## 2 Methodology
### Governing Equations  
In this research, the Level-Set (LS) method coupled with the Volume of Fluid (VOF) method is used to simulate air-water two-phase flow in a horizontal pipe. Both LS and VOF methods belong to one fluid method, where a single set of conservation equations are solved. Both methods have their advantages and disadvantages, but when coupled together, the results are superior in comparison to the standalone LS or VOF method [30]. Since both methods belong to the single fluid method, there is no separated gas and liquid velocity and the continuity and momentum conservation equations are given by Equations (2) and (3), respectively. Equation (4) is used to calculate volume fraction of gas, \(\alpha_{G}\). Therefore, the volume fraction of the liquid can be computed as \(\alpha_{L}=1-\alpha_{G}\).  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{U})=0 \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho\mathbf{U})+\
abla\cdot(\rho\mathbf{U}\otimes \mathbf{U})=-\
abla p+\
abla\cdot\left[\mu(\
abla\mathbf{U}+\
abla\mathbf{U}^ {T})\right]+\rho\mathbf{g}+\mathbf{F} \tag{3}\]  
\[\frac{\partial\alpha_{G}}{\partial t}+\mathbf{U}\cdot\
abla\alpha_{G}=0 \tag{4}\]  
where \(t\) is time, \(\mathbf{U}\) is the fluid velocity vector, \(\rho\) is fluid density, \(p\) is static pressure, \(\mu\) is fluid viscosity, \(\mathbf{g}\) is the gravitational acceleration, and \(\mathbf{F}\) represents external body forces. Fluent evaluates body force as continuum surface force (CSF). Density and viscosity can be calculated using Equations (5) and (6).  
\[\rho=\sum_{k=1}^{n}\alpha_{k}\rho_{k} \tag{5}\]  
\[\mu=\sum_{k=1}^{n}\alpha_{k}\mu_{k} \tag{6}\]  
For mixture turbulence equations, the realizable \(k\)-\(\epsilon\) model was used where \(k\) denotes the turbulent kinetic energy and \(\epsilon\) is its rate of dissipation. The realizable \(k\)-\(\epsilon\) model differs from the standard \(k\)-\(\epsilon\) model in two important manners: it offers modified equations to evaluate dissipation rate \"\(\epsilon\)\" and it contains a new formulation for turbulent viscosity. The ANSYS fluent theory guide [31] reported that the realizable \(k\)-\(\epsilon\) model shows considerable improvements over other \(k\)-\(\epsilon\) models. Governing equations for the realizable turbulence model are shown in Equations (7) and (8).  
\[\frac{\partial}{\partial t}(\rho k)+\
abla\cdot(\rho k\mathbf{U})=\
abla\cdot \left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\
abla k\right]+G_{k}+G_{b}- \rho_{\epsilon}-Y_{M}+S_{k} \tag{7}\]  
\[\frac{\partial}{\partial t}(\rho\epsilon)+\
abla\cdot(\rho\mathbf{U}\epsilon )=\
abla\cdot\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\
abla\epsilon \right]+\rho C_{1}S_{\epsilon}-\rho C_{2}\frac{\epsilon^{2}}{k+\sqrt{\
u_{ \epsilon}}}+C_{1\epsilon}\frac{\epsilon}{k}C_{3\epsilon}G_{b}+S_{\epsilon} \tag{8}\]  
\[C_{1}=\max\left[0.43\frac{\eta}{\eta+5}\right] \tag{9}\]  
\[\eta=S\frac{k}{\epsilon} \tag{10}\]  
\[S=\sqrt{2S_{ij}S_{ij}} \tag{11}\]  
\"\(G_{b}\)\" and \"\(G_{k}\)\" is the generation of turbulence kinetic energy due to the buoyancy and mean velocity gradients, respectively, \"\(Y_{M}\)\" is the contribution of dilatation to the overall dissipation rate, \"\(C_{1},C_{2},C_{1\epsilon},C_{2\epsilon}\)\" are constants, while \(\sigma_{k}\) and \(\sigma_{\epsilon}\) are turbulent Prandtl numbers, \(S_{k}\) and \(S_{\epsilon}\) are user defined source terms, \(\eta\) is an expansion parameter defined by Equation (10) [31], and \(S_{ij}\) represents the strain rate tensor.