zbMATH — the first resource for mathematics

Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. (English) Zbl 1105.76021
Summary: Considering the three-dimensional Navier-Stokes equations under a free moving surface boundary condition and hydrostatic approximation, we perform the derivation, with asymptotic analysis, of a new two-dimensional viscous shallow water model in rotating framework, with irregular topography, linear and quadratic bottom friction terms and capillary effects. A new formulation of viscous effects, consistent with a previous one-dimensional analysis, is obtained. Finally, we describe some simple numerical experiments in order to validate the proposed model.

76D33 Waves for incompressible viscous fluids
76U05 General theory of rotating fluids
86A05 Hydrology, hydrography, oceanography
Full Text: DOI
[1] Gerbeau, J.F.; Perthame, B., Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation, Discrete and continuous dynamical systems ser. B, 1, 89-102, (2001) · Zbl 0997.76023
[2] de Saint Venant, A.J.C., Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l’introduction des marèes dans leur lit, C. R. acad. sci. Paris, 73, 147-154, (1871) · JFM 03.0482.04
[3] Dingemans, M.A., Water wave propagation over uneven bottoms – parts 1, 2, (1997), World Scientific · Zbl 0908.76002
[4] Whitham, G.B., Linear and nonlinear waves, (1999), John Wiley and Sons New York · Zbl 0940.76002
[5] E. Audusse, A multilayer Saint-Venant model, DCDS-B (2004), in press · Zbl 1075.35030
[6] Bouchut, F.; Mangeney-Castelnau, A.; Perthame, B.; Vilotte, J.-P., A new model of Saint-Venant savage – hutter type for gravity driven shallow water flows, C. R. acad. sci. Paris, ser. I, 336, 531-536, (2003) · Zbl 1044.35056
[7] Bouchut, F.; Westdickenberg, M., Gravity driven shallow water models for arbitrary topography, Comm. math. sci., 2, 359-389, (2004) · Zbl 1084.76012
[8] Bresch, D.; Desjardins, B.; Lin, C.-K., On some compressible fluid models: Korteweg, lubrication and shallow water systems, Comm. partial differential equations, 28, 3,4, 843-868, (2003) · Zbl 1106.76436
[9] Bresch, D.; Desjardins, B., Existence of weak solutions for 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. math. phys., 238, 1,2, 211-223, (2003) · Zbl 1037.76012
[10] Pedloski, J., Geophysical fluid dynamics, (1987), Springer-Verlag
[11] Levermore, D.; Sammartino, M., A shallow water model with eddy viscosity for basins with varying bottom topography, Nonlinearity, 14, 1493-1515, (2001) · Zbl 0999.76033
[12] Ferrari, S.; Saleri, F., A new two dimensional shallow water model including pressure effects and slow varying bottom topography, M2an, 38, 211-234, (2004) · Zbl 1130.76329
[13] Viollet, P.L.; Chabard, J.P.; Esposito, P.; Laurance, D., Mécanique des fluides appliquées, (1998), Presse des Ponts et Chaussées Paris
[14] Slinn, D.N.; Allen, J.S.; Newberger, P.A.; Holman, R.A., Nonlinear shear instabilities of alongshore currentsover barred beaches, J. geophys. res., 103, 18, 357-379, (1998)
[15] Aronica, G.; Hankin, B.; Beven, K., Uncertainty and equifinality in calibrating distributed roughness coefficients in a flood propagation model with limited data, Adv. water resources, 22, 4, 349-365, (1998)
[16] Grenier, E., On the derivation of homogeneous hydrostatic equations, ESIAM: math. model. numer. anal., 33, 5, 965-970, (1999) · Zbl 0947.76013
[17] Zeytounian, R.K., Modélisation asymptotique en mécanique des fluides newtoniens, (1994), Springer-Verlag Berlin · Zbl 0797.76001
[18] F. Marche, Theoretical and numerical studies of shallow water models. Applications to nearshore hydrodynamics, Thèse de Doctorat, Université Bordeaux 1, 2005
[19] F. Marche, P. Bonneton, P. Fabrie, N. Seguin, Evaluation of well-balanced bore-capturing schemes for 2D wetting and drying processes, Int. J. Numer. Meth. Fluid, in press · Zbl 1229.76063
[20] Katsaounis, T.; Simeoni, C., Second order approximation of the viscous Saint-Venant system and comparison with experiments, () · Zbl 1134.76397
[21] Dressler, R.F., Comparison of theories and experiments for the hydraulic dam-break wave, Int. assoc. sci. hydraul., 38, 319-328, (1954)
[22] Bresch, D.; Desjardins, B., Some diffusive capillary models of Korteweg type, C. R. mecanique, 332, 11, 881-886, (2004) · Zbl 1386.76070
[23] Peregrine, D.H., Large-scale vorticity generation by breakers in shallow and deep water, Eur. J. mech. B fluids, 18, 403-408, (1999) · Zbl 0939.76016
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.