A semi-implicit approach for sediment transport models with gravitational effects. (English) Zbl 1510.35234
Summary: In this work efficient semi-implicit methods for sediment bedload transport models with gravitational effects under subcritical regimes is proposed. Several families of models with gravitational effects are presented and rewritten under a general formulation that allows us to apply the semi-implicit method. In the numerical tests we focus on the application of a generalization of the Ashida-Michiue model, which includes the gradient of both the bedload and the fluid surface. Analytical steady states solutions (both lake at rest and non vanishing velocity) are deduced an approximated with the proposed scheme. In all the presented tests, the computational efforts are notably reduced thanks to the proposed method without losing the accuracy in the results.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76T20 | Suspensions |
| 86A05 | Hydrology, hydrography, oceanography |
Keywords:
sediment bedload transport; shallow water exner model; gravitational effects; semi-implicit schemeReferences:
| [1] | Ashida, K.; Michiue, M., Study on hydraulic resistance and bed-load transport rate in alluvial streams, Proc. Jpn. Soc. Civil Eng., 1972, 206, 59-69 (1972) |
| [2] | Bagnold, R. A., The flow of cohesionless grains in fluids, Royal Society of London Philosophical transactions. Series A. Mathematical and Physical Sciences, no. 964. Royal Society of London (1956) · Zbl 0072.20201 |
| [3] | Benkhaldoun, F.; Seaïd, M.; Sahmim, S., Mathematical development and verification of a finite volume model for morphodynamic flow applications, Adv. Appl. Math. Mech., 3, 4, 470-492 (2011) · Zbl 1262.65110 |
| [4] | Bilanceri, M.; Beux, F.; Elmahi, I.; Guillard, H.; Salvetti, M. V., Linearised Implicit Time-Advancing Applied to Sediment Transport Simulations, Research Report RR-7492 (2010), INRIA |
| [5] | Bonaventura, L.; Fernández-Nieto, E. D.; Garres-Díaz, J.; Narbona-Reina, G., Multilayer shallow water models with locally variable number of layers and semi-implicit time discretization, J. Comput. Phys., 364, 209-234 (2018) · Zbl 1392.76012 |
| [6] | Díaz, M. J.C.; Fernández-Nieto, E. D.; Ferreiro, A. M., Sediment transport models in Shallow Water equations and numerical approach by high order finite volume methods, Comput. Fluids, 37, 3, 299-316 (2008) · Zbl 1237.76082 |
| [7] | Castro Díaz, M. J.; Fernández-Nieto, E. D.; Ferreiro, A. M.; Parés, C., Two-dimensional sediment transport models in shallow water equations. A second order finite volume approach on unstructured meshes, Comput. Methods Appl. Mech. Eng., 198, 33-36, 2520-2538 (2009) · Zbl 1228.76091 |
| [8] | Casulli, V., Numerical simulation of three-dimensional free surface flow in isopycnal coordinates, Int. J. Numer. MethodsFluids, 25, 645-658 (1997) · Zbl 0893.76050 |
| [9] | Casulli, V., A semi-implicit numerical method for the free-surface Navier-Stokes equations, Int. J. Numer. Methods Fluids, 74, 8, 605-622 (2013) · Zbl 1455.65127 |
| [10] | Casulli, V.; Cattani, E., Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow, Comput. Math. Appl., 27, 4, 99-112 (1994) · Zbl 0796.76052 |
| [11] | Casulli, V.; Cheng, R. T., Semi-implicit finite difference methods for three-dimensional shallow water flow, Int. J. Numer. Methods Fluids, 15, 6, 629-648 (1992) · Zbl 0762.76068 |
| [12] | Casulli, V.; Stelling, G. S., Numerical simulation of 3D quasi-hydrostatic, free-surface flows, J. Hydraul. Eng., 124, 7, 678-686 (1998) |
| [13] | Casulli, V.; Walters, R. A., An unstructured grid, three-dimensional model based on the shallow water equations, Int. J. Numer. MethodsFluids, 32, 331-348 (2000) · Zbl 0965.76061 |
| [14] | Casulli, V.; Zanolli, P., Semi-implicit numerical modelling of non-hydrostatic free-surface flows for environmental problems, Math. Comput. Model., 36, 1131-1149 (2002) · Zbl 1027.76034 |
| [15] | Charru, F., Selection of the ripple length on a granular bed sheared by a liquid flow, Phys. Fluids, 18, 121508 (2006) · Zbl 1146.76347 |
| [16] | Exner, F., Über die Wechselwirkung zwischen Wasser und Geschiebe in Flüssen, Sitzungsber. d. Akad. d. Wiss. pt. IIa. Bd., 134 (1925) · JFM 52.1000.19 |
| [17] | Fernandez-Luque, R.; van Beek, R., Erosion and transport of bedload sediment, J. Hydraul. Res., 14, 2, 127-144 (1976) |
| [18] | Fernández-Nieto, E. D.; de Luna, T. M.; Narbona-Reina, G.; Zabsonré, J. D., Formal deduction of the Saint-Venant-Exner model including arbitrarily sloping sediment beds and associated energy, ESAIM Math. Model. Numer.Anal., 51, 1, 115-145 (2017) · Zbl 1360.35176 |
| [19] | Fowler, A. C.; Kopteva, N.; Oakley, C., The formation of river channel, SIAM J. Appl. Math., 67, 1016-1040 (2007) · Zbl 1117.86002 |
| [20] | Fredsœ, J., On the development of dunes in erodible channels, J. Fluid Mech., 64, 1, 1-16 (1974) · Zbl 0357.76037 |
| [21] | Garegnani, G.; Rosatti, G.; Bonaventura, L., Free surface flows over mobile bed: mathematical analysis and numerical modeling of coupled and decoupled approaches, Commun. Appl. Ind.Math., 1, 3 (2011) · Zbl 1329.76360 |
| [22] | Garegnani, G.; Rosatti, G.; Bonaventura, L., On the range of validity of the Exner-based models for mobile-bed river flow simulations, J. Hydraulic Res., 51, 380-391 (2013) |
| [23] | Garres-Díaz, J.; Bonaventura, L., Flexible and efficient discretizations of multilayer models with variable density, Appl. Math. Comput., 402, 126097 (2021) · Zbl 1510.65192 |
| [24] | Giraldo, F. X.; Kelly, J. F.; Constantinescu, E. M., Implicit-explicit formulations of a three-dimensional nonhydrostatic unified model of the atmosphere (NUMA), SIAM J. Sci. Comput., 35 (2013) · Zbl 1280.86008 |
| [25] | A.J. Grass, Sediment transport by waves and currents, 1981. |
| [26] | Gunawan, P. H.; Eymard, R.; Pudjaprasetya, S. R., Staggered scheme for the Exner-Shallow Water equations, Comput. Geosci., 19, 6, 1197-1206 (2015) · Zbl 1391.76725 |
| [27] | Hascoët, L.; Pascual, V., TAPENADE 2.1 User’s Guide, Technical Report RT-0300 (2004), INRIA |
| [28] | R. Herbin, J.-C. Latché, Y. Nasseri, N. Therme, A consistent quasi-second order staggered scheme for the two-dimensional shallow water equations, 2021. · Zbl 1471.65116 |
| [29] | Hosea, M. E.; Shampine, L. F., Analysis and implementation of TR-BDF2, Appl. Numer. Math., 20, 21-37 (1996) · Zbl 0859.65076 |
| [30] | Kennedy, C. A.; Carpenter, M. H., Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44, 139-181 (2003) · Zbl 1013.65103 |
| [31] | Kovacs, A.; Parker, G., A new vectorial bedload formulation and its application to the time evolution of straight river channels, J. Fluid Mech., 267, 153-183 (1994) · Zbl 0800.76485 |
| [32] | Lysne, D. K., Movement of sand in tunnels, J. Hydraulics Div., 95, 6, 1835-1846 (1969) |
| [33] | Meyer-Peter, E.; Müller, R., Formulas for bed-load transport, Rep. 2nd Meet. Int. Assoc. Hydraul. Struct. Res., Stockolm, 39-64 (1948) |
| [34] | Morales de Luna, T.; Díaz, M. J.C.; Madroñal, C. P., A duality method for sediment transport based on a modified Meyer-Peter & Müller model, J. Sci. Comput., 48, 1-3, 258-273 (2010) · Zbl 1426.76408 |
| [35] | Nielsen, P., Coastal Bottom Boundary Layers and Sediment Transport (1992), World Scientific |
| [36] | Parker, G.; Seminara, G.; Solari, L., Bed load at low shields stress on arbitrarily sloping beds: alternative entrainment formulation, Water Resour. Res., 39, 7 (2003) |
| [37] | Rosatti, G.; Bonaventura, L.; Deponti, A.; Garegnani, G., An accurate and efficient semi-implicit method for section-averaged free-surface flow modelling, Int. J. Numer. Methods Fluids, 65, 448-473 (2011) · Zbl 1428.76138 |
| [38] | Seminara, G.; Solari, L.; Parker, G., Bed load at low shields stress on arbitrarily sloping beds: failure of the bagnold hypothesis, Water Resour. Res., 38, 11, Article 31-1-31-16 (2002) |
| [39] | Tassi, P. A.; Rhebergen, S.; Vionnet, C. A.; Bokhove, O., A discontinuous Galerkin finite element model for river bed evolution under shallow flows, Comput. Methods Appl. Mech. Eng., 197, 33-40, 2930-2947 (2008) · Zbl 1194.76143 |
| [40] | Rijn, L. C.V., Sediment transport, Part I: bed load transport, J. Hydraul. Eng., 110, 10, 1431-1456 (1984) |
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