Interaction of surface water waves with floating or submerged horizontal porous barrier(s) over trench-type bottom topography: A mathematical study

Abstract

This thesis presents a comprehensive analysis of a class of water wave problems pertinent to ocean and marine engineering, particularly focusing on the interaction of water waves with thin horizontal porous plates over trench-type topography where these plates can be used to protect coastal infrastructure, such as floating bridges/tunnels and sea walls, where the horizontal porous plate may be placed either at free surface or submerged inside the water. The thin horizontal porous plates are important and effective because of their additional properties like being lighter, economical, easy to maintain, and environmental friendly. The physical phenomena related to the above wave structure interaction problems are mathematically modeled, assuming that the fluid is homogeneous, inviscid, incompressible, and exhibits irrotational and harmonic motion over time. Additionally, the wave motion is considered to be influenced by gravity, with the free surface deviations from its horizontal position assumed to be small enough to justify the application of linearized water wave theory. The objective of this thesis is to focus on a specific class of wave-structure interaction problems, such as the interaction of water waves with the floating/submerged horizontal porous plate(s) over trench-type bottom topography and the interaction of water waves by the floating bridge/submerged tunnel in the presence of floating/submerged horizontal porous plate over a trench-type bottom. The emphasize is given in the following key areas: (i) reducing the wave impact at the lee side regions by employing thin porous plate(s) over trench(es), (ii) minimizing wave impact on floating bridge/tunnel by the f loating/submerged horizontal porous plate, and (iii) investigating the role of submerged trenches as bottom topography. In these cases, the horizontal porous plate is either a f loating or a submerged porous plate with finite length and negligible thickness. When formulating the physical problems, the governing partial differential equation becomes the Laplace’s equation for the case of normal incidence of surface waves, while it is the Helmholtz equation for the oblique incidence of surface water waves. The combined dynamic and kinematic boundary condition at the free surface is of the Robin type, and the impermeable boundary condition at the bottom is of the Neumann type. The flow past the porous plate is modeled by using the Sollit and Cross model. Furthermore, far-field conditions are imposed at infinite fluid boundaries to ensure the uniqueness of the solution. The resulting boundary value problems are linearized using small amplitude water wave theory. The boundary value problem is cast into a system of algebraic equations by employing eigenfunction expansions and leveraging the orthogonality of eigenfunctions and/or the algebraic least-square method. These equations are then solved numerically using the Gauss-elimination method with the help of MATLAB. For some of the physical problems, the energy identity is derived using Green’s integral theorem, and verifying this identity ensures the accuracy of the numerical results obtained for the physical quantities. Also, the present numerical results are compared with those available in the literature to validate each model. Additionally, in some problems, the convergence on the number of evanescent modes in the eigenfunction series expansions is evaluated numerically. To study the effectiveness of the above thin porous plate(s) in reducing the wave load on the seashore areas or coastal infrastructures like bridges and tunnels, the quantities such as reflection, transmission and dissipation coefficients, force on the porous plates, force on the floating bridge, force on the tunnel, and free surface elevation are calculated numerically. The variations of these quantities with various system and wave parameters are analyzed and illustrated through different graphs. These problems provide information to safeguard essential coastal structures or seashore areas. Hence, the study in this thesis plays an essential role in the field of ocean and marine engineering, particularly in the protection of coastal infrastructure.