# Numerical Calculation of Slosh Dissipation
## 5 Results
### Finite Volume Method Results with Elemental\({}^{\circledR}\)
#### 5.2.2 Oil 2D Results  
In Figure 17, the oil interface is shown at different time instants as well as the vorticity in the fluid for a \(N200\) resolution case. The first frame on the left at \(t=1.20T\) shows the interface just after the central oil structures have impacted the roof for the first time. The asymmetry in the interface structure is a result of the randomized interface initialization explained earlier. Compared to the single-phase results in Figure 8, the liquid looks much more dispersed, or there is a higher degree of mixing between phases. Since the spatial distribution in the tank differs from very early between single and two-phase results, it is difficult to perform a precise comparison of the vorticity. However, the overall vorticity magnitude in the liquid is similar between the approaches. Note that for the Elemental\({}^{\mathbb{D}}\) finite volume results, we use a scaled vorticity \(\tilde{\mathbf{\omega}}\) throughout  
\[\tilde{\mathbf{\omega}}=\frac{\mathbf{\omega}}{2\pi f_{0}}\frac{\mu}{\mu_{\ell}}= \frac{\mathbf{\
abla}\times\mathbf{u}}{2\pi f_{0}}\frac{\mu}{\mu_{\ell}}\,. \tag{41}\]  
The last term is a viscosity ratio scaling factor, which is equal to 1 in the liquid phase and \(\mu_{g}/\mu_{\ell}\) in the air. Without this scaling, the vorticity in the air phase is dominant and does not allow for a comparison with the SPH results in Figure 8.  
Figure 16: The interface position showing the initial Rayleigh–Taylor instability forming. A VOF iso-contour at 0.5 is represented by four different colours for four (out of ten) different simulations at \(N50\) resolution at the same time. Oil is on the left and water on the right. To illustrate the effect of mesh resolution, the turbulent viscosity ratio is shown in Figure 18 at the same time for different mesh resolutions. The areas where the turbulent viscosity ratios are largest are mostly in the air. It was observed for all resolutions that there is large vorticity generated in the air phase as it is accelerated around in the tank by the liquid. As the resolution increases, it is evident that the inertial range for turbulence is increasingly resolved. The energy distribution for a \(N200\) case is represented in Figure 19, along with a magnification of the first five cycles. The total energy dissipation is shown in the light peach shading. In the zoomed plot (bottom), it can be seen that the dissipation starts to increase gradually after \(t=1.20T\), which corresponds with the first roof impact of the oil. The sharp increase in dissipation just after \(t=2.05T\) corresponds to the second roof impact. All the contributions to the total dissipated energy are presented in the plot, with the total viscous dissipation having the largest contribution. The total work done on the fluid, \(W_{NF}\), consists of the work done by the pressure, \(W_{NF}^{P}\) at the boundary, as well as the viscous work, \(W_{NF}^{V}\). It is clear that the viscous component is negligible, and the majority of the work is performed by the pressure. The compressive \(E_{C}\) and surface work \(E_{v}\) are not plotted here as both were negligible. The impact-related loss term is significant, yet slightly less than the total viscous dissipation. The influence of mesh resolution on the total dissipation is shown in Figure 20. The difference between mesh resolutions is significantly more than in the SPH case, Figure 7, but the average dissipation has a convergent trend with increasing resolution. The individual dissipation components, as described by Equation (39), are presented in Figure 21. The impact loss is very similar for the \(N50\) and \(N200\) resolutions, yet the \(N100\) resolution shows slightly more dissipation. There is no clear explanation for this difference and further  
Figure 17: Two-dimensional oil Elemental® results on the \(N200\) resolution at various time instants. The (**top**) row shows the VOF with oil black and air white. The (**bottom**) row shows the non-dimensionalized vorticity.