## 3 CFD Modelling as Digital Twin
### CFD Models for Filling and Emptying Processes
#### 3.3.1 Turbulence Equations
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Research**} & \multirow{2}{*}{**Standard \(k\)-\(e\)**} & \multicolumn{3}{c}{**Turbulence Models**} \\  & & **RNG \(k\)-\(e\)** & **Realizable \(k\)-\(e\)** & **SST \(k\)-\(\omega\)** \\ \hline [PERSON] and [PERSON] [26] & ✓ & & & \\ \hline [PERSON] et al. [27] & ✓ & & & \\ \hline [PERSON] et al. [28] & & & ✓ & \\ \hline [PERSON] et al. [30] & & & ✓ & \\ \hline [PERSON] et al. [4] & & ✓ & & \\ \hline [PERSON] et al. [32] & ✓ & & & \\ \hline [PERSON] et al. [33] & ✓ & & & \\ \hline [PERSON] et al. [24] & ✓ & & & \\ \hline [PERSON] et al. [5] & ✓ & & & \\ \hline [PERSON] et al. [34] & ✓ & & & \\ \hline [PERSON] et al. [29] & & & ✓ & \\ \hline [PERSON] et al. [55] & & & ✓ \\ \hline [PERSON] et al. [56] & & ✓ & \\ \hline [PERSON] et al. [35] & & ✓ & & ✓ \\ \hline [PERSON] et al. [54] & & ✓ & \\ \hline [PERSON] et al. [36] & & ✓ & \\ \hline [PERSON] et al. [6] & & ✓ & \\ \hline [PERSON] et al. [57] & ✓ & & \\ \hline [PERSON] et al. [58] & & ✓ & \\ \hline [PERSON] et al. [59] & & ✓ & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summary of turbulence models for CFD models. carried out using the 2D CFD model of [PERSON] et al. [54], which consists of an emptying process through a branched pipe that contains an air valve. The initial conditions of this emptying process are based on a small air pocket (approximately zero (0) metres) in the upper zone of the pipeline, a total valve opening at both ends of the pipe, and a ball valve opening time of 1.6 s.  
Figure 5 shows the upper zone of the pipeline during an emptying process of the water phase (Test No. 12--Type V (Emptying procedures with a D040 air valve)). Initially, no turbulence processes occur because the fluids (air and water) are at rest. After opening the ball valves gradually, the water starts to drain, the air pocket begins to expand, and the air is admitted through the air valve. Although drainage occurs in the pipe, the turbulent kinetic energy occurs to a lesser degree in the critical zone of the pipe with values between 0 and 0.2 m\({}^{2}\)/s\({}^{2}\) in the study area of the network and between \(t\) = 0.00 s and \(t\) = 2.00 s. On the other hand, the dissipation rate generated in the air admission zone gradually increased from the instant \(t\) = 0.00 s as the air was admitted into the pipe, reaching values of 0.10 m\({}^{2}\)/s\({}^{3}\). Meanwhile, in the pipe's fluid domain, the dissipation rate's value remained between 0 and 0.02 m\({}^{2}\)/s\({}^{3}\) from \(t\) = 0.00 s to \(t\) = 2.00 s.  
Figure 6 shows the turbulent kinetic energy and the turbulence dissipation frequency contours associated with the SST \(k-\omega\) model during the emptying process of a water phase (Test No. 12--Type V (Emptying procedures with a D040 air valve)). In this case, an increase in the turbulent kinetic energy is noticeable in the air admission zone and throughout the air phase, with a value greater than 1.0 m\({}^{2}\)/s\({}^{2}\). The turbulent kinetic energy  
Figure 5: Spatio-temporal evolution of turbulent kinetic energy (\(k\)) and dissipation rate (\(\epsilon\)) during an emptying process. generated during the emptying procedure increases in spatial distribution as the air pocket expands. This expansion is driven by the mixing of the aerodynamic airflow inside the pipe during air admission, which causes an increase in the kinetic energy of the air mixture as the water is drained out of the pipe. Furthermore, the dissipation frequency is noticeable throughout the air phase, especially in the pipe wall areas, where turbulence dissipates due to fluid-pipe interaction, with values greater than 3000 s\({}^{-1}\). Similar values are located in the air admission zone.