The motion of an incompressible fluid confined to a shallow basin with a slightly varying bottom topography is considered. Coriolis force, surface wind and pressure stresses, together with bottom and lateral friction stresses are taken into account. We introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model; after averaging on the vertical direction and considering some asymptotic assumptions, we obtain a two-dimensional model, which approximates the three-dimensional model at the second order with respect to the ratio between the vertical scale and the longitudinal scale. The derived model is shown to be symmetrizable through a suitable change of variables. Finally, we propose some numerical tests with the aim to validate the proposed model.
Mots-clés : Navier-Stokes equations, Saint Venant equations, free surface flows
@article{M2AN_2004__38_2_211_0,
author = {Ferrari, Stefania and Saleri, Fausto},
title = {A new two-dimensional shallow water model including pressure effects and slow varying bottom topography},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {211--234},
publisher = {EDP-Sciences},
volume = {38},
number = {2},
year = {2004},
doi = {10.1051/m2an:2004010},
mrnumber = {2069144},
zbl = {1130.76329},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2004010/}
}
TY - JOUR AU - Ferrari, Stefania AU - Saleri, Fausto TI - A new two-dimensional shallow water model including pressure effects and slow varying bottom topography JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2004 SP - 211 EP - 234 VL - 38 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2004010/ DO - 10.1051/m2an:2004010 LA - en ID - M2AN_2004__38_2_211_0 ER -
%0 Journal Article %A Ferrari, Stefania %A Saleri, Fausto %T A new two-dimensional shallow water model including pressure effects and slow varying bottom topography %J ESAIM: Modélisation mathématique et analyse numérique %D 2004 %P 211-234 %V 38 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2004010/ %R 10.1051/m2an:2004010 %G en %F M2AN_2004__38_2_211_0
Ferrari, Stefania; Saleri, Fausto. A new two-dimensional shallow water model including pressure effects and slow varying bottom topography. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 2, pp. 211-234. doi : 10.1051/m2an:2004010. http://www.numdam.org/articles/10.1051/m2an:2004010/
[1] ,,, and, Mathematical and numerical modelling of shallow water flow. Comput. Mech. 11 (1993) 280-299. | Zbl
[2] , and, Recent developments in the numerical simulation of shallow water equations. Boundary conditions. Appl. Numer. Math. 15 (1994) 175-200. | Zbl
[3] ,,, and, New method for tidal current computation. J. Waterway, Port, Coastal and Ocean Division, ASCE 108 (1982) 396-417.
[4] , and, Numerical methods in environmental fluid mechanics. M.B. Abbot and J.A. Cunge Eds., Eng. Appl. Comput. Hydraulics II (1982) 1-10.
[5] , A new two-dimensional Shallow Water model: physical, mathematical and numerical aspects Ph.D. Thesis, a.a. 2002/2003, Dottorato M.A.C.R.O., Università degli Studi di Milano.
[6] , Convergence analysis of a space-time approximation to a two-dimensional system of Shallow Water equations. Internat. J. Appl. Analysis (to appear). | MR | Zbl
[7] and, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89-102. | Zbl
[8] , and, Modulations in the leading edges of midlatitude storm tracks. SIAM J. Appl. Math. 62 (2002) 746-776. | Zbl
[9] , Boundary layers for parabolic regularizations of totally characteristic quasilinear parabolic equations. J. Math. Pures Appl. 76 (1997) 965-990. | Zbl
[10] and, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Differential Equations 143 (1998) 110-146. | Zbl
[11] , Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites. Grenoble Ann. Inst. Fourier 45 (1995) 973-1006. | Numdam | Zbl
[12] , An introduction to continuum mechanics. Academic Press, New York (1981). | MR | Zbl
[13] and, FreeFem++:Manual version 1.23, 13-05-2002. FreeFem++ is a free software available at: http://www-rocq.inria.fr/Frederic.Hecht/freefem++.htm
[14] and, Code TELEMAC (système ULYSSE) : Résolution et mise en œuvre des équations de Saint-Venant bidimensionnelles, Théorie et mise en œuvre informatique, Rapport EDF HE43/87.37 (1987).
[15] , A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631-645. | Numdam | Zbl
[16] and, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397-425. | Numdam | Zbl
[17] , and, Linear and quasilinear equations of parabolic type. Providence, Rhode Island. Amer. Math. Soc. (1968).
[18] and, A shallow water model with eddy viscosity for basins with varying bottom topography. Nonlinearity 14 (2001) 1493-1515. | Zbl
[19] , and, Finite element approximation of a quasi-3D shallow water equation. Comput. Methods Appl. Mech. Engrg. 174 (1999) 355-369. | Zbl
[20] and, Differentiability of solutions to hyperbolic initial-boundary value problems. Trans. Amer. Math. Soc. 189 (1974) 303-318. | Zbl
[21] and, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space. I. Existence for Euler and Prandtl Equations; II. Construction of the Navier-Stokes solution. Comm. Math. Physics 192 (1998) 433-461 and 463-491. | Zbl
[22] , Sytems of conservation laws. I and II, Cambridge University Press, Cambridge (1996). | Zbl
[23] , Linear and nonlinear waves. John Wiley & Sons, New York (1974). | MR | Zbl
Cité par Sources :
