# Numerical Calculation of Slosh Dissipation
## 5 Results
### SPH-Flow Solver Results
#### 5.1.4 Three-Dimensional SPH Water Results  
Figure 13 shows the 3D free surface for the water test-case at the first roof impact. The 3D flow behavior is similar to the 2D one in the sense that during the first upward acceleration of the tank, a Rayleigh-Taylor instability is influenced by free-surface waves, which are generated due to the collapse of the menisci at the lateral walls.  
The waves move from the lateral walls towards the center of the tank. The further flow evolution is characterized by a complex 3D free surface, as well as a complex interaction of three-dimensional vortex tubes, which makes physics much more complex compared to the two-dimensional case. In addition to these differences, the dimensionless energy dissipation \(E_{diss}/\Delta E\) remains quite close, as reported in Table 1 (for more details, see also [42]). We need to underline again that the 2D results are ensemble averages of ten simulations, while 3D is given by single simulations. Because of the CPU costs, the spatial resolutions adopted for the 3D simulations are \(N=22\), \(33\), \(50\) and \(75\). As expected for the water-case, \(E_{diss}\) varies significantly with \(N\) and, conversely to the 2D case, there is not a clear convergence trend in the results. In the water test-case, the highest resolution is still too coarse to obtain a convergent result and an effective LES simulation. A spatial resolution as high as \(N=400\) corresponds to about 430 million particles and would require about one year of computing time on a cluster with 100 nodes. Even with the above constraints, the dissipated energy given by the 3\(D\) simulations at resolution \(N=75\) is closest to the experimental data by [4].  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \multicolumn{3}{c}{**Single-Phase \(\delta\)-LES-SPH Results (SPH-Flow Solver)**} \\ \hline  & Test-Case & **N = 50** & **N = 100** & **N = 200** & **N = 400** \\ \hline \(E_{diss}(t_{f})/\Delta E_{w}\) & 2D test with water & \(-10.3\) & \(-12.4\) & \(-13.2\) & \(-13.6\) \\ \hline \(E_{diss}(t_{f})/\Delta E_{o}\) & 2D test with oil & \(-14.0\) & \(-14.4\) & \(-14.6\) & \(-\) \\ \hline  & & **N = 22** & **N = 33** & **N = 50** & **N = 75** & **Exp. Data** \\ \hline \(E_{diss}(t_{f})/\Delta E_{w}\) & 3D test with water & \(-13.1\) & \(-14.5\) & \(-15.3\) & \(-15.9\) & \(-16.3\) \\ \hline \(E_{diss}(t_{f})/\Delta E_{o}\) & 3D test with oil & \(-17.8\) & \(-17.7\) & \(-17.6\) & \(-17.9\) & \(-16.7\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Total energy fluid dissipation, \(E_{diss}(t_{f})\) for 2D/3D SPH simulations with water and with oil using different spatial resolution \(N=H/\Delta x\). Numerical outputs are compared with the experimental data of [4]. The 2D results are ensemble averages of ten different simulations, while 3D simulations are given by single simulations.  
Figure 11: Test with water: time histories of the dissipated energy \(E_{diss}\), mechanical energy \((E_{k}+E_{p})\), work completed by the non-inertial forces \(W_{NF}\) and work completed by the tank wall to slosh the fluid \(W_{dyn}^{ext}\).  
Figure 12: Test with water: time histories of the tank acceleration \(a_{tank}\), the fluid center of mass \(y_{G}\), and the power of the non-inertial forces \(P_{NF}\).