## 3 CFD Modelling as Digital Twin
### CFD Models for Filling and Emptying Processes
#### 3.3.1 Turbulence Equations  
The motion of fluids in closed ducts results in a physical interaction between the fluid and the pipe walls, determining the fluid velocity distribution and mixing processes between fluids. CFD models have the advantage of representing the effects of fluid-pipe wall interactions from turbulence models [41]. In the literature, authors of different studies have highlighted the use of the RANS methodology, based on a statistical version of the instantaneous Navier-Stokes equations, obtained by applying a Reynolds average [42]. The turbulence models commonly used in CFD modelling are related to the eddy viscosity models, among which the \(k\)-\(\epsilon\) turbulence model is one of the most applied in two-phase transient flows. This turbulence model is based on turbulent kinetic energy generation (\(k\)), which consists of the energy associated with velocity fluctuations in turbulent flow and represents the amount of energy stored in the eddies or the eddies that are part of the turbulent regime of a fluid. On the other hand, the dissipation rate (\(\epsilon\)) represents the development of turbulent phenomena in a far zone from the walls, where vorticity processes are evident [43]. The turbulence model \(k\)-\(\epsilon\) describes two transport equations for \(k\) (see Equation (8)) and for \(\epsilon\) (see Equation (9)) [44; 45]. \[\frac{D(\rho k)}{Dt}=\
abla\cdot(\rho D_{k}\
abla k)+\rho G-\frac{2}{3}\cdot \rho k(\
abla\cdot\mathbf{u})-\rho\epsilon \tag{8}\]  
\[\frac{D(\rho\epsilon)}{Dt}=\
abla\cdot(\rho D_{\epsilon}\
abla\epsilon)+c_{1} \rho G\frac{\epsilon}{k}-\frac{2}{3}\rho c_{1}\cdot\epsilon(\
abla\cdot \mathbf{u})-c_{2}\rho\frac{\epsilon^{2}}{k} \tag{9}\]  
where the term \(G\) generates turbulence kinetic energy; \(D_{k}\) and \(D_{\epsilon}\) are the diffusion terms concerning to \(k\) and \(\epsilon\), respectively; and coefficients \(c_{1}=1.44\), and \(c_{2}=1.92\)[43]. Most researchers use this turbulence model in its standard form and RNG models. Authors such as [PERSON] et al. [4] used the \(k\)-\(\epsilon\) model to simulate a rapid filling process, in which turbulence is perceptible and well developed. [PERSON] et al. [5] used this turbulence model due to good performance, adequate accuracy, and efficient computational time for simulating the drainage of water pipelines. In addition to these investigations, other authors have used this turbulence model to study different hydraulic scenarios with trapped air  
Some authors have opted for alternative turbulence models to the standard \(k-\epsilon\), such as the RNG \(k-\epsilon\) and the Realisable \(k-\epsilon\) models, which have several significant differences concerning the standard \(k-\epsilon\) model. In the case of (\(k\)) in the RNG model, it is included as an additional term that considers the curvature of the flow, improving the representation in non-isotropic flows, while the dissipation equation (\(\epsilon\)) incorporates adjustments to reflect additional effects such as pressure and velocity gradient. On the other hand, the Realisable \(k-\epsilon\) turbulence model includes an improvement in the restriction of \(\epsilon\) values thus providing more realistic turbulence in the CFD models by omitting dissipation rates with exaggerated or otherwise negative values. Table 3 shows the main features of the standard, Realisable, and RNG variants of the \(k-\epsilon\) model. Authors such as [PERSON] et al. [48] performed CFD simulations on a pipeline using a three-dimensional model, in which it was shown that the standard \(k-\epsilon\) model offered more accurate numerical results for pressure and velocity compared to the numerical results using the RNG and Realizable \(k-\epsilon\) models. In that sense, of the 11 investigations that used at least one variant of the \(k-\epsilon\) turbulence model, 7 of them applied the standard \(k-\epsilon\) model (64%) thus demonstrating the confidence of some authors in using the standard model.