Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-04-30T20:47:10.423Z Has data issue: false hasContentIssue false
  • English
  • Français

A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography

Published online by Cambridge University Press:  15 March 2004

Stefania Ferrari  and
Fausto Saleri
Stefania Ferrari
Affiliation:
MOX, Dipartimento di Matematica , Politecnico di Milano, via Bonardi 9, 20133 Milano, Italy, Stefania.Ferrari@polimi.it.
Fausto Saleri
Affiliation:
MOX, Dipartimento di Matematica , Politecnico di Milano, via Bonardi 9, 20133 Milano Italy, Fausto.Saleri@polimi.it.
Get access

Abstract

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.

Keywords

Navier-Stokes equationsSaint Venant equationsfree surface flows.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 2 , March 2004 , pp. 211 - 234
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

V.I. Agoshkov, D. Ambrosi, V. Pennati, A. Quarteroni and F. Saleri, Mathematical and numerical modelling of shallow water flow. Comput. Mech. 11 (1993) 280–299.
V.I. Agoshkov, A. Quarteroni and F. Saleri, Recent developments in the numerical simulation of shallow water equations. Boundary conditions. Appl. Numer. Math. 15 (1994) 175–200.
J.P. Benque, J.A. Cunge, J. Feuillet, A. Hauguel and F.M. Holly, New method for tidal current computation. J. Waterway, Port, Coastal and Ocean Division, ASCE 108 (1982) 396–417.
J.P. Benque, A. Haugel and P.L. Viollet, Numerical methods in environmental fluid mechanics. M.B. Abbot and J.A. Cunge Eds., Eng. Appl. Comput. Hydraulics II (1982) 1–10.
S. Ferrari, 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.
S. Ferrari, Convergence analysis of a space-time approximation to a two-dimensional system of Shallow Water equations. Internat. J. Appl. Analysis (to appear).
Gerbeau, J.F. and Perthame, B., Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89–102.
R.H. Goodman, A.J. Majda and D.W. Mclaughlin, Modulations in the leading edges of midlatitude storm tracks. SIAM J. Appl. Math. 62 (2002) 746–776.
E. Grenier, Boundary layers for parabolic regularizations of totally characteristic quasilinear parabolic equations. J. Math. Pures Appl. 76 (1997) 965–990.
Grenier, E. and Guès, O., Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Differential Equations 143 (1998) 110–146. CrossRef
Guès, O., Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites. Grenoble Ann. Inst. Fourier 45 (1995) 973–1006. CrossRef
M.E. Gurtin, An introduction to continuum mechanics. Academic Press, New York (1981).
F. Hecht and O. Pironneau, FreeFem++:Manual version 1.23, 13-05-2002. FreeFem++ is a free software available at: http://www-rocq.inria.fr/Frederic.Hecht/freefem++.htm
J.M. Hervouet and A. Watrin, 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).
Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631–645. CrossRef
Kurganov, A. and Doron, L., Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397–425. CrossRef
O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and quasilinear equations of parabolic type. Providence, Rhode Island. Amer. Math. Soc. (1968).
Levermore, D. and Sammartino, M., A shallow water model with eddy viscosity for basins with varying bottom topography. Nonlinearity 14 (2001) 1493–1515. CrossRef
Miglio, E., Quarteroni, A. and Saleri, F., Finite element approximation of a quasi–3D shallow water equation. Comput. Methods Appl. Mech. Engrg. 174 (1999) 355–369. CrossRef
Rauch, J. and Massey, F., Differentiability of solutions to hyperbolic initial-boundary value problems. Trans. Amer. Math. Soc. 189 (1974) 303–318.
Sammartino, M. and Caflisch, R.E., 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. CrossRef
D. Serre, Sytems of conservation laws. I and II, Cambridge University Press, Cambridge (1996).
G.B. Whitham, Linear and nonlinear waves. John Wiley & Sons, New York (1974).