A unified formulation for the three-dimensional shallow water equations using orthogonal co-ordinates: theory and application |
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| Authors: | Herman W J Kernkamp Henri A H Petit Herman Gerritsen Erik D de Goede |
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| Institution: | (1) WL/Delft Hydraulics, P.O. Box 177, 2600 MH Delft, The Netherlands |
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| Abstract: | In this paper, the formulations of the primitive equations for shallow water flow in various horizontal co-ordinate systems
and the associated finite difference grid options used in shallow water flow modelling are reviewed. It is observed that horizontal
co-ordinate transformations do not affect the chosen co-ordinate system and representation in the vertical, and are the same
for the three- and two-dimensional cases. A systematic derivation of the equations in tensor notation is presented, resulting
in a unified formulation for the shallow water equations that covers all orthogonal horizontal grid types of practical interest.
This includes spherical curvilinear orthogonal co-ordinate systems on the globe. Computational efficiency can be achieved
in a single computer code. Furthermore, a single numerical algorithmic code implementation satisfies. All co-ordinate system
specific metrics are determined as part of a computer-aided model grid design, which supports all four orthogonal grid types.
Existing intuitive grid design and visual interpretation is conserved by appropriate conformal mappings, which conserve spherical
orthogonality in planar representation. A spherical curvilinear co-ordinate solution of wind driven steady channel flow applying
a strongly distorted grid is shown to give good agreement with a regular spherical co-ordinate model approach and the solution
based on a β-plane approximation. Especially designed spherical curvilinear boundary fitted model grids are shown for typhoon
surge propagation in the South China Sea and for ocean-driven flows through Malacca Straits. By using spherical curvilinear
grids the number of grid points in these single model grid applications is reduced by a factor of 50–100 in comparison with
regular spherical grids that have the same horizontal resolution in the area of interest. The spherical curvilinear approach
combines the advantages of the various grid approaches, while the overall computational effort remains acceptable for very
large model domains. |
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| Keywords: | Shallow water equations Co-ordinate systems Spherical curvilinear co-ordinates Numerical grid design |
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