Novel Numerical Approach for 1D Variable Density Shallow Flows over Uneven Rigid and Erodible Beds
Publication: Journal of Hydraulic Engineering
Volume 140, Issue 3
Abstract
The numerical modeling of hyperconcentrated shallow flows is a challenging task because they exhibit special features, such as propagation over dry beds, profound bed elevation modifications owing to erosion or deposition phenomena, and flow discontinuities. In this paper, a novel depth-positivity preserving Harten, Lax, and van Leer—contact (HLLC) Riemann solver is devised in order to approximate the solution of the Riemann problem for the 1D (one-dimensional) hyperconcentrated shallow flows equations over horizontal beds. The solver is used as a building block for the construction of hyperconcentrated shallow flows (HCSF), a well-balanced finite-volume scheme for the solution of the hyperconcentrated shallow flows equations with variable elevation. HCSF is able to handle the case of dry beds, to take into account the variability of the topography also in the presence of bed discontinuities, considering the flow resistance and the mass exchange between the flowing mixture and the mobile bed. The numerical tests carried out confirm the well-balancing property of the scheme proposed, the robustness in the presence of dry beds, the ability to approximate the analytic solution of problems with smooth or discontinuous beds, and the ability to reproduce reasonably the results of a laboratory experiment.
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Acknowledgments
The experimental data employed in this paper derive from a campaign of laboratory experiments carried out at the Laboratory of the Civil and Environmental Engineering Department, Université Catholique de Louvain by Dr. B. Spinewine, Prof. Y. Zech, Prof. S. Soares-Frazão, and Dr. N. Le Grelle. They are all gratefully acknowledged for having made this data set available. Finally, the authors want to acknowledge the two anonymous reviewers, the Associate Editor and the Editor, whose constructive comments and suggestions contributed to improve the paper.
References
Abderrazak, K. E.-K., Paquier, A., and Gay, B. (2008). “One-dimensional numerical modelling of dam-break waves over movable beds: Application to experimental and field cases.” Environ. Fluid Mech., 8(2), 169–198.
Armanini, A., Fraccarollo, L., and Rosatti, G. (2009). “Two-dimensional simulation of debris flows in erodible channels.” Comput. Geosci., 35(5), 993–1006.
Batten, P., Clarke, N., Lambert, C., and Causon, M. (1997). “On the choice of wavespeeds for the HLLC Riemann solver.” J. Sci. Comp., 18(6), 1553–1570.
Begnudelli, L., and Rosatti, G. (2011). “Hyperconcentrated 1D shallow flows on fixed bed with geometrical source term due to a bottom step.” J. Sci. Comp., 48(1–3), 319–332.
Bouchut, F. (2004). Nonlinear stability of finite volume methods for hyperbolic conservation laws, Birkhäuser, Basel, Switzerland.
Burguete, J., Garcìa-Navarro, P., Murillo, J., and Garcìa-Palacìn, I. (2007). “Analysis of the friction term in the one-dimensional Shallow-water model.” J. Hydraul. Eng., 1048–1063.
Cao, Z., Pender, G., Wallis, S., and Carling, P. (2004). “Computational dam-break hydraulics over erodible sediment bed.” J. Hydraul. Eng., 689–703.
Capart, H., and Young, D. L. (1998). “Formation of a jump by the dam-break wave over a granular bed.” J. Fluid Mech., 372, 165–187.
Castro, M. J., Milanés, A. P., and Parés, C. (2007). “Well-balanced numerical schemes based on a Generalized hydrostatic reconstruction technique.” Math. Model Meth. Appl. Sci., 17(12), 2055–2113.
Cheng, N. S. (1997). “Simplified settling velocity formula for sediment particle.” J. Hydraul. Eng., 149–152.
Cozzolino, L., Della Morte, R., Covelli, C., Del Giudice, G., and Pianese, D. (2011). “Numerical solution of the discontinuous-bottom shallow-water equations with hydrostatic pressure distribution at the step.” Adv. Water Resour., 34(11), 1413–1426.
Cozzolino, L., Della Morte, R., Del Giudice, G., Palumbo, A., and Pianese, D. (2012). “A well-balanced spectral volume scheme with the wetting–drying property for the shallow-water equations.” J. Hydroinform., 14(3), 745–760.
Dal Maso, G., LeFloch, P. G., and Murat, F. (1995). “Definition and weak stability of nonconservative products.” J. Math. Pures Appl. IX Ser., 74(6), 483–548.
Einfeldt, B., Munz, C. D., Roe, P. L., and Sjogreen, B. (1991). “On Godunov-type methods near low densities.” J. Comput. Phys., 92(2), 273–295.
Fraccarollo, L., and Capart, H. (2002). “Riemann wave description of erosional dam-break flows.” J. Fluid Mech., 461, 183–228.
Fraccarollo, L., and Toro, E. F. (1995). “Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems.” J. Hydraul. Res., 33(6), 843–864.
Goutière, L., Soares-Frazão, S., Savary, C., Laraichi, T., and Zech, Y. (2008). “One-dimensional model for transport flows involving bed-load sediment transport and changes in flow regimes.” J. Hydraul. Eng., 726–735.
Hou, T. Y., and LeFloch, P. G. (1994). “Why nonconservative schemes converge to wrong solutions: Error analysis.” Math. Comput., 62(206), 497–530.
Iverson, R. M. (1997). “The physics of debris flows.” Rev. Geophys., 35(3), 245–296.
Kim, D.-H., and Lee, S. O. (2012). “Stable numerical model for transcritical flow and sediment transport on uneven bathymetry.” J. Hydraul. Eng., 46–56.
Kubatko, E. J., and Westerink, J. J. (2007). “Exact discontinuous solutions of Exner’s bed evolution model: Simple theory for sediment bores.” J. Hydraul. Eng., 305–311.
Leighton, F. Z., Borthwick, A. G. L., and Taylor, P. H. (2010). “1-D numerical modelling of shallow flows with variable horizontal density.” Int. J. Num. Meth. Fluids, 62(11), 1209–1231.
Li, S., and Duffy, C. J. (2011). “Fully coupled approach to modelling shallow water flow, sediment transport, and bed evolution in rivers.” Water Resour. Res., 47(3), W03508.
Parès, C. (2006). “Numerical methods for nonconservative hyperbolic systems: A theoretical framework.” J. Numer. Anal., 44(1), 300–321.
Pelanti, M., Bouchut, F., and Mangeney, A. (2008). “A Roe-type scheme for two-phase shallow granular flows over variable topography.” Math. Model. Numer. Anal., 42(5), 851–885.
Pianese, D. (1993). Influenza della non stazionarietà e non uniformità del trasporto solido sui processi di evoluzione d’alveo, Dept. of Hydraulic and Environmental Engineering G. Ippolito, Univ. of Napoli Federico II, Napoli (in Italian).
Pianese, D. (1994). Comparison of different mathematical models for river dynamics analysis, Dept. of Hydraulic and Environmental Engineering G. Ippolito, Univ. of Napoli Federico II, Napoli.
Pitman, E. B., and Le, L. (2005). “A two-fluid model for avalanche and debris flows.” Phil. Trans. R. Soc. A, 363(1832), 1573–1601.
Sanders, B. F. (2002). “Non-reflecting flux function for finite volume shallow-water models.” Adv. Water Resour., 25(2), 195–202.
Simpson, G., and Castelltort, S. (2006). “Coupled model of surface water flow, sediment transport and morphological evolution.” Comput. Geosci., 32(10), 1600–1614.
Spinewine, B., and Zech, Y. (2007). “Small-scale laboratory dam-break waves on movable beds.” J. Hydraul. Res., 45(Sup1), 73–86.
Toro, E. F. (2001). Shock-capturing methods for free-surface shallow flows, Wiley, Chichester, U.K.
Toro, E. F., Spruce, M., and Speares, W. (1994). “Restoration of the contact surface in the HLL Riemann solver.” Shock Waves, 4(1), 25–34.
Wu, W., and Wang, S. Y. (2007). “One-dimensional modelling of dam-break flow over movable beds.” J. Hydraul. Eng., 48–58.
Wu, W., and Wang, S. Y. (2008). “One-dimensional explicit finite-volume model for sediment transport.” J. Hydraul. Res., 46(1), 87–98.
Xia, J., Lin, B., Falconer, R. A., and Wang, G. (2010). “Modelling dam-break flows over mobile beds using a 2D coupled approach.” Adv. Water Resour., 33(2), 171–183.
Ying, X., Khan, A. A., and Wang, S. S. (2004). “Upwind conservative scheme for the Saint Venant equations.” J. Hydaul. Eng., 977–987.
Zhang, S., and Duan, J. G. (2011). “1D finite volume model of unsteady flow over mobile bed.” J. Hydrol., 405(1–2), 57–68.
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© 2013 American Society of Civil Engineers.
History
Received: Nov 22, 2012
Accepted: Sep 5, 2013
Published online: Sep 7, 2013
Discussion open until: Feb 7, 2014
Published in print: Mar 1, 2014
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