## 3 Numerical Modelling in Single Pipelines
### 3D CFD Model vs. Mathematical Model
Figure 16 showed that the displacement of the air-water interface in the 2D and 3D CFD models was similar at \(t\) = 0, 1, 5, and 10 s, and a difference in the location of the air-water interface occurred at the instant \(t\) = 15 s, just before total drainage of the water in this test. In this sense, the prediction of the air-water interface can be carried out adequately in both the two- and three-dimensional conditions. On the other hand, in the 2D and 3D CFD models of Test C2-1, an evaluation of the spatial distribution of the velocity at the upper end of the pipe was carried out, corresponding to the air phase. In order to make an adequate comparison of the velocity gradients between the 2D and 3D CFD models, a cut of the geometry of the 3D CFD model was made on the XY plane to obtain the information of the two-dimensional profile of the pipe axis, as shown in Figure 17.  
After cutting on the XY plane, the comparison of the velocity gradients presented in the air admission zone of the single pipe of both CFD models was performed. Figure 18a shows the distribution of velocity produced by the 2D CFD model, where it can be seen that the air admission velocity reached its maximum value at \(t\) = 1.0 s and gradually decreased over time, as evidenced at \(t\) = 5.0, 10.0 and 15.0 s, a similar trend to the results presented by the 3D CFD model (Figure 18b). During air admission, three fronts of flow velocity were observed at the upper end of the pipe profile. These fronts reached velocity values between 7 and 10 m/s at the top wall, bottom wall and the central axis of the pipe (see Figure 18a at \(t\) = 0.5 and 1.0 s). On the other hand, in the 3D CFD model scenario, a predominant flow velocity front generated by the air admission orifice was observed. This front was attenuated towards the bottom of the upstream end of the pipe, reaching a magnitude greater than 10 m/s. In addition, a small velocity front was generated, which was seen at \(t\) = 0.5 and 1.0 s in the upper zone of the upstream end of the pipe (see Figure 18b). In that sense, the velocity contour results of both CFD models presented some discrepancies in terms of the generation of velocity gradients inside the geometrical domain, especially in complex areas such as the air admission orifice. From the above comparison, it was possible to verify some elements that are important to take into account to identify the behaviour of velocity gradients in 2D and 3D CFD models. In particular, 2D CFD models require a geometric aspect ratio to simulate the diameter of the air admission orifice to adjust the mass flow rate conditions, as has been used in previous research [22; 34; 35], whereas the 3D CFD model allows the generation of an orifice with a diameter equal to the study cases, being an appropriate condition in the evaluation  
Figure 15: Temperature patterns of the trapped air pocket of CFD model (Tests C1-1 and C1-2). of diameter changes in pipes. In this regard, the simulation of orifices in 2D CFD models requires a geometric aspect ratio that has been favoured in the representation of emptying processes with admitted air; however, it is an element that must be suitably adjusted during the development of the computational model, depending on the geometric conditions of the pressurised pipes. Figure 16: Comparison of air–water interaction of Test C2-1: (**a**) 2D CFD Model [23] and (**b**) 3D CFD Model. Figure 17: Plane for evaluation of velocity profile in single pipe—Test C2-1: (**a**) Normal 3D CFD model and (**b**) Cut-out of 3D CFD model. Figure 18: Comparison of velocity contours at upper end of Test C2-1: (**a**) 2D CFD Model [23] and (**b**) 3D CFD Model.