## 3 Methodology Description
### Experimental Approach  
In this section, the vertical sloshing experimental setup, which is an improved version of the one used in [8], is briefly described. The sloshing rig is a 1 degree of freedom system composed of a mechanical guide that allows only vertical motion. This guide is attached to a C-shaped wooden structure that holds the tank. This C-shaped wooden structure is attached to a set of 6 springs (3 on the upper side and 3 on the lower side), each of them having an individual spring constant of \(k_{i}=708.3\) N/m. The lower springs are mechanically embedded to the floor and on the opposite side, the upper set of springs is attached to a metallic plate that acts as a joint between them and the embedded load cell. A photograph of the 1/5 scale experimental setup at the Model Basin Research group sloshing laboratory of the UPM and a simplified outline are both shown in Figure 1.  
The tank dimensions of a scaled experiment with a scale parameter \(\lambda=H/H_{\text{W}}\) are: \(H\) for the height and depth, \(L\) for the length and \(h\) for the liquid height. The length  
Figure 1: Snapshot of the experimental setup for the \(\lambda=1/5\) scale (**left**) and outline (**right**). (1) Load cell, (2) metallic plate with mass \(m_{p}=0.06\) kg, (3) upper set of springs with stiffness constant \(k_{1}=2125\) N/m, (4) lower set of springs with stiffness constant \(k_{2}=2125\) N/m, (5) laser sensor, (6) mechanical guide, (7) accelerometer, (8) methacrylate tank and C-shaped wooden structure and (9) pair of solenoids acting as release mechanism. represents the tank height of the aircraft fuel tank. The system is displaced by an initial amplitude \(y_{0}\) until it is fixed by the action of the pair of solenoids located at the bottom of the setup at \(y=-y_{0}\). The permanent solenoids produce a magnetic force until the electrical current is turned on, triggering the release of the structure and the beginning of the oscillatory movement. In Figure 2, the experimental snapshots of a 50% fill level \(f=0.5\) water test for \(\lambda=1/5\) and \(y_{0}/H=1\) are presented. The timeline of the snapshots has been referenced to the wing scale according to the Froude scaling laws (\(t/\sqrt{(}\lambda)\)). Once the system is released and accelerated upwards, the initial flat free surface is rippled; see, for example, Figure 2a,b. While several effects cause the initial free surface deformation, the main cause is the meniscus deformation generated at the boundaries but also in those cases where the tank ceiling contains droplets of liquid attached. These droplets can cause a relevant perturbation when falling down from the upper wall and impacting the free surface. These perturbations in the free surface grow as Rayleigh-Taylor instabilities; see Figure 2c,d. The instabilities grow and form liquid jets that impact the upper wall of the tank (Figure 2e) producing the first sloshing force peak. The linear growth of these instabilities has been deeply studied in [23], where the initial stages of an SPH computation are compared to the linear stability analysis of the Navier-Stokes equations in terms of growth rate and mode shape. For the rest of the test, a violent and turbulent regime is developed where the free surface is fragmented, small bubbles form, splashing occurs and liquid-to-liquid and liquid-to-wall impacts are seen (Figure 2f). When the sloshing force is lower than the liquid weight, the violence of the sloshing decreases (Figure 2g), bigger bubbles form and fewer impacts occur. The liquid starts to return to its initial state with moderate liquid impacts to the lateral walls, and it starts to develop a standing wave-like regime ( Figure 2h). It finally restores its initial state displaying a rotary motion that is attenuated over time (Figure 2i). This general description will be particularized for the two liquids studied in this paper in Section 4. In this experimental setup, there are 3 magnitudes that are being measured: acceleration, position and force.