# Numerical Calculation of Slosh Dissipation
## 5 Results
### SPH-Flow Solver Results
#### 5.1.3 Mechanisms of Fluid Dissipation  
The plots in Figure 9 show the energy dissipation due to real and turbulent viscosity at three different spatial resolutions for both oil and water cases. Looking at the water (left of Figure 9), it is clear that the viscous dissipation due to the sub-scale model is predominant compared to the real viscosity. This is expected, considering the high Reynolds number. However, it is interesting that the absolute value of \(E_{V}^{turb}\) does not increase with \(N\) while for the real viscosity component, it does. This indicates that the viscous dissipation related  
Figure 8: Test with oil: vorticity fields at the fourth bottom impact for different spatial resolutions \(N=50\), 100, 200, 400.  
Figure 7: Test with oil: ensemble average of the energy dissipation \(E_{diss}\) time history for different spatial resolutions \(N=50,100,200\). Error bars refer to the computed standard deviation. Resolution \(N=400\), the data refers to a single simulation.  
to the real viscosity can be resolved directly beyond the sub-scale turbulent model, which is something expected when a good LES is performed. For the oil cases, \(E_{V}^{turb}\) largely decreases with \(N\) and remains lower than \(E_{V}\), indicating that the LES simulation is much closer to being resolved, compared to water. This is again expected with the large difference in Reynolds number between the two liquid cases.  
Figure 10 depicts the energy dissipated by the numerical diffusion of the scheme, which acts mainly during the liquid impact events. Because of the complexities of the flow studied, this component--like the \(E_{V}^{turb}\) component--is not negligible, even for the lower Reynolds oil case.  
In Figure 11, the time history of the energy dissipation \(E_{diss}\) for the water test-cases and for one of the runs at resolution \(N=400\) is compared with the histories of the mechanical energy (kinetic plus potential energy) in the Ni-FoR, along with \(W_{NF}\) and \(W_{dyn}^{ext}\). As already commented in Section 2.2, the mechanical energy of the fluid in the Ni-FoR remains quite limited and small during the experiments with respect to the dissipated energy. Furthermore, in this plot, it can be seen how both the work containers \(W_{NF}\) and \(W_{dyn}^{ext}\) tend to \(E_{diss}\) at the end of the experiment \(t_{f}=30\)T, indeed already after fifteen periods they are practically the same.  
Figure 12 depicts the comparison between the time histories of the tank acceleration \(a_{tank}\), the vertical velocity of the fluid center of mass \(\dot{y}_{G}(t)\) and the power exerted by the non-inertial forces on the fluid \(P_{NF}\). The latter is characterized by large peaks when the first two are in counter phase giving an indication of when intense fluid dissipation mechanisms are taking place.  
Figure 10: Energy dissipation due to numerical viscosity for different spatial resolutions. (**Left**) 2D water test-case. (**Right**) 2D oil test-case.  
Figure 9: Energy dissipation due to the real and turbulent viscosity for different spatial resolutions. (**Left**) water test-case. (**Right**) oil test-case.