# Numerical Calculation of Slosh Dissipation
## 5 Results
### SPH-Flow Solver Results
#### 5.1.1 Two-Dimensional SPH Water Results  
We start to consider the test with water. Figure 4 shows the free-surface particles just prior to the first roof impact for the first four runs. Clearly, at this stage, the interface position from four different solutions are starting to deviate. The spatial resolution adopted is \(N:=H/\Delta x=200\), corresponding to about 133,000 particles in the liquid domain.  
Figure 5 depicts the vorticity field of four simulations, doubling the spatial resolution from \(N=50\) up to 400. The time instant considered is related to the fourth bottom liquid impact event. The chaotic nature of the flow is also clearly visible in this plot, where the free  
Figure 4: Test with water: free-surface particles for the first four simulations at the first roof impact.  
surface of the four fluid domains present large differences from each other. Furthermore, it is quite evident in the case of water that, by increasing the resolution, the number of eddies increases considerably. This is an indication that we are still resolving the inertial range of the turbulence regime. This is also confirmed by the fact that for the first three resolutions (\(N=50\), \(100\) and \(200\)), the turbulent viscosity is one order of magnitude greater than the material or laminar viscosity. In a more in depth-analysis shown in [12; 13], a resolution of \(N=800\) is used for obtaining good LES simulations (i.e., for resolving the Taylor microscales).  
However, because of this chaotic behavior, ten runs are performed for the four different spatial resolutions (\(N=50\), \(100\), \(200\) and \(400\)) in order to perform a statistical analysis for controlling the convergence of the numerical results. On the other hand, the CPU costs for resolution \(N=800\) is too large to perform ten runs, and even for a 2D framework, a cluster machine would be required. The limitation is not mainly linked to the number of particles, which are about two million for \(N=800\), but to the fact that more than \(1.5\) million time iterations are required. Figure 6 reports the ensemble average of the dissipated energy of the forty simulations performed for the spatial resolution \(N=50\), \(100\), \(200\), \(400\). A convergence of the curves is obtained, where the error bars refer to the standard deviation measured between the set of the ten simulations. Even if a resolution \(N=800\) would be required, the result trend for \(N=200\) and \(400\) indicates that the average dissipation will likely remain within the standard deviation of the \(N=400\) case. It is worth noting that the results for the oil test case presented here are slightly different with respect to the ones reported in [13] since a different acceleration profile was used for the tank motion.  
Figure 5: Test with water: vorticity fields at the fourth bottom impact for different spatial resolutions \(N=50\), \(100\), \(200\), \(400\).  
Figure 6: Test with water: ensemble average of the energy dissipation \(E_{diss}\) time history for different spatial resolutions. Error bars refer to the computed standard deviation.