## 4 Results and Discussion
### Flow Kinematics
#### 4.2.1 First Stage  
As explained in Section 4.1, this first stage starts with the relative flow movement inside the tank and finishes with the first ceiling impact. In the case of water, the initial free surface deformation can be studied in terms of impelling jets, which are created symmetrically to the middle vertical plane. In Figure 10, the free surface obtained by the SPH computation and the one obtained by image processing from experimental snapshots are compared at different time steps before the ceiling impact. The free surface position from  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Liquid** & \(\xi\) [\%] & **t(\(a=0.1\) - \(a_{0}\)) (s)** \\ \hline Water & 2.5 & 2.29 \\ \hline Glycerine & 1.7 & 3.19 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Damping ratio and characteristic decaying time for the water and glycerine cases.  
Figure 9: Comparison between \(\delta\)-SPH simulations and experimental data of the nondimensional vertical sloshing force over time for water and glycerine decay tests at 10 g of maximum acceleration.  
the experimental snapshots is extracted by taking into account the decrease in luminosity close to the free surface. The gradient of the luminosity of the snapshot is computed, and the points with the most negative gradient values are extracted. Complex dynamic phenomena are present in the flow, such as traveling wave formation, free surface deformation, central jet formation and the boundary layer evolution on the lateral walls. These are all well captured qualitatively. Although the 2D computational flow shows a very realistic representation of the experimental one, it was observed that the computational region allows the formation of more ripples on the free surface than the real flow is able to diffuse. Although this code has been validated in a typical free surface evolution problem such as the Rayleigh-Taylor one [23], including surface tension and viscosity, the results obtained here show that when higher velocities and accelerations are present in the free surface, the viscosity and surface tension models should be able to damp these short wavelength instabilities.  
The authors would like to underline that in the comparison of the free surface evolution in Figure 10, a 2D simulation is being compared with a 3D flow; therefore, some differences in terms of flow morphology should be expected. Furthermore, since the tank is moving and the camera is at a fixed position, the angle of view varies with time, and the free surface represented in the experimental snapshots presents some perspective which is not present in 2D simulations.  
The symmetric dominant central jets that appear in the free surface have been monitored in terms of relative velocity magnitude for both methodologies, and the quantitative results have been compared in Figure 11, where the modulus of the velocity has been non-dimensionalized with \(v_{0}=\omega_{0}y_{0}\). During tank deceleration, a linear growth is observed in the relative jet velocity, and this tendency is well captured by the computations. It is particularly interesting to see that before the final impact, during the whole jet acceleration process, the tank is decelerating. The dispersion of the experimental results when compared to the computational ones is due to the image processing technique, where jet velocity is approximated graphically.  
Figure 10: Comparison between the free surface position obtained by the SPH computation and the image processing obtained from experimental snapshots.