## 2 Numerical Modelling in Irregular Pipelines
### Two-Dimensional CFD Model  
A two-dimensional CFD model was developed by [PERSON] et al. [22] for the numerical resolution of compressible multiphase fluids using OpenFOAM software v2012 of CFD Direct (ICL, Saltburn-by-the-Sea, UK) [26]. Different physical equations were used, such as (i) fluid equations (continuity for compressible fluids, momentum and energy) based on Navier-Stokes equations for compressible flows, (ii) transport equations (Partial Volume Fluid Model (PVoF)) [27; 28], and (iii) thermodynamic equations for air  
\begin{table}
\begin{tabular}{l l} \hline Mass Oscillation No. 1 & \(\frac{du_{e}}{dt}=\frac{p_{a}-p_{atm}}{\rho_{a}-L_{e,1}}+g\frac{\Delta z_{e,1} }{L_{e,1}}-f_{1}\frac{|u_{e,1}|}{2\phi}-\frac{R_{v}A_{1}^{2}u_{e,1}|u_{e,1}|}{ L_{e,1}}\) \\ \multirow{2}{*}{Emptying Column No. 1} & \(\frac{dL_{e,1}}{dt}=-u_{e,1}(L_{e,1}0-\int_{0}^{t}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!phase. Equations (1) and (2) show the formulation of mixing density and viscosity using a phase fraction of water (\(\alpha_{w}\)), and Equation (3) corresponds to the transport equation of PVoF:  
\[\rho=\alpha_{w}\rho_{w}+(1-\alpha_{w})\rho_{a} \tag{1}\]  
\[\mu=\alpha_{w}\mu_{w}+(1-\alpha_{w})\mu_{a} \tag{2}\]  
\[\frac{\partial\alpha_{w}}{\partial t}+\
abla\cdot(\alpha_{w}\mathbf{u})+\
abla \cdot((1-\alpha_{w})\alpha_{w}u_{r})=0 \tag{3}\]  
where \(\mu\) = dynamic viscosity, \(\mathbf{u}\) = velocity vector, and \(u_{r}\) = velocity source. Fluid equations are fundamental to represent the fluid dynamics during their physical interaction in the geometrical domain, with Equation (4) being the principle of conservation of mass and Equation (5) being the principle of conservation of momentum:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{u})=0 \tag{4}\]  
\[\frac{\partial(\rho\mathbf{u})}{\partial t}+\
abla\cdot(\rho\mathbf{u} \mathbf{u})=-\
abla p+\
abla\cdot(\mu\
abla\mathbf{u})+\rho\mathbf{g}-F_{s} \tag{5}\]  
where \(p\) = static pressure and \(F_{s}\) = surface tension. To guarantee the thermodynamic and mechanical interaction of the fluids, the energy conservation equation (Equation (6)) and the equation of state of air (Equation (7)) were added. On the other hand, the density of water was simulated as a constant value, because changes in density with respect to physical changes in mechanical and thermodynamic origin can be considered negligible in this fluid:  
\[\frac{\partial(\rho e)}{\partial t}+\
abla\cdot(\rho e\mathbf{u})=\
abla \cdot\mathbf{q}+p(\
abla\cdot\mathbf{u})+S_{T} \tag{6}\]  
\[\rho_{a}=\frac{p_{a}}{RT_{a}} \tag{7}\]  
where \(e\) = specific energy, \(\mathbf{q}\) = heat flux and \(S_{T}\) = energy source term. Turbulence equations were based on the SST \(k\)-\(\omega\) model of [PERSON] [29; 30] complemented with wall functions [31]. The turbulence equations of SST \(k\)-\(\omega\) were used based on the advantages offered by these equations in the simulation of turbulent effects, especially those of admitted air, i.e., (i) adequacy in aerodynamic flows, (ii) presence of adverse pressure and velocity gradient, and (iii) adequate representation of the boundary layer (logarithmic region and viscous sub-layer), being an analysis with low sensitivity to dimensionless distance (\(y^{+}\)).