The shallow water equations and their application to realistic cases

Abstract

The numerical modelling of 2D shallow flows in complex geometries involving transient flow and movable boundaries has been a challenge for researchers in recent years. There is a wide range of physical situations of environmental interest, such as flow in open channels and rivers, tsunami and flood modelling, that can be mathematically represented by first-order non-linear systems of partial differential equations, whose derivation involves an assumption of the shallow water type. Shallow water models may include more sophisticated terms when applied to cases of not pure water floods, such as mud/debris floods, produced by landslides. Mud/debris floods are unsteady flow phenomena in which the flow changes rapidly, and the properties of the moving fluid mixture include stop and go mechanisms. The present work reports on a numerical model able to solve the 2D shallow water equations even including bed load transport over erodible bed in realistic situations involving transient flow and movable flow boundaries. The novelty is that it offers accurate and stable results in realistic problems since an appropriate discretization of the governing equations is performed. Furthermore, the present work is focused on the importance of the computational cost. Usually, the main drawback is the high computational effort required for obtaining accurate numerical solutions due to the high number of cells involved in realistic cases. However, the proposed model is able to reduce computer times by orders of magnitude making 2D applications competitive and practical for operational flood prediction. Moreover our results show that high performance code development can take advantage of general purpose and inexpensive Graphical Processing Units, allowing to run almost 100 times faster than old generation codes in some cases.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

(Adapted from [17])

Fig. 11

References

  1. 1.

    Bates PD, De Roo APJ (2000) A simple raster-based model for flood inundation simulation. J Hydrol 236(1–2):54–77

    Article  Google Scholar 

  2. 2.

    Briggs MJ, Synolakis CE, Harkins GS, Green DR (1995) Laboratory experiments of tsunami runup on a circular island. Pure Appl Geophys 144(3–4):569–593

    Article  Google Scholar 

  3. 3.

    Brodtkorb AR, Saetra ML, Altikanar M (2012) Efficient shallow water simulations on GPUs: implementation, visualization, verification and validation. Comput Fluids 55:1–12

    Article  Google Scholar 

  4. 4.

    Bouchut F, Fernández-Nieto ED, Mangeney A, Lagrée PY (2008) On new erosion models of Savage–Hutter type for avalanches. Acta Mech 199(1–4):181–208

    Article  Google Scholar 

  5. 5.

    Caviedes-Voullième D, García-Navarro P, Murillo J (2012) Influence of mesh structure on 2D full shallow water equations and SCS curve number simulation of rainfall/runoff events. J Hydrol 448, 449:39–59

    Article  Google Scholar 

  6. 6.

    Caviedes-Voullième D, Morales-Hernández M, López-Marijuán IP, García-Navarro P (2014) Reconstruction of 2D river beds by appropriate interpolation of 1D cross-sectional information for flood simulation. Environ Model Softw 61:206–228

    Article  Google Scholar 

  7. 7.

    Juez C, Murillo J, García-Navarro P (2013) 2D simulation of granular flow over irregular steep slopes using global and local coordinates. J Comput Phys 255:166–204

    Article  Google Scholar 

  8. 8.

    Juez C, Murillo J, García-Navarro P (2014) A 2D weakly-coupled and efficient numerical model for transient shallow flow and movable bed. Adv Water Resour 71:93–109

    Article  Google Scholar 

  9. 9.

    Juez C, Lacasta A, Murillo J, García-Navarro P (2016) An efficient GPU implementation for a faster simulation of unsteady bed-load transport. J Hydraul Res 54(3):275–288

    Article  Google Scholar 

  10. 10.

    Lacasta A, Morales-Hernández M, Murillo J, García-Navarro P (2014) An optimized gpu implementation of a 2D free surface simulation model on unstructured meshes. Adv Eng Softw 78:1–15

    Article  Google Scholar 

  11. 11.

    Lacasta A, Juez C, Murillo J, García-Navarro P (2015) An efficient solution for hazardous geophysical flows simulation using GPUs. Comput Geosci 78:63–72

    Article  Google Scholar 

  12. 12.

    Meyer-Peter E, Müller R (1948) Formulas for Bed-Load Transport. In: Report on 2nd meeting on international association on hydraulic structures research, pp 39–64

  13. 13.

    Murillo J, García-Navarro P, Burguete J (2009) Time step restrictions for wellbalanced shallow water solutions in non-zero velocity steady states. Int J Numer Methods Fluids 60:1351–1377

    Article  Google Scholar 

  14. 14.

    Murillo J, García-Navarro P (2010) Weak solutions for partial differential equations with source terms: application to the shallow water equations. J Comput Phys 229:4327–4368

    Article  Google Scholar 

  15. 15.

    Murillo J, García-Navarro P (2012) Wave Riemann description of friction terms in unsteady shallow flows: application to water and mud/debris floods. J Comput Phys 231:1963–2001

    Article  Google Scholar 

  16. 16.

    NVIDIA (2011) NVIDIA CUDA C programming guide

  17. 17.

    Qian H, Cao Z, Liu H, Pender G (2018) New experimental dataset for partial dam-break floods over mobile beds. J Hydraul Res 56(1):124–135

    Article  Google Scholar 

  18. 18.

    Sanders BF, Schubert JE, Detwiler RL (2010) Parbrezo: a parallel, unstructured grid, Godunov type, shallow water code for high resolution flood inundation modeling at the regional scale. Adv Water Resour 33(12):1456–1467

    Article  Google Scholar 

  19. 19.

    Schubert JE, Sanders BF, Smith MJ, Wright NG (2008) Unstructured mesh generation and landcover-based resistance for hydrodynamic modeling of urban flooding. Adv Water Resour 31:1603–1621

    Article  Google Scholar 

  20. 20.

    Toro EF (2001) Shock-capturing methods for free-surface shallow flows. Wiley, New York

    Google Scholar 

  21. 21.

    UK Environment Agency (2010) Benchmarking of 2D hydraulic modelling packages

  22. 22.

    Vacondio R, Aureli F, Mignosa P, Dal Pal A (2016) Simulation of the January 2014 flood on the Secchia River using a fast and high-resolution 2D parallel shallow-water numerical scheme. Nat Hazards 80(1):103–125

    Article  Google Scholar 

  23. 23.

    Vázquez-Cendón ME (1999) Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J Comput Phys 148:497–498

    Article  Google Scholar 

Download references

Acknowledgements

This work was partially funded by the MINECO/FEDER under research project CGL2015-66114-R and by Diputacion General de Aragon, DGA, through Fondo Europeo de Desarrollo Regional, FEDER. The third author also wants to thank to the MINECO for his Research Grant DI-14-06987.

Author information

Affiliations

Authors

Corresponding author

Correspondence to P. García-Navarro.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

García-Navarro, P., Murillo, J., Fernández-Pato, J. et al. The shallow water equations and their application to realistic cases. Environ Fluid Mech 19, 1235–1252 (2019). https://doi.org/10.1007/s10652-018-09657-7

Download citation

Keywords

  • Unsteady shallow flows
  • Wetting/drying fronts
  • Finite volumes
  • River flows