## 4 Discussion  
Numerical models have been useful over the years to facilitate the prediction of hydraulic phenomena in pipelines, serving as an adequate solution to hydraulic engineering problems. The use of these models depends on different user-determined factors, such as (i) physical equations, (ii) level of detail of the information, (iii) use of computational resources, (iv) calibration process and (v) computational simulation time. For example, mathematical models have provided good numerical results for pressure oscillations and discharge water flow in emptying processes, which have also been used in large-scale pipelines through the manipulation or control of the associated physical terms that may be considered in the water and air phases [19; 36]. However, these models present restrictions in their use, taking into account that some complex hydraulic and thermodynamic phenomena cannot be predicted in detail or must be assumed in these hydraulic events.  
On the other hand, some authors who have developed CFD models to study two-phase transient flows (including emptying operations) highlight the relevant information that these complex models can predict and that the quality of the visual and numerical information is more consistent with what occurs in different experimental cases [8; 21; 33; 37; 38]. However, authors who have used CFD models mentioned the complexity of the definition and calibration of these models, as well as the demanding process and computational time required, especially in three-dimensional models, where elements such as (i) the order of solution of numerical schemes, (ii) the algorithm of solution, (iii) turbulence models, (iv) meshing quality, (v) time step and (vi) parallel processing conditions influence the achievement of good numerical and physical results, along with optimisation of computational time.  
Detailed knowledge of the advantages and disadvantages of using the different existing numerical models for the study of emptying processes in pressurised pipes could contribute to the appropriate selection of these tools for the solution of hydraulic engineering problems in real pipelines, in addition to verifying and validating possible operational scenarios, including water distribution network design activities. Table 8 defines the advantages and disadvantages of the use of mathematical, 2D CFD and 3D CFD models for the solution of hydraulic engineering problems, focused on the study of emptying processes in pressurised pipes with trapped air, considering the presence or absence of air valves.  
Some criteria should be taken into account for the selection of the appropriate numerical model for different studies at the hydraulic engineering level. For example, mathematical models can be applied in cases where hydraulic engineers are only interested in understanding the evolution of certain hydraulic and thermodynamic parameters over time, as well as to identify maximum or minimum values for design and operational efficiency purposes. On the other hand, 2D CFD models can be useful in hydraulic scenarios where it is necessary to identify the motion of the air-water interface and the spatial variation in different variables such as pressure, velocity and temperature in the form of contours on a pipe profile plane (2D plane), in addition to the numerical evolution of these variables at various specific points of the geometric domain over time. Furthermore, 3D CFD models can be adequate in cases where hydraulic engineers require in-detail study of the behaviour of the trapped air volume and its interaction with water installations, in order to obtain a better capture of different hydraulic and thermodynamic phenomena, as well as to observe the evolution of inlet and outlet flows and visualisation of the physical evolution of the variables associated with these fluids using the contours of CFD.