Analytical Solution to the Zero-Inertia Problem for Surge Flow Phenomena in Nonprismatic Channels
Publication: Journal of Hydraulic Engineering
Volume 128, Issue 6
Abstract
Surge flow phenomena, e.g., as a consequence of a dam failure or a flash flood, represent free boundary problems. The extending computational domain together with the discontinuities involved renders their numerical solution a cumbersome procedure. This contribution proposes an analytical solution to the problem. It is based on the slightly modified zero-inertia (ZI) differential equations for nonprismatic channels and uses exclusively physical parameters. Employing the concept of a momentum-representative cross section of the moving water body together with a specific relationship for describing the cross sectional geometry leads, after considerable mathematical calculus, to the analytical solution. The hydrodynamic analytical model is free of numerical troubles, easy to run, computationally efficient, and fully satisfies the law of volume conservation. In a first test series, the hydrodynamic analytical ZI model compares very favorably with a full hydrodynamic numerical model in respect to published results of surge flow simulations in different types of prismatic channels. In order to extend these considerations to natural rivers, the accuracy of the analytical model in describing an irregular cross section is investigated and tested successfully. A sensitivity and error analysis reveals the important impact of the hydraulic radius on the velocity of the surge, and this underlines the importance of an adequate description of the topography. The new approach is finally applied to simulate a surge propagating down the irregularly shaped Isar Valley in the Bavarian Alps after a hypothetical dam failure. The straightforward and fully stable computation of the flood hydrograph along the Isar Valley clearly reflects the impact of the strongly varying topographic characteristics on the flow phenomenon. Apart from treating surge flow phenomena as a whole, the analytical solution also offers a rigorous alternative to both (a) the approximate Whitham solution, for generating initial values, and (b) the rough volume balance techniques used to model the wave tip in numerical surge flow computations.
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References
Aitken, A. C.(1926). “On Bernoulli’s numerical solution of algebraic equations.” Proc. R. Soc. Edinburgh, 46, 289–305.
Henderson, F. M. (1966). Open channel flow, Macmillan, New York.
Jaynes, D. B.(1986). “Simple model of border irrigation.” J. Irrig. Drain. Eng., 112(2), 172–184.
Katopodes, N. D., and Strelkoff, T.(1977). “Hydrodynamics of border-irrigation: Complete model.” J. Irrig. Drain. Eng., 103(3), 309–324.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992). Numerical recipes, 2nd Ed., Cambridge University Press, Cambridge.
Schmitz, G. H. (1989). “Strömungsvorgänge auf der Oberfläche und im Bodeninnern beim Bewässerungslandbau [Surface and subsurface flow processes during surface irrigation].” Institut für Wasserbau and Wassermengenwirtschaft und Versuchsanstalt für Wasserbau, TU München, Report 60 (in German).
Schmitz, G. H., Edenhofer, J., and Czirwitzky, H.-J. (1982). “An analytical and numerical solution of Saint-Venant equations.” Symposium Int. Sur la Modélisation Fine des Écoulements, Paris.
Schmitz, G. H., Haverkamp, R., and Palacios Velez, O. (1985). “A coupled surface-subsurface model for shallow water flow over initially dry soil.” Proc., 21st Int. IAHR Congress, Melbourne, Vol. 1, 23–30.
Schmitz, G. H., and Seus, G. J.(1987). “Analytical solution of simplified surge flow equations.” J. Irrig. Drain. Eng., 113(4), 605–610.
Schmitz, G. H., and Seus, G. J.(1990). “Mathematical zero-inertia modeling of surface irrigation: Advance in borders.” J. Irrig. Drain. Eng., 116(5), 603–615.
Strelkoff, T., and Katopodes, N. D.(1977). “Border-irrigation hydraulics with zero inertia.” J. Irrig. Drain. Eng., 103(3), 325–342.
Whitham, G. B.(1955). “The effects of hydraulic resistance in the dam-break problem.” Proc. R. Soc. London, Ser. A, 227, 399–407.
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Copyright © 2002 American Society of Civil Engineers.
History
Received: Jun 14, 1999
Accepted: Nov 28, 2001
Published online: May 15, 2002
Published in print: Jun 2002
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