinuity (a simple shock wave in a nozzle). The next chapter is dealing with
isothermal flow with and without external forces (the moving of the choking point),
again under construction. The next three chapters are dealing with three models of
choked flow: Isothermal flow5 , Fanno flow and Rayleigh flow. First, the Isothermal flow
is introduced because of the relative ease of the analytical treatment. Isothermal flow
provides useful tools for the pipe systems design. These chapters are presented almost
independently. Every chapter can be “ripped” out and printed independently. The
topics of filling and evacuating of gaseous chambers are presented, normally missed from
traditional textbooks. There are two advanced topics which included here: oblique shock
wave, and properties change effects (ideal gases and real gases) (under construction).
In the oblique shock, for the first time analytical solution is presented, which is excellent
tool to explain the strong, weak and unrealistic shocks. The chapter on one-dimensional
unsteady state, is currently under construction.
The last chapter deals with the computer program, Gas Dynamics Calculator
(CDC-POTTO). The program design and how to use the program are described (briefly).
Discussions on the flow around bodies (wing, etc), and Prandtl–Meyer expansion
will be included only after the gamma version unless someone will provide discussion(s)
(a skeleton) on these topics.
It is hoped that this book will serve the purposes that was envisioned for the
book. It is further hoped that others will contribute to this book and find additional
use for this book and enclosed software.
4 If you have better and different examples or presentations you are welcome to submit them.
5 It is suggested to referred to this model as Shapiro flow
xiv PROLOGUE
Contents
Please Update v
Abstract vii
Why Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Short Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Long Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Prologue ix
Version 0.5.2 491 pages, 5.6M bytes April 18, 2022 . . . . . . . . . . . . . . ix
Version 0.5.0 525 pages, 5.5M bytes July 13, 2021 . . . . . . . . . . . . . . ix
Version 0.4.9 pp. ? Feb ?, 2012 . . . . . . . . . . . . . . . . . . . . . . . . ix
Version 0.4.8.5a . July 21, 2009 . . . . . . . . . . . . . . . . . . . . . . . . ix
Version 0.4.8 Jan. 23, 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Version 0.4.3 Sep. 15, 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Version 0.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Version 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Version 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Nomenclature xxxi
GNU Free Documentation License . . . . . . . . . . . . . . . . . . . . . . . xi
1. APPLICABILITY AND DEFINITIONS . . . . . . . . . . . . . . . . xii
2. VERBATIM COPYING . . . . . . . . . . . . . . . . . . . . . . . . . xiii
3. COPYING IN QUANTITY . . . . . . . . . . . . . . . . . . . . . . . xiii
4. MODIFICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
5. COMBINING DOCUMENTS . . . . . . . . . . . . . . . . . . . . . xvi
6. COLLECTIONS OF DOCUMENTS . . . . . . . . . . . . . . . . . . xvi
7. AGGREGATION WITH INDEPENDENT WORKS . . . . . . . . . . xvi
8. TRANSLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
9. TERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
10. FUTURE REVISIONS OF THIS LICENSE . . . . . . . . . . . . . . xvii
ADDENDUM: How to use this License for your documents . . . . . . . xviii
How to contribute to this book . . . . . . . . . . . . . . . . . . . . . . . . xix
Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
John Martones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Grigory Toker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
xv
xvi CONTENTS
Ralph Menikoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
Domitien Rataaforret . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
Gary Settles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
Your name here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
Typo corrections and ”minor” contributions . . . . . . . . . . . . . . . xx
Potto Prologue xxv

Version 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi
Version 0.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi
Version 0.4.1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi
Speed of Sound [beta] . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv
Stagnation effects [advance] . . . . . . . . . . . . . . . . . . . . . . . xxxvi
Nozzle [advance] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi
Normal Shock [advance] . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi
Minor Loss [NSV] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi
Isothermal Flow [advance] . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii
Fanno Flow [advance] . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii
Rayleigh Flow [beta] . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii
Add mass [NSY] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii
Evacuation and filling semi rigid Chambers [alpha] . . . . . . . . . . . . xxxvii
Evacuating and filling chambers under external forces [alpha] . . . . . . xxxvii
Oblique Shock [advance] . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii
Prandtl–Meyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii
Transient problem [NYP] . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii
General 1-D flow [NYP] . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii
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 . . . . . . . . . . . . 13
1.3.6 Biographies of Major Figures . . . . . . . . . . . . . . . . . . . 14
2 Review of Thermodynamics 25
2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.1 Maximum Work of Expansion Process . . . . . . . . . . . . . . 33
2.2 The Velocity–Temperature Diagram . . . . . . . . . . . . . . . . . . . 35
CONTENTS xvii
3 Basic of Fluid Mechanics 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Kinds of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.3 Kinematic Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.4 Bulk Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.1 Control Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.3 Reynolds Transport Theorem . . . . . . . . . . . . . . . . . . . 48
3.4 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.1 Momentum Governing Equation . . . . . . . . . . . . . . . . . 53
3.4.2 Conservation Moment of Momentum . . . . . . . . . . . . . . 54
3.5 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5.1 Approximation of Energy Equation . . . . . . . . . . . . . . . 59
3.6 Limitations of Integral Approach . . . . . . . . . . . . . . . . . . . . . 63
3.7 Differential Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . 64
3.7.2 Momentum Equations or N–S equations . . . . . . . . . . . . . 65
3.7.3 Boundary Conditions and Driving Forces . . . . . . . . . . . . . 67
4 Speed of Sound 69
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Speed of Sound in Ideal and Perfect Gases . . . . . . . . . . . . . . . . 71
4.4 Speed of Sound in Real Gases . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Speed of Sound in Almost Incompressible Liquid . . . . . . . . . . . . . 77
4.5.1 Sound in Variable Compressible Liquids . . . . . . . . . . . . . 79
4.6 Speed of Sound in Solids . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7 Sound Speed in Two Phase Medium . . . . . . . . . . . . . . . . . . . 84
4.8 The Dimensional Effect of the Speed of Sound . . . . . . . . . . . . . . 86
4.8.1 Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.8.2 Acoustic Wave Equation – Derivation . . . . . . . . . . . . . . 89
4.8.3 Hearing and Sound Waves . . . . . . . . . . . . . . . . . . . . 93
4.8.4 Sound Wave in Three Dimensions . . . . . . . . . . . . . . . . 95
5 Isentropic Flow 101

5.1 Stagnation State for Ideal Gas Model . . . . . . . . . . . . . . . . . . . 101
5.1.1 General Relationship . . . . . . . . . . . . . . . . . . . . . . . . 101
5.1.2 Relationships for Small Mach Number . . . . . . . . . . . . . . 104
5.2 Isentropic Converging-Diverging Flow in Cross Section . . . . . . . . . 105
5.2.1 The Properties in the Adiabatic Nozzle . . . . . . . . . . . . . . 106
5.2.2 Isentropic Flow Examples . . . . . . . . . . . . . . . . . . . . . 110
5.2.3 Mass Flow Rate (Number) . . . . . . . . . . . . . . . . . . . . 113
xviii CONTENTS
5.3 Isentropic Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.3.1 Isentropic Isothermal Flow Nozzle . . . . . . . . . . . . . . . . 125
5.4 The Impulse Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.4.1 Impulse in Isentropic Adiabatic Nozzle . . . . . . . . . . . . . . 133
5.4.2 The Impulse Function in Isothermal Nozzle . . . . . . . . . . . 135
5.5 Isothermal Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.6 The effects of Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.7 Isentropic Relationships for Real Gases . . . . . . . . . . . . . . . . . . 142
6 Normal Shock 145

6.1 Solution of the Governing Equations . . . . . . . . . . . . . . . . . . . 147
6.1.1 Informal Model . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.1.2 Formal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.1.3 Prandtl’s Condition . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2 Operating Equations and Analysis . . . . . . . . . . . . . . . . . . . . 153
6.2.1 The Limitations of the Shock Wave . . . . . . . . . . . . . . . 154
6.2.2 Small Perturbation Solution . . . . . . . . . . . . . . . . . . . . 155
6.2.3 Shock Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.2.4 Shock Drag or Wave Drag . . . . . . . . . . . . . . . . . . . . 155
6.3 The Moving Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3.1 Shock or Wave Drag Result from a Moving Shock . . . . . . . . 159
6.3.2 Shock Result from a Sudden and Complete Stop . . . . . . . . 161
6.3.3 Moving Shock into Stationary Medium (Suddenly Open Valve) . 163
6.3.4 Partially Open Valve . . . . . . . . . . . . . . . . . . . . . . . 174
6.3.5 Partially Closed Valve . . . . . . . . . . . . . . . . . . . . . . 176
6.3.6 Worked–out Examples for Shock Dynamics . . . . . . . . . . . 176
6.4 Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.4.1 Special Shock Dynamics . . . . . . . . . . . . . . . . . . . . . 192
6.4.2 Shock Tube Thermodynamics Considerations . . . . . . . . . . 195
6.5 Shock with Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.6 Shock in Wet Steam . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.7 Normal Shock in Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.8 Additional Examples for Moving Shocks . . . . . . . . . . . . . . . . . 206
6.9 Tables of Normal Shocks, k = 1.4 Ideal Gas . . . . . . . . . . . . . . . 208
7 Normal Shock in Variable Duct Areas 215

7.1 Nozzle efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
7.2 Diffuser Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8 Nozzle Flow With External Forces 233

8.1 Isentropic Nozzle (Q = 0) . . . . . . . . . . . . . . . . . . . . . . . . . 234
8.2 Isothermal Nozzle (T = constant) . . . . . . . . . . . . . . . . . . . . 236
CONTENTS xix
9 Isothermal Flow 237
9.1 The Control Volume Analysis/Governing equations . . . . . . . . . . . 238
9.2 Dimensionless Representation . . . . . . . . . . . . . . . . . . . . . . 238
9.3 The Entrance Limitation of Supersonic Branch . . . . . . . . . . . . . 243
9.4 Comparison with Incompressible Flow . . . . . . . . . . . . . . . . . . 244
9.5 Supersonic Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
9.6 Figures and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
9.7 Isothermal Flow Examples . . . . . . . . . . . . . . . . . . . . . . . . . 247
9.8 Unchoked Situations in Fanno Flow . . . . . . . . . . . . . . . . . . . . 253
9.8.1 Reynolds Number Effect . . . . . . . . . . . . . . . . . . . . . 254
10 Fanno Flow 257

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10.2 Fanno Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
10.3 Non–Dimensionalization of the Equations . . . . . . . . . . . . . . . . 259
10.4 The Mechanics and Why the Flow is Choked? . . . . . . . . . . . . . . 262
10.5 The Working Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 263
10.6 Examples of Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . . . 267
10.7 Supersonic Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
10.8 Maximum Length for the Supersonic Flow . . . . . . . . . . . . . . . . 273
10.9 Working Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
10.9.1 Variations of The Tube Length ( 4fDL ) Effects . . . . . . . . . . 275
10.9.2 The Pressure Ratio, P2 / P1 , effects . . . . . . . . . . . . . . . 279
10.9.3 Entrance Mach number, M1 , effects . . . . . . . . . . . . . . . 283
10.10Practical Examples for Subsonic Flow . . . . . . . . . . . . . . . . . . 288
10.10.1 Subsonic Fanno Flow for Given 4 D fL
and Pressure Ratio . . . . 288
10.10.2 Subsonic Fanno Flow for a Given M1 and Pressure Ratio . . . . 291
10.11The Approximation of the Fanno Flow by Isothermal Flow . . . . . . . 295
10.12 The Table for Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . . 297
10.13 Appendix – Reynolds Number Effects . . . . . . . . . . . . . . . . . . 298
11 Rayleigh Flow 301

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
11.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
11.3 Rayleigh Flow Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
11.4 Examples For Rayleigh Flow . . . . . . . . . . . . . . . . . . . . . . . 308
12 Evacuating SemiRigid Chambers 315

12.1 Governing Equations and Assumptions . . . . . . . . . . . . . . . . . . 316
12.2 General Model and Non–Dimensionalization . . . . . . . . . . . . . . . 318
12.2.1 Isentropic Process . . . . . . . . . . . . . . . . . . . . . . . . . 320
12.2.2 Isothermal Process in The Chamber . . . . . . . . . . . . . . . 321
12.2.3 A Note on the Entrance Mach number . . . . . . . . . . . . . . 321
12.3 Rigid Tank with A Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . 321
12.3.1 Adiabatic Isentropic Nozzle Attached . . . . . . . . . . . . . . . 322
xx CONTENTS
12.3.2 Isothermal Nozzle Attached . . . . . . . . . . . . . . . . . . . . 323
12.4 Rapid evacuating of a rigid tank . . . . . . . . . . . . . . . . . . . . . 324
12.4.1 Assuming Fanno Flow Model . . . . . . . . . . . . . . . . . . . 324
12.4.2 Filling Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
12.4.3 The Isothermal Process . . . . . . . . . . . . . . . . . . . . . . 327
12.4.4 Simple Semi Rigid Chamber . . . . . . . . . . . . . . . . . . . 328
12.4.5 The “Simple” General Case . . . . . . . . . . . . . . . . . . . . 328
12.5 Advance Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
12.6 Remark on Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
13 Evacuating under External Volume Control 333

13.1 General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
13.1.1 Rapid Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
13.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
13.1.3 Direct Connection . . . . . . . . . . . . . . . . . . . . . . . . . 340
13.2 Non–Linear Functions Effects . . . . . . . . . . . . . . . . . . . . . . . 340
13.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
14 Oblique Shock 343

14.1 Preface to Oblique Shock . . . . . . . . . . . . . . . . . . . . . . . . . 343
14.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
14.2.1 Introduction to Oblique Shock . . . . . . . . . . . . . . . . . . 344
14.2.2 Introduction to Prandtl–Meyer Function . . . . . . . . . . . . . 344
14.2.3 Introduction to Zero Inclination . . . . . . . . . . . . . . . . . . 345
14.3 Oblique Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
14.4 Solution of Mach Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 348
14.4.1 Upstream Mach Number, M1 , and Deflection Angle, δ . . . . . 348
14.4.2 When No Oblique Shock Exist or the case of D > 0 . . . . . . 351
14.4.3 Upstream Mach Number, M1 , and Shock Angle, θ . . . . . . . 359
14.4.4 Given Two Angles, δ and θ . . . . . . . . . . . . . . . . . . . 361
14.4.5 Flow in a Semi–2D Shape . . . . . . . . . . . . . . . . . . . . . 362
14.4.6 Flow in a Semi-2D Shape . . . . . . . . . . . . . . . . . . . . . 363
14.4.7 Small δ “Weak Oblique shock” . . . . . . . . . . . . . . . . . . 364
14.4.8 Close and Far Views of the Oblique Shock . . . . . . . . . . . . 365
14.4.9 Maximum Value of Oblique shock . . . . . . . . . . . . . . . . 365
14.5 Detached Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
14.5.1 Issues Related to the Maximum Deflection Angle . . . . . . . . 367
14.5.2 Oblique Shock Examples . . . . . . . . . . . . . . . . . . . . . 369
14.5.3 Application of Oblique Shock . . . . . . . . . . . . . . . . . . . 370
14.5.4 Optimization of Suction Section Design . . . . . . . . . . . . . 382
14.5.5 Retouch of Shock Drag or Wave Drag . . . . . . . . . . . . . . 382
14.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
14.7 Appendix: Oblique Shock Stability Analysis . . . . . . . . . . . . . . . 385
CONTENTS xxi
15 Prandtl–Meyer Function 387
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
15.2 Geometrical Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . 388
15.2.1 Alternative Approach to Governing Equations . . . . . . . . . . 389
15.2.2 Comparison And Limitations between the Two Approaches . . . 393
15.3 The Maximum Turning Angle . . . . . . . . . . . . . . . . . . . . . . . 393
15.4 The Working Equations for the Prandtl-Meyer Function . . . . . . . . . 394
15.5 d’Alembert’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
15.6 Flat Body with an Angle of Attack . . . . . . . . . . . . . . . . . . . . 395
15.7 Examples For Prandtl–Meyer Function . . . . . . . . . . . . . . . . . . 395
15.8 Combination of the Oblique Shock and Isentropic Expansion . . . . . . 398
A Computer Program 403

A.1 About the Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
A.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
A.3 Program listings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
B Oblique Shock History 407
Bibliography 411
Index 413
xxii CONTENTS
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 Flow rate as a function of the back pressure . . . . . . . . . . . . . . . 11
1.4 Portrait of Galileo Galilei . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Photo of Ernest Mach . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 The photo of a bullet in a supersonic flow taken by Mach . . . . . . . . 15
1.7 Lord Rayleigh portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.8 Portrait of Rankine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.9 The photo of Gino Fanno approximately in 1950 . . . . . . . . . . . . . 18
1.10 Photo of Prandtl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.11 The photo and famous diagram of Theodor Meyer . . . . . . . . . . . . 20
1.12 The photo of Ernst Rudolf George Eckert with Bar-Meir’s family . . . . 22
2.1 Cylinder and piston configuration of maximum work . . . . . . . . . . . 33

2.2 Dimensionless work available in a cylinder piston configuration . . . . . 35
2.3 Temperature Velocity Pressure . . . . . . . . . . . . . . . . . . . . . . 36
a The pressure lines . . . . . . . . . . . . . . . . . . . . . . . . . 36
b The pressure lines . . . . . . . . . . . . . . . . . . . . . . . . . 36
c The energy lines . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 The velocity temperature diagram . . . . . . . . . . . . . . . . . . . . 38
3.1 Schematics of flow in a pipe with varying density . . . . . . . . . . . . 44

3.2 Pressure lines a static fluid with a constant density . . . . . . . . . . . 50
3.3 The explanation for the direction relative to surface . . . . . . . . . . . 55
3.4 The work on the control volume . . . . . . . . . . . . . . . . . . . . . 57
3.5 A long pipe exposed to a sudden pressure difference . . . . . . . . . . . 61
3.6 The mass balance on the infinitesimal control volume . . . . . . . . . . 64
4.1 A very slow moving piston in a still gas . . . . . . . . . . . . . . . . . . 70

4.2 Stationary sound wave and gas moves relative to the pulse . . . . . . . 70
4.3 The compressibility chart . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 The Correction Factor for Time of Sound Wave . . . . . . . . . . . . . 81
4.5 Doppler effect schematic . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.6 Moving object at three relative velocities . . . . . . . . . . . . . . . . . 87
a Object travels at 0.005 of the speed of sound . . . . . . . . . . 87
xxiii
xxiv LIST OF FIGURES
b Object travels at 0.05 of the speed of sound . . . . . . . . . . . 87
c Object travels at 0.15 of the speed of sound . . . . . . . . . . . 87
4.7 The hearing as a function of the frequency . . . . . . . . . . . . . . . . 94
4.8 Schematic of the spherical waves . . . . . . . . . . . . . . . . . . . . . 94
5.1 Flow through a converging diverging nozzle . . . . . . . . . . . . . . . 101

5.2 Perfect gas flows through a tube . . . . . . . . . . . . . . . . . . . . . 103
5.3 Station properties as f (M ) . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4 Control volume inside a converging-diverging nozzle . . . . . . . . . . . 105
5.5 The relationship between the cross section and the Mach number . . . 109
5.6 Various ratios as a function of Mach number for isothermal Nozzle . . . 128
5.7 The comparison of nozzle flow . . . . . . . . . . . . . . . . . . . . . . 129
a Comparison between the isothermal nozzle and adiabatic nozzle
in various variables . . . . . . . . . . . . . . . . . . . . . . . . 129
b The comparison of the adiabatic model and isothermal model . 129
5.8 Comparison of the pressure and temperature drop (two scales) . . . . . 131
5.9 Schematic to explain the significances of the Impulse function . . . . . 133
5.10 Schematic of a flow through a nozzle example (5.10) . . . . . . . . . . 134
6.1 A shock wave inside a tube . . . . . . . . . . . . . . . . . . . . . . . . 145

6.2 The intersection of Fanno flow and Rayleigh flow . . . . . . . . . . . . 147
6.3 The Mexit and P0 as a function Mupstream . . . . . . . . . . . . . . . 151
6.4 The ratios of the static properties of the two sides of the shock. . . . . 153
6.5 The shock drag diagram . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.6 Stationary and moving coordinates for the moving shock . . . . . . . . 157
a Stationary coordinates . . . . . . . . . . . . . . . . . . . . . . . 157
b Moving coordinates . . . . . . . . . . . . . . . . . . . . . . . . 157
6.7 The shock drag diagram for moving shock . . . . . . . . . . . . . . . . 159
6.8 The diagram for the common explanation for shock drag . . . . . . . . 160
6.10 A shock from close Valve in two coordinates . . . . . . . . . . . . . . . 161
6.11 The moving shock a result of a sudden stop . . . . . . . . . . . . . . . 162
6.12 A shock move into still medium: open valve case . . . . . . . . . . . . 164
6.13 The number of iterations to achieve convergence. . . . . . . . . . . . . 165
0
a My = 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
0
b My = 1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.14 Schematic of showing the piston pushing air . . . . . . . . . . . . . . . 166
6.15 Time the pressure at the nozzle for the French problem . . . . . . . . . 168
6.16 Max Mach number as a function of k. . . . . . . . . . . . . . . . . . . 169
6.17 Time the pressure at the nozzle for the French problem . . . . . . . . . 172
6.18 Moving shock as a result of valve opening . . . . . . . . . . . . . . . . 174
LIST OF FIGURES xxv
6.19 The results of the partial opening of the valve . . . . . . . . . . . . . . 175
6.20 A shock as a result of partially a valve closing . . . . . . . . . . . . . . 176
6.21 Schematic of a piston pushing air in a tube . . . . . . . . . . . . . . . 181
6.22 Figure for Example 6.11 . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.23 The shock tube schematic with a pressure “diagram” . . . . . . . . . . 183
6.24 Maximum Mach number that can be obtained for given specific heats . 188
6.25 The Mach number obtained with various parameters . . . . . . . . . . 190
6.26 Differential element to describe the isentropic pressure . . . . . . . . . 191
6.27 Porous piston pushing gas . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.28 Initial Shock tube schematic for thermodynamics consideration . . . . . 196
6.29 The final or equilibrium stage in the shock tube . . . . . . . . . . . . . 198
6.30 Dimensionless work of shock tube . . . . . . . . . . . . . . . . . . . . 200
a Dimensionless work as a function for various ξe , k = 1.4 . . . . 200
b Dimensionless work as a function for various k, ξe = 0.4 . . . . 200
6.31 The equilibrium length as a function of the initial dimensionless length 202
6.32 The equilibrium pressure as a function of the initial dimensionless length 203
6.33 Explanation why the ruptured diaphragm cannot reach maximum tem-
perature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.34 Ames Research Center Shock Tube with Thomas N. Canning . . . . . . 205
6.35 Figure for Example (6.19) . . . . . . . . . . . . . . . . . . . . . . . . . 207
6.36 The results for Example (6.19) . . . . . . . . . . . . . . . . . . . . . . 208
7.1 The flow in the nozzle with different back pressures. . . . . . . . . . . 215
7.2 A nozzle with a normal shock . . . . . . . . . . . . . . . . . . . . . . . 216
7.3 Clarify the definitions of diffuser efficiency . . . . . . . . . . . . . . . . 222
7.4 Schematic of a supersonic tunnel for Fig. 7.4 . . . . . . . . . . . . . . 223
7.5 Exit Mach number for non–ideal nozzle general solution . . . . . . . . . 230
7.6 The ratio of the exit temperature and stagnation temperature . . . . . 231
9.1 Control volume for isothermal flow . . . . . . . . . . . . . . . . . . . . 237

9.2 Working relationships for isothermal flow . . . . . . . . . . . . . . . . . 243
9.3 The entrance Mach for isothermal flow for 4 DfL
. . . . . . . . . . . . . 253
10.1 Control volume of the gas flow in a constant cross section . . . . . . . 257
10.2 Various parameters in fanno flow . . . . . . . . . . . . . . . . . . . . . 267
10.3 Schematic of Example 10.1 . . . . . . . . . . . . . . . . . . . . . . . . 268
10.4 The schematic of Example (10.2) . . . . . . . . . . . . . . . . . . . . . 269
10.5 The maximum length as a function of specific heat, k . . . . . . . . . . 274
10.6 The effects of increase of 4 D
fL
on the Fanno line . . . . . . . . . . . . 274
4f L
10.7 The effects of the increase of D on the Fanno Line . . . . . . . . . . 275
xxvi LIST OF FIGURES
10.8 Min and ṁ as a function of the 4fDL . . . . . . . . . . . . . . . . . . . 275
10.9 M1 as a function M2 for various 4fDL . . . . . . . . . . . . . . . . . . . 277
10.10 M1 as a function M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
10.11 The pressure distribution as a function of 4 D
fL
. . . . . . . . . . . . . 280
4f L
10.12Pressure as a function of long D . . . . . . . . . . . . . . . . . . . . 281
10.13 The effects of pressure variations on Mach number profile . . . . . . . 282
10.14 Pressure ratios as a function of 4 D
fL
when the total 4 D
fL
= 0.3 . . . . 283
10.15 Schematic of a “long” tube in supersonic branch . . . . . . . . . . . 283
10.16 The extra tube length as a function of the shock location . . . . . . . 284
10.17 The maximum entrance Mach number as a function of 4fDL . . . . . 285
10.18Pressure ratio obtained for fix 4 D
fL
for k=1.4 . . . . . . . . . . . . . . 288
4f L
10.19Conversion of solution for given = 0.5 and pressure ratio . . . . . 290
D
10.20 Unchoked flow showing the hypothetical “full” tube . . . . . . . . . . 290
10.21 The results of the algorithm showing the conversion rate . . . . . . . . 292
10.22 Solution to a missing diameter . . . . . . . . . . . . . . . . . . . . . 294
10.23 M1 as a function of 4 DfL
comparison with Isothermal Flow . . . . . . 295
10.24 “Moody” diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
11.1 The control volume of Rayleigh Flow . . . . . . . . . . . . . . . . . . . 301

11.2 The temperature entropy diagram for Rayleigh line . . . . . . . . . . . 303
11.3 The basic functions of Rayleigh Flow (k=1.4) . . . . . . . . . . . . . . 307
11.4 Schematic of the combustion chamber . . . . . . . . . . . . . . . . . . 312
12.1 The two different classifications of models . . . . . . . . . . . . . . . . 315

12.2 Comparison direct and reduced connection . . . . . . . . . . . . . . . . 316
a Reduced connection . . . . . . . . . . . . . . . . . . . . . . . . 316
b Direct connections . . . . . . . . . . . . . . . . . . . . . . . . . 316
12.3 Comparison direct and reduced connection . . . . . . . . . . . . . . . . 317
a Reduced connection . . . . . . . . . . . . . . . . . . . . . . . . 317
b Direct connections . . . . . . . . . . . . . . . . . . . . . . . . . 317
12.4 The pressure assumptions in the chamber . . . . . . . . . . . . . . . . 317
12.5 The reduced time as a function of the modified reduced pressure . . . . 325
12.6 The reduced time as a function of the modified reduced pressure . . . . 327
13.1 The control volume of the “Cylinder” . . . . . . . . . . . . . . . . . . 334

13.2 The pressure ratio as a function of the dimensionless time . . . . . . . 337
13.3 P as a function of t for choked condition . . . . . . . . . . . . . . . . . 338
13.4 The pressure ratio as a function of the dimensionless time . . . . . . . 339
13.5 Energy transfer in cycle piston cylinder Assembly . . . . . . . . . . . . 341
14.1 A view of a normal shock as a limited case for oblique shock . . . . . . 343
14.2 The oblique shock or Prandtl–Meyer function regions . . . . . . . . . . 344
14.3 A typical oblique shock schematic . . . . . . . . . . . . . . . . . . . . 345
14.4 Flow around spherically blunted 30◦ cone-cylinder . . . . . . . . . . . . 350
LIST OF FIGURES xxvii
14.5 The different views of a large inclination angle . . . . . . . . . . . . . . 352
14.6 The three different Mach numbers . . . . . . . . . . . . . . . . . . . . 353
14.7 The various coefficients of three different Mach numbers . . . . . . . . 357
14.8 The “imaginary” Mach waves at zero inclination . . . . . . . . . . . . . 358
14.9 The D, shock angle, and My for M1 = 3 . . . . . . . . . . . . . . . . . 359
14.10 The possible range of solutions . . . . . . . . . . . . . . . . . . . . . 361
14.11 Two dimensional wedge . . . . . . . . . . . . . . . . . . . . . . . . . . 362
14.12 Schematic of finite wedge with zero angle of attack. . . . . . . . . . . 364
14.13 A local and a far view of the oblique shock. . . . . . . . . . . . . . . 365
14.14 The schematic for a round–tip bullet in a supersonic flow . . . . . . . 366
14.15 The schematic for a symmetrical suction section with reflection. . . . . 367
14.16The “detached” shock in a complicated configuration . . . . . . . . . . 368
14.17 Oblique shock around a cone . . . . . . . . . . . . . . . . . . . . . . 369
14.18 Maximum values of the properties in an oblique shock . . . . . . . . . 371
14.19 Two variations of inlet suction for supersonic flow . . . . . . . . . . . 371
14.20Schematic for Example (??) . . . . . . . . . . . . . . . . . . . . . . . . 372
14.21Schematic for Example (14.5) . . . . . . . . . . . . . . . . . . . . . . . 373
14.22 Schematic of two angles turn with two weak shocks . . . . . . . . . . 374
14.23Schematic for Example (14.10) . . . . . . . . . . . . . . . . . . . . . . 377
14.24Illustration for Example (14.13) . . . . . . . . . . . . . . . . . . . . . . 380
14.25 Revisiting of shock drag diagram for the oblique shock. . . . . . . . . 383
14.26Oblique δ − θ − M relationship figure . . . . . . . . . . . . . . . . . . 384
14.27 Typical examples of unstable and stable situations. . . . . . . . . . . . 385
14.28 The schematic of stability analysis for oblique shock. . . . . . . . . . . 385
15.1 Definition of the angle for the Prandtl–Meyer function . . . . . . . . . 387

15.2 The angles of the Mach line triangle . . . . . . . . . . . . . . . . . . . 387
15.3 The schematic of the turning flow . . . . . . . . . . . . . . . . . . . . 388
15.4 The mathematical coordinate description . . . . . . . . . . . . . . . . . 389
15.5 Prandtl-Meyer function after the maximum angle . . . . . . . . . . . . 393
15.6 The angle as a function of the Mach number . . . . . . . . . . . . . . 394
15.7 Diamond shape for supersonic d’Alembert’s Paradox . . . . . . . . . . 394
15.8 The definition of attack angle for the Prandtl–Meyer function . . . . . 395
15.9 Schematic for Example (15.1) . . . . . . . . . . . . . . . . . . . . . . . 395
15.10 Schematic for the reversed question of Example 15.2 . . . . . . . . . . 397
15.11 Schematic of the nozzle and Prandtl–Meyer expansion. . . . . . . . . 400
A.1 Schematic diagram that explains the structure of the program . . . . . 404
xxviii LIST OF FIGURES
List of Tables
1 Books Under Potto Project . . . . . . . . . . . . . . . . . . . . . . . . xxvii
2.1 Properties of Various Ideal Gases at [300K] . . . . . . . . . . . . . . . 30
3.1 Bulk modulus for selected materials . . . . . . . . . . . . . . . . . . . 42
4.1 Speed of sound in water . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2 Liquids speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Solids speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1 Fliegner’s number a function of Mach number . . . . . . . . . . . . . . 119

5.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Isentropic Table k = 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.3 Isothermal Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3 Isothermal Table (continue) . . . . . . . . . . . . . . . . . . . . . . . . 137
6.1 Table of maximum values of the shock-choking phenomenon . . . . . . 171

6.2 The shock wave table for k = 1.4 . . . . . . . . . . . . . . . . . . . . 208
6.2 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.2 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.3 Table for a Reflective Shock suddenly closed valve . . . . . . . . . . . . 210
6.3 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.4 Table for shock suddenly opened valve (k=1.4) . . . . . . . . . . . . . 211
6.4 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.5 Table for shock from a suddenly opened valve (k=1.3) . . . . . . . . . 212
6.5 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
6.5 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
9.1 The Isothermal Flow basic parameters . . . . . . . . . . . . . . . . . . 247

9.2 The flow parameters for unchoked flow . . . . . . . . . . . . . . . . . 253
10.1 Fanno Flow Standard basic Table k=1.4 . . . . . . . . . . . . . . . . . 297

10.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
xxix
xxx LIST OF TABLES
11.1 Rayleigh Flow k=1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
11.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
11.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
14.1 Table of maximum values of the oblique Shock k=1.4 . . . . . . . . . 365

14.1 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
Nomenclature
R̄ Universal gas constant, see equation (2.26), page 29

τ The shear stress Tenser, see equation (3.33), page 52
` Units length., see equation (2.1), page 25
M Angular Momentum, see equation (3.43), page 54
F ext External forces by non–fluids means, see equation (3.36), page 53
ρ Density of the fluid, see equation (4.1), page 68
B bulk modulus, see equation (4.38), page 76
Bf Body force, see equation (2.9), page 27
c Speed of sound, see equation (4.1), page 68
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 (4.50), page 81
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
kT Fluid thermal conductivity, see equation (3.47), page 56
M Mach number, see equation (5.8), page 100
n The polytropic coefficient, see equation (4.35), page 75
P Pressure, see equation (4.3), page 68
xxxi
LIST OF TABLES
q Energy per unit mass, see equation (2.6), page 26
Q12 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
Rmix The universal gas constant for mixture, see equation (4.62), page 83
S Entropy of the system, see equation (2.13), page 28

t Time, see equation (4.18), page 71
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 (4.22), page 72
P Power or energy per time, see equation (4.90), page 91

The Book Change Log
Version 0.5.2 new branch

On 18th April 2022 (5.6M pp. 491)
• Utilizing tcolorbox (exBox) to color to the examples
• Reference fixing
• Added two examples
Version 0.5.0 new branch

On 12th July 2021 (5.5M pp. 524)
• Kyu–Seop Kim senior researcher in Hyundai Cooperation Korea to Rayleigh flow
equation (stagnation pressure).
• Add examples and minor things.
• Upgrade to today (2021) index making procedure.
• Other minor corrections.
Version 0.4.9.8 (second beautified version)

On 17th March 2013 (3.7M pp. 465)
• Real gas issues correction by Guy de Carufel - Canada
• Continue enhance the figures in various chapters.
• Add short dimensional analysis for sound chapter.

On 26th Feb 2013 (3.7M pp. 465)
• Real gas issues in isentropic flow and filling chambers
i
ii LIST OF TABLES
• Add short appendix for oblique shock history.
• Enhance the indexes of the book.

On 14th Feb 2013 (3.7M pp. 462)
• Provide additional algorithm for fld and the pressure ratio Fanno Flow.
Version 0.4.9.5 (beautified version)

On 20th Jan 2013 (3.7M pp. 458)
• Enhance the discussion and examples on the nozzle efficiency.
• Enhance the figures in various chapters.
Version 0.4.9.4
On 24th Dec 2012 (3.7M pp. 452)
• Enhance the discussion and examples on the maximum temperature in shock tube.
Version 0.4.9.3
On 11th Dec 2012 (3.7M pp. 442)
• Add two examples to the Isentropic chapter.
• Fixing the format and graphics in Isentropic chapter.
Version 0.4.9.2
On 10th Dec 2012 (3.7M pp. 440)
• Add the section on speed of sound in variable density liquid due the gravity with
constant gravity.
• English and typo corrections.

LIST OF TABLES iii
Version 0.4.9.1
On 29th Nov 2012 (3.6M pp. 438)
• Maximum work that can be obtained from a cylinder piston configuration
• English and typos corrections.
Version 0.4.9.0
On 13rd Feb 2012 (3.6M pp. 432)
• Significant Enhancement the shock tube section.
• Update the book to compile with the current potto.sty.
• insert the introduction to fluid mechanics.
Version 0.4.8.8
On 29th Dec 2011 (3.6M pp. 386)
• Add two figures explain the maximum Mach number limits in the shock tube.
Version 0.4.8.7
On 29th Dec 2011 (3.6M pp. 386)
• Significantly improved the shock tube section.
• Improvements of the structure to meed to the standard.
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).
iv LIST OF TABLES
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.
• Improve the multilayer sound traveling example (Heru’s suggestion)
Version 0.4.8.5a
On 04th July 2009 (3.3M pp. 380)
• Correction to the gases properties table (Michael Madden and Heru Reksoprodjo)
• 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).
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.

LIST OF TABLES v
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.
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.
vi LIST OF TABLES
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.
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.
LIST OF TABLES vii
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.
Version 0.4.3.2rc1
Dec-04-2006 version
• Add new algorithm for Fanno Flow calculation of the shock location in the super-
sonic 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 chap-
ter.
• 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
viii LIST OF TABLES
• 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)
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 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.

LIST OF TABLES ix
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.
Version 0.4.1.8rc2
Apr-11-2006
• Added the Maximum Deflection Mach number’s equation
• Added several examples to oblique shock
x LIST OF TABLES
Notice of Copyright For This Book
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document must themselves be free in the same sense. It complements the GNU General
Public License, which is a copyleft license designed for free software.
We have designed this License in order to use it for manuals for free software,
because free software needs free documentation: a free program should come with
manuals providing the same freedoms that the software does. But this License is not
limited to software manuals; it can be used for any textual work, regardless of subject
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xi
xii LIST OF TABLES
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Imprisoned lay the giant snake,
With naught his sullen sleep to break.”
Thor’s Fishing, Oehlenschläger (Pigott’s tr.).
Baiting his powerful hook with the ox head, Thor angled for
Iörmungandr, while the giant drew up two whales, which seemed
enough for an early morning’s meal.
As Hymir was about to propose a return, Thor suddenly felt a jerk,
and began pulling as hard as he could, for he knew by the resistance
of his prey, and the terrible storm lashed up by its writhings, that he
had hooked the Midgard snake. In his determined efforts to force him
to rise to the surface, Thor braced his feet so strongly against the
bottom of the boat that he went through it and stood on the bed of the
sea.
After an indescribable struggle, the monster’s terrible venom-
breathing head appeared, and Thor, seizing his hammer, was about to
annihilate it when the giant, frightened by the proximity of
Iörmungandr, and fearing lest the boat should sink and he become its
prey, drew his knife, cut the fishing line, and thus allowed the monster
to drop back like a stone to the bottom of the sea.
“The knife prevails: far down beneath the main

The serpent, spent with toil and pain,
To the bottom sank again.”
Thor’s Fishing, Oehlenschläger (Pigott’s tr.).
Angry with Hymir for his inopportune interference, Thor dealt him
a blow with his hammer which knocked him overboard; but Hymir,
undismayed, waded ashore, and met him as he returned to the
beach. Hymir then took both whales, his share of the fishing, upon his
back, to carry them to the house; and Thor, wishing to show his
strength also, shouldered boat, oars, and fishing tackle, and followed
him.
Breakfast being disposed of, Hymir challenged Thor to show his
strength by breaking his goblet; but although the thunder-god threw it
with irresistible force against stone pillars and walls, it remained whole
and was not even bent. In obedience to a whisper from Tyr’s mother,
however, Thor suddenly hurled it against the giant’s forehead, the
only substance tougher than itself, where it was shivered to pieces.
Hymir, having thus seen what Thor could do, told him he might have
the required kettle, which Tyr vainly tried to lift, and which Thor could
raise from the floor only after he had drawn his belt of strength up to
the very last hole.
“Tyr twice assayed

To move the vessel,
Yet at each time
Stood the kettle fast.
Then Môdi’s father
By the brim grasped it,
And trod through
The dwelling’s floor.”
Lay of Hymir (Thorpe’s tr.).
The wrench with which he pulled it up, however, greatly shattered

the giant’s house and broke his floor to pieces. As Tyr and Thor were
departing, the latter having clapped the huge pot on his head in the
guise of a hat, Hymir summoned the other frost giants, and proposed
that they should slay their inveterate foe. Before they could overtake
him, Thor, turning around, became aware of their pursuit, and, hurling
Miölnir repeatedly at them, slew them all ere he carried the kettle in
triumph to Ægir to enable him to brew enough ale for the harvest
feast.
The physical explanation of this myth is, of course, a thunder
storm (Thor), in conflict with the raging sea (the Midgard snake), and
the breaking up of the polar ice (Hymir’s goblet and floor) in the heat
of summer.
The gods now joyfully accepted Ægir’s invitation to be present at
his feast, went there in festive array, and were ever after wont to
celebrate the harvest home in his coral caves.
“Then Vans and Æsir, mighty gods,

Of earth and air, and Asgard, lords,—
Advancing with each goddess fair,
A brilliant retinue most rare,—
Attending mighty Odin, swept
Up wave-worn aisle in radiant march.”
Valhalla (J. C. Jones).
Ægir, as we have seen, ruled over all the sea with the help of the
treacherous Ran. Both of these divinities were considered cruel by the
Northern nations, who had much to suffer from the sea, which,
surrounding them on all sides, ran far into the heart of their countries
by means of the numerous fiords, and often swallowed the ships of
their vikings, with all the men on board.
“We Goth-folk know indeed

That the sea is a foe full deadly, and a friend that fails at need,
And that Ran, who dwells thereunder, will many a man beguile.”
Sigurd the Volsung (William Morris)
Besides these principal divinities of the sea, the Northern nations

believed in mermen and mermaids, the latter having
Other
divinities of swan plumage or seal garments, which they sometimes
the sea. laid for a moment upon the beach, and if a mortal
secured them he could compel the fair maidens to
remain ashore.
“She came through the waves when the fair moon shone
(Drift o’ the wave and foam o’ the sea);
She came where I walked on the sands alone,
With a heart as light as a heart may be.”
L. E. R.
There were also malignant marine monsters who were known as

Nicors, from whose name has been derived the proverbial Old Nick.
Many of the lesser water divinities had fish tails; the females bore the
name of Undines, and the males of Stromkarls, Nixies, Necks, or
Neckar. These water spirits often left their native streams, especially
during the middle ages, to appear at village dances, where they were
recognized by the wet hem of their garments. They often sat beside
the flowing brook or river, playing on a harp, or sang alluring songs
while combing out their long golden or green hair.
“The Neck here his harp in the glass castle plays,

And mermaidens comb out their green hair always,
And bleach here their shining white clothes.”
Stagnelius (Keightley’s tr.).
The Nixies, Undines, and Stromkarls were particularly gentle and

lovable beings, and were very anxious indeed to obtain repeated
assurances of their ultimate salvation.
Many stories are therefore told of priests or children meeting
these spirits playing by a stream, and taunting them with future
damnation, which threat turned the joyful music to pitiful wails. But
when priest or children, discovering their mistake, hastened back to
the stream and assured the green-toothed water sprites of future
redemption, they invariably resumed their happy strain.
“Know you the Nixies, gay and fair?

Their eyes are black, and green their hair—
They lurk in sedgy shores.”
Mathisson.
Besides Elf or Elb, the water sprite who gave its name to the Elbe
River nymphs. River in Germany, the Neck, from whom the Neckar
derives its name, and old Father Rhine, with his
numerous daughters (tributary streams), the most famous of all the
lesser water divinities is the Lorelei, the siren maiden who sits upon
the Lorelei rock near St. Goar, on the Rhine, and whose alluring song
has enticed many a mariner to death. The legends concerning this
siren are very numerous indeed, one of the most ancient being as
follows:
Lorelei was an immortal, a water nymph, daughter of old Father
Rhine; during the day she dwelt in the cool depths of the river bed, but
late at night she appeared in the moonlight, sitting aloft upon a
pinnacle of rock, in full view of all who passed up or down the stream.
At times, the evening breeze wafted some of the notes of her song to
the boatmen’s ears, when, forgetting time and place in listening to
these enchanting melodies, they drifted upon the sharp and jagged
rocks, where they invariably perished.
“Above the maiden sitteth,

A wondrous form, and fair;
With jewels bright she plaiteth
Her shining golden hair:
With comb of gold prepares it,
The task with song beguiled;
A fitful burden bears it—
That melody so wild.
“The boatman on the river

Lists to the song, spell-bound;
Oh! what shall him deliver
From danger threat’ning round?
The waters deep have caught them,
Both boat and boatman brave;
’Tis Loreley’s song hath brought them
Beneath the foaming wave.”
Song, Heine (Selcher’s tr.).
One person only is reported to have seen the Lorelei close by, a
young fisherman from Oberwesel, who met her every
The Lorelei evening by riverside, and spent a few delightful hours
and the
fisherman. with her, drinking in her beauty and listening to her
entrancing song. Tradition further relates that ere they
parted the Lorelei invariably pointed out the places where the youth
must cast his nets on the morrow—instructions which he always
obeyed, and which invariably brought him success.
One night the young fisherman was seen going towards the river,
but as he never returned search was made for him. No clew to his
whereabouts being found, the credulous Germans finally reported that
the Lorelei had dragged him down to her coral caves that she might
enjoy his companionship forever.
LORELEI AND THE FISHERMAN.—Paul Thumann.
According to another version, the Lorelei, perching on the rocks

above, and luring the fishermen by her songs, caused so many
deaths that an armed force was once sent out at nightfall to surround
and seize her. But the water nymph used her magic to lay such a
powerful spell upon the captain and his men that they could move
neither hand nor foot. While they stood motionless around her, the
Lorelei divested herself of all her ornaments, which she flung into the
waves below; then, chanting a spell, she lured the waters up to the
top of the rock, and the soldiers saw her spring into a sea-green
chariot drawn by white-maned steeds, and drive rapidly away. A few
moments later the Rhine had subsided to its usual level, the spell was
broken, and the men recovered the power of motion, and retreated to
announce how their efforts had been baffled. Since then, however, the
Lorelei has never been seen, and the peasants declare that she still
resents the insult offered her and will no longer leave her coral caves.
CHAPTER XXI.
BALDER.
Odin and Frigga, we are told, were parents of twin sons as

dissimilar in character and physical appearance as it was possible to
be; for while Hodur, god of darkness, was somber, taciturn, and
blind, like the obscurity of sin, which he was supposed to symbolize,
Balder, the beautiful, was the pure and radiant god of innocence and
light. The snowy brow and golden locks of this Asa seemed to send
out beams of sunshine to gladden the hearts of gods and men, by
whom he was equally beloved.
“Of all the twelve round Odin’s throne,

Balder, the Beautiful, alone,
The Sun-god, good, and pure, and bright,
Was loved by all, as all love light.”
Valhalla (J. C. Jones).
Balder, attaining his full growth with marvelous rapidity, was

admitted to the council of the gods, and married
Nanna.
Nanna (blossom), the daughter of Nip (bud), a
beautiful and charming young goddess, with whom he lived in
perfect unity and peace. He took up his abode in the palace of
Breidablik, whose silver roof rested upon golden pillars, and whose
purity was such that nothing common or unclean was ever allowed
within its precincts.
The god of light was well versed in the science of runes which
were carved on his tongue; he knew the various virtues of the
simples, one of which, the camomile, was always called “Balder’s
brow,” because its flower was just as immaculately pure as his
forehead. The only thing hidden from Balder’s radiant eyes, at first,
was the perception of his own ultimate fate.
“His own house

Breidablik, on whose columns Balder graved
The enchantments that recall the dead to life.
For wise he was, and many curious arts,
Postures of runes, and healing herbs he knew;
Unhappy! but that art he did not know,
To keep his own life safe, and see the sun.”
Balder Dead (Matthew Arnold).
As Balder the beautiful was always smiling and happy, the gods
were greatly troubled when they finally saw the light die out of his
blue eyes, a careworn look come into his face, and his step grow
heavy and slow. Odin and Frigga, seeing their beloved son’s evident
depression, tenderly implored him to reveal the cause of his silent
grief. Balder, yielding at last to their anxious entreaties, confessed
that his slumbers, instead of being peaceful and restful as of yore,
had been strangely troubled of late by dark and oppressive dreams,
which, although he could not clearly remember them when he
awoke, constantly haunted him with a vague feeling of fear.
“To that god his slumber

Was most afflicting;
His auspicious dreams
Seemed departed.”
Lay of Vegtam (Thorpe’s tr.).
When Odin and Frigga heard this, they were troubled indeed, but
declared they were quite sure nothing would harm their son, who
was so universally beloved. Yet, when the anxious father and mother
had returned home, they talked the matter over, acknowledged that
they also were oppressed by strange forebodings, and having
learned from the giants that Balder really was in danger, they
proceeded to take measures to avert it.
Frigga, therefore, sent out her servants in every direction,
bidding them make all living creatures, all plants, metals, stones—in
fact, every animate and inanimate thing—register a solemn vow not
to do any harm to Balder. All creation readily took the oath, for all
things loved the radiant god, and basked in the light of his smile. So
the servants soon returned to Frigga, telling her that all had been
duly sworn except the mistletoe, growing upon the oak stem at the
gate of Valhalla, which, they added, was such a puny, inoffensive
thing that no harm could be feared from it.
“On a course they resolved:

That they would send
To every being,
Assurance to solicit,
Balder not to harm.
All species swore
Oaths to spare him;
Frigg received all
Their vows and compacts.”
Sæmund’s Edda (Thorpe’s tr.).
Frigga now resumed her spinning with her usual content, for she
knew no harm could come to the child she loved best
The Vala’s
prophecy. of all. Odin, in the mean while, also sorely troubled,
and wishing to ascertain whether there was any cause
for his unwonted depression, resolved to consult one of the dead
Valas or prophetesses. He therefore mounted his eight-footed steed
Sleipnir, rode over the tremulous bridges Bifröst and Giallar, came to
the entrance of Nifl-heim, and, passing the Hel-gate and the dog
Garm, penetrated into Hel’s dark abode.
“Uprose the king of men with speed,

And saddled straight his coal-black steed;
Down the yawning steep he rode,
That leads to Hela’s drear abode.”
Descent of Odin (Gray).
To his surprise, he noticed that a feast was being spread in this

dark realm, and that the couches had all been covered with tapestry
and rings of gold, as if some highly honored guest were expected
before long. Hastening on, Odin finally reached the grave where the
Vala had rested undisturbed for many a year, and solemnly began to
chant the magic spell and trace the runes which had the power of
raising the dead.
“Thrice pronounc’d, in accents dread,

The thrilling verse that wakes the dead:
Till from out the hollow ground
Slowly breath’d a sullen sound.”
Suddenly the grave opened, and the prophetess slowly rose,

inquiring who he was and why he thus came to trouble her long rest.
Odin, not wishing her to know that he was king of the gods, replied
that he was Vegtam, Valtam’s son, and that he had awakened her to
inquire for whom Hel was spreading her couches and preparing a
festive meal. In hollow tones, the prophetess now confirmed all his
fears by telling him that the expected guest was Balder, who would
shortly be slain by Hodur, his brother, the blind god of darkness.
“Hodur will hither

His glorious brother send;
He of Balder will
The slayer be,
And Odin’s son
Of life bereave.
By compulsion I have spoken;
Now I will be silent.”
Sæmund’s Edda (Thorpe’s tr.).
But in spite of these sad tidings, and of the Vala’s evident

reluctance to answer any other questions, Odin was not yet satisfied,
and forced her to tell him who would avenge the murdered man by
calling his assassin to account—a spirit of revenge and retaliation
being considered a sacred duty among the races of the North.
Then the prophetess told him, as Rossthiof had predicted before,
that Rinda, the earth-goddess, would bear a son to Odin, and that
this divine emissary, Vali, would neither wash his face nor comb his
hair until he had avenged Balder and slain Hodur.
“In the caverns of the west,

By Odin’s fierce embrace comprest,
A wondrous boy shall Rinda bear,
Who ne’er shall comb his raven hair,
Nor wash his visage in the stream,
Nor see the sun’s departing beam,
Till he on Hoder’s corse shall smile
Flaming on the fun’ral pile.”
Having discovered this from the reluctant Vala, Odin, who,

thanks to his visit to the Urdar fountain, already knew much of the
future, now incautiously revealed some of his knowledge by inquiring
who would refuse to weep at Balder’s death. When the prophetess
heard this question, she immediately knew that it was Odin who had
called her out of her grave, and, refusing to speak another word, she
sank back into the silence of the tomb, declaring that none would
ever be able to lure her out again until the end of the world had
come.
“Hie thee hence, and boast at home,

That never shall inquirer come
To break my iron sleep again,
Till Lok has burst his tenfold chain;
Never, till substantial Night
Has reassum’d her ancient right:
Till wrapt in flames, in ruin hurl’d,
Sinks the fabric of the world.”
Descent of Odin (Gray)
Odin had questioned the greatest prophetess the world had ever
known, and had learned Orlog’s (fate’s) decrees, which he knew
could not be set aside. He therefore remounted his steed, and sadly
wended his way back to Asgard, thinking of the time, no longer far
distant, when his beloved son would no more be seen in the
heavenly abodes, and when the light of his presence would have
vanished forever.
On entering Glads-heim, however, Odin was somewhat cheered
when he heard of the precautions taken by Frigga to insure their
darling’s safety, and soon, feeling convinced that if nothing would
slay Balder he would surely continue to gladden the world with his
presence, he cast aside all care and ordered games and a festive
meal.
The gods resumed their wonted occupations, and were soon
The gods at casting their golden disks on the green plain of Ida,
play. which was called Idavold, the playground of the gods.
At last, wearying of this pastime, and knowing that no
harm could come to their beloved Balder, they invented a new game
and began to use him as a target, throwing all manner of weapons
and missiles at him, certain that no matter how cleverly they tried,
and how accurately they aimed, the objects, having sworn not to
injure him, would either glance aside or fall short. This new
amusement was so fascinating that soon all the gods were
assembled around Balder, at whom they threw every available thing,
greeting each new failure with prolonged shouts of laughter. These
bursts of merriment soon excited the curiosity of Frigga, who sat
spinning in Fensalir; and seeing an old woman pass by her dwelling,
she bade her pause and tell what the gods were doing to provoke
such great hilarity. The old woman, who was Loki in disguise,
immediately stopped at this appeal, and told Frigga that all the gods
were throwing stones and blunt and sharp instruments at Balder,
who stood smiling and unharmed in their midst, daring them to touch
him.
The goddess smiled, and resumed her work, saying that it was
quite natural that nothing should harm Balder, as all things loved the
light, of which he was the emblem, and had solemnly sworn not to
injure him. Loki, the personification of fire, was greatly disappointed
upon hearing this, for he was jealous of Balder, the sun, who so
entirely eclipsed him and was generally beloved, while he was
feared and avoided as much as possible; but he cleverly concealed
his chagrin, and inquired of Frigga whether she were quite sure that
all objects had joined the league.
Frigga proudly answered that she had received the solemn oath
of all things, except of a harmless little parasite, the mistletoe, which
grew on the oak near Valhalla’s gate, and was too small and weak to
be feared. Having obtained the desired information, Loki toddled off;
but as soon as he was safely out of sight, he resumed his wonted
form, hastened to Valhalla, found the oak and mistletoe indicated by
Frigga, and by magic arts compelled the parasite to assume a
growth and hardness hitherto unknown.
From the wooden stem thus produced he deftly fashioned a
Death of shaft ere he hastened back to Idavold, where the gods
Balder. were still hurling missiles at Balder, Hodur alone
leaning mournfully against a tree, and taking no part in
the new game. Carelessly Loki approached him, inquired the cause
of his melancholy, and twitted him with pride and indifference, since
he would not condescend to take part in the new game. In answer to
these remarks, Hodur pleaded his blindness; but when Loki put the
mistletoe in his hand, led him into the midst of the circle, and
indicated in what direction the novel target stood, Hodur threw his
shaft boldly. Instead of the loud shout of laughter which he expected
to hear, a shuddering cry of terror fell upon his ear, for Balder the
beautiful had fallen to the ground, slain by the fatal blow.
“So on the floor lay Balder dead; and round
Lay thickly strewn swords, axes, darts, and spears,
Which all the Gods in sport had idly thrown
At Balder, whom no weapon pierced or clove;
But in his breast stood fixed the fatal bough
Of mistletoe, which Lok, the Accuser, gave
To Hoder, and unwitting Hoder threw—
’Gainst that alone had Balder’s life no charm.”
BALDER
Anxiously the gods all crowded around him, but alas! life was
quite extinct, and all their efforts to revive the fallen sun-god were
vain. Inconsolable at their loss, they turned angrily upon Hodur,
whom they would have slain had they not been restrained by the
feeling that no willful deed of violence should ever desecrate their
peace steads. At the loud sound of lamentation the goddesses came
in hot haste, and when Frigga saw that her darling was dead, she
passionately implored the gods to go to Nifl-heim and entreat Hel to
release her victim, for the earth could not live happy without him.
As the road was rough and painful in the extreme, none of the
Hermod’s gods at first volunteered to go; but when Frigga added
errand. that she and Odin would reward the messenger by
loving him most of all the Æsir, Hermod signified his
readiness to execute the commission. To help him on his way, Odin
lent him Sleipnir, and bade him good speed, while he motioned to
the other gods to carry the corpse to Breidablik, and directed them to
go to the forest and cut down huge pines to make a worthy pyre for
his son.
“But when the Gods were to the forest gone,

Hermod led Sleipnir from Valhalla forth
And saddled him; before that, Sleipnir brook’d
No meaner hand than Odin’s on his mane,
On his broad back no lesser rider bore;
Yet docile now he stood at Hermod’s side,
Arching his neck, and glad to be bestrode,
Knowing the God they went to seek, how dear.
But Hermod mounted him, and sadly fared
In silence up the dark untravel’d road
Which branches from the north of Heaven, and went
All day; and daylight waned, and night came on.
And all that night he rode, and journey’d so,
Nine days, nine nights, toward the northern ice,
Through valleys deep-engulph’d by roaring streams.
And on the tenth morn he beheld the bridge
Which spans with golden arches Giall’s stream,
And on the bridge a damsel watching, arm’d,
In the straight passage, at the further end,
Where the road issues between walling rocks.”
While Hermod was traveling along the cheerless road to Nifl-

heim, the gods hewed and carried down to the shore a vast amount
of fuel, which they placed upon the deck of Balder’s favorite vessel,
Ringhorn, constructing an elaborate funeral pyre, which, according to
custom, was decorated with tapestry hangings, garlands of flowers,
vessels and weapons of all kinds, golden rings, and countless
objects of value, ere the immaculate corpse was brought and laid
upon it in full attire.
One by one, the gods now drew near to take a last farewell of
their beloved companion, and as Nanna bent over him, her loving
heart broke, and she fell lifeless by his side. Seeing this, the gods
reverently laid her beside her husband, that she might accompany
him even in death; and after they had slain his horse and hounds
and twined the pyre with thorns, the emblems of sleep, Odin, the last
of the gods, drew near.
In token of affection for the dead and of sorrow for his loss, all
The funeral laid their most precious possessions upon his pyre,
pyre. and Odin, bending down, now added to the offerings
his magic ring Draupnir. The assembled gods then
perceived that he was whispering in his dead son’s ear, but none
were near enough to hear what word he said.
These preliminaries ended, the gods now prepared to launch the
ship, but found it so heavily laden with fuel and treasures that their
combined efforts could not make it stir an inch. The mountain giants,
witnessing the sad scene from afar, and noticing their quandary, said
that they knew of a giantess called Hyrrokin, who dwelt in Jötun-
heim, and was strong enough to launch the vessel without any other
aid. The gods therefore bade one of the storm giants hasten off to
summon Hyrrokin, who soon appeared, riding a gigantic wolf, which
she guided by a bridle made of writhing live snakes. Riding down to
the shore, the giantess dismounted and haughtily signified her
readiness to give them the required aid, if in the mean while they
would but hold her steed. Odin immediately dispatched four of his
maddest Berserkers to fulfill this task; but, in spite of their
phenomenal strength, they could not hold the monstrous wolf until
the giantess had thrown and bound it fast.
Hyrrokin, seeing them now able to manage her refractory steed,
marched down the beach, set her shoulder against the stern of
Balder’s ship Ringhorn, and with one mighty shove sent it out into
the water. Such was the weight of the burden she moved, however,
and the rapidity with which it shot down into the sea, that all the earth
shook as if from an earthquake, and the rollers on which it glided
caught fire from the friction. The unexpected shock almost made the
gods lose their balance, and so angered Thor that he raised his
hammer and would have slain the giantess had he not been
restrained by his fellow gods. Easily appeased, as usual—for Thor’s
violence, although quick, was evanescent—he now stepped up on
the vessel once more to consecrate the funeral pyre with his sacred
hammer. But, as he was performing this ceremony, the dwarf Lit
managed to get into his way so provokingly that Thor, still slightly
angry, kicked him into the fire, which he had just kindled with a thorn,
where the dwarf was burned to ashes with the corpses of the faithful
pair.
As the vessel drifted out to sea, the flames rose higher and
higher, and when it neared the western horizon it seemed as if sea
and sky were all on fire. Sadly the gods watched the glowing ship
and its precious freight, until it suddenly plunged into the waves and
disappeared; nor did they turn aside and go back to their own homes
until the last spark of light had vanished, and all the world was
enveloped in darkness, in token of mourning for Balder the good.
“Soon with a roaring rose the mighty fire,

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how ridiculous design and research can be.

Potto Prologue xxv

5 Isentropic Flow 101

6 Normal Shock 145

7 Normal Shock in Variable Duct Areas 215

8 Nozzle Flow With External Forces 233

10 Fanno Flow 257

11 Rayleigh Flow 301

12 Evacuating SemiRigid Chambers 315

13 Evacuating under External Volume Control 333

14 Oblique Shock 343

A Computer Program 403

B Oblique Shock History 407

1.1 The shock as a connection of Fanno and Rayleigh lines . . . . . . . . . 7

2.1 Cylinder and piston configuration of maximum work . . . . . . . . . . . 33

3.1 Schematics of flow in a pipe with varying density . . . . . . . . . . . . 44

4.1 A very slow moving piston in a still gas . . . . . . . . . . . . . . . . . . 70

5.1 Flow through a converging diverging nozzle . . . . . . . . . . . . . . . 101

6.1 A shock wave inside a tube . . . . . . . . . . . . . . . . . . . . . . . . 145

9.1 Control volume for isothermal flow . . . . . . . . . . . . . . . . . . . . 237

11.1 The control volume of Rayleigh Flow . . . . . . . . . . . . . . . . . . . 301

12.1 The two different classifications of models . . . . . . . . . . . . . . . . 315

13.1 The control volume of the “Cylinder” . . . . . . . . . . . . . . . . . . 334

15.1 Definition of the angle for the Prandtl–Meyer function . . . . . . . . . 387

1 Books Under Potto Project . . . . . . . . . . . . . . . . . . . . . . . . xxvii

2.1 Properties of Various Ideal Gases at [300K] . . . . . . . . . . . . . . . 30

3.1 Bulk modulus for selected materials . . . . . . . . . . . . . . . . . . . 42

4.1 Speed of sound in water . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.1 Fliegner’s number a function of Mach number . . . . . . . . . . . . . . 119

6.1 Table of maximum values of the shock-choking phenomenon . . . . . . 171

9.1 The Isothermal Flow basic parameters . . . . . . . . . . . . . . . . . . 247

10.1 Fanno Flow Standard basic Table k=1.4 . . . . . . . . . . . . . . . . . 297

14.1 Table of maximum values of the oblique Shock k=1.4 . . . . . . . . . 365

R̄ Universal gas constant, see equation (2.26), page 29

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

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