## 2 Physical Modelling and Numerical Methodology
### Numerical Validation Study  
Since the calculation of the ad hoc Stokes numbers (see Section 2.3) is based on parameters obtained by simulation, the numerical models were thoroughly validated with previously published experimental data [4; 6; 8; 10]. It must be taken into account that the Stokes numbers were formulated under characteristic macroscales, so the important thing is to solve well the large turbulent structures, that is, the Dean vortices and the smaller but considerable scale vortices that form near the nucleus of the pipe because of the local imbalance between the centrifugal force and the pressure gradient [11]. The fluid phase was first validated by replicating the pressure drop in the work of [PERSON] et al. for gas flow through a curved pipe with \(\mathrm{Rc}=2\mathrm{D}\)--\(\mathrm{Re}=61\),500 [4] and then with the velocity profiles exposed by [PERSON] [8] and [PERSON] [10]. However, as mentioned, the RNG \(k\)-\(\varepsilon\) model was used as turbulence closure, due to the better performance shown in the studies by [PERSON] et al. compared to other linear and nonlinear parasitic viscosity turbulence models for curved pipe flows [8], and although there is an inherent problem with models of this type because they do not capture the anisotropic nature of turbulence in this type of flow, this can be circumvented by the benefits of the RNG \(k\)-\(\varepsilon\) model that allows precise closure of large structures for the specific problem, where the RNG \(k\)-\(\varepsilon\) model adds an additional term in its equation \(\varepsilon\) that improves the accuracy of rapidly deformed flows, as well as including  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Properties** & **C11** & **C12** & **C13** & **C21** & **C22** & **C23** \\ \hline \(t_{e}\) [s] & 1 & 1 & 1 & 0.75 & 0.75 & 0.75 \\ \(t_{r}\) [s] & 1.5 & 1.5 & 1.5 & 1 & 1 & 1 \\ \(t_{m}\) [s] & 2 & 2 & 2 & 2 & 2 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation times. \(t_{e}\) = time for a statistically steady flow, \(t_{r}\) = travel time for a steady flow of particles y \(t_{m}\) = sampling time.  
improvements in the precision of parasitic flows and provides an analytical formula for turbulent Prandtl numbers. Finally, as turbulence is generally affected by rotation or eddy in the medium flow, the RNG model in ANSYS Fluent provides an option (Eddy Modification) to account for the effects of eddy or rotation by modifying the turbulent viscosity of properly.  
The discrete phase was validated by comparing the experimental concentration results obtained by [PERSON] et al. in gas-solid flows with Rc = 1.8D and Re = 75,327 for mass loads of 0.35 and 0.42. The particle size distribution was carried out as detailed in reference ([PERSON] et al., 2018). In the same way that was commented for the fluid phase, there is the problem that by not considering the turbulent fluctuations of the scales not captured by the model, the process of turbulent Lagrangian particle diffusion is significantly affected. However, as explained in the work of ([PERSON] et al., 2018), that although for a long-term diffusion, we cannot avoid the possibility that the particle escapes from its originally \"surrounding\" fluid, resulting in a slightly eddy diffusion coefficient reduced particle relative to fluid. This effect is reduced for an increase in \(\rho_{p}/\rho_{f}\), becoming insignificant for a density ratio of 1000. Furthermore, it should be noted that from the delivery results by [PERSON] et al., it is seen that, although there are variations between the SLE and the RANS results, both can predict the general trends of the string formation observed during the experiments. Thus confirming that the simulations account for the physics of the flow as observed in the experiments.