A fluid partially occupying a moving tank undergoes wave motions (sloshing). These motions generate severe hydrodynamic loads that can be dangerous for structural integrity and stability of rockets, satellites, LNG ships, trucks and even stationary petroleum containers. Free surface motions of the liquid in partially filled tanks under gravity are of practical significance particularly in marine and road transportation applications. For this reason, liquid sloshing has always been a research subject attracting great concern during the last several decades. In this paper, a fully non-linear finite difference model has been developed based on the inviscid flow equations, and a simple mapping function was used to remove the time-dependence of the free surface in the fluid domain. The time-varying fluid surface can be mapped onto a rectangular domain by the σ-transformation. This method is a simple way to simulate non-breaking waves quickly and accurately especially that has a low steepness. The fluid motion is solved in a unit square mesh in the transformed flow domain (i.e., computational domain). The fourth order central difference scheme and the Gauss–Seidel point successive over-relaxation iterative procedure are used to capture the free surface wave profiles and the free surface elevation plots of the fluid domain. Difference between the peaks and troughs of waves are discussed for the case of vertical excitation of first three natural frequency of the tank. Phase-plane diagrams are drawn to show the non-linearity of the motion of time dependent free surface. The results agree well with the previously published results.

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