## 2 Physical Modelling and Numerical Methodology
### Numerical Validation Study
#### 2.2.1 Validation of the Fluid Phase  
We compared the pressure distribution (through the pressure coefficient), axial velocity profiles (upstream of the bend) and velocity iso-contours in cross sections through the bend, with the results reported ([PERSON] et al., 2018; [PERSON] et al., 2018). * Pressure Coefficient (Cp)  
The pressure coefficient values are shown in Figure 2. The maximum error found by the RNG \(k\)-\(\epsilon\) model is 9.7% (on the inner side of the curve), much lower than the 26% of the RSM and standard \(k\)-\(\epsilon\) models, and there is a relative \(<\)1% in the maxima of the Cp located at 30\({}^{\circ}\) on the inner side and 60\({}^{\circ}\) on the outer side of the bend. The results obtained for the pressure field were also satisfactory, with the RNG \(k\)-\(\epsilon\) model performing better than the models RSM and \(k\)-\(\epsilon\) reported by [PERSON] et al. (2018). Figure 2: Comparison between experimental (**markers**) and estimated (**lines**) pressure drop values using different models. The experimental values, RSM model and \(k\)-\(\epsilon\) are those reported by [PERSON] et al. (2018). Figure 3 shows the axial velocity profiles. Uz, r, and z are the axial flow velocity, the pipe radius, and the distance along the axial direction, respectively. For comparison, the coordinates were normalized with respect to the pipe diameter (D) and the average velocity (Ub). The numerical values are consistent with experimental results [8; 10]. A small underestimation of the profile is present on the adjacent side of the inner face of the upstream curve between z/D = 0 and z/D = 2 (see lower part of the graph). * Secondary Flow  
A comparison of the flow patterns obtained from the numerical simulations and experiments reported by [PERSON] et al. (Re = 60,000--Rc = 2D) are presented in Figure 4. Velocity contours normalized by the average flow velocity (\(\overline{W}\)/Wa) are shown as isolines that allow visualizing the pattern and behavior of the secondary flow (Dean vortices), at different angles of the curve, as well as at z/D distance upstream and downstream of the elbow. Because the flow is symmetrical to the midplane, only the lower contour of the velocity contour cross section is shown. Figure 3: Longitudinal flow velocity comparisons at z/D = 0, z/D = 1, z/D = 2, z/D = 3.5, z/D = 10 and z/D = 50. (**a**) Vertical plane (\(\theta\) = 90\({}^{\circ}\)); (**b**) horizontal plane (\(\theta\) = 0\({}^{\circ}\)). Solid line: RNG \(k\)-\(\varepsilon\) turbulence model results. Experimental data from [PERSON] et al. (Re = 60,000) [10] and [PERSON] et al. (Re = 50,800) [8]. Figure 4: _Cont._  
In the simulation gravity acts against the flow in the inlet section, unlike the test performed by [PERSON] et al. where gravity always acts perpendicular to the flow [10]. Nevertheless, this change of orientation does not have a significant effect on the flow structure (Figure 4b). Regardless the orientation, a strong recirculation of the primary flow in the inner curve is noticeable, as the fluid passes through the closed curve. Secondary flow patterns can be seen in several cross sections of the curve (at \(\varphi=60^{\circ}\), \(90^{\circ}\)). [PERSON] et al. report a pair of counter-rotating vortices already defined at \(45^{\circ}\) deflection (not shown in the image), but in our simulation using the RNG \(k\)-\(\varepsilon\) model it is clearly visible beginning at \(\varphi=50^{\circ}\) deflection (also not shown in the image). However, in both cases the vortices are clearly formed at \(\varphi=90^{\circ}\) (z/D = 0) [10]. Typically, recirculation leads to a significant pressure drop in the bend, so it is especially important to adequately capture the secondary flow inside a bent pipe. Dean vortices are observed to be fully developed downstream at z/D = 2 and persist until z/D = 10, then propagating to the entire cross section.