Hello all,
I have come across the following relation at many places reg. water hammer and acoustic in compressible flow
i.e Pressure = density x fluid velocity x sound speed in fluid.
I am looking for the research paper wherein this proof may have been presented. I could find in standard textbooks but I am looking for research article regarding the same.
Thanks
Deewakar
This relation is commonly known as the “Joukowsky equation. It is the first explicit statement in waterhammer and usually is attributed to Joukowsky (1898). Please access the link for more detail.
www.win.tue.nl/analysis/reports/rana06-08.pdf
Dear Deewakar,
Originally, the relationship was determined by Rankine.(1870).
Rankine, W.J.M. (1870). “On the thermodynamic theory of waves of finite longitudinal disturbance.” Philosophical Transactions of the Royal Society (London) 160, 277-288. Reprinted (1881) in Miscellaneous Scientific Papers by W.J. Macquorn Rankine, 32, 530-543, W.J. Millar, ed., Charles Griffith, London, UK.
However, the equation is called "equation of Zhukovsky" (or Joukowsky)
Here is the original article reference
Joukowsky, N. (1898). “Über den hydraulischen Stoss in Wasserleitungsröhren.” (“On the hydraulic hammer in water supply pipes.”) Mémoires de l'Académie Impériale des Sciences de St.-Pétersbourg (1900), Series 8, 9(5), 1-71 (in German). [Sections presented to the Division of Physical Sciences of O.L.E., 26 September 1897, to the Physical-Mathematical Commission of that Society, 30 January 1898, to the Polytechnic Society of the Moscow Imperial Institute, 21 February 1898; complete paper to the Russian Technical Society, 24 April 1898, to the PhysicalMathematical Division of the Academy of Sciences, 13 May 1898.] [English translation, partly, by O. Simin (1904); French translation, partly, by M. Goupil (1907).] [Also: Жуковский, Н.Е. (1899). “О гидравлическом ударе в водопроводных трубах.” (“On hydraulic hammer in water mains.”), Proc., 4th Russian Water Pipes Congress, April 1899, Odessa, Russia, 78-173, printed in Moscow (1901); Bulletin of the Polytechnic Society of the Imperial Technical School 8(5), 255-290, Moscow (1899); Reprinted (1948) in “Selected works.” Vol. 2, 3-73, Gostechizdat, Moscow, Russia; Reprinted (1949) in: “Classics of science,” 1-105, State Publisher of Technical-Theoretical Literature, Moscow/Leningrad, Russia (all in Russian).]
With my best regards
Prof Bachir ACHOUR
Dear Deewakar,
This relation $\Delta P=\rho c \Delta v$ is known as the Joukowsky relation. As mentionned in a previous answer the original paper is:
Joukowsky, N. (1898). “Über den hydraulischen Stoss in Wasserleitungsröhren.” (“On the hydraulic hammer in water supply pipes.”) Mémoires de l'Académie Impériale des Sciences de St.-Pétersbourg (1900), Series 8, 9(5), 1-71 (in German).
You can have a look also to the report of Arris S. Tijsseling
www.win.tue.nl/analysis/reports/rana06-08.pdf
In practice, the Joukowsky relation is a kind linearization of the Rankine-Hugoniot relation (associated to shock waves) in the case of quasi-incompressible fluid, i.e. \rho=cste and c=cste.
Strating from the Euler equations modeling compressible fluid flows through a duct of variable cross–section, you can retrieve this relation. See for example,
http://hydraulics.unibs.it/hydraulics/wp-content/uploads/2012/04/A_review_of_water_hammer_theory_and_practice.pdf
Do not hesitate to contact me if you need more details or explanations concerning the way to obtain the Joukowsky relation from the Euler equations.
Good luck,
Frederic
Please do not forget that this relation showes only pressures on first amplitude after a pump failure or quick instanteneus valve closure. But if cavitation occurs in the pipe the pressure on the second amplitude can be up to twice that big as calculated from this equation! S have it in mind and check the articles about water hammer with cavitation. Cheers
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