Page 46 - 48Fundamentals of Compressible Fluid Mechanics

8 CHAPTER 1. INTRODUCTION
22
famous report of NACA 1135 that there is not explicit analytical solution .
The question whether the oblique shock is stable or which root is stable
was daunting since the early discovery that there are more than one possible so-
lution. It is amazing that early research concluded that only the weak solution is
possible or stable as oppose the reality. The first that attempt this question where
23
in 1931 by Epstein . His analysis was based on Hamilton’s principle when he ig-
nore the boundary condition. The results of that analysis was that strong shock is
unstable. The researchers understood that flow after a strong shock governed by
elliptic equation while the flow after weak shock governed by hyperbolic equations.
This difference probably results in not recognizing that The boundary conditions
24
play important role in the stability of the shock . In fact analysis based on Hamil-
ton’s principle isn’t suitable for stability because entropy creation was recognized
25
1955 by Herivel .
Carrier 26 was first to recognize that strong and weak shock stable. If
fact the confusion on this issue was persistent until now. Even all books that pub-
lished recently claimed that no strong shock ever was observed in flow around cone
(Taylor–Maccoll flow). In fact, even this author sinned in this erroneous conclusion.
The real question isn’t if they exist rather under what conditions these shocks exist
which was suggested by Courant and Friedrichs in their book “Supersonic Flow
and Shock Waves,” published by Interscience Publishers, Inc. New York, 1948, p.
317.
The effect of real gases was investigated very early since steam was
used move turbines. In general the mathematical treatment was left to numerical
investigation and there is relatively very little known on the difference between ideal
gas model and real gas. For example, recently, Henderson and Menikoff 27 dealt
with only the procedure to find the maximum of oblique shock, but no comparison
between real gases and ideal gas is offered there.
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 . 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
to
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 just redone 36 years work (how many
times works have to be redone in this field). In a way part of analysis of this book 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.
23 Epstein, P. S., “On the air resistance of Projectiles,” Proceedings of the National Academy of Sci-
ence, Vol. 17, 1931, pp. 532-547.
24 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.
25 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.
26 Carrier, G.F., “On the Stability of the supersonic Flows Past as a Wedge,” Quarterly of Applied
Mathematics, Vol. 6, 1949, pp. 367–378.
27 Henderson and Menikoff, ”Triple Shock Entropy Theorem,” Journal of Fluid Mechanics 366 (1998)
pp. 179–210.