Abstract
Using one-dimensional Beji & Nadaoka extended Boussinesq equation, a numerical study of solitary waves over submerged breakwaters has been conducted. Two different obstacles of rectangular as well as circular geometries over the seabed inside a channel have been considered in view of solitary waves passing by. Since these bars possess sharp vertical edges, they cannot directly be modeled by Boussinesq equations. Thus, sharply sloped lines over a short span have replaced the vertical sides, and the interactions of waves including reflection, transmission, and dispersion over the seabed with circular and rectangular shapes during the propagation have been investigated. In this numerical simulation, finite element scheme has been used for spatial discretization. Linear elements along with linear interpolation functions have been utilized for velocity components and the water surface elevation. For time integration, a fourth-order Adams-Bashforth-Moulton predictor-corrector method has been applied. Results indicate that neglecting the vertical edges and ignoring the vortex shedding would have minimal effect on the propagating waves and reflected waves with weak nonlinearity.
Highlights
Boussinesq type equations are among the most practical mathematical models used in offshore engineering
These equations were derived directly from 2-dimensional continuity and Euler equations on a seabed with variable depth. Another type of extended Boussinesq equations was obtained by Beji and Nadaoka [6] using an algebraic manipulation of the classical Peregrine equations
For Beji and Nadaoka’s extended Boussinesq equations with the consideration of the above mentioned value for β, the error of the dispersion relation with μ ≈ 0.5 would be less than 0.5 percent
Summary
Boussinesq type equations are among the most practical mathematical models used in offshore engineering. Other forms of the extended Boussinesq equations were derived by Nwogu [5] where the velocity components in an arbitrary distance from the calm free surface were used. These equations were derived directly from 2-dimensional continuity and Euler equations on a seabed with variable depth. Another type of extended Boussinesq equations was obtained by Beji and Nadaoka [6] using an algebraic manipulation of the classical Peregrine equations. This means that, while their algebraic representation differs from each other, the dispersive properties of these equations could be indicated to be equivalent after performing a respective change of variable
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
