## 2 Numerical Modelling in Irregular Pipelines
### 2D CFD Model vs. Mathematical Model
Concerning the phase fraction of the 2D CFD model (Figure 5), the separation between the two fluids presented an adequate stratification between the trapped air pocket and the water phase, since there was no collapse of this interface. Different authors in the literature developed mathematical models to study the effect of air admission orifices in different emptying processes to predict the evolution of subatmospheric pressures as a function of the sizing of these protection devices. In addition to predicting the evolution of air pocket pressure patterns under these scenarios, CFD modelling allows us to visualise the dynamic behaviour of the admitted air and its aerodynamic ratio with the entrapped air pocket that expands during drainage events. Figure 6 shows the behaviour of the air pocket and the air-flow admitted by the air valve through streamlines, corresponding to Test 1-1. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Test** & \(\mathbf{u_{max}}\) **(Exp.)** & \(\mathbf{R_{e}}\) **(Exp.)** & \(\mathbf{u_{max}}\) **(Math.)** & \(\mathbf{R_{e}}\) **(Math.)** & \(\mathbf{u_{max}}\) **(CFD)** & \(\mathbf{R_{e}}\) **(CFD)** \\ \hline
1-1 & 0.391 & 20,097 & 0.398 & 20,467 & 0.412 & 21,198 \\
1-2 & 0.317 & 16,294 & 0.358 & 18,395 & 0.358 & 18,410 \\
1-3 & 0.313 & 16,088 & 0.325 & 16,681 & 0.323 & 16,622 \\
1-4 & 0.261 & 13,415 & 0.286 & 14,705 & 0.282 & 14,486 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum water drainage velocities and Reynolds number obtained in the emptying tests of an irregular pipe (\(u_{max}\) in m/s). Figure 4: Comparison of velocity 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). Initially, the pipe is filled with water (there is a small air pocket in the upper zone of the hydraulic system). During the emptying process, air is admitted through the air valve. Subsequently, the air-flow is damped by its impact with the air-water interface at \(t\) = 0.25 s. From \(t\) = 0.75 s, the air-flow is dispersed towards the pipe branches as a consequence of the impact with the lower vertex of the branch junction. On the right branch of the hydraulic system, an air vorticity phenomenon occurs near the air-water interface from \(t\) = 1.0 s. On the other hand, in the left branch, the air-flow is displaced over the lower zone of the pipe cross-section and then damped at the air-water interface. The air-flow velocity inside the pipe tends to decrease as it approaches the air-water interface. At \(t\) = 1.25 s, the air-flow velocity admitted in the air valve zone tends to values higher than 20 m/s. Inside the vertex  
Figure 5: Air–water interface location in 2D CFD model and experimental test (Test 1-1). Figure 6: Evolution of the air inflow during the emptying process of the irregular pipeline. of the hydraulic system, the air-flow velocity attains values between 5 and 10 m/s, and in the branches of the pipeline, the value of the air-flow velocity decreases to values less than 5 m/s. In this sense, aerodynamic flow in emptying processes of pipelines with air valves represents a significant contribution to verify the design of air valves and their influence on the durability of the hydraulic system, being a physical phenomenon that cannot be identified by mathematical models. In general, 2D CFD models show velocity, pressure and temperature contours as additional information to the visual information of air-water interface. The 2D CFD model is a more robust model compared to the mathematical model; therefore, it requires more computational time, on the order of minutes up to hours, depending on the conditions of the processor performing such simulations. It is important that the user uses a multi-core processor to facilitate parallel processing. This procedure is one of the determining factors to reduce computational time.