Page 44 - 48Fundamentals of Compressible Fluid Mechanics

6 CHAPTER 1. INTRODUCTION

need to achieve powerful/long–distance shooting rifles/guns. At that time many
inventions dealt with machine guns which where able to shoot more bullets per
minute. At the time, one anecdotal story suggests a way to make money by invent-
ing 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
backs. 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 15 equations or explanations results in of this 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 supper sonic flow and the effects
on bullet movement (super sonic 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
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
proof to discontinuity. He further showed that the discontinuity occurs at
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 Poission’s work or the works of others.).
Back to the pencil and paper, the jump conditions were redeveloped
17
and now named after Rankine 16 and Hugoniot . Rankine and Hugoniot, redevel-
oped independently the equation that governs the relationship of the shock wave.
Shock was assumed to be one dimensional and mass, momentum, and energy
equations 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

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