Author(s): Lucas Calvo; Diana De Padova; Michele Mossa
Keywords: Depth-integrated; Galerkin finite element method; Non-hydrostatic; Wave propagation
Abstract: In recent decades, due to the occurrence of a high number of coastal catastrophes, partly increased by the rise in sea level, the importance of research on wave propagation mechanisms in coastal areas has increased. Depth-integrated models using the Boussinesq-type equations have been traditionally utilized to simulate wave propagation, but the high order partial derivative terms included in the Boussinesq equations difficult discretization, cause numerical instabilities, and has a non-negligible computational cost. Additionally, the Boussinesq-type equations arise from the supposition of no rotation and no viscosity. The shallow water models with non-hydrostatic pressure distribution have shown their capacity for accurate modelling of nonlinear and dispersive waves since their introduction. The vertical momentum is considered in these models by the introduction of a non-hydrostatic pressure term into the Reynolds-averaged Navier–Stokes equations. The non-hydrostatic models for water waves are classified as either single-layer models (two-dimensional depth-integrated non-hydrostatic) or multi-layer models (three-dimensional non-hydrostatic), depending on the number of layers in the vertical discretization. The capabilities of the non-hydrostatic models for water waves with single or multiple layers in the vertical direction have been demonstrated by many researchers. Due to the linearization of the vertical momentum equation, it is well known that the depth-integrated, non-hydrostatic models are only applicable to the intermediate water depth and for weakly nonlinear cases. Very recently, researchers have focused on increasing the order of the non-hydrostatic pressure interpolation from a linear to a quadratic vertical pressure profile, showing significant improvements for dispersion. With the use of a quadratic vertical non-hydrostatic pressure profile, instead of a linear profile, the model can achieve a good correspondence to existing fully non-linear weakly dispersive Boussinesq models. The new method is an interesting alternative for modelling shallow water waves, while avoiding the numerical instabilities caused by the higher-order terms in a Boussinesq-type model, as well as the increased computational costs arising from a larger number of vertical layers in a multi-layer non-hydrostatic model. The objective of the proposed work is to develop a new two-dimensional depth-integrated non-hydrostatic model for the simulation of wave propagation using a quadratic vertical non-hydrostatic pressure profile. This objective can be readily achieved by modifying an existing non-hydrostatic discontinuous/continuous Galerkin finite element model. The new model will have his dispersive characteristics highly improved by a quadratic vertical non-hydrostatic pressure profile, with the additional advantage of using unstructured meshes that easily allow local refinement and complex boundaries. Additionally, the utilization of linear quadrilateral finite elements presents the advantage of requiring less nodal variables for a given area than linear triangular finite elements in a discontinuous Galerkin context and with a higher order of interpolation than their triangular counterparts.