# Numerical Calculation of Slosh Dissipation  
[PERSON]  
1 Industrial CFD Research Group, Department Mechanical Engineering, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa 14 University of Cape Town, Private Bag  
fixed rigidly to an elastic beam, which is deflected to induce a heaving oscillatory motion with accelerations up to 10 g. In [8; 9], experiments of slosh-induced energy dissipation in partially filled tanks subject to vertical, harmonic excitation is investigated. In [10] the decaying harmonic excitation is investigated experimentally, along with an equivalent mechanical model and a weakly compressible single-phase SPH simulations.  
In the present work, we perform numerical simulations of the single degree of freedom vertical slosh experiments carried out in [4]. This experimental work has recently been extended [11] with comments on repeatability as well as a study on scaling effects. Previous results on the experiments in [4] are published [12; 13], but new results are added in this work. Similar to [12; 13], we employ an energetic approach to study the dissipation mechanisms in the tank flow. This is completed for two different numerical approaches. In the first approach, only the liquid is solved, using a weakly compressible formulation with a Smoothed Particle Hydrodynamics (SPH) method. The second approach uses the multi-physics code Elemental(r)[14; 15; 16; 17; 18; 19], which employs a vertex-centered, Finite Volume Method (FVM) with Volume-of-Fluid (VOF) tracking of the liquid phase. Both the liquid and gas (air) are therefore solved. The liquid is assumed incompressible and the gas weakly compressible [20]. Surface tension effects are accounted for using a Continuum Surface Force (CSF) method, where the SPH approach does not include capillary effects. As explained in [21; 22], the latter does not have a significant role in the fluid energy dissipation mechanisms (apart from the initial liquid meniscus). This is due to the high Weber numbers of the simulated flows. As noted, surface tension is indeed relevant during the initial upward acceleration of the tank when a Rayleigh-Taylor instability develops (see, e.g., [23]). The instability is influenced by the fluid meniscus formed at the intersection between the lateral walls and the liquid surface as the accelerating meniscus perturbs the interface, leading to the growth of a Rayleigh-Taylor instability during the upward acceleration of the tank from rest. For this reason, the initial meniscus free-surface deformation is taken into account in the simulations (see also, e.g., [12; 22]).  
Test cases are set up and solved using similar initial conditions and tank acceleration profiles for both approaches. The energy distribution in both cases is studied, as well as other metrics, such as the center of mass position. In this work, we prescribe the tank acceleration to match experimental measurements. This allows us a more direct comparison to the experiment and ensures that both methods are subject to the same acceleration. There has been research with coupled simulation, where the fluid-structure interaction is resolved [13], but that is outside the scope of this work. First, two-dimensional simulations are performed using random interface perturbations to study the variation in energy metrics with the slight difference in interface shape. Three-dimensional experiments are also performed to compare the 2D results of each approach as well as an overall comparison between approaches. The results of these cases provide insights into several aspects. These include the suitability of the modeling approach (single or two-phase), the chaotic nature of the flow (in 2D) with random initial perturbations, mesh resolution effects, and the difference in 3D results compared to 2D.  
The experiment will be described next, followed by the mathematical formulation and numerical method details of each approach.