Skip to main content
Book cover

Advances in Hydroinformatics pp 227–239Cite as

Numerical Scheme for a Viscous Shallow Water System Including New Friction Laws of Second Order: Validation and Application

  • 1144 Accesses

Part of the Springer Water book series (SPWA)

Abstract

In this work, we are interested in the derivation of a new shallow water model with a diffusion source term. Analytical solutions for steady flow regimes are first presented to validate a numerical method designed to solve this new model. Then this model is applied on real data and seems to give better results than the classical shallow water system.

Keywords

  • Shallow water system
  • Model derivation
  • Finite volume scheme
  • Well-balanced method
  • Hydrostatic reconstruction
  • Friction law
  • Manning friction law
  • Darcy-Weisbach friction law

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-981-287-615-7_16
  • Chapter length: 13 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   169.00
Price excludes VAT (USA)
  • ISBN: 978-981-287-615-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   219.99
Price excludes VAT (USA)
Hardcover Book
USD   249.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  1. de Saint-Venant, A. J.-C. (1871). Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lit. Comptes Rendus de l’Académie des Sciences, 73, 147–154.

    MATH  Google Scholar 

  2. Delestre, O. (2010). Simulation du ruissellement d’eau de pluie sur des surfaces agricoles. PhD thesis University of Orléans, in french. http://tel.archives-ouvertes.fr/INSMI/tel-00531377/fr.

  3. Delestre, O., Cordier, S., Darboux, F., Du, M., James, F., & Laguerre, C., et al. (2014). FullSWOF: A software for overland flow simulation. In P.Gourbesville, J. Cunge, & G. Caignaert, (Eds.), Advances in Hydroinformatics, Springer Hydrogeology (pp. 221–231). Springer: Singapore.

    Google Scholar 

  4. Esteves, M., Faucher, X., Galle, S., & Vauclin, M. (2000). Overland flow and infiltration modelling for small plots during unsteady rain: numerical results versus observed values. Journal of Hydrology, 228, 265–282.

    CrossRef  Google Scholar 

  5. Tatard, L., Planchon, O., Wainwright, J., Nord, G., Favis-Mortlock, D., Silvera, N., et al. (2008). Measurement and modelling of high-resolution flow-velocity data under simulated rainfall on a low-slope sandy soil. Journal of Hydrology, 348(1–2), 1–12.

    CrossRef  Google Scholar 

  6. Goutal, N., & Maurel, F. (2002). A finite volume solver for 1D shallow-water equations applied to an actual river. International Journal for Numerical Methods in Fluids, 38, 1–19.

    CrossRef  MATH  Google Scholar 

  7. Caleffi, V., Valiani, A., & Zanni, A. (2003). Finite volume method for simulating extreme flood events in natural flood events in natural channels. Journal of Hydraulic Research, 41(2), 167–177.

    CrossRef  Google Scholar 

  8. Alcrudo, F., & Gil, E. (1999). The Malpasset dam break case study. Proceedings of the 4th CADAM Workshop, Zaragoza (pp. 95–109).

    Google Scholar 

  9. Valiani, A., Caleffi, V., & Zanni, A. (2002). Case study: Malpasset dam-break simulation using a two-dimensional finite volume methods. Journal of Hydraulic Engineering, 128(5), 460–472.

    CrossRef  Google Scholar 

  10. Popinet, S. (2011). Quadtree-adaptive tsunami modelling. Ocean Dynamics, 61(9), 1261–1285.

    CrossRef  Google Scholar 

  11. Gerbeau, J.-F., & Perthame, B. (2001). Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete and Continuous Dynamical Systems---Series S, 1, 89–102.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Marche, F. (2007). Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. European Journal of Mechanics B/Fluids, 26, 49–63.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. MacDonald, I., Baines, M. J., Nichols, N. K., & Samuels, P. G. (1997). Journal of Hydraulic Engineering, 123, 1041–1045.

    CrossRef  Google Scholar 

  14. Chow, V. T. (1959). Open-channel hydraulics. New York: McGraw-Hill.

    Google Scholar 

  15. Delestre, O., Darboux, F., James, F., Lucas, C., Laguerre, C., & Cordier, S. (Submitted). FullSWOF: A free software package for the simulation of shallow water flows. arxiv.org/abs/1401.4125.

    Google Scholar 

  16. Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., & Perthame, B. (2004). A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. Journal of Scientific Computing, 25(6), 2050–2065.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Bouchut, F. (2004). Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources. Frontiers in Mathematics. Basel: Birkhauser.

    Google Scholar 

  18. Bristeau, M.-O., & Coussin, B. (2001). Boundary conditions for the shallow water equations solved by kinetic schemes. Inria report RR-4282.

    Google Scholar 

  19. Fiedler, R. F., & Ramirez, J. A. (2000). A numerical method for simulating discontinuous shallow flow over an infiltrating surface. International Journal for Numerical Methods in Fluids, 32, 219–240.

    CrossRef  MATH  Google Scholar 

  20. Anderson, E., Bai, Z., Bischof, C., Blackford, L.S., Demmel, J., & Dongarra, J., et al. (1999). LAPACK Users’ guide (3rd ed.). Philadelphia: Society for Industrial and Applied Mathematics.

    Google Scholar 

  21. Delestre, O., Lucas, C., Ksinant, P.-A., Darboux, F., Laguerre, C., Vo, T. N. T., et al. (2013). SWASHES: A compilation of Shallow-Water analytic solutions for hydraulic and environmental studies. International Journal for Numerical Methods in Fluids, 72, 269–300. doi:10.1002/fld.3741.

    CrossRef  MathSciNet  Google Scholar 

  22. Delestre, O., Lucas, C., Ksinant, P.-A., Darboux, F., Laguerre, C., & James, F., et al. (2014). SWASHES: A library for benchmarking in hydraulic. In p. Gourbesville, J. Cunge & G. Caignaert, (Eds.), Advances in Hydroinformatics, Springer Hydrogeology (pp. 233–243). Springer: Singapore.

    Google Scholar 

  23. Legout, C., Darboux, F., Nédélec, Y., Hauet, A., Esteves, M., Renaux, B., et al. (2012). High spatial resolution mapping of surface velocities and depths for shallow overland flow. Earth Surface Processes and Landforms, 37(9), 984–993. doi:10.1002/esp.3220.

    CrossRef  Google Scholar 

Download references

Acknowledgements

The authors whish to thanks the ANR-11-JS01-006-01 project CoToCoLa (Contemporary Topics on Conservation Laws), Carine Lucas for her advices and Frédéric Darboux for the data used in Sect. 4.2.

Author information

Authors and Affiliations

  1. Laboratory J.A. Dieudonné CNRS UMR 7351 & Polytech Nice – Sophia, UNSA, Arequipa, Peru

    Olivier Delestre

  2. Laboratoire de Mathématiques, CNRS UMR 6623, Université de Franche-Comté, 16 Route de Gray, 25030, Besançon Cedex, France

    Ulrich Razafison

Authors
  1. Olivier Delestre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ulrich Razafison
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ulrich Razafison .

Editor information

Editors and Affiliations

  1. University of Nice-Sophia Antipolis, Sophia Antipolis, France

    Philippe Gourbesville

  2. Society Hydrotechnique of France, La Tronche, France

    Jean A. Cunge

  3. ARTS et METIERS PARISTECH, LILLE cedex, Paris, France

    Guy Caignaert

Rights and permissions

Reprints and Permissions

Copyright information

© 2016 Springer Science+Business Media Singapore

About this chapter

Cite this chapter

Delestre, O., Razafison, U. (2016). Numerical Scheme for a Viscous Shallow Water System Including New Friction Laws of Second Order: Validation and Application. In: Gourbesville, P., Cunge, J., Caignaert, G. (eds) Advances in Hydroinformatics. Springer Water. Springer, Singapore. https://doi.org/10.1007/978-981-287-615-7_16

Download citation