## 3 Validation of the CFD Approach
### Imposed Motion Studies  
The series of studies was designed using certain parameters from [35], such that the outcome supports the application case of a TSD, as described in [36; 37]. We present the sloshing response in the form of the horizontal sloshing force, the shape of the hysteresis curve and the dissipated energy. These values are compared to the reference results in [35]. The comparison is supported by using the dimensionless expressions in Equations (8) and (10). The effect per vibration cycle is of focus, where the enclosed area in the hysteresis loop represents the dissipated energy. We calculated the relative deviation of the assessed values from the results in [35].  
Whereas the detailed results are contained in [21], here, we particularly focused on representing the characteristic outcome relevant for further design steps. The deviations observed in Figures 3 and 4 are low. Accompanying quantitative results are presented in Table 1. Not only are the areas of the hysteresis loops very close, but also the shape of the curves depict a comparable behaviour. For cases with a high water depth and small amplitude ratios, the simulations tend to slightly overestimate the sloshing response. We attribute this to additional internal numerical damping for such cases. Overall, the CFD-based approach reproduces the nonlinear sloshing forces very reliably. It can be seen that even higher harmonics are captured by the model, despite the hysteresis loops not identically overlapping with the reference ones.  
Figure 3: Validation results for varying the water height.  
Such studies can be designed in various ways, depending on the exact aim. For our use case, we limited them to a two-dimensional domain, without extensive focus on varying the excitation frequency. The results are mainly related to the effect of varying depth and amplitude ratios at a frequency characteristic for our building. It is expected of the TSD to  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{\(\Delta E^{\prime}\)} & \multicolumn{2}{c}{\(a=5\) mm} & \multicolumn{2}{c}{\(a=20\) mm} & \(a=60\) mm \\ \cline{2-5}  & \(h_{0}=20\) mm & \(h_{0}=40\) mm & & \(h_{0}=60\) mm & \\ \hline
**Experiment** & 36.00 & 29.90 & 8.30 & 19.40 & 10.20 \\
**CFD** & 32.01 & 24.92 & 9.12 & 17.55 & 11.82 \\ _Diff. to exp._ & \([\%]\) & \(-11.1\) & \(-16.7\) & 9.9 & \(-9.5\) & _15.9_ \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dimensionless energy per hysteresis cycle.  
Figure 4: Validation results for varying the amplitude of excitation.  
have to mitigate motions that are unidirectional, so this assumption is deemed appropriate. For other cases, three-dimensional simulations might be the only viable alternative, which is possible with CFD simulations.  
Computational methods make it possible to visualize various results. In Figure 5, we include one of these, as a selection of variables and properties can be captured. Here, we see the clear distinction between air and water based upon density (similarly possible based upon viscosity). Additionally, the velocity and pressure fields are also available to aid in various discussions and design purposes.