# CFD Based Non-Dimensional Characterization of Energy Dissipation Due to Verticle Slosh
## 4 Dimensional Analysis
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Quantity** & & & **Units** & \\  & & \(M\) & \(L\) & \(T\) \\ \hline \(\mu\) & Liquid Viscosity & [Pa\(\cdot\)s] & 1 & \(-1\) & \(-1\) \\ \(\rho\) & Liquid Density & [kg\(\cdot\)m\({}^{-3}\)] & 1 & \(-3\) & 0 \\ \(\rho_{air}\) & Gas Density & [kg\(\cdot\)m\({}^{-3}\)] & 1 & \(-3\) & 0 \\ \(\gamma\) & Liquid Surface Tension & [N\(\cdot\)m\({}^{-1}\)] & 1 & 0 & \(-2\) \\ \(\theta\) & liquid–solid Contact Angle & [rad] & 0 & 0 & 0 \\ \(v\) & Fluid speed & [m\(\cdot\)s\({}^{-1}\)] & 0 & 1 & \(-1\) \\ \(g\) & Gravity & [m\(\cdot\)s\({}^{-2}\)] & 0 & 1 & \(-2\) \\ \(m\) & Solid Mass & [kg] & 1 & 0 & 0 \\ \(k\) & Structural Stiffness & [N\(\cdot\)m\({}^{-1}\)] & 1 & 0 & \(-2\) \\ \(c\) & Structural Damping & [Ns\(\cdot\)m\({}^{-1}\)] & 1 & 0 & \(-1\) \\ \(h\) & Tank Height & [m] & 0 & 1 & 0 \\ \(l\) & Tank Length & [m] & 0 & 1 & 0 \\ \(\eta_{0}\) & Height of fluid & [m] & 0 & 1 & 0 \\ \(y_{0}\) & Initial tank offset & [m] & 0 & 1 & 0 \\  & (spring displacement) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimensional Matrix. where \(Re\), \(Fr\) and \(We\) are the excitation frequency-based Reynolds, Froude and Weber numbers, respectively. The Froude number as a function of the frequency parameter, therefore, infers a reduced frequency or a Strouhal number which represents the ratio of the characteristic time of the mechanical system and the fluid-characteristic time. Finally, the manipulation of \(\pi_{6}\) and \(\pi_{7}\) by combining with \(\pi_{1}\) provides:  
\[ \begin{split}\pi_{6}^{\prime}&=\sqrt{\frac{\pi_{6}} {\pi_{1}}}=\sqrt{\frac{k}{m}}\frac{h}{v}=\frac{\omega_{n}}{\omega}=\omega_{r} \\ \pi_{7}^{\prime}&=\frac{\pi_{7}}{2\pi_{1}\pi_{6}^{ \prime}}=\frac{c}{2m\omega_{n}}=\xi_{d}\end{split}  \tag{16}\]  
Here, we employ the standard definitions of the structural natural frequency of excitation \(\omega_{n}\) and the structural damping ratio \(\xi_{d}\) for harmonic oscillators, where:  
\[\omega_{n}=\sqrt{\frac{k}{m}} \tag{17}\]  
and  
\[\xi_{d}=\frac{c}{2m\omega_{n}} \tag{18}\]  
Further, \(\omega_{r}\) denotes the reduced frequency of the system (structure and slosh). We are justified to choose \(h\) (tank height) as the characteristic length, as \(\pi_{9}\), \(\pi_{10}\) and \(\pi_{11}\) are derived by dividing all other relevant geometrical quantities by the characteristic length. The resulting \(\tilde{A}\) is the tank aspect ratio, \(\tilde{F}\) the liquid fill ratio and \(\tilde{Y_{0}}\) the dimensionless initial deflection of the mass-spring system. For the slosh case considered in this article, these quantities are \(O(1)\) and as a result the liquid centre of gravity displacements and wave height. The latter is a consequence of the slosh being within the physical bounds of the tank. The derivation of a functional relationship to characterise the fluid-structure-system-damping will employ the resulting \(\pi\) groups:  
\[\xi=f\big{(}\bar{m},\omega_{r},\bar{p},Re,Fr,We,\theta,\tilde{A},\tilde{F}, \tilde{Y}_{0}\big{)} \tag{19}\]  
The total system damping \(\xi\) can be decomposed into its two parts:  
\[\xi=\xi_{d}+\Delta\xi_{s} \tag{20}\]  
where \(\xi_{d}\) and \(\Delta\xi_{s}\) are the damping due the structure (dry) and slosh damping. To focus the analysis on the slosh induced damping via prescribed tank motion, we shall prescribe \(\xi\) (the chosen value will be motivated when used in the relevant sections to follow) and such that the derivation of a functional relationship characterising the slosh damping results in the form:  
\[\Delta\xi_{s}=F\big{(}\bar{m},\omega_{r},\bar{p},Re,Fr,We,\theta,\tilde{A}, \tilde{F},\tilde{Y}_{0},\xi\big{)} \tag{21}\]  
From the above, we can identify three distinct groups of dimensionless numbers influencing the slosh damping within the SDOF system. The three groups are flow physics (fluid specific non-dimensional numbers such as \(\bar{\rho}\), \(Re\), \(Fr\), \(We\) and \(\theta\)), fluid-structure interactions (FSI) and geometric non-dimensional factors.