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Numerical modelling of free-surface shallow flows over irregular topography with complex geometry. (English) Zbl 1427.76163
Summary: A well-balanced Godunov-type finite volume algorithm is developed for modelling free-surface shallow flows over irregular topography with complex geometry. The algorithm is based on a new formulation of the classical shallow water equations in hyperbolic conservation form. Unstructured triangular grids are used to achieve the adaptability of the grid to the geometry of the problem and to facilitate localised refinement. The numerical fluxes are calculated using HLLC approximate Riemann solver, and the MUSCL-Hancock predictor-corrector scheme is adopted to achieve the second-order accuracy both in space and in time where the solutions are continuous, and to achieve high-resolution results where the solutions are discontinuous. The novelties of the algorithm include preserving well-balanced property without any additional correction terms and the wet/dry front treatments. The good performance of the algorithm is demonstrated by comparing numerical and theoretical results of several benchmark problems, including the preservation of still water over a two-dimensional hump, the idealised dam-break flow over a frictionless flat rectangular channel, the circular dam-break, and the shock wave from oblique wall. Besides, two laboratory dam-break cases are used for model validation. Furthermore, a practical application related to dam-break flood wave propagation over highly irregular topography with complex geometry is presented. The results show that the algorithm can correctly account for free-surface shallow flows with respect to its effectiveness and robustness thus has bright application prospects.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[1] Zoppou, C.; Roberts, S., Numerical solution of the two-dimensional unsteady dam break, Appl. Math. Model., 24, 457-475, (2000) · Zbl 1004.76064
[2] Singh, J.; Altinakar, M. S.; Ding, Y., Two-dimensional numerical modeling of dam-break flows over natural terrain using a central explicit scheme, Adv. Water Resour., 34, 1366-1375, (2011)
[3] Song, L.; Zhou, J.; Li, Q.; Yang, X.; Zhang, Y., An unstructured finite volume model for dam-break floods with wet/dry fronts over complex topography, Int. J. Numer. Meth. Fluids, 67, 960-980, (2011) · Zbl 1316.76060
[4] Benkhaldoun, F.; Sari, S.; Seaid, M., A flux-limiter method for dam-break flows over erodible sediment beds, Appl. Math. Model., 36, 4847-4861, (2012) · Zbl 1252.76050
[5] Xia, J.; Lin, B.; Falconer, R. A.; Wang, G., Modelling dam-break flows over mobile beds using a 2D coupled approach, Adv. Water Resour., 33, 171-183, (2010)
[6] Yue, Z.; Cao, Z.; Che, T.; Li, X., Two-dimensional mathematical modeling of glacier lake outburst flood, J. Glaciol. Geocryol., 29, 756-763, (2007), [in Chinese]
[7] Wang, X.; Cao, Z.; Yue, Z., Numerical modeling of shallow flows over irregular topography, Chinese J. Hydrodyn., 24, 56-62, (2009), [in Chinese]
[8] Brown, J. D.; Spencer, T.; Moeller, I., Modeling storm surge flooding of an urban area with particular reference to modeling uncertainties: a case study of canvey island, united kingdom, Water Resour. Res., 43, W06402, (2007)
[9] Gallegos, H. A.; Schubert, J. E.; Sanders, B. F., Two-dimensional, high-resolution modeling of urban dam-break flooding: a case study of Baldwin hills, California, Adv. Water Resour., 32, 1323-1335, (2009)
[10] Pan, C. H.; Dai, S. Q.; Chen, S. M., Numerical simulation for 2D shallow water equations by using Godunov-type scheme with unstructured mesh, J. Hydrodyn., 18, 475-480, (2006) · Zbl 1203.76019
[11] Pan, C. H.; Lin, B. Y.; Mao, X. Z., Case study: numerical modeling of the tidal bore on the qiantang river, China, ASCE J. Hydraul. Eng., 133, 130-138, (2007)
[12] Qi, D. M.; Ma, G. F.; Gu, F. F.; Mou, L., An unstructured grid hydrodynamic and sediment transport model for changjiang estuary, J. Hydrodyn., 22, 1015-1021, (2010)
[13] Alcrudo, F.; Garcia-Navarro, P., A high-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equations, Int. J. Numer. Meth. Fluids, 16, 489-505, (1993) · Zbl 0766.76067
[14] Toro, E. F., Shock-capturing methods for free-surface shallow flows, (2001), John Wiley & Sons Ltd Chichester, England · Zbl 0996.76003
[15] Liu, H.; Zhou, J. G.; Burrows, R., Lattice Boltzmann simulations of the transient shallow water flows, Adv. Water Resour., 33, 387-396, (2010)
[16] Liang, Q.; Borthwick, A. G.L., Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography, Comput. Fluids, 38, 221-234, (2009) · Zbl 1237.76017
[17] Begnudelli, L.; Sanders, B. F., Unstructured grid finite-volume algorithm for shallow-water flow and scalar transport with wetting and drying, ASCE J. Hydraul. Eng., 132, 371-384, (2006)
[18] Song, L.; Zhou, J.; Guo, J.; Zou, Q.; Liu, Y., A robust well-balanced finite volume model for shallow water flows with wetting and drying over irregular terrain, Adv. Water Resour., 34, 915-932, (2011)
[19] Hubbard, M. E., Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids, J. Comput. Phys., 155, 54-74, (1999) · Zbl 0934.65109
[20] Liang, Q.; Marche, F., Numerical resolution of well-balanced shallow water equations with complex source terms, Adv. Water Resour., 32, 873-884, (2009)
[21] Erpicum, S.; Dewals, B. J.; Archambeau, P.; Pirotton, M., Dam break flow computation based on an efficient flux vector splitting, J. Comput. Appl. Math., 234, 2143-2151, (2010) · Zbl 1402.76079
[22] Erpicum, S.; Dewals, B.; Archambeau, P.; Detrembleur, S.; Pirotton, M., Detailed inundation modelling using high resolution dems, Eng. Appl. Comput. Fluid Mech., 4, 196-208, (2010)
[23] Bellos, C. V.; Sakkas, J. G., 1-D dam-break flood-wave propagation on dry bed, ASCE J. Hydraul. Eng., 113, 1510-1524, (1987)
[24] Garcia-Navarro, P.; Fras, A.; Villanueva, I., Dam-break flow simulation: some results for one-dimensional models of real cases, J. Hydrol., 216, 227-247, (1999)
[25] S. Soares Frazao, Dam-break induced flows in complex topographies-Theoretical, numerical and experimental approaches, PhD Thesis, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, 2002.
[26] Anastasiou, K.; Chan, C. T., Solution of the 2D sallow water equations using the finite volume method on unstructured triangular meshes, Int. J. Numer. Meth. Fluids, 24, 1225-1245, (1997) · Zbl 0886.76064
[27] Lai, W.; Khan, A. A., A discontinuous Galerkin method for two-dimensional shallow water flows, Int. J. Numer. Meth. Fluids, 70, 939-960, (2012) · Zbl 1412.76026
[28] Hager, W. H.; Schwalt, M.; Jimenez, O.; Hanif Chaudhry, M., Supercritical flow near an abrupt wall deflection, J. Hydraul. Res., 32, 103-118, (1994)
[29] Kesserwani, G.; Liang, Q., Well-balanced RKDG2 solutions to the shallow water equations over irregular domains with wetting and drying, Comput. Fluids, 39, 2040-2050, (2010) · Zbl 1245.76068
[30] Brufau, P.; Vazquez-Cendon, M. E.; Garcia-Navarro, P., A numerical model for the flooding and drying of irregular domains, Int. J. Numer. Meth. Fluids, 39, 247-275, (2002) · Zbl 1094.76538
[31] Gottardi, G.; Venutelli, M., Central scheme for two-dimensional dam-break flow simulation, Adv. Water Resour., 27, 259-268, (2004)
[32] Brufau, P.; Garcia-Navarro, P., Two-dimensional dam break flow simulation, Int. J. Numer. Meth. Fluids, 33, 35-57, (2000) · Zbl 0974.76535
[33] Soares Frazão, S.; Zech, Y., Dam-break in channels with 90 bend, ASCE J. Hydraul. Eng., 128, 956-968, (2002)
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