Shock-capturing Boussinesq Model And Its Application In Modeling Waves Generated By Seabed Movement

Waves are not only the important factor to be considered in coastal engineering and offshore engineering, but also the main object of marine disaster research. Boussinesq-type wave equations contain nonlinear and dispersive terms, which can effectively describe the wave motion in the offshore area, and their computational cost is acceptable due to the depth-integration nature. As a result, the numerical model based on Boussinesq-type wave equations is widely used in port, coastal and offshore engineering. From the point of view of depth-integrated equations, the second order fully nonlinearity in Green-Naghdi wave equations is completely consistent with the order of dispersion. Hence, Green-Naghdi wave equations after improved can be regarded as classical Boussinesq wave equations. Ignoring the dispersive and high order nonlinear terms in Boussinesq-type equations, nonlinear shallow water equations can be obtained and they are suitable to describe the waves of long wavelength. Numerous high resolution in the framework of the finite volume method have been developed to solve nonlinear shallow water equations, which is highly beneficial for capturing wave breaking and coastal moving shoreline. The studies in recent decades show that Boussinesq theories develop rapidly and have been widely used in simulating wave propagation on fixed seabed, and waves generated by sea bottom movement as well.Based on the weakly nonlinear Boussinesq equations and fully nonlinear Green-Naghdi equations, a shock-capturing Boussinesq model which can model propagation and runup of offshore wave is proposed using a hybrid finite volume and finite difference method on structured grid system. The governing equations are firstly rewritten into conservative form and the interface fluxes are then discretized by finite volume method. The left and right variables of the cell interface are reconstructed with the Fourth-order Compact MUSCL Total Variation Diminishing (TVD) method (FCMT). In order to ensure non-negative water depth, the hydrostatic reconstruction technique is used for in the reconstruction process. The center upwind scheme, which is efficient and simple than the conventional upwind schemes, is utilized to compute the numerical fluxes with reconstructed variables. By introducing the local bed modification method, models have the ability to deal with the coastal moving shoreline and ensure the well-balance property. The wave breaking is efficiently simulated by locally switching off the high dispersive and nonlinear terms after a breaking index is reached. The third-order Runge-Kutta method with TVD property is adopted to perform time marching. Based on the above-mentioned Boussinesq model, which is originally used for simulating coastal water waves over fixed seabed, the extension is made to include the effect of time-varying bed for simulating landslide-generated waves. A series of validation of the developed models are conducted through simulating solitary wave propagation and runup, regular wave propagating over the submerged dike and classic examples of waves generated by seabed movement. Computational results show a good agreement with analytical solutions, experimental and other models'results, which shows that the Boussinesq-type models have advantages in strongly stability and shock-capturing capability, which can reproduce the coastal wave dynamics for both fixed and time-varying seabed.