Calculation of Solitary Wave Shoaling on Plane Beaches by Extended Boussinesq Equations

Abstract

In this study, shoaling phenomenon is analyzed using Galerkin finite element approach. This numerical scheme is applied to the extended Boussinesq equations derived by Beji and Nadaoka (1996) for simulation of shoaling on plane beaches. For spatial discretization, quadratic elements with three-station Lagrange interpolation polynomials are used for horizontal velocity and the water surface elevation. However, for time discretization, two different numerical schemes are used. The first method is a combination of semi-implicit schemes with low-order backward finite difference for time integration and the second method is high-order AdamBashforth-Moulton predictor-corrector strategy. Based on this numerical approach, shoaling phenomenon caused by propagation of a solitary wave on sloped beaches is modeled and the results are compared with the available results from the fully nonlinear potential flow model. Considering the fact that the extended Boussinesq equations are affected by nonlinear effects, a non-dimensional parameter called “Asymmetric Parameter” is introduced. This parameter expresses the effects of the travelled distance of the solitary wave as well as the relative wave height on the resulting wave asymmetry. Finally, using this parameter, shoaling coefficient has been computed in an appropriate range.