# The Effect of Porous Media on Wave-Induced Sloshing in a Floating Tank
## 4 Results and Discussion
### Frequency Response Analysis  
To complete the frequency response curves, the near-resonance condition was emphasized in the range of \(\beta=0.85\)-\(1.15\), where \(\beta\) is the ratio between the frequencies of the wavemaker's movement and the body's pitch motion. The small wave excitation was applied to avoid unstable numerical results in uncontrolled cases. For the first application of structural control, the water tank was installed as a TLD. The frequency responses of the root-mean-square (RMS) of the steady-state angular displacement and acceleration of the body are shown in Figure 34. TLDs can enhance the workability and comfort of the workstation. The average reduction in the RMS displacement and acceleration by the porous-media TLD was 85%, while that by the fresh-water TLD (TLD without porous media) was 80%. However, the porous-media TLD showed at least 10% better performance than the fresh-water one in the high-frequency range (\(\beta\geq 1.075\)). This is because the high nonlinearity of the fresh-water TLD can cause a strong mistuning effect. Once the natural frequency of the water tank deviates from the tuning frequency too much, the force cancellation will be weakened. In terms of the tuning condition, the porous-media TLD can be optimized more easily according to a simple design plan [22]. A single peak occurred at \(\beta=1\) in uncontrolled cases, while there were two peaks at \(\beta=0.925\) and \(1.1\)  
Figure 32: The wave elevations on the left-side wall of the tank with and without porous media. when the TLD was installed. Theoretically, the sloshing fluid had infinite degrees of freedom. Therefore, the body-tank system was a multi-degree-of-freedom system. Since the fundamental sloshing mode dominated (as was concluded in the previous example), the frequency response curves only had two peaks that represented two major vibration modes of the TLD-body system. Figure 34: The frequency responses of the RMS responses of the controlled and uncontrolled floating bodies for the small-wave case: (**a**) angular displacement; (**b**) angular acceleration. Figure 33: The wave elevations in the channel at the locations of: (**a**) \(x=2\) m; (**b**) \(x=4\) m; (**c**) \(x=6\) m; (**d**) \(x=8\) m.  
For a liquid carrier on the ocean, its stability is related to the amplitude of the wave-induced sloshing resultants. The frequency responses of the maximum horizontal force and overturning moment due to sloshing fluid are shown in Figure 35. In Figure 35a, the porous media can reduce the horizontal force by 30-70% because the extra damping effect limits the sloshing responses, hence reducing the hydrodynamic force. In Figure 35b, the reduction in the overturning moment was low. This is because, for a shallow-water tank used in this example, the overturning moment did not mainly come from the horizontal component. Therefore, the reduction by the porous media was not obvious. The integrity of a liquid container is another important issue that is related to pressure concentration. The frequency responses of the maximum dynamic pressure on the wetted wall and the normalized height of the local maximum pressure are shown in Figure 36. Note that \(h_{p}\) denotes the elevation of the fluid particle that has the maximum dynamic pressure. In Figure 36a, the maximum dynamic pressure in the tank without porous media was 485 Pa, which was higher than the maximum hydrostatic pressure (460 Pa). The porous media reduced 25-45% of the dynamic pressure. The averages were 343 versus 228 Pa. In Figure 36b, the elevations of the local maximum pressure were similar, which were below the still-water line (\(h_{p}/h<1\)) in most cases. As the wave traveled back and forth during the vibration, the greatest impact occurred at the instant when it reached the walls. The contact location was usually near but below the still-water line. This phenomenon was visualized in the past article [48]. Overall, the porous media will not change the location of the pressure concentration but can effectively reduce the hydrodynamic pressure.