# CFD Based Non-Dimensional Characterization of Energy Dissipation Due to Verticle Slosh
## 8 Dissipated Energy Scaling Laws  
We next focus on quantifying the change in slosh induced dissipation as a function of the fluid physics non-dimensional numbers. As mentioned before, we now set \(\xi=0\) resulting in a vertical displacement and acceleration as defined by:  
\[y(t)= -\frac{a_{1}}{\omega^{2}}cos(\omega t) \tag{31}\] \[a(t)= a_{1}cos(\omega t) \tag{32}\]  
where it is instructive to note that the magnitude of tank displacement \(y_{0}=a_{1}/\omega^{2}\). It is important that this value remain fixed when varying the fluid physics non-dimensional numbers. Setting \(\xi=0\) simplifies the calculation of slosh induced dissipation to:  
\[E_{Disp}\approx\int_{0}^{t}\Gamma_{y}u_{dy}dt \tag{33}\]  
The % change in computed slosh induced dissipation energy due a change in a non-dimensional parameter (from the Protospace value \(E_{Disp}(\text{Protospace})\)) is expressed as:  
\[||\Delta E_{Disp}||=\frac{E_{Disp}-E_{Disp}(\text{Protospace})}{|E_{Disp}( \text{Protospace})|} \tag{34}\]  
Figure 8 depicts the discrete \(||\Delta E_{Disp}||\) values for individually scaled non-dimensional numbers as well as the resulting trend line. Also delineated on the figures are the properties for the Protospace (black line) and full aircraft (green line). While no statistical trend was  
Figure 7: Influence of flow physics on slosh induced damping, work and loads. found for \(We\) and \(Re\) (as a result not shown), the following cubic trend for \(\theta\) was found around \(60^{\circ}\):  
\[ \begin{split}||\Delta E_{Disp}(\theta)||=& 4.14\times 10^{-4}\theta^{3}-0.091\theta^{2}+6.702\theta-161.7 \\ r^{2}=& 0.82\end{split}  \tag{35}\]  
Here, \(r^{2}\) is the weighted sum of the squares of the difference between the fitted curve and the data such that \(r^{2}\in[0,1]\) with \(r^{2}=1\) being a perfect fit. Outliers to this trend occur due to lack of Rayleigh Taylor instability for \(\theta=90^{\circ}\) which leads to \(P_{Disp}\to 0\), while an enlarged meniscus results as \(\theta\to 0\) (to be investigated in future work). A very well-defined trend can however be found for the scaling of \(\bar{\rho}\) as:  
\[ \begin{split}||\Delta E_{Disp}(\bar{\rho})||=&-0.5 4\bar{\rho}^{-0.802}+117.39\\ r^{2}=& 0.98\end{split}  \tag{36}\]  
Similarly, a very well defined trend is visible for the effect of the Froude number on slosh induced dissipation. The resulting trend line reads:  
\[ \begin{split}||\Delta E_{Diss}||(Fr)=&-6.57 Fr^{2}-12.42 Fr+114.3\\ r^{2}=& 0.99\end{split}  \tag{37}\]  
which is clearly quadratic in nature. This quadratic relationship between slosh induced dissipation and \(Fr\) number is indeed a new finding and a key contribution of this article. Also note that the difference in dissipated energy between the Protospace experiment \(Fr\) number and that of the actual aircraft (or indeed ideal experiment) are by far the most significant of all quantities. This necessitates the need to use the scaling law to estimate the expected slosh induced damping for the full scale aircraft tank. This is dealt with the in next section. Figure 8: Change in Dissipated Energy due to Scaling of Non-dimensional Properties. From the above it is evident that \(\Delta E_{Disp}\) scales circa linearly with \(\rho\) (due Equation (36)) and with \(v^{2}\) (due Equation (37)). The latter is as the maximum tank velocity \(v\propto\omega\) due Equation (14). From this, we postulate that it is possible to express dissipated energy as a linear function of tank kinetic energy viz. \[E_{Disp}\propto\rho v^{2} \tag{38}\]  
for the slosh under consideration. We digress briefly to investigate the difference in fitted scaling curves for each of the three cycles of oscillation. This is to confirm that scaling laws obtained per cycle are consistent with the experimental observation that the slosh damping which occurs over the first two to three cycles of oscillation is representative of the energy dissipation potential.