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 (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 45.1 (2011): 169-200. <http://eudml.org/doc/197421>.

@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},
keywords = {Navier-Stokes equations; Saint-Venant equations; free surface; multilayer system; kinetic scheme},
language = {eng},
month = {1},
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/197421},
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
DA - 2011/1//
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/197421
ER -

References

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  1. E. Audusse, A multilayer Saint-Venant system: Derivation and numerical validation. Discrete Contin. Dyn. Syst. Ser. B5 (2005) 189–214.  
  2. E. Audusse and M.O. Bristeau, Transport of pollutant in shallow water flows: A two time steps kinetic method. ESAIM: M2AN37 (2003) 389–416.  
  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.  
  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.  
  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.  
  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.  
  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. Paris73 (1871) 147–154.  
  8. F. Bouchut, An introduction to finite volume methods for hyperbolic conservation laws. ESAIM: Proc.15 (2004) 107–127.  
  9. F. Bouchut and T. Morales de Luna, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment. ESAIM: M2AN42 (2008) 683–698.  
  10. F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography. Commun. Math. Sci.2 (2004) 359–389.  
  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.  
  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.  
  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: M2AN35 (2001) 107–127.  
  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.  
  15. A. Decoene, L. Bonaventura, E. Miglio and F. Saleri, Asymptotic derivation of the section-averaged shallow water equations for river hydraulics. M3AS19 (2009) 387–417.  
  16. S. Ferrari and F. Saleri, A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography. ESAIM: M2AN38 (2004) 211–234.  
  17. FreeFem++ home page, (2009).  URIhttp://www.freefem.org/ff++/index.htm
  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.  
  19. P.L. Lions, Mathematical Topics in Fluid Mechanics, Incompressible models, Vol. 1. Oxford University Press, UK (1996).  
  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, Fluids26 (2007) 49–63.  
  21. B. Mohammadi, O. Pironneau and F. Valentin, Rough boundaries and wall laws. Int. J. Numer. Methods Fluids27 (1998) 169–177.  
  22. O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port Coast. Ocean Eng. ASCE119 (1993) 618–638.  
  23. D.H. Peregrine, Long waves on a beach. J. Fluid Mech.27 (1967) 815–827.  
  24. B. Perthame, Kinetic formulation of conservation laws. Oxford University Press, UK (2002).  
  25. B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo38 (2001) 201–231.  
  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. 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. F. Ursell, The long wave paradox in the theory of gravity waves. Proc. Cambridge Phil. Soc.49 (1953) 685–694.  
  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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