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Non-linear shallow water flow modelling over topography with depth-averaged potential equations

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Abstract

Hunter Rouse was the father of the Fluid Mechanics of Open Channel Flow. In this review article the fundamental advances on the non-hydrostatic modeling of shallow open channel flow over topography, since the publication of his 1938 book, are discussed. Flows over uneven topography, like at open channel controls, occurs in a short length, thus rendering ideal fluid flow theory an adequate mathematical tool. This paper emphasis is on depth-averaged modeling of these flows using Boussinesq equations, given the significant advances basically since the 80’s. The one-dimensional steady flow model from Picard iteration is reviewed in detail, including energy and momentum formulations, flow at a weir crest, and on a slope. A new numerical scheme is developed for the solution of transcritical steady weir flow, showing excellent match with experiments. A new derivation for the two-dimensional depth-averaged unsteady ideal fluid model is presented based on a continuum mechanics formulation. The result is the Serre–Green–Naghdi equations (SGNE). It is demonstrated that the extended Bernoulli equation derived from Picard iteration is an integral form of the SGNE for 1D steady flow. A robust MUSCL-Hancock finite volume model is developed to solve the unsteady 1D SGNE, showing excellent agreement with the steady transcritical flow solutions previously constructed. The accuracy of the MUSCL-Hancock solver is critically assessed using a higher-order solver for the SGNE.

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Acknowledgements

The first Author would like to thank Prof. Willi H. Hager, ETH-Zurich, Switzerland, and Prof. Sergio Montes, University of Hobart, Tasmania, for the long-time discussion on the topic and comments. This work was supported by the Spanish project CTM2017-85171-C2-1-R.

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  1. University of Cordoba, Spain, Campus Rabanales, Leonardo Da Vinci Building, 14071, Cordoba, Spain

    Oscar Castro-Orgaz

  2. University of Southampton, Southampton, SO17 1BJ, UK

    Francisco N. Cantero-Chinchilla

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Appendix: Matlab scripts as supplementary material

Appendix: Matlab scripts as supplementary material

Available as supplementary material to this article, there is a Matlab_codes.rar file. It contains 3 folders, each with a main program and linked functions. The folders are:

  • Folder MatthewSteady: go to script “MatthewSteady.m” to execute or edit functions linked to the main program.

  • Folder MUSCLHancock: go to script “ MUSCLHancock.m” to execute or edit functions linked to the main program.

  • Folder rk3MUSCL4: go to script “rk3MUSCL4.m” to execute or edit functions linked to the main program.

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Castro-Orgaz, O., Cantero-Chinchilla, F.N. Non-linear shallow water flow modelling over topography with depth-averaged potential equations. Environ Fluid Mech 20, 261–291 (2020). https://doi.org/10.1007/s10652-019-09691-z

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