A new two-dimensional finite element model for the shallow water equations using a Lagrangian framework constructed along fluid particle trajectories

In this paper we describe a new finite element model for the tidal hydrodynamics in estuaries. The mathematical model is based on the solution of the two-dimensional shallow water equations in a Lagrangian framework which is defined along the trajectories of fluid particles. This method gives a flexible and robust numerical scheme for moving boundary flows encountered in tidal water systems. In order to validate the developed model we have, at first instance, compared our numerical results with analytical solutions obtained for domains with simple geometries. Further tests are then conducted to demonstrate the model's ability to cope with conditions such as hydraulic shock, abrupt changes in the flow domain geometry and gradual changes of water surface breadth. The change in the water surface breadth corresponds to the drying and wetting of the plains along the banks of a typical tidal river/estuary reach. The drying and wetting of flood plains result in the existence of very shallow depth of water at some sections of the flow domain during a tidal cycle; The flow equations under these conditions are strongly convection dominated. Previously published tidal models rely on either, some form of upwinding or the use of extremely fine meshes to give stable results for the convection dominated very shallow depth computations in estuaries. We show that our model can yield stable and accurate results for very shallow depths in the tidal how domains without using any kind of artifical damping or excessive mesh refinement. Computational costs of simulating hydrodynamical conditions in a natural water course, even using a depth averaged two-dimensional approach, can be very high; The ability of our scheme to cope with convection dominated conditions has enabled us to economize the computational efforts by using coarse meshes in our finite element calculations. After the validation stage, the developed model is applied to simulate the tidal conditions in a real estuary. The comparison of the model results with the field observations shows a close agreement between these sets of data