# CFD Based Non-Dimensional Characterization of Energy Dissipation Due to Verticle Slosh
## 3 Energy Analysis  
We next derive the relations employed to estimate slosh induced energy dissipation. For the system under consideration the mechanical energy (\(E_{M}\)) budget is considered for this purpose. This is composed of the work done on the fluid by the domain boundary surfaces or surface work (\(W_{\rm s}\)), deformation work (\(E_{Defo}\)), kinetic energy (\(E_{Kine}\)), potential energy (\(E_{Pote}\)), viscous dissipation (\(E_{visc}\)), and surface tension energy (\(E_{ST}\)). The full energy budget is defined by  
\[ \begin{split} E_{M}=W_{\rm s}=& E_{Defo}+E_{Kine}+E_{Pote}+E_{visc}+E_{ST}\\ =& E_{Disp}+E_{Kine}+E_{Pote}+E_{ST}\end{split}  \tag{2}\]  
where \(E_{Disp}=E_{Defo}+E_{visc}\) and \(E_{Defo}\) includes losses due liquid impact (as described next). Consider an enclosed fluid domain containing an immiscible liquid-gas mixture (gas is weakly compressible) undergoing a rigid body linear acceleration of \(\mathbf{a}_{d}\). The resulting absolute fluid velocity \(\mathbf{u}_{a}\) (velocity relative to earth) can be defined as the sum of the domain velocity (\(\mathbf{u}_{d}\)) and the fluid velocity (\(\mathbf{u}\)) relative to the non-inertial reference frame (CFD mesh reference frame), therefore \(\mathbf{u}_{a}=\mathbf{u}+\mathbf{u}_{d}\). The momentum conservation of the fluid is given by:  
\[ \begin{split}\frac{D}{Dt}(\rho\mathbf{u}_{a})&= \left[\frac{\partial\rho\mathbf{u}}{\partial t}+\mathbf{\
abla}\cdot(\rho\mathbf{u }\otimes\mathbf{u})+\rho\mathbf{a}_{d}\right]\\ &=-\mathbf{\
abla}p+\mathbf{\
abla}\cdot\mu\Big{(}\mathbf{\
abla}\mathbf{u}+ \mathbf{\
abla}\mathbf{u}^{T}\Big{)}+\rho g+\gamma\kappa\mathbf{\
abla}\alpha\end{split}  \tag{3}\]  
where \(p\) is pressure, \(t\) is time, \(\alpha\) is VoF volume fraction and the fluid properties are defined in Table 2. Mechanical energy conservation results from taking the dot product of Equation (3) with the volume averaged \(\bar{\mathbf{u}}_{a}\) followed by integration over space and time (includes integration by parts of the pressure term):  
\[ \begin{split}\int_{0}^{t}\bar{\mathbf{u}}_{a}\cdot\int_{\mathcal{ V}}&\bigg{[}\frac{\partial\rho\mathbf{u}}{\partial t}+\mathbf{\
abla} \cdot(\rho\mathbf{u}\otimes\mathbf{u})+\rho\mathbf{a}_{d}\bigg{]}dVdt=\\ &\qquad-\int_{0}^{t}\mathbf{u}_{d}\cdot\int_{\mathcal{A}_{d}}pn \mathit{Ad}dt\\ &\qquad+\int_{0}^{t}\bar{\mathbf{u}}_{a}\cdot\int_{\mathcal{V}} \Big{[}\mathbf{\
abla}\cdot\mu\Big{(}\mathbf{\
abla}\mathbf{u}+\mathbf{\
abla}\mathbf{u}^ {T}\Big{)}\Big{]}dVdt\\ &\qquad+\int_{0}^{t}\bar{\mathbf{u}}_{a}\cdot\int_{\mathcal{V}} \rho gdVdt+\int_{0}^{t}\bar{\mathbf{u}}_{a}\cdot\int_{\mathcal{V}}\gamma \kappa\mathbf{\
abla}adVdt\end{split}  \tag{4}\]  
where volume averaged quantities are defined as \(\overline{\phi}=\frac{1}{\mathcal{V}}\int_{\mathcal{V}}\phi d\mathcal{V}\), \(\mathcal{A}\) is the surface of all mesh finite volumes \(\mathcal{V}\), and \(\mathbf{n}\) is the outwards pointing normal. Further \(\mathcal{A}_{d}\) is the fluid  
Figure 5: Comparison of CFD predicted and experimentally derived slosh induced energy dissipation. domain bounding surface and \(\
abla\cdot\bar{\mathbf{u}}_{a}=0\) due the domain being assumed rigid.