## 4 Results and Discussion
### Comparison of the Flow Evolution: Experimental Visualization versus Numerical Computation
#### 4.1.2 Glycerine Case \(Re_{g}\sim O(10^{2})\)  
For the glycerine case, the Reynolds number is three orders of magnitude lower, but it can also be distinguished three different stages in the flow evolution, where the same time notation will be used. * As displayed in Figure 6, during the first instants, no free surface deformation is observed with the exception of the lateral wall meniscus; see a snapshot in Figure 6a (\(t=0.09\) s). As demonstrated in [47], the increment of viscosity decreases the growth rate of the Rayleigh-Taylor instability. It could be assumed that this initial stage finishes with the first fluid-wall impact (\(t_{1}\equiv t_{fi}=0.41\) s), which happens at the third tank cycle (first in the case of water). It is worth commenting that three dimensional effects are more present at this first stage in the case of glycerine compared to water, and this can be interpreted by the more dominant role of the viscous stress tensor in this flow. * The second regime of the glycerine is less chaotic and turbulent when compared to the water regime; in this case, jets and traveling boundary layers are clearly observable, and fewer droplets and bubbles appear in the flow. As expected, the higher viscosity makes the liquid more compact and less fragmented. This stage takes approximately the same amount of time compared to the water and also dominates during the longest part of the experiment. As already measured in [21] comparing water and oil, the damping capacity of a fluid is related to the number of violent impacts, and high viscosity plays an opposite role in this behavior. As a consequence, the more viscous  
Figure 5: Experimental snapshots (**top**) of a vertical sloshing experiment with water at a 50% filling level and 10 g of maximum acceleration. Numerical counterparts (**bottom**) simulated the \(\delta\)-SPH method. the liquid, the less damping generated. The last ceiling impact is observed at time \(t_{li}\sim 2.46\) s\(\ll 3.56\), which corresponds to the case of water. * After a similar duration to the water \(t_{2}=2.58\) s, the flow transitions to a standing-wave-like behavior. This stage takes less time to relax toward a final flat free surface, with very small vertical sloshing force variations. A typical snapshot of this stage can be observed in Figure 6d (\(t=2.87\) s). It could also be observed that in contrast to the water case, the final low-amplitude oscillations of the glycerine case have no impact in the free surface, and the glycerine remains as a block at the bottom of the tank. The videos corresponding to the cases of water and glycerine can be downloaded from [[http://canal.etsin.upm.es/files/SLOWD/videos](http://canal.etsin.upm.es/files/SLOWD/videos)]([http://canal.etsin.upm.es/files/SLOWD/videos](http://canal.etsin.upm.es/files/SLOWD/videos)), Last access 1 August 2022. An interesting comparison between these two cases can be made in terms of the relative kinetic energy, and this time-dependant number is defined as:  
\[\varepsilon_{k}=\sum_{i}\frac{1}{2}m_{i}(v_{i}-v_{t})^{2} \tag{21}\]  
where \(v_{i}\) and \(m_{i}\) are the absolute velocity and mass of a fluid particle, respectively, and \(v_{t}\) is the tank velocity. When the evolution of this relative kinetic energy is plotted for both fluids (see Figure 7), it is observed that they both start with a similar energy value, but the energy growth for the water is completely different from the glycerine, especially during the first half second of the experiment. The relative energy has been normalized by the initial elastic potential energy introduced in the system \(\varepsilon_{0}=\frac{1}{2}Ky_{0}^{2}\), where \(K\) is the spring constant of the experimental setup. Despite the fact that both flows produce impulsive impacts with the upper and lower tank walls, the higher fragmentation of the water produces larger levels of relative kinetic energy. This high energy stage takes approximately the same time for both fluids \(t_{2}-t_{1}\sim 2\) s, after which the impacts are clearly smoothed.