Despite the increasing use of non-hydrostatic models in the study of wave processes in coastal regions, there is still limited understanding of the non-linear properties of the governing equations and how they improve with increased vertical resolution. In this study, the governing equations of the non-hydrostatic wave model SWASH are analysed and the linear and non-linear solutions up to third-order of all dependent variables are derived, considering one to four vertical layers. The analysis concludes that the model can achieve excellent non-linear properties with respect to the Stokes theory over a large range of water depths using only a few vertical layers. Furthermore, deriving solutions for all variables enables the formulation of improved wave generation and absorption boundary conditions for non-hydrostatic models. A well-known issue of non-linear wave models is related to the generation and propagation of spurious free waves, resulting to non-homogeneous wave fields. In this study, it is proven that by imposing the derived exact mathematical solutions of the governing equations at the model's boundaries, the target first- and second-order wave profiles can be generated with high accuracy, while the spurious waves can be entirely eliminated.

Vegetation meadows in coastal waters are a key constituent of a future green defense package due to the ecosystem services they provide and the potential to attenuate wave energy. To numerically describe the vegetation dynamics under wave action, this paper presents a novel application of a numerical coupling for solving fluid–elastic structure interactions (FSI) problems involving ultra-thin elements in a 3-D environment. The extended two-way coupling employed in this work combines the mesh-free Smoothed Particle Hydrodynamics (SPH) method in the DualSPHysics code to solve the fluid flow, and the Finite Element Analysis (FEA) structural solver in Project Chrono to solve the structural dynamics. To represent the vegetation, a flexible structure based on the Euler–Bernoulli beam model is used. The beam element is embedded into the SPH domain using an envelope subdomain that is discretized using dummy boundary particles. As such, this dummy envelope serves as a decoupling interface for the geometrical properties of the structure, allowing for ultra-thin structures smaller than the initial inter-particle distance (dp). The numerical approach is validated against an experimental setup including a flexible blade swaying under the action of an oscillatory flow. The results demonstrate that the numerical model is able to resolve the wave–vegetation interaction problem. Furthermore, additional insights into the blade dynamics reveal that the swaying velocity increases linearly along the length, with the upper part swaying at a speed comparable to the fluid velocity while the stem remains relatively stationary. Additionally, the findings indicate that rigid vegetation experiences higher forces per unit length, and in systems with substantial swaying motion, energy dissipation predominantly occurs around the lower base of the vegetation.

The weakly reflective wave generation is a wave generation and absorption method in phase-resolving models, based on the assumption that the waves propagating towards the wave generation boundary are small amplitude shallow water waves with direction perpendicular to the boundary. This assumption makes the method weakly reflective for dispersive and directional waves. The internal wave generation method was proposed by Vasarmidis et al. (2019b) as an alternative, for the non-hydrostatic wave model, SWASH, to avoid reflections. In this study, a comparison is made between the performance of the new internal wave generation method and the weakly reflective wave generation method. It is shown that using the internal wave generation leads to a significantly more accurate prediction of the resulting wave field in case of waves reflected back to the numerical boundary. Additionally, the internal wave generation method is extended to short-crested waves and SWASH is validated for the first time with experimental data for the cases of wave propagation over a shoal and wave diffraction around a wall. The proposed extended internal wave generation method increases the capability of SWASH towards the study of wave propagation of highly dispersive short-crested waves in coastal environments with minimal reflection from the boundaries.