Eigenfunction expansion method for analysis of mitigating wave load on an elastic plate and a sea wall in the presence of thick porous structure(s)

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 thick porous structures designed to protect coastal infrastructure, such as VLFS and sea walls. The physical phenomena related to water wave propagation 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, emphasizing the following key areas: (i) reducing the wave impact on an elastic plate by employing thick porous structure(s), (ii) minimizing wave impact on sea wall when an elastic plate and thick porous structure are present, and (iii) investigating the role of submerged porous structure in reducing wave load on sea wall in a step type bottom topography. In Case (i) and Case (ii), the porous structure is a vertical porous structure extended from top to bottom or bottom-standing or surface-piercing. When formulating the physical problems, the governing partial differential equation is the Laplace 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 elastic plate is modeled by using the thin plate theory, while the flow past the thick porous structure 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 transformed into a system of algebraic equations by employing eigenfunction expansions and leveraging the orthogonality of eigenfunctions. These equations are then solved numerically using the Gauss-Elimination method with the help of MATLAB. For each physical problem, 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 thick porous structure(s) in reducing the wave load on the elastic plate/sea wall, the quantities such as reflection, transmission and dissipation coefficients, force on the porous structure, force on the sea wall, and free surface elevation, plate deflection, shear force and strain 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 such as VLFS and sea walls. Hence, the study in this thesis play an essential role in the field of ocean and marine engineering, particularly towards the protection of coastal infrastructure.