Compressible Flow Free Textbook

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The shock wave puzzle

Here is where the politics of science was a major obstacle to achieving an advancement^1.10. not giving the due credit to Rouse. In the early 18xx, conservation of energy was a concept that was applied only to mechanical energy. On the other side, a different group of scientists dealt with calorimetry (internal energy). It was easier to publish articles about the second law of thermodynamics than to convince anyone of the first law of thermodynamics. Neither of these groups would agree to ``merge'' or ``relinquish'' control of their ``territory'' to the other. It took about a century to establish the first law^1.11.

At first, Poisson found a ``solution'' to the Euler's equations with certain boundary conditions which required discontinuity^1.12which had obtained an implicit form in 1808. Poisson showed that solutions could approach a discontinuity by using conservation of mass and momentum. He had then correctly derived the jump conditions that discontinuous solutions must satisfy. Later, Challis had noticed contradictions concerning some solutions of the equations of compressible gas dynamics^1.13. Again the ``jumping'' conditions were redeveloped by two different researchers independently: Stokes and Riemann. Riemann, in his 1860 thesis, was not sure whether or not discontinuity is only a mathematical creature or a real creature. Stokes in 1848 retreated from his work and wrote an apology on his ``mistake.''^1.14Stokes was convinced by Lord Rayleigh and Lord Kelvin that he was mistaken on the grounds that energy is conserved (not realizing the concept of internal energy). At this stage some experimental evidence was needed. Ernst Mach studied several fields in physics and also studied philosophy. He was mostly interested in experimental physics. The major breakthrough in the understanding of compressible flow came when Ernest Mach ``stumbled'' over the discontinuity. It is widely believed that Mach had done his research as purely intellectual research. His research centered on optic aspects which lead him to study acoustic and therefore supersonic flow (high speed, since no Mach number was known at that time). However, it is logical to believe that his interest had risen due to the need to achieve powerful/long-distance shooting rifles/guns. At that time many inventions dealt with machine guns which were able to shoot more bullets per minute. At the time, one anecdotal story suggests a way to make money by inventing a better killing machine for the Europeans. While the machine gun turned out to be a good killing machine, defense techniques started to appear such as sand bags. A need for bullets that could travel faster to overcome these obstacles was created. Therefore, Mach's paper from 1876 deals with the flow around bullets. Nevertheless, no known^1.15equations or explanations resulted from these experiments.

Mach used his knowledge in Optics to study the flow around bullets. What makes Mach's achievement all the more remarkable was the technique he used to take the historic photograph: He employed an innovative approach called the shadowgraph. He was the first to photograph the shock wave. In his paper discussing "Photographische Fixierung der durch Projektile in der Luft eingeleiten Vorgange" he showed a picture of a shock wave (see Figure 1.7). He utilized the variations of the air density to clearly show shock line at the front of the bullet. Mach had good understanding of the fundamentals of supersonic flow and the effects on bullet movement (supersonic flow). Mach's paper from 1876 demonstrated shock wave (discontinuity) and suggested the importance of the ratio of the velocity to the speed of sound. He also observed the existence of a conical shock wave (oblique shock wave).

Mach's contributions can be summarized as providing an experimental proof to discontinuity. He further showed that the discontinuity occurs at M=1 and realized that the velocity ratio (Mach number), and not the velocity, is the important parameter in the study of the compressible flow. Thus, he brought confidence to the theoreticians to publish their studies. While Mach proved shock wave and oblique shock wave existence, he was not able to analyze it (neither was he aware of Poisson's work or the works of others.).

Back to the pencil and paper, the jump conditions were redeveloped and now named after Rankine^1.16 and Hugoniot^1.17. Rankine and Hugoniot, redeveloped independently the equation that governs the relationship of the shock wave. Shock was assumed to be one dimensional and mass, momentum, and energy equations^1.18 lead to a solution which ties the upstream and downstream properties. What they could not prove or find was that shock occurs only when upstream is supersonic, i.e., direction of the flow. Later, others expanded Rankine-Hugoniot's conditions to a more general form^1.19.

Here, the second law has been around for over 40 years and yet the significance of it was not was well established. Thus, it took over 50 years for Prandtl to arrive at and to demonstrate that the shock has only one direction^1.20. Today this equation/condition is known as Prandtl's equation or condition (1908). In fact Prandtl is the one who introduced the name of Rankine-Hugoniot's conditions not aware of the earlier developments of this condition. Theodor Meyer (Prandtl's student) derived the conditions for oblique shock in 1908^1.21 as a byproduct of the expansion work.

**Figure:** The shock as connection of Fanno and Rayleigh lines.
$\begin{figure}\centerline{\includegraphics{scaned/shock}} \end{figure}$

It was probably later that Stodola (Fanno's adviser) realized that the shock is the intersection of the Fanno line with the Rayleigh line. Yet, the supersonic branch is missing from his understanding (see Figure (1.1)). In fact, Stodola suggested the graphical solution utilizing the Fanno line.

The fact that the conditions and direction were known did not bring the solution to the equations. The ``last nail'' of understanding was put by Landau, a Jewish scientist who worked in Moscow University in the 1960's during the Communist regimes. A solution was found by Landau & Lifshitz and expanded by Kolosnitsyn & Stanyukovich (1984).

Since early in the 1950s the analytical relationships between the oblique shock, deflection angle, shock angle, and Mach number was described as impossible to obtain. There were until recently (version 0.3 of this book) several equations that tied various properties/quantities for example, the relationship between upstream Mach number and the angles. The first full analytical solution connecting the angles with upstream Mach number was published in this book version 0.3. The probable reason that analytical solution was not published because the claim in the famous report of NACA 1135 that explicit analytical solution isn't possible^1.22. 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 solution. It is amazing that early research concluded that only the weak solution is possible or stable as opposed to the reality. The first that attempt this question where in 1931 by Epstein^1.24. His analysis was based on Hamilton's principle when he ignore the boundary condition. The results of that analysis was that strong shock is unstable. The researchers understood that flow after a strong shock was governed by elliptic equation while the flow after a weak shock was governed by hyperbolic equations. This difference probably results in not recognizing that The boundary conditions play an important role in the stability of the shock^1.25. In fact analysis based on Hamilton's principle isn't suitable for stability because entropy creation was recognized 1955 by Herivel^1.26.

Carrier^1.27was first to recognize that strong and weak shocks stable. If fact the confusion on this issue was persistent until now. Even all books that were published recently claimed that no strong shock was ever 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^1.28 dealt with only the procedure to find the maximum of oblique shock, but no comparison between real gases and ideal gas is offered there.

The moving shock and shock tube were study even before World War Two. The realization that in most cases the moving shock can be analyzed as steady state since it approaches semi steady state can be traced early of 1940's. Up to this version 0.4.3 of this book (as far it is known, this book is first to publish this tables), trial and error method was the only method to solve this problem. Only after the dimensionless presentation of the problem and the construction of the moving shock table the problem became trivial. Later, an explicit analytical solution for shock a head of piston movement (special case of open valve) was originally published in this book for the first time.

Next: Choking Flow Up: Historical Background Previous: Speed of Sound Index

Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21