Highly accurate Boussinesq-type equations in terms of velocity potential are used for the simulation of shallow-water sloshing in a rectangular tank. The finite difference method is used for the spatial discretization of derivatives. The total velocity potential is separated into two parts: a particular solution satisfying the Laplace equation and the no-flow condition on the walls while the other part is solved by the Boussinesq-type model. Distinct features of the free surface can be observed under the selected external exciting frequencies. Comparisons are made between different cases, the results are analyzed and discussed.
Sloshing is defined as the movement of fluid in a partially filled container, and has been studied by numerous scholars with different methods. A series of analyses about sloshing in spherical and cylindrical tanks including linear and nonlinear situations were given in the pioneering work of Abramson (1966). A study of the sloshing model and frequency in a spherical tank in two dimensions, as well as a multidimensional modal method of establishing the mode and calculating the frequency of natural sloshing was developed by Faltinsen(2000). A complete discussion of various aspects of liquid sloshing was given by Ibrahim (2005). Most analytical methods are based on potential flow theory and focus on the mechanism of fluid motion.
Because of the limitations of the theoretical method, computational fluid dynamic methods are more widely used for researching sloshing phenomena. In Gedikli and Erguven (2003), a Variational Boundary Element Method (VBEM) based on the Hamilton's principle was presented to investigate the effect of a rigid baffle on the natural frequencies of the sloshing in a cylindrical container. Firouz et al. (2010) developed a modal approach for the nonlinear analysis of sloshing in a tank of arbitrary-shape under both horizontal and vertical excitations. An empirical formula for calculating the natural frequency of sloshing in a rectangular tank with baffle is presented in Hu et al. (2018) by using the Boundary Element Method (BEM) with linear free surface condition considered. The boundary element method has the advantages of less meshes and reducing dimension.