Galilean-Invariant Expression for Bernoulli’s Equation
Publication: Journal of Hydraulic Engineering
Volume 146, Issue 2
Abstract
The Bernoulli principle is one of the basic and most famous concepts in fluid mechanics, and the related equation is an extremely useful and effective tool in the solution of a wide range of practical engineering problems. However, the Bernoulli equation is not Galilean-invariant (i.e., it does not remain the same as a result of a change in the inertial frame of reference when the new frame moves with constant velocity relative to the original one). In this paper, a frame-independent expression of the Bernoulli equation is obtained from the Euler equation of inviscid motion for the case of steady flow, for the two classic versions valid along a streamline, and for irrotational flow. Compared to the conventional formulation of the equation, an additional term is present in the definition of the Bernoulli constant. An interpretation of the new equation is provided on the basis of the first law of thermodynamics. Some classic applications are presented, and the results found are in accordance with the consolidated knowledge of fluid mechanics.
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Data Availability Statement
No data, models, or code were generated or used during the study.
Acknowledgments
Professor Paolo Mignosa (University of Parma, Italy) and Professor Massimo Tomirotti (University of Brescia, Italy) are kindly acknowledged for their useful suggestions and thorough reviews. The editors and reviewers are acknowledged too for their valuable and constructive comments, which have greatly contributed to the improvement of the paper.
References
Anderson, J. D. 2017. Fundamentals of aerodynamics. 6th ed. New York: McGraw-Hill.
Aref, H. 2005. “Bernoulli’s equation.” In Encyclopedia of nonlinear science, edited by A. Scott, 44–45. New York: Taylor & Francis.
Aureli, F., A. Maranzoni, P. Mignosa, and C. Ziveri. 2008. “A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography.” Adv. Water Resour. 31 (7): 962–974. https://doi.org/10.1016/j.advwatres.2008.03.005.
Batchelor, G. K. 1967. An introduction to fluid dynamics. Cambridge, UK: Cambridge University Press.
Bauman, R. P. 2000. “An alternative derivation of Bernoulli’s principle.” Am. J. Phys. 68 (3): 288–289. https://doi.org/10.1119/1.19423.
Bauman, R. P., and R. Schwaneberg. 1994. “Interpretation of Bernoulli’s equation.” Phys. Teach. 32 (8): 478–488. https://doi.org/10.1119/1.2344087.
Bernoulli, D. 1738. Hydrodynamica. Strassburg, Austria: Johann Reinhold Dulsecker.
Carmody, T., and H. Kobus. 1968. Hydrodynamics by Daniel Bernoulli and hydraulics by Johann Bernoulli. New York: Dover.
Clamond, D. 2017. “Remarks on Bernoulli constants, gauge conditions and phase velocities in the context of water waves.” Appl. Math. Lett. 74 (Dec): 114–120. https://doi.org/10.1016/j.aml.2017.05.018.
Delestre, O., C. Lucas, P. A. Ksinant, F. Darboux, C. Laguerre, T. N. T. Vo, F. James, and S. Cordier. 2013. “SWASHES: A compilation of shallow water analytic solutions for hydraulic and environmental studies.” Int. J. Numer. Methods Fluids 72 (3): 269–300. https://doi.org/10.1002/fld.3741.
Katz, J., and A. Plotkin. 2001. Low-speed aerodynamics. 2nd ed. Cambridge, UK: Cambridge University Press.
Kelley, J. B. 1950. “The extended Bernoulli equation.” Am. J. Phys. 18 (4): 202–204. https://doi.org/10.1119/1.1932534.
Landau, L. D., and E. M. Lifshitz. 1987. Fluid mechanics. 2nd ed. Oxford, UK: Pergamon Press.
Malcherek, A. 2016. “History of the Torricelli principle and a new outflow theory.” J. Hydraul. Eng. 142 (11): 02516004. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001232.
McDonald, K. T. 2009. Is Bernoulli’s equation relativistically invariant? Princeton, NJ: Joseph Henry Laboratories, Princeton Univ.
McDonald, K. T. 2012. Oscillating fluid in a U-tube. Princeton, NJ: Joseph Henry Laboratories, Princeton Univ.
Miller, R. W. 1996. Flow measurement engineering handbook. 3rd ed. New York: McGraw-Hill.
Mungan, C. E. 2011. “The Bernoulli equation in a moving reference frame.” Eur. J. Phys. 32 (2): 517–520. https://doi.org/10.1088/0143-0807/32/2/022.
Munson, B. R., D. F. Young, and T. H. Okiishi. 2002. Fundamentals of fluid mechanics. 4th ed. New York: Wiley.
Pritchard, P. J., and J. W. Mitchell. 2015. Fox and McDonald’s introduction to fluid mechanics. 9th ed. New York: Wiley.
Replogle, J., and B. Wahlin. 2000. “Pitot-static tube system to measure discharges from wells.” J. Hydraul. Eng. 126 (5): 335–346. https://doi.org/10.1061/(ASCE)0733-9429(2000)126:5(335).
Segletes, S. B., and W. P. Walters. 1999. “A note on the application of the extended Bernoulli equation.”. Aberdeen, MD: US Army Research Laboratory.
Sharp, Z. B., M. C. Johnson, and S. L. Barfuss. 2018. “Optimizing the ASME Venturi recovery cone angle to minimize head loss.” J. Hydraul. Eng. 144 (1): 04017057. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001387.
Synolakis, C. E., and H. S. Badeer. 1989. “On combining the Bernoulli and Poiseuille equation–A plea to authors of college physics texts.” Am. J. Phys. 57 (11): 1013–1019. https://doi.org/10.1119/1.15812.
White, F. M. 2011. Fluid mechanics. 7th ed. New York: McGraw-Hill.
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Published In
Journal of Hydraulic Engineering
Volume 146 • Issue 2 • February 2020
Copyright
©2019 American Society of Civil Engineers.
History
Received: Nov 29, 2018
Accepted: Jun 27, 2019
Published online: Dec 9, 2019
Published in print: Feb 1, 2020
Discussion open until: May 9, 2020
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