# Two-Phase Flow Simulations Using 1D Centerline-Based C- and U-Shaped Pipe Meshes
## 2 Numerical Methods
### Governing Equations  
A two-fluid model is mostly used to simulate dispersed flows. It permits multiple bubbles or droplets in a grid; hence, it can predict bubbly flows with a small number of grids [15]. The continuous Eulerian phases are treated as dispersed systems, and each phase has its own density and viscosity properties. The governing equations are as follows:  
\[\frac{\partial\alpha_{\rm i}}{\partial t}+\
abla\cdot(\alpha_{\rm i}\mathbf{u}_ {\rm i})=0, \tag{1a}\] \[\frac{\partial(\alpha_{\rm i}\rho_{\rm i}\mathbf{u}_{\rm i})}{\partial t}+\
abla \cdot(\alpha_{\rm i}\rho_{\rm i}\mathbf{u}_{\rm i}\mathbf{u}_{\rm i})=-\alpha_ {\rm i}\
abla p+\
abla\cdot(\alpha_{\rm i}\tau)+\alpha_{\rm i}\rho_{\rm i}\mathbf{ g}+\mathbf{f}_{\sigma\rm i}+\mathbf{M}_{\rm i}, \tag{1b}\]  
where \(\alpha_{\rm i}\), \(\mathbf{u}\), \(\rho_{\rm i}\), \(\rho\), \(\tau\), \(\mathbf{g}\), \(\mathbf{f}_{\sigma\rm i}\), and \(\mathbf{M}_{\rm i}\) are the volume fraction, flow velocity vector, fluid density, static pressure, viscous stress tensor, gravitational acceleration, surface tension force, and momentum transfer, respectively. Subscript \(i\) indicates the respective phases for liquid: 1 and gas: 2. On account of independent momentum equations, the velocity of each phase can be different at the cells, and the momentum transfer term \(\mathbf{M}_{\rm i}\) is necessary. We considered the drag, lift, and virtual mass force effects:  
\[\mathbf{M}_{\rm i}=\mathbf{M}_{\rm i,drag}+\mathbf{M}_{\rm i,lift}+\mathbf{M}_{ \rm i,VM}, \tag{2}\]  
where \(\mathbf{M}_{\rm i,drag},\mathbf{M}_{\rm i,lift}\), and \(\mathbf{M}_{\rm i,VM}\) are the momentum transfers caused by the effects of drag, lift, and virtual mass force, respectively. The momentum transfer caused by the Basset force was neglected, because the particle diameter herein was greater than 1 \(\mu\)m [16]. The drag force term is described as follows  
\[\mathbf{M}_{\rm i,drag}=\frac{3}{4}\alpha_{\rm i}\alpha_{\rm i}\left(\alpha_{ \rm j}\frac{C_{D,i}\rho_{i}}{d_{p,i}}+\alpha_{\rm i}\frac{C_{D,j}\rho_{i}}{d_{p,j}}\right)|\mathbf{u}_{\rm r}|u_{\rm r}, \tag{3}\]  
where \(C_{D,i}\), \(d_{p,i}\), and \(\mathbf{u}_{\rm r}\) are the drag coefficient, particle diameter, and relative velocity vector at the \(i\)th phase, respectively. Subscript \(j\) indicates the fluid phase counterpart of the phase indicated by \(i\). The lift force term is defined as follows:  
\[\mathbf{M}_{\rm i,lift}=\alpha_{\rm i}\alpha_{\rm i}(\alpha_{\rm i}C_{\rm L,i }\rho_{i}+\alpha_{\rm i}C_{\rm L,i}\rho_{i})\mathbf{u}_{\rm r}(\
abla\times \mathbf{u}_{\rm i}), \tag{4}\]  
where \(C_{L,i}\) and \(\mathbf{u}_{\rm i}\) are the lift force coefficient and the velocity vector of the liquid phase, respectively. The virtual mass force term is evaluated as  
\[\mathbf{M}_{\rm i,VM}=\alpha_{\rm i}\alpha_{\rm j}C_{VM}\rho_{j}\bigg{\{}\left( \frac{\partial\mathbf{u}_{\rm i}}{\partial t}+\mathbf{u}_{\rm j}\cdot\
abla \mathbf{u}_{\rm j}\right)-\left(\frac{\partial\mathbf{u}_{\rm i}}{\partial t} +\mathbf{u}_{\rm i}\cdot\
abla\mathbf{u}_{\rm i}\right)\bigg{\}}. \tag{5}\]  
The virtual mass coefficient \(C_{VM}\) was set to 0.5.