A new two-dimensional shallow water model including pressure effects and slow varying bottom topography

Stefania Ferrari; Fausto Saleri

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 2, page 211-234
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Ferrari, Stefania, and Saleri, Fausto. "A new two-dimensional shallow water model including pressure effects and slow varying bottom topography." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.2 (2004): 211-234. <http://eudml.org/doc/245994>.

@article{Ferrari2004,
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.},
author = {Ferrari, Stefania, Saleri, Fausto},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Navier-Stokes equations; Saint Venant equations; free surface flows},
language = {eng},
number = {2},
pages = {211-234},
publisher = {EDP-Sciences},
title = {A new two-dimensional shallow water model including pressure effects and slow varying bottom topography},
url = {http://eudml.org/doc/245994},
volume = {38},
year = {2004},
}

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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 2
SP - 211
EP - 234
AB - 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.
LA - eng
KW - Navier-Stokes equations; Saint Venant equations; free surface flows
UR - http://eudml.org/doc/245994
ER -

References

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  1. [1] 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. Zbl0771.76032
  2. [2] 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. Zbl0833.76008
  3. [3] 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. 
  4. [4] 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. 
  5. [5] 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. 
  6. [6] S. Ferrari, Convergence analysis of a space-time approximation to a two-dimensional system of Shallow Water equations. Internat. J. Appl. Analysis (to appear). Zbl1122.76053MR2072307
  7. [7] J.F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89–102. Zbl0997.76023
  8. [8] 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. Zbl0989.86007
  9. [9] E. Grenier, Boundary layers for parabolic regularizations of totally characteristic quasilinear parabolic equations. J. Math. Pures Appl. 76 (1997) 965–990. Zbl0914.35032
  10. [10] E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Differential Equations 143 (1998) 110–146. Zbl0896.35078
  11. [11] O. Guès, Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites. Grenoble Ann. Inst. Fourier 45 (1995) 973–1006. Zbl0831.34023
  12. [12] M.E. Gurtin, An introduction to continuum mechanics. Academic Press, New York (1981). Zbl0559.73001MR636255
  13. [13] 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 
  14. [14] 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). 
  15. [15] S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631–645. Zbl1001.35083
  16. [16] A. Kurganov and L. Doron, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397–425. Zbl1137.65398
  17. [17] 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). 
  18. [18] D. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography. Nonlinearity 14 (2001) 1493–1515. Zbl0999.76033
  19. [19] E. Miglio, A. Quarteroni and F. Saleri, Finite element approximation of a quasi–3D shallow water equation. Comput. Methods Appl. Mech. Engrg. 174 (1999) 355–369. Zbl0958.76046
  20. [20] J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial-boundary value problems. Trans. Amer. Math. Soc. 189 (1974) 303–318. Zbl0282.35014
  21. [21] M. Sammartino and R.E. Caflisch, 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. Zbl0913.35102
  22. [22] D. Serre, Sytems of conservation laws. I and II, Cambridge University Press, Cambridge (1996). Zbl0930.35001
  23. [23] G.B. Whitham, Linear and nonlinear waves. John Wiley & Sons, New York (1974). Zbl0373.76001MR483954

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