## 5 Conclusions  
In this work, an experimental methodology and a computational methodology are presented in order to study a simplified version of the sloshing phenomenon that takes place inside aircraft fuel tanks. The single degree of freedom numerical model based on a fluid-structure interaction coupling contains the most relevant non-linear phenomena in terms of fluid-wall impacts and free-surface breaking that dominate the most real-world versions of this problem. In order to test the influence of the viscosity on this kind of flow, two fluids have been tested which are very different in terms of viscosity, but similar in terms of density. A clear conclusion is that there is a critical Reynolds number \(Re_{c}\sim O(10^{4})\) where, depending on whether the flow is below or above this critical number, important differences in terms of the characteristics of the sloshing regimes, free surface fragmentation and relative kinetic energy distribution are observed. It was also concluded that the less viscous fluid is able to obtain a larger damping in terms of tank movement, something that might challenge our intuition. The comparison between the experiments and numerical simulation has been performed both on the tank kinematics and on the fluid magnitudes. In regard to the fluid study, some integral fluid magnitudes have been compared, including the sloshing force, liquid center of mass position and liquid kinetic energy; however, local measurements such as free-surface location and the jet velocities were also examined. There are three well-separated stages during the flow evolution for both fluids studied, the first and the last being less chaotic and particularly appropriate for numerical validations. The 2D numerical model is able to compute either the evolution of the impelling jets at the initial stage or the kinetic energy of the remaining standing wave in the last part the flow evolution. An important conclusion is that despite some differences appearing when local values such as free-surface position or jet velocities are compared, the numerical model approximates the results very well in terms of global magnitudes such as tank acceleration, sloshing force and liquid center of mass. A direct conclusion coming from the application of the momentum conservation to these apparently complex flows is that there is a direct relation between the vertical component of the sloshing force and the vertical position of the liquid center of mass, showing in this work that this relation is satisfied by both fluids. Other important conclusions of this article are based on the study of the intrinsic properties of the sloshing force. First is that a simplified version of the sloshing force can be modeled as a shifted cosine function with decaying amplitude. The projection of the force on the tank velocity and acceleration confirms that one of the key magnitudes of the sloshing phenomenon is the shift angle between the force and the tank position maxima. Second, the sloshing force can be interpreted as a sum of the inertial force and a second part that depends on the relative movement of the liquid inside the tank. As a result, the dissipative part of the sloshing force has been related to the product of the liquid mass times the center of mass of the liquid. Third, the viscous and pressure components of the sloshing force and the derived dissipated energy have been compared for both liquids, and the relevance of the viscous part is not negligible when high viscosity fluids such as glycerine are used. Finally, the pressure fields computed with the SPH methodology on the horizontal walls of the tank have been discretized using several pressure sensors on each wall. It was proven that with the pressure field recorded, a minimum number of sensors are necessary to obtain an accurate sloshing force reconstruction, with this number varying depending on the fluid contained within.