# Numerical Calculation of Slosh Dissipation
## 4 Volume-of-Fluid Finite Volume Method: The Elemental Solver
### Governing Equations for the Elemental Vertex-Centered FVM  
In this section, we present the governing equations used for the Elemental(r) vertex-centered, finite volume VOF solver [19]. They consist of equations for the conservation of mass, momentum and an advection equation for the VOF, \(\alpha\). The equations are written in integral form for a volume \(V_{i}\), which represents a finite volume cell in tank volume \(V\)  
\[\frac{1}{V_{i}}\int_{V_{i}}\left[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{\
abla }\cdot\rho\mathbf{u}\otimes\mathbf{u}\right]\mathrm{d}V=\frac{1}{V_{i}}\int_{V_{i}}[ \mathbf{\
abla}\cdot\mathbf{\tau}+\rho(\mathbf{g}+\mathbf{a})+\sigma\kappa\mathbf{u}_{a}\delta_{ \alpha}]\,\mathrm{d}V \tag{15}\]  
Figure 3: Schematic illustrating a non-inertial simulation domain in reference frame \(x,y,z\), undergoing linear acceleration \(\mathbf{a}_{d}\). The simulation domain is represented by volume \(V\) with bounding surface \(\Omega\). There is a two-phase flow inside the domain, with \(\delta_{\alpha}\) the interface between fluids.  
\[\frac{1}{V_{i}}\int_{V_{i}}\mathbf{\
abla}\cdot u\,\mathrm{d}V=\frac{1}{V_{i}}\int_{V_{ i}}\frac{(\alpha-1)}{\rho_{g}}\frac{1}{c_{g}^{2}}\frac{\partial p}{\partial t}\, \mathrm{d}V\,, \tag{16}\]  
with \(\rho\) the density, \(\mathbf{u}\) the fluid velocity relative to the tank, \(\mathbf{g}\) gravitational acceleration and \(\mathbf{a}\) the acceleration of the tank. The last term in (15) is the estimation for the surface force due to surface tension, with \(\delta_{a}\) a Dirac delta function defined on the VOF interface, \(\mathbf{n}_{a}\) the interface normal, \(\sigma\) the surface tension coefficient and \(\kappa\) the interface curvature. In (16), a weakly compressible gas formulation [20] is used, with \(\alpha\) the VOF, which is defined in volume \(V_{i}\) as a volume weighted tracer of the liquid phase \(H_{\alpha}\), so that \(H_{\alpha}=1\) is in the liquid and \(H_{\alpha}=0\) is in the gas phase  
\[\alpha_{i}=\frac{1}{V_{i}}\int_{V_{i}}H_{\alpha}\,\mathrm{d}V\,. \tag{17}\]  
Further, \(c_{g}\) is the acoustic velocity in the gas phase. The fluid density and viscosity are represented as \(\rho\) and \(\mu\). Throughout this work, a harmonic mean is used for the fluid viscosity, assuming constant values in each phase. The density within a finite volume \(V_{i}\) is calculated as an average using the VOF  
\[\rho=\alpha\rho_{l}+(1-\alpha)\rho_{g}\,, \tag{18}\]  
where the liquid density \(\rho_{l}\) is a constant in this work, and the gas density \(\rho_{g}\) is calculated assuming an ideal gas law with adiabatic compression such that  
\[\rho_{g_{2}}=\rho_{g_{1}}\bigg{(}\frac{p_{2}}{p_{1}}\bigg{)}^{(1/\gamma)}\,. \tag{19}\]  
Here, \(\gamma=\frac{c_{p}}{c_{v}}=1.4\) for air, with \(c_{p}\) and \(c_{v}\), respectively, the specific heat capacity at constant pressure and volume. The gas acoustic velocity is  
\[c^{2}=\frac{\gamma p}{\rho_{g}}\,, \tag{20}\]  
where for dry air at Normal Temperature and Pressure (NTP) \(\rho=1.205\) kg\(\cdot\)m\({}^{-3}\) and \(p=101.325\) kPa, hence \(c_{g}=343\) m\(\cdot\)s\({}^{-1}\).