A generalised Serre-Green-Naghdi equations for variable rectangular open channel hydraulics and its finite volume approximation. (English) Zbl 1481.76028

Muñoz-Ruiz, María Luz (ed.) et al., Recent advances in numerical methods for hyperbolic PDE systems. NumHyp 2019. Selected papers based on the presentations at the 6th international conference on numerical methods for hyperbolic problems, Málaga, Spain, June 17–21, 2019. Cham: Springer. SEMA SIMAI Springer Ser. 28, 251-268 (2021).
Summary: We present a non-linear dispersive shallow water model which enters in the framework of section-averaged models. These new equations are derived up to the second order of the shallow water approximation starting from the three-dimensional incompressible and irrotational Euler system. The derivation is carried out in the case of non-uniform rectangular section and it generalises the well-known one-dimensional Serre-Green-Naghdi (SGN) equations on uneven bottom. The section-averaged model is asymptotically consistent with the Euler system in terms of mass, momentum, and energy equation which provides the richness of content for this model. We propose a well-balanced finite volume approximation and we present some numerical results to show the influence of the section variation.
For the entire collection see [Zbl 1470.65004].


76B07 Free-surface potential flows for incompressible inviscid fluids
76M12 Finite volume methods applied to problems in fluid mechanics
Full Text: DOI HAL


[1] Seabra-Santos, F.J., Renouard, D.P., Temperville, A.M.: Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle. J. Fluid Mech
[2] de Saint-Venant, A.J.C.B.: Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lit. C. R. Acad. Sci. 73, 147-154 (1871) · JFM 03.0482.04
[3] Peregrine, D.: Calculations of the development of an undular bore. J. Fluid Mech. 25, 321-330 (1966)
[4] Lannes, D., Marche, F.: Nonlinear wave-current interactions in shallow water. Stud. Appl. Math. 136, 382-423 (2016) · Zbl 1341.35119
[5] Lannes, D., Marche, F.: A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations. J. Comput. Phys. 282, 238-268 (2015) · Zbl 1351.76114
[6] Lannes, D., Bonneton, P.: Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids 21, 016601 (2009) · Zbl 1183.76294
[7] Lannes, D.: The Water Waves Problem: Mathematical Analysis and Asymptotics, vol. 188. American Mathematical Society, Providence (2013) · Zbl 1410.35003
[8] Green, A.E., Naghdi, P.M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237-246 (1976) · Zbl 0351.76014
[9] Gouta, N., Maurel, F.: A finite volume solver for 1D shallow-water equations applied to an actual river. Int. J. Numer. Methods. Fluids. 38, 1-19 (2002) · Zbl 1115.76355
[10] Gerbeau, J.F., Perthame, B.: Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Continuous Dyn. Syst. Ser. B 1, 89-102 (2001) · Zbl 0997.76023
[11] Fedotova, Z.I., Khakimzyanov, G.S., Dutykh, D.: Energy equation for certain approximate models of long-wave hydrodynamics. Russ. J. Numer. Anal. Math. Model. 29, 167-178 (2014) · Zbl 1291.76064
[12] Ersoy, M.: Dimension reduction for incompressible pipe and open channel flow including friction. In: Brandts, J., Korotov, S., Krizek, M., Segeth, K., Sistek, J., Vejchodsky, T. (eds.) Conference Applications of Mathematics 2015, in Honor of the 90th Birthday of Ivo Babuska and 85th Birthday of Milan Práger and Emil Vitásek , pp. 17-33, Prague, France (2015). https://hal.archives-ouvertes.fr/hal-00908961 · Zbl 1329.00187
[13] Decoene, A., Bonaventura, L., Miglio, E., Saleri, F.: Asymptotic derivation of the section-averaged shallow water equations for natural river hydraulics. Math. Models Methods Appl. Sci. 19, 387-417 (2009) · Zbl 1207.35092
[14] Debyaoui, M.A., Ersoy, M.: Generalised Serre-Green-Naghdi equations for open channel and for natural river hydraulics (2020). https://hal.archives-ouvertes.fr/hal-02444355. Working paper or preprint
[15] Cienfuegos, R., Barthélemy, E., Bonneton, P.: A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive boussinesq-type equations. Part II: boundary conditions and validation. Int. J. Numer. Methods. Fluids. 53, 1423-1455 (2007) · Zbl 1370.76090
[16] Chazel, F., Lannes, D., Marche, F.: Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model. J. Sci. Comput. 48, 105-116 (2011) · Zbl 1419.76454
[17] Bourdarias, C., Gerbi, S., Lteif, R.: A numerical scheme for an improved Green-Naghdi model in the Camassa-Holm regime for the propagation of internal waves. Comput. Fluids 156, 283-304 (2017) · Zbl 1390.76400
[18] Bourdarias, C., Ersoy, M., Gerbi, S.: A mathematical model for unsteady mixed flows in closed water pipes. Sci. China Math. 55, 221-244 (2012) · Zbl 1387.35484
[19] Audusse, E., Bouchut, F., Bristeau, M.O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050-2065 (2004) · Zbl 1133.65308
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.