## 2 Numerical Modelling in Irregular Pipelines
### 2D CFD Model vs. Mathematical Model  
The CFD model represented a two-dimensional analysis of the emptying process of an irregular pipeline with air valves, for which it was numerically validated with experimental measurements. Additionally, the calibration of a geometric aspect ratio used for the simulation of the air admission orifice was performed, and the influence of the air valve sizing on the subatmospheric pressures, emptying velocity and water column \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Inlet** & **Outlet** & **Walls** \\ \hline \(p\) (Pa) & \(p_{inlet}=p_{atan}-0.5\mu^{2}\) & \(p_{outlet}=p_{atan}-0.5\mu^{2}-p\beta\Delta h\) & \(p_{walls}=p_{atm}\) \\ \(u\) (m/s) & \(u_{inlet}=u_{inlet,0}-\frac{p_{inlet}-p_{outlet}}{p_{initial}}\) & \(u_{outlet}=u_{outlet,0}-\frac{p_{outlet}-p_{outlet}}{p_{initial}}\) & \(u_{walls}=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mathematical equations for boundaries of the CFD model ([PERSON], 2013). Figure 2: CFD computational domain of an irregular pipeline. length was analysed. Numerical information of pressure and velocity was extracted with a point located in the upper end of the geometric domain of the 2D CFD model to evaluate pressure oscillations (similar to the location of the pressure transducer) and a point located in the right horizontal branch of the pipeline to evaluate water drainage velocity (similar to the location of the UDV device). This numerical information presented an outstanding numerical error margin, according to the contribution of [PERSON] et al. [22], where numerical errors ranged from 0.07% to 3.7%, which are acceptable values from the numerical point of view, as evidenced in the contributions made by [PERSON] et al. [20]. Mathematical models and 2D CFD models have in common the capability of obtaining good numerical results, which is evidenced by the pressure patterns in Figure 3 and in the drainage velocity patterns in Figure 4, corresponding to the simulation in tests presented in Table 1.  
With the information from the pressure patterns of the experimental model and the numerical models (mathematical and CFD models), it was possible to identify the minimum pressure value identified in the hydraulic system, which allowed us to observe the influence of the size of the trapped air pockets in the generation of negative pressures that can threaten the hydraulic system. In this case, Tests 1-1, 1-2, 1-3 and 1-4 reach minimum pressure values of 10.15, 10.19, 10.21 and 10.24 m of H\({}_{2}\)O (m-H\({}_{2}\)O), respectively. Additionally, it was possible to validate the use of the turbulence equations in the CFD model with the data on the water drainage velocity, taking into account the peak velocity values presented in all tests. Maximum values of the Reynolds number were identified in this hydraulic system using the Reynolds formula (Equation (10)):  
\[R_{e}=\frac{u_{max}\phi}{\
u_{w}} \tag{10}\]  
where \(\phi\) = 0.0514 m and \(\
u_{w}\) = 1 \(\times\) 10\({}^{-6}\) m\({}^{2}\)/s at \(T\) = 20 \({}^{\circ}\)C. Table 4 shows the Reynolds number results for the different tests of the irregular pipe, in which maximum values  
Figure 3: Comparison of pressure patterns of (**a**) Test 1-1, (**b**) Test 1-2, (**c**) Test 1-3 and (**d**) Test 1-4 (Experimental model, 2D CFD model and mathematical model). between 13,415 and 21,198 were present, which are part of the turbulent flow regime, validating the use of a turbulence model for the CFD modelling. On the other hand, the 2D CFD model shows information concerning air-water interface deformation, such as the air-water interface location, which can be obtained at different time instants, as opposed to the mathematical model. This component is corroborated and validated with video data from experimental tests. Figure 5 shows the air-water interface location of the 2D CFD model of Test 1-1, where a good agreement is shown between the 2D CFD model contours and the screenshots of the video data at different time instants.