# CFD Based Non-Dimensional Characterization of Energy Dissipation Due to Verticle Slosh
## 9 Scaling-Laws Application  
The scaling between the full aircraft model and Protospace experiment employs an attempted Froude scaling due being limited by experimental design and fluid availability (as discussed previously). In contrast high-fidelity CFD enables the modelling of both the ideal (employing an ideal fluid which achieves perfect similarity for all physics non-dimensional numbers) and practical (employing kerosene) Froude scaled experiments. These are denoted as Protospace* and Protospace*, respectively with the resulting non-dimensional numbers listed in Table 8. Note that by using kerosene (practical \(Fr\) scaling or Protospace*) both density ratio and \(Fr\) numbers equal that of the full scale Aircraft tank. Further, by manipulating fluid properties in the CFD model it is possible to achieve perfect correlations for all shown non-dimensional numbers as shown for ideal scaling (Protospace*).  
We now seek to quantify the difference in computed dissipated energy for the different cases and have two options in which to achieve this viz. CFD or using our scaling laws. Starting with CFD first, both practical and ideal scaled cases were simulated and the resulting computed dissipated energy \(E_{Disp}\) compared in Table 9. As expected, the actual Protospace experiment overestimates the slosh induced dissipation while practical \(Fr\) scaling (Protospace*) yields a slosh induced dissipation which is within 5% of the ideal experiment.  
Considering finally the use of the scaling laws, we aim to scale the actual Protospace experiment to the ideal experiment. As both \(\bar{\rho}\) and \(Fr\) are to be accounted for, a response surface of \(||\Delta E_{Disp}||\) is created using Kriging interpolation which is shown in Figure 10. The resulting scaling computed dissipated power is 4.83 W which is within 14% of the CFD computed value.  
\begin{table}
\begin{tabular}{l l l l l} \hline  & _Protospace_ & _Protospace_ * & _Protospace_ * & _Aircraft_ \\  & _(Water)_ & _(Cold Kerosene)_ & _(dden)_ & _(Cold Kerosene)_ \\ \hline \(\bar{\rho}\) & \(1.21\times 10^{-3}\) & \(1.39\times 10^{-3}\) & \(1.39\times 10^{-3}\) & \(1.39\times 10^{-3}\) \\ \(Re\) & \(161.7\times 10^{3}\) & \(4.67\times 10^{3}\) & \(52.2\times 10^{3}\) & \(52.2\times 10^{3}\) \\ \(We\) & \(5.73\times 10^{3}\) & \(2.78\times 10^{3}\) & \(69.4\times 10^{3}\) & \(69.4\times 10^{3}\) \\ \(Fr\) & 3.44 & 1.65 & 1.65 & 1.65 \\ \hline \end{tabular}
\end{table}
Table 8: Fluid Properties due to Froude Scaling. (Nomenclature for scaling methods as defined above).  
\begin{table}
\begin{tabular}{l l l l} \hline  & _Protospace_ & _Protospace_ * & _Protospace_ * \\ \hline \(E_{Disp}\) (\(W\)) & \(-37.21\) & \(-5.32\) & \(-5.59\) \\ \hline \end{tabular}
\end{table}
Table 9: CFD simulated dissipated energy. (Nomenclature for scaling methods as defined above).  
Please note that the above derived scaling laws allow for scaling of a SDOF system with a geometric scaling \(\lambda=1\). This is as the excitation velocity is affected directly by geometric scaling as per Table 3, i.e., velocity is expected to scale by \(\lambda^{\frac{1}{2}}\). For a larger aircraft size tank this would increase the impact velocity of the slosh and hence also the energy dissipation.