## 3 Numerical Modelling in Single Pipelines
### 3D CFD Model vs. Mathematical Model  
In order to compare the 3D CFD model with the information provided by the mathematical model of [PERSON] et al. [18], a single pipe without air admission orifice was used as a case study. Tests C1-1 and C1-2 were simulated for 3 s, and Tests C2-1 and C2-2 were simulated until the pipes were drained. Case 1 tests were run up to an instant \(t=3\) s since, experimentally, the water column could not be drained due to subatmospheric pressures. In addition, numerical information of pressure oscillations of 3D CFD models were extracted with an internal point located at 0.105 m from the upper end of the pipeline (similar to the location of pressure transducer). The 3D CFD model was performed, which showed a variety of relevant information that these models offer under a three-dimensional domain, providing an approximation to real conditions. In this model, different hydraulic and thermodynamic phenomena, such as (i) deformation of the air-water interface, (ii) backflow air (DAPs), (iii) transient flows, (iv) velocity distribution curves in different sectors of the pipe due to the influence of backflow air and (v) velocity and temperature distribution in cross-sections were seen in detail (see these in more detail in the contribution of [PERSON] et al. [8]). The pressure oscillations of the mathematical model and the 3D CFD model were compared to show the similarity between the results, as shown in Figure 10.  
Figure 10: Comparison of pressure patterns of (**a**) Test C1-1, (**b**) Test C1-2, (**c**) Test C2-1 and (**d**) Test C2-2 (3D CFD model vs. mathematical model). In Test C1-1 (Figure 10a), a critical subatmospheric pressure occurred at \(t=0.64\) s, where the minimum value reached by the mathematical model was 8.34 m-H\({}_{2}\)O, and the value of minimum pressure in the CFD model reached a maximum value of 8.29 m-H\({}_{2}\)O. Subsequently, it was identified that the CFD model predicted the behaviour of the pressure oscillations recorded by the mathematical model which occurred after the minimum pressure value. It was possible to identify that the pressure oscillations in the CFD model presented a delay in their peaks and valleys with respect to the oscillations detected by the mathematical model, an effect that is mainly influenced by the backflow air phenomenon. On the other hand, in Test C1-2 (Figure 10b), with a lower valve opening degree than Test C1-1, a good agreement with the pressure oscillations of the mathematical model was presented, reaching a minimum pressure of 8.20 m-H\({}_{2}\)O. Tests C2-1 and C2-2 (scenarios with air admission orifices) were also compared, where CFD model results were similar to pressure patterns of the mathematical model. In Test C2-1 (Figure 10c), the emptying process resulted in an expansion of the trapped air, reaching a minimum pressure of 9.55 m-H\({}_{2}\)O at \(t=0.58\) s and then gradually rising until it reached atmospheric pressure again at \(t=16\) s, which coincided with the instant of time at which the pipe had been completely drained. On the other hand, Test C2-2 (Figure 10d) showed a decrease in trapped air pressure, down to a minimum value of 9.87 m-H\({}_{2}\)O at \(t=0.60\) s, then returning to atmospheric pressure after \(t=18\) s.  
In addition to pressure patterns, the water flow patterns obtained from the 3D CFD model were compared with the results of the mathematical model, using all tests of this scenario. Figure 11a shows the flow patterns of Test C1-1 of the CFD model, compared with the results of the mathematical model associated with the test in reference. Transient flow with higher amplitudes compared to the mathematical model were observed, except for the first flow peak, for which there was a maximum value of 0.29 L/s in the mathematical model, and a value of 0.22 L/s in the 3D CFD model. There were maximum peaks in the flow patterns of the mathematical model and the 3D CFD model of 0.08 and 0.11 L/s, respectively, in Test C1-2 (Figure 11b).