A finite element model for depth integrated form of Boussinesq equations is presented. The equations are solved on an unstructured triangular mesh using standard Galerkin method with mixed interpolation scheme. The elemental integrals are calculated analytically and time-stepping is done using Runge–Kutta 4th order method. It is extended to simulate ship-generated waves using moving pressure fields. The unstructured formulation provides the flexibility of mesh refinement as needed, for capturing wave transformation or moving pressure field. The model is verified against experimental and numerical results for wave transformation over the Whalin shoal. The results for moving pressure field are compared against numerical results from FUNWAVE. Further, a simulation of ship navigating a curved path is presented. Finally, a real-life application and validation against field measurements is provided for waves generated by a fast ferry moving along a GPS tracked path in Tallinn Bay, Estonia.

The paper presents coupling between a mesh-based finite-element model for Boussinesq equations (FEBOUSS Agarwal et al., 2022) with a meshless local Petrov–Galerkin model for the Navier–Stokes equations (MLPG_R Agarwal et al., 2021) in 3D. Boussinesq equation models are widely used for simulating wave-propagation over large domains with uneven topography using a 2D surface mesh. Mesh-less models inherently capture large free-surface deformations and have shown promise in simulating wave-structure interaction, run-up and breaking phenomenon. The hybrid approach in this paper assumes a 3D MLPG_R sub-domain surrounded by the 2D mesh of FEBOUSS. The coupling interface in MLPG_R consists of relaxation zones that can be placed along multiple boundaries of the sub-domain for exchanging particle velocity from FEBOUSS. This hybrid model is therefore capable of simulating directional waves, that has not been reported previously.
The paper first presents the procedure for calculating the depth-resolved velocities in 3D from the Boussinesq model. The resultant velocities are compared against theory, experiments and other models. The following sections present the coupling algorithm along a single and multiple coupling interfaces in MLPG_R. Validation results for this hybrid model are provided using surface elevation and velocity measurements for regular waves, including directional cases. In general, the results from the hybrid model are reported to have marginal over-prediction of peaks compared to purely MLPG_R simulation. Finally, the interaction of a vertical cylinder with direction regular wave is simulated using the 3D hybrid model.

This paper presents new developments in meshless local Petrov–Galerkin with Rankine source (MLPG_R) particle based method for studying interaction of waves with fixed structures in a numerical wave-tank. A new 3D formulation of the Lagrangian flow problem for incompressible fluid with optimised solution strategy is presented. The pressure Poisson equation is solved in local weak-form with integration done semi-analytically using a new symmetric expression. The wave-generation is done using one-way coupling with a 2D fully-nonlinear potential theory based finite-element model. Further a simple identification method for free-surface particles is proposed, which is shown to work well in vicinity of the structure. The solid-wall boundary condition is treated using ghost and mirror particles for accurate calculation of gradients. The waterline on domain boundary faces is treated using a tangentially moving side-wall approach which makes this particle based scheme capable of capturing small amplitude waves and focusing waves. The paper briefly presents experimental setup used for studying the interaction of a fixed emergent cylinder with uni-directional regular and focusing waves in 3D. The numerical model is validated against results from this experiment. An analysis is conducted on parameters related to local integration domain, wave-making coupling algorithm, particle distribution and time-step. This work highlights the use of hybrid approach for efficient and accurate simulation of waves-structure interaction.

The paper presents a numerical model of the two-dimensional enhanced Boussinesq equations to simulate wave transformations in the near-shore region. The finite element-based discretisation over unstructured mesh with triangular elements uses mixed linear and quadratic shape functions. The domain integrals are calculated analytically. The model is extended to study flow through porous structures using Darcy velocity, with the energy dissipation within the porous medium modelled through additional laminar and turbulent resistance terms. A single set of empirical constants gives accurate prediction for various stone sizes and porosity. This paper reports the model development and its validation using existing experimental studies. Application of the model is demonstrated by studying the interaction between ship-generated waves in a narrow channel and the porous walls of the channel.

This paper presents a 2D/3D hybrid numerical model for studying the interaction of non-breaking waves with cylindrical structure. The work combines the strengths of mesh-based and particle-based methods, where the wave propagation is solved using 2D mesh-based potential theory model and the interaction with the structure is solved using 3D particle-based Navier-Stokes model. The paper presents the formulation of the two models and the weak-coupling methodology, along with recent improvements in the 3D MLPG_R (Mesh-less Petrov-Galerkin based on Rankine source solution) particle based method. The numerical results from interaction of fixed cylinder with solitary and focussed waves are compared with experimental data. The work demonstrates a significant reduction in simulation time for wave-structure interaction problems achieved using this hybrid approach without compromising on accuracy.