# Numerical Calculation of Slosh Dissipation
## 5 Results
### Finite Volume Method Results with Elemental\({}^{\circledR}\)  
#### 5.2.1 Simulation Setup  
For 2D and 3D simulations in Elemental\({}^{\circledR}\), a structured, Cartesian mesh is used. The mesh resolutions used are similar to the particle resolution in the SPH simulations, as shown in Table 2.  
The interface is initialized as a flat, horizontal plane. To model the effect of liquid meniscii at the walls, liquid wedges are initialized as equilateral triangles with a height of 1.5 mm, as explained in [12] and used in the SPH simulations. As mentioned earlier, interface perturbations are applied to the interface in 2D simulations. These perturbations are introduced as sinusoidal waves on the interface, all with an amplitude of 0.15 mm, or 1/10  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Mesh Resolution:** & \\
**Mesh Description and Dimension** & \(N_{x}\times N_{y}\) (2D) or & **Cell Dimension (\(\upmu\)m)** \\  & \(N_{x}\times N_{y}\times N_{z}\) Nodes (3D) & \\ \hline \(N50\) 2D & \(160\times 96\) & 625.00 \\ \(N100\) 2D & \(320\times 192\) & 312.50 \\ \(N200\) 2D & \(640\times 384\) & 156.25 \\ \(N33\) 3D & \(125\times 75\times 75\) & 800.00 \\ \(N50\) 3D & \(160\times 96\times 96\) & 625.00 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mesh resolutions for different simulations.  
Figure 15: Time history of the vertical position of the fluid center of mass \(y_{G}(t)\) evaluated by the SPH-Flow solver for the 2D and the 3D simulation. The 2D results are given by an ensemble average of ten simulations. Error bars refer to the computed standard deviation. (**Top**) water test-case. (**Bottom**) oil test-case.  
the height of the meniscus. Five different wavelengths with randomized phase angles for each are used, resulting in differences in the interface shape once the Rayleigh-Taylor instability is formed with the first upward acceleration of the tank. Since the amplitudes of these perturbations were so small, the total difference in mass between simulations was negligible. Figure 16 shows the interface position for four different simulations at the \(N50\) resolution for oil and water to illustrate the effect of the initial interface perturbations.  
All simulations are run using the weakly compressible gas formulation for air and treating the liquid as an incompressible liquid. A continuum surface force (CSF) surface tension model is used to model surface tension. Like the SPH method, a Smagorinsky-Lilly large eddy simulation (LES) turbulence model is applied, with a Smagorinsky constant of 0.18 used throughout. For all simulations, the turbulent viscosity is capped to twice the laminar viscosity. The resulting linear system of Equation (23) is solved using a bi-conjugate gradient (BiCG) solver with an incomplete lower-upper (ILUT) preconditioner.  
The measured tank acceleration is applied as a body force to all fluids in the tank, along with gravitational acceleration, exactly the same as in the single-phase approach. The evolution of the energy, as described in Section 4.4, is calculated by integrating (in time) the different energy components each time step. The total velocity at each time step is calculated as per Equation (14), where \(\mathbf{u}\) is the latest fluid velocity relative to the domain, and \(u_{d}\) is the domain velocity as measured experimentally. Note that due to experimental measurement errors, there are slight inconsistencies between the measured velocity and acceleration of the tank. This can be seen, for example, in the \"drift\" of the potential energy. However, for the total dissipation, it can be shown that the inconsistency does not noticeably affect the calculated dissipation.