A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation

Emmanuel Audusse; Marie-Odile Bristeau; Benoît Perthame; Jacques Sainte-Marie

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2011)

  • Volume: 45, Issue: 1, page 169-200
  • ISSN: 0764-583X

Abstract

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The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighboring layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy and hyperbolicity properties of the model. We also give a kinetic interpretation leading to effective numerical schemes with positivity and energy properties. Numerical tests show the versatility of the approach and its ability to compute recirculation cases with wind forcing.

How to cite

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Audusse, Emmanuel, et al. "A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 45.1 (2011): 169-200. <http://eudml.org/doc/273147>.

@article{Audusse2011,
abstract = {The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighboring layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy and hyperbolicity properties of the model. We also give a kinetic interpretation leading to effective numerical schemes with positivity and energy properties. Numerical tests show the versatility of the approach and its ability to compute recirculation cases with wind forcing.},
author = {Audusse, Emmanuel, Bristeau, Marie-Odile, Perthame, Benoît, Sainte-Marie, Jacques},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Navier-Stokes equations; Saint-Venant equations; free surface; multilayer system; kinetic scheme},
language = {eng},
number = {1},
pages = {169-200},
publisher = {EDP-Sciences},
title = {A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation},
url = {http://eudml.org/doc/273147},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Audusse, Emmanuel
AU - Bristeau, Marie-Odile
AU - Perthame, Benoît
AU - Sainte-Marie, Jacques
TI - A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2011
PB - EDP-Sciences
VL - 45
IS - 1
SP - 169
EP - 200
AB - The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighboring layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy and hyperbolicity properties of the model. We also give a kinetic interpretation leading to effective numerical schemes with positivity and energy properties. Numerical tests show the versatility of the approach and its ability to compute recirculation cases with wind forcing.
LA - eng
KW - Navier-Stokes equations; Saint-Venant equations; free surface; multilayer system; kinetic scheme
UR - http://eudml.org/doc/273147
ER -

References

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  1. [1] E. Audusse, A multilayer Saint-Venant system: Derivation and numerical validation. Discrete Contin. Dyn. Syst. Ser. B5 (2005) 189–214. Zbl1075.35030MR2129374
  2. [2] E. Audusse and M.O. Bristeau, Transport of pollutant in shallow water flows: A two time steps kinetic method. ESAIM: M2AN 37 (2003) 389–416. Zbl1137.65392MR1991208
  3. [3] E. Audusse and M.O. Bristeau, A well-balanced positivity preserving second-order scheme for shallow water flows on unstructured meshes. J. Comput. Phys.206 (2005) 311–333. Zbl1087.76072MR2135839
  4. [4] E. Audusse and M.O. Bristeau, Finite-volume solvers for a multilayer Saint-Venant system. Int. J. Appl. Math. Comput. Sci.17 (2007) 311–319. Zbl1152.35305MR2356890
  5. [5] E. Audusse, F. Bouchut, M.O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for Shallow Water flows. SIAM J. Sci. Comput.25 (2004) 2050–2065. Zbl1133.65308MR2086830
  6. [6] E. Audusse, M.O. Bristeau and A. Decoene, Numerical simulations of 3d free surface flows by a multilayer Saint-Venant model. Int. J. Numer. Methods Fluids56 (2008) 331–350. Zbl1139.76036MR2378555
  7. [7] A.J.C. Barré de Saint-Venant, Théorie du mouvement non permanent des eaux avec applications aux crues des rivières et à l'introduction des marées dans leur lit. C. R. Acad. Sci. Paris 73 (1871) 147–154. Zbl03.0482.04JFM03.0482.04
  8. [8] F. Bouchut, An introduction to finite volume methods for hyperbolic conservation laws. ESAIM: Proc. 15 (2004) 107–127. Zbl1083.76046MR2441319
  9. [9] F. Bouchut and T. Morales de Luna, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment. ESAIM: M2AN 42 (2008) 683–698. Zbl1203.76110MR2437779
  10. [10] F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography. Commun. Math. Sci.2 (2004) 359–389. Zbl1084.76012MR2118849
  11. [11] M.O. Bristeau and J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems. Discrete Contin. Dyn. Syst. Ser. B10 (2008) 733–759. Zbl1155.35405MR2434908
  12. [12] M.J. Castro, J.A. García-Rodríguez, J.M. González-Vida, J. Macías, C. Parés and M.E. Vázquez-Cendón, Numerical simulation of two-layer shallow water flows through channels with irregular geometry. J. Comput. Phys.195 (2004) 202–235. Zbl1087.76077
  13. [13] M.J. Castro, J. Macías and C. Parés, A q-scheme for a class of systems of coupled conservation laws with source term. application to a two-layer 1-D shallow water system. ESAIM: M2AN 35 (2001) 107–127. Zbl1094.76046MR1811983
  14. [14] A. Decoene and J.-F. Gerbeau, Sigma transformation and ALE formulation for three-dimensional free surface flows. Int. J. Numer. Methods Fluids59 (2009) 357–386. Zbl1154.76031MR2488293
  15. [15] A. Decoene, L. Bonaventura, E. Miglio and F. Saleri, Asymptotic derivation of the section-averaged shallow water equations for river hydraulics. M3AS 19 (2009) 387–417. Zbl1207.35092MR2502468
  16. [16] S. Ferrari and F. Saleri, A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography. ESAIM: M2AN 38 (2004) 211–234. Zbl1130.76329MR2069144
  17. [17] FreeFem++ home page, http://www.freefem.org/ff++/index.htm (2009). 
  18. [18] J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation. Discrete Contin. Dyn. Syst. Ser. B1 (2001) 89–102. Zbl0997.76023MR1821555
  19. [19] P.L. Lions, Mathematical Topics in Fluid Mechanics, Incompressible models, Vol. 1. Oxford University Press, UK (1996). Zbl1264.76002MR1422251
  20. [20] F. Marche, Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. Eur. J. Mech. B, Fluids 26 (2007) 49–63. Zbl1105.76021MR2281291
  21. [21] B. Mohammadi, O. Pironneau and F. Valentin, Rough boundaries and wall laws. Int. J. Numer. Methods Fluids27 (1998) 169–177. Zbl0904.76031MR1602088
  22. [22] O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port Coast. Ocean Eng. ASCE119 (1993) 618–638. 
  23. [23] D.H. Peregrine, Long waves on a beach. J. Fluid Mech.27 (1967) 815–827. Zbl0163.21105
  24. [24] B. Perthame, Kinetic formulation of conservation laws. Oxford University Press, UK (2002). Zbl1030.35002MR2064166
  25. [25] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo38 (2001) 201–231. Zbl1008.65066MR1890353
  26. [26] M.J. Salençon and J.M. Thébault, Simulation model of a mesotrophic reservoir (Lac de Pareloup, France): Melodia, an ecosystem reservoir management model. Ecol. model. 84 (1996) 163–187. 
  27. [27] N.J. Shankar, H.F. Cheong and S. Sankaranarayanan, Multilevel finite-difference model for three-dimensional hydrodynamic circulation. Ocean Eng.24 (1997) 785–816. 
  28. [28] F. Ursell, The long wave paradox in the theory of gravity waves. Proc. Cambridge Phil. Soc.49 (1953) 685–694. Zbl0052.43107MR57667
  29. [29] M.A. Walkley, A numerical Method for Extended Boussinesq Shallow-Water Wave Equations. Ph.D. Thesis, University of Leeds, UK (1999). 

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