# CFD Based Non-Dimensional Characterization of Energy Dissipation Due to Verticle Slosh
## 7 Non-Dimensional Study Results and Analysis  
To enable direct comparison (plotting on the same graph) of computed energy budget and slosh loads for different excitation frequencies and liquid masses, normalisation is required. Hence, the computed energy dissipated \((E_{Disp})\) and work in \((W_{\rm s})\) are normalised  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Property Range** \\ \hline \(\theta\in[0.52,1.51]\) & – \\ \(\tilde{\rho}\in[6.49\times 10^{-4},6.18\times 10^{-3}]\) & \(\because\rho\in[199,1894]\) kg\(\cdot\)m\({}^{-3}\) \\ \(Re\in[42.56\times 10^{3},\infty)\) & \(\because\mu\in[0,3.7\times 10^{-3}]\) Pa\(\cdot\)s \\ \(We\in[4.92\times 10^{3},1.37\times 10^{5}]\) & \(\because\gamma\in[0.003,0.085]\) N\(\cdot\)m\({}^{-1}\) \\ \(Fr\in[0.69,5.16]\) & \(\because f\in[1.4,10.5]\) Hz \\ \hline \hline \end{tabular}
\end{table}
Table 6: Non-dimensional property parameter space.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Actual** & **Froude Scaled** & **Aircraft** \\  & **Protospace** & **Protospace** & **Aircraft** \\ \hline \(\theta\) & \(1.05\) & \(0\) & \(0\) \\ \(\tilde{\rho}\) & \(1.21\times 10^{-3}\) & \(1.39\times 10^{-3}\) & \(1.39\times 10^{-3}\) \\ \(Re\) & \(161.7\times 10^{3}\) & \(4.67\times 10^{3}\) & \(52.2\times 10^{3}\) \\ \(We\) & \(5.73\times 10^{3}\) & \(2.78\times 10^{3}\) & \(69.4\times 10^{3}\) \\ \(Fr\) & \(3.44\) & \(1.65\) & \(1.65\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Protospace vs Full Aircraft Non-dimensional Properties.  
to the maximum change in potential energy of a simulation \((E_{u})\). Further, computed vertical slosh force \((F_{y})\) is normalised and oscillation count employed as follows:  
\[\begin{split}||E_{Disp}||&=\frac{E_{Disp}}{E_{u}}\\ ||W_{s}||&=\frac{W_{s}}{E_{u}}\\ ||F_{y}||&=\frac{F_{y}}{|F_{y}(t=0)|}\\ n_{oscillation}&=\frac{t}{\omega}\end{split} \tag{30}\]  
Figures 6 and 7 presents comparative ranges for the influence of fluid properties on the normalised sloshing fluid dissipation energy (\(||E_{Disp}||\)), surface work (\(||W_{s}||\)) and vertical sloshing force (\(||F_{y}||\)). The influence range of each non-dimensional number on the aforementioned quantities are presented via the shaded region.  
Table 7 lists the percentage root-mean-square difference due to the scaling of each non-dimensional property as compared to the actual Protospace values. The analysis reveals the relatively low influence of almost all fluid properties on \(||E_{Disp}||\) where the RMSE \(<15\%\) and negligible on \(||W_{s}||\) as RMSE \(<10\%\). The exceptions are contact angle and \(Fr\) number. The former is due a contact angle of \(90^{\circ}\) resulting in no liquid-gas surface instability (Rayleigh-Taylor) and hence no slosh damping (note that the reported 94.06% variation as opposed to 100% is due numerically induced free-surface instability which occurs late in the analysis). At this point in the scaling analysis, the choice of Froude scaling (as opposed to \(Re\) etc.) is vindicated (and consistent with industrial practice) as all other fluid properties are shown to have a relatively small effect. This was further also demonstrated by a recent experimental campaign [25].  
We finally consider the effect that Froude scaling (via changing the excitation frequency) has on the system, with the results shown in Figure 7. Interestingly, for the Froude number range tested, the major effect is on slosh damping as shown, i.e., 77.5%. This concludes the important insight that slosh induced dissipation is highly sensitive to Froude number, and to a far greater degree as compared to slosh forces.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline  & \(||E_{Disp}||\) & \(||W_{s}||\) & \(||F_{y}||\) \\ \hline \(Re\) & 7.80 & 4.85 & 13.47 \\ \(\bar{\  
Figure 6: Influence of flow physics on slosh induced damping, work and loads. (**a**) Density Ratio and Reynolds Number Scaling; (**b**) Fluid Surface Tension Properties.