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Shallow fluid flow over an obstacle: higher-order non-hydrostatic modeling and breaking waves
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Shallow fluid flow over an obstacle: higher-order non-hydrostatic modeling and breaking waves
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Journal Article
Shallow fluid flow over an obstacle: higher-order non-hydrostatic modeling and breaking waves
2022
Overview
The simulation of shallow flows over obstacles is an important problem in environmental fluid dynamics, including exchange flows over seabed sills, atmospheric flows past steep mountains and water flows over river bedforms. A common mathematical treatment consists in using vertically-averaged models instead of vertically-resolved ones by introducing a suitable shallow water approximation. The dispersionless Saint Venant equations are a useful tool, albeit accuracy is not enough in many circumstances. The next approach consists in resorting to the Serre–Green–Naghdi theory, which is well known to produce good solutions for long non-breaking waves. However, a common feature of flows over obstacles is the generation of breaking waves at its lee side, which are important to model, given their role in the mixing and transport of passive scalars downstream of the terrain barrier. The Serre–Green–Naghdi theory fails to model these flows, producing unrealistic trains of undular waves. A widely used practice consist in resorting to a patching approach in a numerical setting where the solutions of Serre–Green–Naghdi and Saint Venant equations are assembled once wave breaking is detected by case-dependent empirical parameters. In this work an alternative method to dealt with wave breaking over obstacles within the Boussinesq-type approximation is proposed. The exact depth-averaged equations for flows over uneven beds are developed and presented as function of the vertical acceleration and non-uniformity of velocity with elevation. By introducing a suitable kinematic field, a new high-order phase resolving system of non-hydrostatic equations is presented, containing the usual dispersive corrections of Serre–Green–Naghdi theory plus high-order corrections for velocity profile modeling. It is found that the new theory allows the simulation of both breaking and non-breaking waves in shallow flows over obstacles without introducing any case-dependent calibration parameter. The new shallow water approximation is thus an alternative method to deal with wave breaking in Boussinesq type models.
