# Simulating Slosh Induced Damping, with Application to Aircraft Wing-like Structures
## 4 Results and Discussion
### Impact of Initial Excitation Displacement with One Compartment
Figure 5: Time history of non-dimensional (**a**,**c**) tip vertical displacement and (**b**,**d**) vertical fluid force of the system over ten oscillation cycles with one compartment, in which (**c**,**d**) are normalised by the initial excitation amplitude. Figure 6: Snapshots for the sloshing tank with \(D_{i}=1.0\) and \(N_{c}=1\) at (**a**) \(t/T=0.0\) and (**b**) \(t/T=0.5\). Water phase is shown as transparent red with dark red representing the free surface. Blue arrows show the general direction of motion of the sloshing water. From the down stroke of the first cycle onwards (i.e., \(t/T>0.5\)), the system with different initial excitation amplitudes begins to show different sloshing patterns in the water tank. A significant difference can be observed among different initial excitation amplitude cases from the third oscillation. Figures 7 and 8 show snapshots for the sloshing tank of three different initial excitation amplitudes with one compartment at \(t/T=2.0\) and \(t/T=2.5\). It can be seen from Figures 7a and 8a that the sloshing pattern for \(D_{i}=0.25\) is dominated by gentle sloshing waves without any contact with the ceiling of the tank during the third cycle. The sloshing in \(D_{i}=0.5\) is dominated by the same model pattern as in \(D_{i}=0.25\), but due to the larger oscillation amplitude, the wave amplitude is larger than that of \(D_{i}=0.25\) and the crest of the waves show impacts with the tank ceiling. Damping due to impact on the ceiling/floor of the tank is discussed in [PERSON] et al. [5] and [PERSON] et al. [21]. When the water impacts the ceiling of the tank, the resultant splash then falls back and impacts the floor. These ceiling and floor impacts introduce significant impact-driven turbulence and affects the fluid force acting on the tank wall. At \(t/T=2.0\), there are three wave crests for both the \(D_{i}=0.25\) and \(D_{i}=0.5\) cases. The general direction of the motion of the water is from the crest of the wave towards the wave trough as shown in Figure 7, i.e., these are falling waves. When the system finishes its upstroke and reaches the highest position, crests and troughs alternate, as shown in Figure 8.  
It can be seen that the amplitude of the wave crests in the front of the tank are larger than those in the rear. This is expected behaviour as the oscillation amplitude in the front of the tank is greater than that in the rear. For \(D_{i}=1.0\), no sloshing modal pattern can be discerned. In this case, the sloshing is dominated by the turbulent vertical motion entirely. Strong impacts between the water and the top/bottom wall of the tank can be observed for both the beginning and the middle of the third oscillation cycle. In Figure 9, the vertical fluid force during the third oscillation cycle is represented by the normalised force-velocity diagram, where the time parameter is eliminated. Similar to the vertical fluid force, the normalised tip oscillation velocity, \(\delta DT/\delta tHD_{i}\), follows the positive upwards \(y\)-direction. The two grey solid lines (\(F_{y}/F_{g}D_{i}=0\) and \(\delta DT/\delta tHD_{i}=0\)) divide the normalised force-velocity diagram into four quadrants. In the second and the fourth quadrants, the vertical fluid force and the tip oscillation velocity are in the opposite sign, which causes the vertical fluid force to decrease the momentum and increase the damping of the free decay in the system. In the first and the third quadrants, the fluid force and the tip oscillation velocity are in the same sign, so the fluid force increases the momentum and decreases damping. In each cycle, the normalised force-velocity loop starts from point \(A\) (the subscript label denotes the \(D_{i}\) value), follows the green dotted arrow towards point \(B\) for its upstroke, and follows the other half of the loop to go back to point \(A\) with its down stroke.