Abstract
The possibility of simulating the process of propagation of discontinuous waves along a dry bed on the basis of the equations of the first approximation of shallow water theory is studied. It is shown that the consistent losses of the free-stream total momentum and energy can be found from the mass conservation law within the framework of the shallow water equations. As an example, solutions of the problem of dam break with a dry bed in the lower pool and the problem of impingement of a discontinuous wave on a coastal shelf are constructed. These exact solutions are compared with the results of laboratory experiments.
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References
J. J. Stoker, Water Waves: the Mathematical Theory with Applications, Wiley (1958).
B. I. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations [in Russian], Nauka, Moscow (1978).
L. V. Ovsyannikov, N. I. Makarenko, V. I. Nalimov et al., Nonlinear Problems of the Theory of Surface and Internal Waves [in Russian], Nauka, Novosibirsk (1985).
A. F. Voevodin and S. M. Shugrin, Methods for Solving One-dimensional Evolutionary Systems [in Russian], Nauka, Novosibirsk (1993).
V. Yu. Lyapidevskii and V. M. Teshukov, Mathematical Models of the Propagation of Long Waves in an Inhomogeneous Fluid [in Russian], Publishing House of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk (2000).
A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Numerical Systems, Chapman & Hall/CRC, Boca Raton, London, New York, Washington, D.C. (2001).
V. V. Ostapenko, Hyperbolic Systems of Conservation Laws and their Application to Shallow Water Theory [in Russian], Novosibirsk University Press, Novosibirsk (2004).
A. Ritter, “Die Fortpflanzung der Wasserwellen,” Z. Ver. Deut. Ing., 36 (1892).
J. C. Martin, W. J. Moyce, W. G. Penney, A. T. Price, and C. K. Thornhill, “An experimental study of the collapse of liquid columns on a rigid horizontal plane,” Phil. Trans. Roy. Soc. London, Ser. A, 244, No. 882, 312–324 (1952).
R. F. Dressler, “Comparison of theories and experiments for the hydraulic dam-break wave,” Intern. Assoc. Sci. Hydrology, 3, No. 38, 319–328 (1954).
P. K. Stansby, A. Chegini, and T. C. D. Barnes, “The initial stages of dam-break flow,” J. Fluid Mech., 374, 407–424 (1998).
G. Collicchio, A. Colagrossi, M. Greco, and M. Landrini, “Free-surface flow after a dam break: a comparative study,” Shiffstechnik (Ship Technology Research), 49, No. 3, 95–104 (2002).
V. I. Bukreev, A. V. Gusev, A. A. Malysheva, and I. A. Malysheva, “Experimental verification of the gas-hydraulic analogy with reference to the dam-break problem,” Fluid Dynamics, 39, No. 5, 801–809 (2004).
V. I. Bukreev and A. V. Gusev, “Initial stage of wave generation in the dam break,” Dokl. Ros. Akad. Nauk, 401, No. 5, 619–622 (2005).
K. O. Friedrichs and P. D. Lax, “Systems of conservation equations with a convex extension,” Proc. Nat. Acad. Sci.: USA, 68, No. 8, 1686–1688 (1971).
P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia (1973).
O. V. Voinov, “Hydrodynamics of wetting,” Fluid Dynamics, 11, No. 5, 714–721 (1976).
F. Alcrudo and F. Benkhaldoun, “Exact solutions to the Riemann problem of the shallow water equations with bottom step,” Comput. and Fluids, 30, No. 6, 643–671 (2001).
A. A. Atavin and O. F. Vasil’ev, “Estimation of possible consequences of failure in a ship lock due to the breakdown of its chamber gates,” in: Abstracts of Intern. Symp. ‘Hydraulic and Hydrologic Aspects of the Reliability and Safety of Waterworks’, St. Petersburg, 2002, Russian Research Institute of Water Engineering, St. Petersburg (2002), p. 121.
V. V. Ostapenko, “Flows above a bottom shelf in dam break,” Zh. Prikl. Mekh. Tekh. Fiz., 44, No. 4, 51–63 (2003).
V. V. Ostapenko, “Flows above a bottom step in dam break,” Zh. Prikl. Mekh. Tekh. Fiz., 44, No. 6, 107–122 (2003).
V. I. Bukreev and A. V. Gusev, “Gravity waves during discontinuity breakdown over a bottom step in an open channel,” Zh. Prikl. Mekh. Tekh. Phys., 44, No. 4, 64 (2003).
V. I. Bukreev, A. V. Gusev, V. V. Ostapenko, “Waves formed during sluice-gate withdrawal in an open channel ahead of an irregular bottom of the shelf type,” Vodnye Resursy, 31, No. 5, 540–545 (2004).
V. I. Bukreev, A. V. Gusev, V. V. Ostapenko, “Breakdown of a discontinuity of the free fluid surface over a bottom step in a channel,” Fluid Dynamics, 38, No. 6, 889–899 (2004).
V. V. Ostapenko, “Discontinuous solutions of the ’shallow water’ equations above a bottomstep,” Zh. Prikl. Mekh. Tekh. Fiz., 43, No. 6, 62–74 (2002).
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Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, 2006, pp. 135–148.
Original Russian Text Copyright © 2006 by Borisova, Gusev, and Ostapenko.
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Borisova, N.M., Gusev, A.V. & Ostapenko, V.V. Propagation of discontinuouswaves along a dry bed. Fluid Dyn 41, 606–618 (2006). https://doi.org/10.1007/s10697-006-0079-y
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DOI: https://doi.org/10.1007/s10697-006-0079-y