# Two-Phase Flow Simulations Using 1D Centerline-Based C- and U-Shaped Pipe Meshes
## 3 Results and Discussion
### Grid Convergence Test and Sensitivity Analysis of Turbulence Model  
This section introduces the grid convergence test for the case of uniform and non-uniform grid. In addition, we carry out a sensitivity analysis of turbulence model to justify the choice of turbulence model in this study.  
With the benchmark case (i.e., Case 1), we performed a grid convergence test with three different grid sizes (i.e., coarse, medium, and dense grids) to ensure that all solutions are satisfied with the given mesh size (Table 2). The tetrahedral grid is constructed in the entire domain, because our semi-automatic meshing algorithm is optimized for the mesh morphology. In addition, the average y+ and maximum skewness are obtained at 12.4, 4.8, and 2.6, and 0.86, 0.84, and 0.83, for the three types of grids, respectively, ensuring a sufficiently fine mesh near the wall to validate the prediction data using LES turbulence model. Figure 8 shows the quantitative results of the axial water velocity at the curve inlet. The numerical simulations using both medium and dense grids provided almost the same results, while the water velocity was overestimated in the coarse non-uniform grid. Based  
Figure 6: Streamwise water velocity at the three cross lines at the (**A**) curve inlet, (**B**) curve center, and (**C**) curve outlet for validating the existing experimental data [19] with \(\alpha_{air}~{}=~{}0.94\%\).  
Figure 7: Iso-surface of the velocity magnitude of water (cyan) = 1.43 m/s and the velocity vectors at three cross-sections in Case 1 with \(\alpha_{air}~{}=~{}0.94\%\).  
on the grid sensitivity test, we decided to use a medium grid to ensure both numerical accuracy and computational cost. Figure 8 also depicts a comparison of the water velocity profiles obtained with the uniform and non-uniform grids. The dense uniform grid case contained 4,855,391 tetrahedral cells with 847,095 nodes. The dense non-uniform grid case contained a similar number of cells and nodes, with 5,021,251 tetrahedral cells and 905,524 nodes, but with finer grids near the wall. Overall, the numerical results obtained with the uniform and non-uniform meshes were in a good agreement. However, a slight difference was observed near the inner and outer walls with a large velocity gradient. This difference can possibly be attributed to the non-uniform case with finer grids near the wall capturing the velocity gradient well relative to the uniform case with a finer grid.  
Three turbulence models including LES, k-\(\omega\), SST, and k-\(\varepsilon\), are carried out for the sensitivity analysis. Figure 9 shows the quantitative results of the axial water velocity at the curve inlet. The numerical simulation using the LES turbulence model is in a good agreement with the experimental data, whereas the axial water velocity obtained from k-\(\omega\) and k-\(\varepsilon\) turbulence models were not aligned with the experimental results. Therefore, we employed the LES turbulence model to perform the numerical simulations.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **Coarse** & **Medium** & **Dense** \\ \hline \multirow{2}{*}{Uniform} & Node & - & - & 847,095 \\  & Cell & - & - & 4,855,391 \\ \hline \multirow{2}{*}{Non-uniform} & Node & 139,436 & 664,278 & 905,524 \\  & Cell & 709,221 & 3,686,309 & 5,021,251 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Number of nodes and cells for coarse, medium, dense uniform, and non-uniform grids.  
Figure 8: Grid convergence test of the axial water velocity at the curve inlet with \(\alpha_{air}\ =\ 0.94\%\).  
Figure 9: Streamwise water velocity at the curve inlet of LES, k-\(\omega\), and k-\(\varepsilon\) turbulence models with \(\alpha_{air}\ =\ 0.94\%\).