Modeling 3D Supercritical Flow with Extended Shallow-Water Approach
This article has a reply.
VIEW THE REPLYPublication: Journal of Hydraulic Engineering
Volume 132, Issue 9
Abstract
Despite the three-dimensional (3D) nature of the flow, the classical shallow-water equations are often used to simulate supercritical flow in channel transitions. A closer comparison with experimental data, however, often shows large discrepancies in the height and pattern of the shock waves that increase with the Froude number. An extension to the classical shallow-water approach is derived considering higher-order distribution functions for pressure and horizontal and vertical velocities, therefore taking nonhydrostatic pressure distribution and vertical momentum into account. The approach is applied to highly supercritical flow in a channel contraction , a channel junction ( for both branches), and a channel expansion . Specific problems of such flows—wetting and drying of computational cells and wave breaking due to steep free-surface gradients—are discussed and solved numerically. The solutions with the extended approach are compared both with experimental data and classical shallow-water computations, and the influence of the additional terms considering the 3D nature of such flows is illustrated.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The research work was performed at the Laboratory of Hydraulics, Hydrology and Glaciology (VAW) of the Swiss Federal Institute of Technology in Zurich (ETH-Z) with a Swiss National science Foundation Grant (SNF20–45631.95).
References
Abbott, M. B. (1979). “Computational hydraulics, elements of the theory of free surface flows.” Monographs and surveys in water resources engineering, Vol. 1, Pitman, London.
Abbot, M. B., McCowan, A. D., and Warren, I. R. (1984). “Accuracy of short-wave numerical models.” J. Hydraul. Eng., 110(10), 1287–1301.
Abbot, M. B., and Verwey, A. (1970). “Four-point method of characteristics.” J. Hydraul. Div., Am. Soc. Civ. Eng., 96(12), 2549–2564.
Bagge, G., and Herbich, J. B. (1967). “Transitions in supercritical open-channel flow.” J. Hydraul. Div., Am. Soc. Civ. Eng., 93(5), 23–41.
Beffa, C. J. (1994). “Praktische Lösung der tiefengemittelten Flachwassergleichungen.” Ph.D. thesis No. 10870, ETH Zurich, Zurich, Switzerland (in German).
Berger, R. C., and Stockstill, R. L. (1995). “Finite-element model for high-velocity channels.” J. Hydraul. Eng., 121(10), 710–716.
Bhallamudi, S. M., and Chaudhry, M. H. (1992). “Computation of flows in open-channel transitions.” J. Hydraul. Res., 30(1), 77–93.
Brooks, A. N., and Hughes, T. J. R. (1982). “Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations.” Comput. Methods Appl. Mech. Eng., 32, 199–259.
Causon, D. M., Mingham, C. G., and Ingram, D. M. (1999). “Advances in calculation methods for supercritical flow in spillway channels.” J. Hydraul. Eng., 125(10), 1039–1050.
Ellis, J. (1970). “Unsteady flow in channel of variable cross section.” J. Hydraul. Div., Am. Soc. Civ. Eng., 96(10), 1927–1945.
Evangelisti, G. (1966). “Mathematical models of open-channel flow.” Surface hydrodynamics, NATO Advanced Study Institute, Bressano, Italy, 198–283.
Fennema, R. F., and Chaudhry, M. H. (1987). “Simulation of one-dimensional dam-break flows.” J. Hydraul. Res., 25(1), 41–51.
Fennema, R. F., and Chaudhry, M. H. (1989). “Implicit methods for two-dimensional unsteady free-surface flows.” J. Hydraul. Res., 27(3), 321–332.
Hager, W. H. (1989). “Supercritical flow in channel junctions.” J. Hydraul. Eng., 115(5), 595–616.
Hirt, C. W., and Nichols, B. D. (1981). “Volume of fluid (VOF) method for the dynamics of free boundaries.” J. Comput. Phys., 39, 201–225.
Hughes, T. J. R., and Brooks, A. N. (1979). “A multidimensional upwind scheme with no crosswind diffusion.” Finite elements for convection dominated flows, T. J. R Hughes, ASME.
Hughes, T. J. R., Franca, L. P., and Hulbert, G. M. (1989). “A new finite element formulation for computational fluid dynamics. VIII: The Galerkin/least-squares method for advective-diffusive systems.” Comput. Methods Appl. Mech. Eng., 73, 173–189.
Hughes, T. J. R., Liu, W. K., and Zimmermann, T. K. (1981). “Lagrangian-Eulerian finite-element formulation for incompressible viscous flows.” Comput. Methods Appl. Mech. Eng., 29, 329–349.
Ippen, A. T., and Dawson, J. H. (1951). “Design of channel contractions.” Symp., High-Velocity Flow in Open Channels, 116, 326–446.
Jiménez, O. F., and Chaudhry, M. H. (1988). “Computation of supercritical free-surface flows.” J. Hydraul. Eng., 114(4), 377–395.
Katopodes, N. D. (1984a). “Dissipative Galerkin scheme for open-channel flow.” J. Hydraul. Eng., 110(4), 450–466.
Katopodes, N. D. (1984b). “Two-dimensional surges and shocks in open channels.” J. Hydraul. Eng., 110(6), 794–812.
Katopodes, N. D., and Wu, C.-T. (1986). “Explicit computation of discontinuous channel flow.” J. Hydraul. Eng., 112(6), 456–475.
Khan, A. A., and Steffler, P. M. (1996). “Vertically averaged and moment equations model for flow over curved beds.” J. Hydraul. Eng., 122(1), 3–9.
Krüger, S. (2000). “Computational contribution to highly supercritical flows.” Ph.D. thesis No. 13838, ETH Zurich, Zurich, Switzerland.
Krüger, S., Bürgisser, M. F., and Rutschmann, P. (1998). “Advances in calculating supercritical flows in spillway contractions.” Hydroinformatics 98, V. Babovic and L. C. Larsen, eds., Vol. 1, 163–170.
Liggett, J. A., and Vasudev, S. U. (1965). “Slope and friction effects in two-dimensional high speed channel flow.” 11th IAHR Congress, Leningrad, Vol. 1, Rep. No. 1.25.
Liggett, J. A., and Woolhiser, D. A. (1967). “Difference solutions of the shallow-water equation.” J. Eng. Mech. Div., Am. Soc. Civ. Eng., 93(2), 39–71.
Mazumder, S. K., and Hager, W. H. (1993). “Supercritical expansion flow in Rouse modified and reversed transitions.” J. Hydraul. Eng., 119(2), 201–219.
Näf, D. (1996). “Numerische Simulationen von Stosswellen in Freispiegelströmungen.” Ph.D. thesis No. 11968, ETH Zurich, Zurich, Switzerland (in German).
Preissman, A. (1971). “Modèles pour le calcul de la propagation des crues.” La Houille Blanche, 26(3), 219–224 (in French).
Rahman, M., and Chaudhry, M. H. (1997). “Computation of flow in open-channel transitions.” J. Hydraul. Res., 35(2), 243–256.
Rutschmann, P. (1993). “Fe solver with 4D finite elements in space and time.” FE in Fluids, Proc., VIII. Int. Conf. On FE in Fluid, Barcelona, K. Morgan, K. Oñate, J. Periaux, J. Peraire and O. C. Zienkiewicz, eds., 136–144.
Rutschmann, P. (1994). “Obtaining higher order accuracy with linear shape functions—A new approach for transient problems.” Comput. Fluid Dyn., 94 Ed., Wagner, Hirschel, Periaux, and Riva, 165–169.
Sethian, J. A. (1996). “Level set methods: Evolving interfaces in geometry, fluid mechanics, computer vision, and materials science.” Cambridge University Press, Cambridge, U.K.
Steffler, P. M., and Jin, Y.-C. (1993). “Depth-averaged and moment equations for moderately shallow free surface flow.” J. Hydraul. Res., 21(1), 5–17.
Von Kármán, T. (1938). “Eine praktische Anwendung der Analogie zwischen Ueberschallströmungen in Gasen und überkritischer Strömung in offenen Gerinnen.” ZAMM, 18 Feb., 49–56.
Vreugdenhil, C. B. (1994). “Numerical methods for shallow-water flow.” Water sci. technol., Vol. 13, Kluwer Academic, Dordrecht.
Zienkiewicz, O. C. (1971). “The finite-element method in engineering science.” 2nd Ed., McGraw-Hill, New York.
Information & Authors
Information
Published In
Journal of Hydraulic Engineering
Volume 132 • Issue 9 • September 2006
Pages: 916 - 926
Copyright
© 2006 ASCE.
History
Received: Jun 1, 2001
Accepted: Oct 26, 2005
Published online: Sep 1, 2006
Published in print: Sep 2006
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.