Compressible Flow Free Textbook

Where the mathematicians were able only to prove that the solution exists.

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... ``infinite ²

After the last decision of the Supreme Court in the case of Eldred v. Ashcroff (see http://cyber.law.harvard.edu/openlaw/eldredvashcroft for more information) copyrights practically remain indefinitely with the holder (not the creator).

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... idea ³

In some sense one can view the encyclopedia Wikipedia as an open content project (see http://en.wikipedia.org/wiki/Main_Page). The wikipedia is an excellent collection of articles which are written by various individuals.

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...)⁴

see also in Franks, Nigel R.; "Army Ants: A Collective Intelligence," American Scientist, 77:139, 1989

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... data ⁵

Data are not copyrighted.

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... originated ⁶

Originally authored by Dr. Schlichting, who passed way some years ago. A new version is created every several years.

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... macros ⁷

One can only expect that open source and readable format will be used for this project. But more than that, only L^ATEX, and perhaps troff, have the ability to produce the quality that one expects for these writings. The text processes, especially L^ATEX, are the only ones which have a cross platform ability to produce macros and a uniform feel and quality. Word processors, such as OpenOffice, Abiword, and Microsoft Word software, are not appropriate for these projects. Further, any text that is produced by Microsoft and kept in ``Microsoft'' format are against the spirit of this project In that they force spending money on Microsoft software.

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... research ⁸

A reader asked this author to examine a paper on Triple Shock Entropy Theorem and Its Consequences by Le Roy F. Henderson and Ralph Menikoff. This led to comparison between maximum to ideal gas model to more general model.

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... shore ⁹

Please read the undersigned's book ``Fundamentals of Die Casting Design,'' which demonstrates how ridiculous design and research can be.

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... topic ¹⁰

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.

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... demonstrated ¹¹

If you have better and different examples or presentations you are welcome to submit them.

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... flow ¹²

It is suggested to referred to this model as Shapiro flow

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... account ¹³

Still in untyped note form.

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... work ¹⁴

If you would like to to help me to write a new spell check user interface, please contact me.

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... skipped.¹⁵

At the present, the book is not well organized. You have to remember that this book is a work in progress.

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... Marshall ¹⁶

Dr. Marshall wrote to this author that the author should review other people work before he write any thing new (well, literature review is always good?). Over ten individuals wrote me about this letter. I am asking from everyone to assume that his reaction was innocent one. While his comment looks like unpleasant reaction, it brought or cause the expansion the oblique shock chapter. However, other email that imply that someone will take care of this author aren't appreciated.

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... introduction ^1.1

This book gradually sliding to include more material that isn't so introductory. But attempt is made to present the material in introductory level.

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... flow ^1.2

It can be argued that in open channel flow there is a hydraulic jump (discontinuity) and in some ranges no effect of downstream conditions on the flow. However, the uniqueness of the phenomena in the gas dynamics provides spectacular situations of a limited length (see Fanno model) and thermal choking, etc. Further, there is no equivalent to oblique shock wave. Thus, this richness is unique to gas dynamics.

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... flow ^1.3

The thermal choking is somewhat different but similarity exists.

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... flow ^1.4

This book is intended for engineers and therefore a discussion about astronomical conditions isn't presented.

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... flow ^1.5

Any search on the web on classes of compressible flow will show this fact and the undersigned can testify that this was true in his first class as a student of compressible flow.

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... Rouse ^1.6

Hunter Rouse and Simon Inc, History of Hydraulics (Iowa City: Institute of Hydraulic Research, 1957)

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... (aeronautic)^1.7

Anderson, J. D., Jr. 1997. A History of Aerodynamics: And Its Impact on Flying Machines, Cambridge University Press, Cambridge, England.

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... book ^1.8

The only remark found about Fanno flow that it was taken from the Fanno Master thesis by his adviser. Here is a challenge: find any book describing the history of the Fanno model.

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... this ^1.9

Who developed the isothermal model? The research so far leads to Shapiro. Perhaps this flow should be named after the Shapiro. Is there any earlier reference to this model?

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... advancement ^1.10

Amazingly, science is full of many stories of conflicts and disputes. Aside from the conflicts of scientists with the Catholic Church and Muslim religion, perhaps the most famous is that of Newton's netscaping (stealing and embracing) Leibniz['s] invention of calculus. There are even conflicts from not giving enough credit, like Moody Even the undersigned encountered individuals who have tried to ride on his work. The other kind of problem is ``hijacking'' by a sector. Even on this subject, the Aeronautic sector ``took over'' gas dynamics as did the emphasis on mathematics like perturbations methods or asymptotic expansions instead on the physical phenomena. Major material like Fanno flow isn't taught in many classes, while many of the mathematical techniques are currently practiced. So, these problems are more common than one might be expected.

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... law ^1.11

This recognition of the first law is today the most ``obvious'' for engineering students. Yet for many it was still debatable up to the middle of the nineteen century.

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... discontinuity ^1.12

Siméon Denis Poisson, French mathematician, 1781-1840 worked in Paris, France. "M'emoire sur la th'eorie du son," J. Ec. Polytech. 14 (1808), 319-392. From Classic Papers in Shock Compression Science, 3-65, High-press. Shock Compression Condens. Matter, Springer, New York, 1998.

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... dynamics ^1.13

James Challis, English Astronomer, 1803-1882. worked at Cambridge, England UK. "On the velocity of sound," Philos. Mag. XXXII (1848), 494-499

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... ``mistake.''^1.14

Stokes George Gabriel Sir, Mathematical and Physical Papers, Reprinted from the original journals and transactions, with additional notes by the author. Cambridge, University Press, 1880-1905.

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... known ^1.15

The words ``no known'' refer to the undersigned. It is possible that some insight was developed but none of the documents that were reviewed revealed it to the undersigned.

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... Rankine ^1.16

William John Macquorn Rankine, Scottish engineer, 1820-1872. He worked in Glasgow, Scotland UK. "On the thermodynamic theory of waves of finite longitudinal disturbance," Philos. Trans. 160 (1870), part II, 277-288. Classic papers in shock compression science, 133-147, High-press. Shock Compression Condens. Matter, Springer, New York, 1998

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... Hugoniot ^1.17

Pierre Henri Hugoniot, French engineer, 1851-1887. "Sur la propagation du mouvement dans les corps et sp'ecialement dans les gaz parfaits, I, II" J. Ec. Polytech. 57 (1887), 3-97, 58 (1889), 1-125. Classic papers in shock compression science, 161-243, 245-358, High-press. Shock Compression Condens. Matter, Springer, New York, 1998

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... equations ^1.18

Today it is well established that shock has three dimensions but small sections can be treated as one dimensional.

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... form ^1.19

To add discussion about the general relationships.

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... direction ^1.20

Some view the work of G. I. Taylor from England as the proof (of course utilizing the second law)

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... 1908 ^1.21

Theodor Meyer in Mitteil. üb. Forsch-Arb. Berlin, 1908, No. 62, page 62.

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... T.^1.23

See for a longer story in www.potto.org/obliqueArticle.php.

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... possible ^1.22

Since writing this book, several individuals point out that a solution was found in book ``Analytical Fluid Dynamics'' by Emanuel, George, second edition, December 2000 (US$ 124.90). That solution is based on a transformation of $\sin\theta$ to $\tan\beta$ . It is interesting that transformation result in one of root being negative. While the actual solution all the roots are real and positive for the attached shock. The presentation was missing the condition for the detachment or point where the model collapse. But more surprisingly, similar analysis was published by Briggs, J. ``Comment on Calculation of Oblique shock waves,'' AIAA Journal Vol 2, No 5 p. 974, 1963. Hence, Emanuel's partial solution just redone 36 years work (how many times works have to be redone in this field). In additonal there was additional publishing of similar works by Mascitti, V.R. and Wolf, T. ^1.23 In a way, part of analysis of this book is also redoing old work. Yet, what is new in this work is completeness of all the three roots and the analytical condition for detached shock and breaking of the model.

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... Epstein ^1.24

Epstein, P. S., ``On the air resistance of Projectiles,'' Proceedings of the National Academy of Science, Vol. 17, 1931, pp. 532-547.

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... shock ^1.25

In study this issue this author realized only after examining a colleague experimental Picture (13.4) that it was clear that the Normal shock along with strong shock and weak shock ``live'' together peacefully and in stable conditions.

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... Herivel ^1.26

Herivel, J. F., ``The Derivation of The Equations of Motion On an Ideal Fluid by Hamilton's Principle,,'' Proceedings of the Cambridge philosophical society, Vol. 51, Pt. 2, 1955, pp. 344-349.

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...Carrier ^1.27

Carrier, G.F., ``On the Stability of the supersonic Flows Past as a Wedge,'' Quarterly of Applied Mathematics, Vol. 6, 1949, pp. 367-378.

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... Menikoff ^1.28

Henderson and Menikoff, "Triple Shock Entropy Theorem," Journal of Fluid Mechanics 366 (1998) pp. 179-210.

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... 186x ^1.29

Fliegner Schweizer Bauztg., Vol 31 1898, p. 68-72. The theoretical first work on this issue was done by Zeuner, ``Theorie die Turbinen,'' Leipzig 1899, page 268 f.

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... Rayleigh ^1.30

Rayleigh was the first to develop the model that bears his name. It is likely that others had noticed that flow is choked, but did not produce any model or conduct successful experimental work.

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... Zeuner ^1.31

Zeuner, ``Theorie der Turbinen, Leipzig 1899 page 268 f.

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... group ^1.32

Some of the publications were not named after Prandtl but rather by his students like Meyer, Theodor. In the literature appeared reference to article by Lorenz in the Physik Zeitshr., as if in 1904. Perhaps, there are also other works that this author did not come crossed.

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... calculations ^1.33

Meyer, Th., Über zweidimensionals Bewegungsvordange eines Gases, Dissertation 1907, erschienen in den Mitteilungen über Forsch.-Arb. Ing.-Wes. heft 62, Berlin 1908.

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... Parenty ^1.34

Parenty, Comptes R. Paris, Vol. 113, 116, 119; Ann. Chim. Phys. Vol. 8. 8 1896, Vol 12, 1897.

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... believed ^1.35

The personal experience of this undersigned shows that even instructors of Gas Dynamics are not aware that the chocking occurs at different Mach number and depends on the model.

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... (1997)^1.36

These researchers demonstrate results between two extremes and actually proposed this idea. However, that the presentation here suggests that topic should be presented case between two extremes.

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... Bendemann ^1.37

Bendemann Mitteil über Forschungsarbeiten, Berlin, 1907, No. 37.

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... al^1.38

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... by^1.39

Romer, I Carl Jr., and Ali Bulent Cambel, ``Analysis of Isothermal Variable Area Flow,'' Aircraft Eng. vol. 27 no 322, p. 398 December 1955.

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... probably ^1.40

As most of the history research has shown, there is also a possibility that someone found it earlier. For example, Piosson was the first one to realize the shock wave possibility.

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... family ^1.41

This material is very important and someone should find it and make it available to researchers.

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...^1.42

Fanning

based radius is only one quarter of the Darcy

which is based on diameter

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... lift ^1.43

The English call this theory the Lanchester-Prandtl theory. This is because the English Astronomer Frederick Lanchester published the foundation for Prandtl's theory in his 1907 book Aerodynamics. However, Prandtl claimed that he was not aware of Lanchester's model when he had begun his work in 1911. This claim seems reasonable in the light that Prandtl was not ware of earlier works when he named erroneously the conditions for the shock wave. See for the full story in the shock section.

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... calculations ^1.44

This undersigned is aware of only one case that these methods were really used to calculations of wing.

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... ``dinosaur ^1.45

It is like teaching using slide ruler in today school. By the way, slide rule is sold for about 7.5$ on the net. Yet, there is no reason to teach it in a regular school.

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... Dynamics.''^1.46

International Textbook Co., Scranton, Pennsylvania, 1964.

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... progress ^1.47

In fact, the emergence of the CFD gave the illusion that there are solutions at hand, not realizing that garbage in is garbage out, i.e., the model has to be based on scientific principles and not detached from reality. As anecdotal story explaining the lack of progress, in die casting conference there was a discussion and presentation on which turbulence model is suitable for a complete still liquid. Other ``strange'' models can be found in the undersigned's book ``Fundamentals of Die Casting Design.

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... Prague ^1.48

It is interesting to point out that Prague provided us two of the top influential researchers[:] E. Mach and E.R.G. Eckert.

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... flow ^1.49

Mach dealt with only air, but it is reasonable to assume that he understood that this ratio was applied to other gases.

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... ``smashing^1.50

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... nationality ^1.51

In some places, the ridicules claims that Jews persecuted only because their religion. Clearly, Fanno was not part of the Jewish religion (see his picture) only his nationality was Jewish.

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... Shapiro ^1.52

Parts taken from Sasha Brown, MIT

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... engineer ^3.1

Aerospace engineer that alumni of University of Minnesota, Aerospace Department.

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... industry ^3.2

Pardon, but a joke is must in this situation. A cat is pursuing a mouse and the mouse escape and hide in the hole. Suddenly, the mouse hear a barking dog and a cat yelling. The mouse go out to investigate, and cat is catching the mouse. The mouse ask the cat I thought I hear a dog. The cat reply, yes you right. My teacher was right, one language is not enough today.

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... table ^3.3

This data is taken form Van Wylen and Sontag ``Fundamentals of Classical Thermodynamics'' 2nd edition

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... shown ^3.4

See Van Wylen p. 372 SI version, perhaps to insert the discussion here.

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... Wilson ^3.5

J. Acoust. Soc. Amer., 1960, vol.32, N 10, p. 1357. Wilson's formula is accepted by the National Oceanographic Data Center (NODC) USA for computer processing of hydrological information.

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...variableArea:eq:mass ^4.1

The momentum equation is not used normally in isentropic process, why?

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... internal ^4.2

This condition does not impose any restrictions for external flow. In external flow, an object can be moved in arbitrary speed.

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... ^4.4

This pressure is about two atmospheres with temperature of $250[K]$

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... here ^4.5

Since version 0.44 of this book.

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... equation ^4.6

The one dimensional momentum equation for steady state is $U{dU/dx} = - dP/dx +0(\hbox{other effects})$ which are neglected here.

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... section ^5.1

Currently under construction.

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... uniform ^5.2

Clearly the change in the shock is so significant compared to the changes in medium before and after the shock that the changes in the mediums (flow) can be considered uniform.

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... range ^5.3

Ireland, K. and Rosen, M. "Cubic and Biquadratic Reciprocity." Ch. 9 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 108-137, 1990.

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... suddenly ^5.4

It will be explained using dimensional analysis what is suddenly open

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... valve ^5.5

According to my son, the difference between these two cases is the direction of the information. Both case there essentially bodies, however, in one the information flows from inside the field to the boundary while the other case it is the opposite.

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... textbooks ^5.6

Similar situation exist in the surface tension area.

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... example ^6.1

The meaning of the word practical is that in reality the engineer does not given the opportunity to determine the location of the shock but rather information such as pressures and temperature.

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... choked ^8.1

This explanation is not correct as it will be shown later on. Close to the critical point (about, $1/\sqrt{k}$ , the heat transfer, is relatively high and the isothermal flow model is not valid anymore. Therefore, the study of the isothermal flow above this point is only an academic discussion but also provides the upper limit for Fanno Flow.

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... factor ^8.2

It should be noted that Fanning factor based on hydraulic radius, instead of diameter friction equation, thus ``Fanning f'' values are only 1/4th of ``Darcy f'' values.

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...)^8.3

Assuming the upstream variables are known.

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... analysis ^8.4

This dimensional analysis is a bit tricky, and is based on estimates. Currently and ashamedly the author is looking for a more simplified explanation. The current explanation is correct but based on hands waving and definitely does not satisfy the author.

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... down ^8.5

see Kays and Crawford ``Convective Heat Transfer'' (equation 12-12).

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... questions ^8.6

The proof questions are questions that ask for proof or for finding a mathematical identity (normally good for mathematicians and study of perturbation methods). These questions or examples will appear in the later versions.

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... exist ^8.7

Those who are mathematically inclined can include these kinds of questions but there are no real world applications to isothermal model with shock.

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... ^8.8

It is unfortunate, but it seems that this standard will be around in USA for some time.

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... ignored ^9.1

Even the friction does not convert into heat

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... examined ^9.2

Not ready yet, discussed on the ideal gas model and the entry length issues.

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... assumed ^9.3

The equation of state is written again here so that all the relevant equations can be found when this chapter is printed separately.

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... algorithm ^9.5

You can use any method you wish, but be careful of second order methods like Newton-Rapson method which can be unstable.

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... center ^9.6

The word information referred to is the shear stress transformed from the wall to the center of the tube.

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... Frossel ^9.7

See on the web http://naca.larc.nasa.gov/digidoc/report/tm/44/NACA-TM-844.PDF

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... exist ^9.8

Many in the industry have difficulties in understanding this concept. The author seeks for a nice explanation of this concept for non-fluid mechanics engineers. This solicitation is about how to explain this issue to non-engineers or engineer without a proper background.

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... experiments ^9.9

If you have experiments demonstrating this point, please provide to the undersign so they can be added to this book. Many of the pictures in the literature carry copyright statements.

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... constant!^9.10

On a personal note, this situation is rather strange to explain. On one hand, the resistance increases and on the other hand, the exit Mach number remains constant and equal to one. Does anyone have an explanation for this strange behavior suitable for non-engineers or engineers without background in fluid mechanics?

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...)^9.11

Note that $\rho_1$ increases with decreases of

but this effect is less significant.

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... flow ^9.12

See more on the discussion about changing the length of the tube.

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... shock ^9.13

It is common misconception that the back pressure has to be at point d.

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... entrance ^9.14

The word ``entrance'' referred to the tube and not to the nozzle. The reference to the tube is because it is the focus of the study.

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... nozzle ^9.15

Strange? Frictionless nozzle has a larger resistance when the throat area decreases

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...)^9.16

It is one of the strange phenomenon that in one way increasing the resistance (changing the throat area) decreases the flow rate while in a different way (increasing the $\frac{4fL}{D}$ ) does not affect the flow rate.

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... side.^9.17

What if the right side is also negative? The flow is choked and shock must occur in the nozzle before entering the tube. Or in a very long tube the whole flow will be subsonic.

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... choke ^9.18

This questions were raised from many who didn't found any book that discuss these practical aspects and send questions to this author.

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... ``star''^10.1

The star is an asterisk.

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... books ^11.1

After completion of these Chapters, the undersigned discover two text books which to include some material related to this topic. These books are OCR, J. A., Fundamentals of Gas Dynamics, International Textbook Co., Scranton, Pennsylvania, 1964. and ``Compressible Fluid Flow,'' 2nd Edition, by M. A. Saad, Prentice Hall, 1985. However, these books contained only limit discussions on the evacuation of chamber with attached nozzle.

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... presented ^11.2

Even if the instructor feels that their students are convinced about the importance of the compressible, this example can further strength and enhance this conviction.

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...^11.3

This notation is used in many industrial processes where time of process referred to sometime as the maximum time.

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... equation ^11.4

To those mathematically included, find the general solution for this equation.

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... flow ^11.5

This work is suggested by Donald Katze the point out that this issue appeared in Shapiro's Book Vol 1, Chapter 4, p. 111 as a question 4.31.

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... linear ^11.6

Some suggested this border point as infinite evocation to infinite time for evacuation etc. This undersigned is not aware situation where this indeed play important role. Therefore, it is waited to find such conditions before calling it as critical condition.

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... tube ^12.1

such reaction are possible and expected to be part of process but the complicates the analysis and do not contribute to understand to the compressibility effects.

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... valve ^12.2

After certain sizes, the possibility of crack increases.

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... models ^13.1

In this chapter, even the whole book, a very limited discussion about reflection shocks and collisions of weak shock, Von Neumann paradox, triple shock intersection, etc are presented. The author believes that these issues are not relevant to most engineering students and practices. Furthermore, these issues should not be introduced in introductory textbook of compressible flow. Those who would like to obtain more information, should refer to J.B. Keller, ``Rays, waves and asymptotics,'' Bull. Am. Math. Soc. 84, 727 (1978), and E.G. Tabak and R.R. Rosales, ``Focusing of weak shock waves and the Von Neuman paradox of oblique shock reflection,'' Phys. Fluids 6, 1874 (1994).

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... disturbance ^13.2

Zero velocity, pressure boundary conditions, and different inclination angle, are examples of forces that create shock. The zero velocity can be found in a jet flowing into a still medium of gas.

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...chap:intro ^13.3

This section is under construction and does not appear in the book yet.

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... direction ^13.4

The author begs for forgiveness from those who view this description as offensive (There was an unpleasant email to the author accusing him of revolt against the holy of the holies.). If you do not like this description, please just ignore it. You can use the traditional explanation, you do not need the author's permission.

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... pairs ^13.5

This issue is due to R. Menikoff, who raised the solution completeness issue.

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... (thermodynamically)^13.6

The solution requires solving the entropy conservation equation. The author is not aware of ``simple'' proof and a call to find a simple proof is needed.

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... extent ^13.7

Actually this term is used from historical reasons. The lesser extent angle is the unstable angle and the weak angle is the middle solution. But because the literature referred to only two roots, the term lesser extent is used.

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... angle ^13.8

This point was pointed out by R. Menikoff. He also suggested that $\theta$ is bounded by $\sin^{-1}{1/M_1}$ and 1.

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... roots ^13.9

The highest power of the equation (only with integer numbers) is the number of the roots. For example, in a quadratic equation there are two roots.

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... exist ^13.10

A call for suggestions, to explain about complex numbers and imaginary numbers should be included. Maybe insert an example where imaginary solution results in no physical solution.

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... dictates ^13.11

This situation is somewhat similar to a cubical body rotation. The cubical body has three symmetrical axes which the body can rotate around. However, the body will freely rotate only around two axes with small and large moments of inertia. The body rotation is unstable around the middle axes. The reader can simply try it.

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... analysis ^13.12

There is no experimental or analytical evidence, that the author has found, showing that it is totally impossible. The ``unstable'' terms can be thermodynamcily stable in unsteady case. Though, those who are dealing with rapid transient situations should be aware that this angle of oblique shock can exist. There is no theoretical evidence that showing that in strong unsteady state this angle is unstable. The shock will initially for a very brief time transient in it and will jump from this angle to the thermodynamically stable angles.

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... root ^13.13

See the hist/rical discussion on the stability. There are those who view this question not as a stability equation but rather as under what conditions a strong or a weak shock will prevail.

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... stability ^13.14

This material is extra and not recommended for standard undergraduate students.

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... researchers ^13.15

A whole discussion on the history of this can be found in ``Open content approach to academic writing'' on http://www.potto.org/obliqueArticle.php

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... someone ^13.16

At first, it was seen as C. J.Chapman, English mathematician to be the creator but later an earlier version by several months was proposed by Bernard Grossman. At this stage it is not clear who was the first to propose it.

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...^13.17

A mathematical challenge for those who like to work it out.

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... exists ^13.18

There are several papers that attempt to prove this point in the past. Once this analytical solution was published, this proof became trivial. But for non ideal gas (real gas) this solution is only an indication.

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... point ^13.19

See for example, paper by Rosles, Tabak, ``Caustics of weak shock waves,'' 206 Phys. Fluids 10 (1) , January 1998.

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... weak ^13.20

It is not a mistake, there are two ``weaks.'' These words mean two different things. The first ``weak'' means more of compression ``line'' while the other means the weak shock.

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... wedge ^13.21

Even finite wedge with limiting wall can be considered as an example for this discussion if the B.L. is neglected.

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... correct ^13.22

At this stage, dimensional analysis is not completed. The author is not aware of any such analysis in literature. The common approach is to carry out numerical analysis. In spite of recent trends, for most engineering applications, a simple tool is sufficient for limit accuracy. Additionally, the numerical works require many times a ``reality check.''

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... Menikoff ^13.23

Henderson and Menikoff "Triple Shock Entropy Theorem" Journal of Fluid Mechanics 366 (1998) pp. 179-210.

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... state ^13.24

The effect of the equation of state on the maximum and other parameters at this state is unknown at this moment and there are more works underway.

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... question ^13.25

See example 13.5.

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...oblique:fig:inlet)^13.26

In fact, there is general proof that regardless to the equation of state (any kind of gas), the entropy is to be minimized through a series of oblique shocks rather than through a single normal shock. For details see Henderson and Menikoff ``Triple Shock Entropy Theorem,'' Journal of Fluid Mechanics 366, (1998) pp. 179-210.

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... results ^13.27

The results in this example are obtained using the graphical interface of POTTO-GDC thus, no input explanation is given. In the past the input file was given but the graphical interface it is no longer needed.

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... solution ^14.1

Not really different from this explanation but shown in more mathematical form, due to Landau and friends. It will be presented in the future version. It isn't present now because the low priority to this issue present for a text book on this subject.

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... obtained ^14.2

This example is for academic understanding. There is very little with practicality in this kind of problem.

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... here.^A.1

when will be written in C++ will be add to this program.

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