# Simulating Slosh Induced Damping, with Application to Aircraft Wing-like Structures
## 4 Results and Discussion
### Impact of Initial Excitation Displacement with One Compartment
From the third oscillation cycle, the effect of slosh-induced damping is evident for all initial excitation amplitudes, as can be seen in Figure 5c. Meanwhile, during the last five oscillation cycles, the amplitude of the tip vertical displacement for \(D_{i}=0.25\) is larger than that of \(D_{i}=1.0\) and \(D_{i}=0.5\). This indicates that the slosh-induced damping for \(D_{i}=0.25\) is smaller than that of \(D_{i}=1.0\) and \(D_{i}=0.5\), which is the same conclusion seen in Figure 4.  
As mentioned in Section 2, each oscillation cycle of the beam-tank system starts from its tip, located at the lowest position for its upstroke in the positive \(y\)-direction. The vertical fluid force reaches its minimum value (i.e., the maximum negative value) at the proximity of the beginning of each oscillation cycle, where the bulk of the water pushes against the floor of the tank. The fluid force reaches the maximum value at around the middle of each cycle as the tip of the system reaches the highest position, where the bulk of fluid has been accelerated in the positive \(y\)-direction and sloshing waves are formed. The spikes of the vertical fluid force are distinct from cycle to cycle, as shown in Figure 5d. This is partially due to the nature of the problem that the sloshing motion of the fluid is subject to rapid development during the first ten oscillations and also due to the non-linear nature of the sloshing regime within this study (reported in [PERSON] et al. [21]). It can be seen in Figure 5d that the vertical fluid force of the sloshing tank with \(D_{i}=0.25\) is smaller than that of the other two initial excitation amplitudes at the beginning of each oscillation cycle. We further investigated the fluid domain for a better understanding of the observations in Figures 4 and 5d. Through this, it was found that the shape of the sloshing waveform in the first half cycle is similar from case to case with one compartment within the parameter range of this study. An example can be seen in Figure 6, which shows the snapshots of the water phase within the sloshing tank with \(D_{i}=1.0\) and \(N_{c}=1\) at the beginning and the half of the first oscillation. The water phase is shown as transparent red with dark red representing the free surface and pink representing the bulk of liquid without free surface. Blue arrows show the general direction of motion of the sloshing water in the global reference frame. It can be seen that, at the very beginning of free decay cycles (i.e., \(t/T=0\)), the tip of the tank is located at the lowest position due to the external force, and is about to be accelerated towards the equilibrium position by the elasticity of the beam. The water inside the tank flows along the slope to the front due to gravity. The water starts to build up in the front of the tank. At the end of the first upstroke (i.e., \(t/T=0.5\)), initial slosh-induced waves start to form in the front and rear of the tank. There is no contact with the ceiling of the tank by the sloshing wave for all simulation cases within the parameter range of this study. It is noticed that the free surface is not visible at \(t/T=0\) but can be seen clearly at \(t/T=0.5\). The reason for this behaviour is mainly due to the \(3-D\) shape of the slosh pattern. More specifically, at \(t/T=0\), the slosh pattern has not yet developed and the elevation of the free surface along the lateral direction has the same value, while at \(t/T=0.5\), the slosh pattern has started to develop and the elevation of the free surface along the lateral direction is no longer equal. This is consistent with reported experimental results [5; 18]. In addition however, the present study applies a meniscus to the initial starting conditions of the fluid simulation in order to replicate the wetting effect of water on the tank wall before release. This perturbation causes the lateral motion of the water and contributes to creating a realistic \(3-D\) slosh pattern.