# The Effect of Porous Media on Wave-Induced Sloshing in a Floating Tank
## 4 Results and Discussion
### Floating Body in Resonant Waves: Controlled vs. Uncontrolled
The kinetic energy and potential energy of the floating body and sloshing fluid can be obtained by:  
\[T_{b}=\frac{1}{2}M_{b}\big{|}\dot{x}_{G}\big{|}^{2}+\frac{1}{2}I_{b}\big{|} \dot{\theta}_{G}\big{|}^{2}\ \text{and}\ T_{s}=\frac{1}{2}\int_{S_{1}}\gamma\rho\dot{\phi} \frac{\partial\phi}{\partial n}dS \tag{52}\]  
\[V_{b}=M_{b}gz_{G}\ \text{and}\ V_{s}=\frac{1}{2}\gamma\rho g\int_{S_{f1}} \eta^{2}dS \tag{53}\]  
where the subscripts \(b\) and \(s\) denote the floating body and sloshing fluid, respectively. Note that \(T_{s}\) is obtained by applying Green's identity to the volume integral of the kinetic energy of a fluid particle [47]. The zero-potential plane is defined at \(z=0\). The changes in the kinetic and potential energy, i.e., the difference between the energy at current and initial states of the floating body are shown in Figure 30. In Figure 30a, the kinetic energy increased rapidly because the body velocity increased when the incident wave came. For the controlled case, the water tank dampened the transient vibrations, so the body's kinetic energy reached the steady state quickly. It is noticeable that the kinetic energy of the body without porous media was lower than that with porous media because of the lower heaving velocity, as shown in Figure 25. However, the porous media dampened more body kinetic energy from the surge and pitch components. Figures 25 and 30b show that the potential energy change was in sync with the vertical displacement. Since the water tank did not mitigate the heaving motion, the potential energy change of the body was irrelevant to the sloshing behavior and only depended on the vertical force equilibrium. Figure 29: The response spectra of the controlled and uncontrolled floating bodies for the large-wave case. The kinetic and potential energy changes of the sloshing fluid are shown in Figure 24:0 For the first 15 s in Figure 31a, the kinetic energy of the sloshing fluid without porous media increased faster than that with porous media. This shows the porous media can quickly dampen the kinetic energy in the transient state, therefore suppressing the fluid velocity and the free-surface deformation. By correlating the force/moment and energy change as shown in Figures 24, 28, 30 and 31, the effect of porous media on the FSI can be characterized. When the incident waves hit the floating body, the work is transferred to the body's kinetic energy. Through the force cancellation, part of the wave-induced work will be taken by the sloshing fluid, and then dissipated by the damping mechanism of the porous media. Therefore, good force cancellation and enough sloshing damping are two key factors for structural vibration control. In Figure 31b, the potential energy of the fluid in the tank without porous media grew higher than that of the fluid in the porous media. Since they had similar elevations, the difference in the potential energy came from different free-surface deformations. Figure 24:1 The energy changes of the sloshing fluid in the water tank: (**a**) kinetic energy; (**b**) potential energy. Figure 24:2 The energy changes of the floating body: (**a**) kinetic energy; (**b**) potential energy. The free-surface elevations on the left-side wall of the tank with and without porous media are shown in Figure 24:3 The wave elevation in the tank without porous media was very nonlinear, while the porous media reduced the degree of nonlinearity due to their extra damping effect. This shows the benefit for the liquid storage tank that not only the dynamic pressure can be reduced, but the roof damage can be alleviated. The wave elevations of four gauges distributed at \(x=2\), 4, 6, and 8 m (two on the left side and two on the right side of the body) along the channel are shown in Figure 24:4 At the beginning (\(t<2\) s), the incident wave had not reached the body, so the body only oscillated vertically. The uncontrolled body had a larger initial acceleration, so it induced a larger disturbance to the waves, as shown in Figure 33b,c.