Analytical Solution of Shallow Water Equations for Ideal Dam-Break Flood along a Wet-Bed Slope
Publication: Journal of Hydraulic Engineering
Volume 146, Issue 2
Abstract
The existing analytical solutions of dam-break flow do not consider simultaneously the effects of wet downstream bottom and bed slope on the dam-break wave propagation. In this study, a new analytical solution for the shallow water equations (SWE) is developed to remove this limitation to simulate the wave caused by an instantaneous dam break. The approach adopts the method of characteristics and has been applied to simulate the dam-break flows with different downstream water depths and slopes. The analytical solutions have been compared with predictions by the lattice Boltzmann method and the agreement is good. Although the proposed analytical solution treats an idealized case, it is nonetheless suitable for assessing the robustness and accuracy of numerical models based on the SWE without the frictional slope.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code generated or used during the study are available from the corresponding author by request (such as the analytical and numerical data).
Acknowledgments
The authors would like to thank for the financial support of the National Natural Science Foundation of China (Grant Nos. 51879179 and 51579166) and Sichuan Science and Technology Program (No. 2019JDTD0007). The research is also supported by the Open Fund from the State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University (SKHL1809, SKHL1601, and SKHL1602). Comments made by reviewers have greatly improved the quality of the final paper.
References
Ancey, C., R. M. Iverson, M. Rentschler, and R. P. Denlinger. 2008. “An exact solution for ideal dam-break floods on steep slopes.” Water Resour. Res. 44 (1): 567–568. https://doi.org/10.1029/2007WR006353.
Batchelor, G. K. 2000. An introduction to fluid dynamics. Cambridge, UK: Cambridge University Press.
Chanson, H. 2010. “Application of the method of characteristics to the dam break wave problem.” J. Hydraul. Res. 47 (1): 41–49. https://doi.org/10.3826/jhr.2009.2865.
Chen, Y. L., C. Wu, and B. Wang. 2011. “Similarity solution of dam-break flow on horizontal frictionless channel.” J. Hydraul. Res. 49 (3): 384–387. https://doi.org/10.1080/00221686.2011.571537.
Chow, V. T. 1959. Open channel hydraulics. New York: McGraw-Hill.
Cozzolino, L., V. Pepe, F. Morlando, L. Cimorelli, A. D’Aniello, R. Della Morte, and D. Pianese. 2017. “Exact solution of the dam-break problem for constrictions and obstructions in constant width rectangular channels.” J. Hydraul. Eng. 143 (11): 04017047. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001368.
Dressler, R. H. 1958. “Unsteady non-linear waves in sloping channels.” In vol. 247 of Proc., Royal Society of London. Series A, Mathematical and Physical Sciences, 186–198. London: Royal Society Publishing. https://doi.org/10.1098/rspa.1958.0177.
Fernandez-Feria, R. 2006. “Dam-break flow for arbitrary slopes of the bottom.” J. Eng. Math. 54 (4): 319–331. https://doi.org/10.1007/s10665-006-9034-5.
Guo, Y. K. 2005. “Numerical modelling of free overfall.” J. Hydraul. Eng. 131 (2): 134–138. https://doi.org/10.1061/(ASCE)0733-9429(2005)131:2(134).
Guo, Y. K., X. Wen, C. G. Wu, and D. Fang. 2010. “Numerical modelling of spillway flow with free drop and initially unknown discharge.” J. Hydraul. Res. 36 (5): 785–801. https://doi.org/10.1080/00221689809498603.
Hunt, B. 1984. “Perturbation solution for dam-break floods.” J. Hydraul. Eng. 110 (8): 1058–1071. https://doi.org/10.1061/(ASCE)0733-9429(1984)110:8(1058).
Peng, Y., J. G. Zhou, and R. Burrows. 2011a. “Modelling the free surface flow in rectangular shallow basins by lattice Boltzmann method.” J. Hydraul. Eng. 137 (12): 1680–1685. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000470.
Peng, Y., J. G. Zhou, and R. Burrows. 2011b. “Modelling solute transport in shallow water with the lattice Boltzmann method.” Comput. Fluids 50 (1): 181–188. https://doi.org/10.1016/j.compfluid.2011.07.008.
Ritter, A. 1892. “Die Fortpflanzung der Wasserwellen.” [In German.] Z. Verein Deutscher Ing. 36(33): 947–954.
Stoker, J. J. 1957. Water waves: The mathematical theory with applications. New York: Interscience.
Wang, B., Y. L. Chen, C. Wu, Y. Peng, X. Ma, and J. J. Song. 2017. “Analytical solution of dam-break flood wave propagation in a dry sloped channel with an irregular-shaped cross-section.” J. Hydro-Environ. Res. 14 (Mar): 93–104. https://doi.org/10.1016/j.jher.2016.11.003.
Wu, C., G. Huang, and Y. Zheng. 1999. “Theoretical solution of dam-break shock wave.” J. Hydraul. Eng. 125 (11): 1210–1215. https://doi.org/10.1061/(ASCE)0733-9429(1999)125:11(1210).
Zhou, J. G. 2004. Lattice Boltzmann methods for shallow water flows. New York: Springer.
Zoppou, C., and S. Roberts. 2003. “Explicit schemes for dam-break simulations.” J. Hydraul. Eng. 129 (1): 11–34. https://doi.org/10.1061/(ASCE)0733-9429(2003)129:1(11).
Information & Authors
Information
Published In
Copyright
©2019 American Society of Civil Engineers.
History
Received: Feb 11, 2019
Accepted: Jul 8, 2019
Published online: Dec 4, 2019
Published in print: Feb 1, 2020
Discussion open until: May 4, 2020
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.