## 3 Computational Methods
### Mesh Sensitivity Study
\begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **M1** & **M2** & **M3** & **M4** \\ \hline Number of circular divisions & & & & 40 \\ Number of radial divisions (outer cylinder) & & & & 14 \\ Number of radial divisions (medium cylinder) & & & & 10 \\ Element size for the inner rectangle (m) & & & \(4\times 10^{-4}\) \\ Number of divisions for the inner rectangle & & & & \\ Inlet region length (m) & & & & 0.07 \\ Injection region length (m) & & & & 0.06 \\ Temperature collection region length (m) & & & & 0.37 \\ Number of divisions in the inlet region in the axial direction & 40 & 50 & 65 & 80 \\ Number of divisions in the injection region in the axial direction & 28 & 40 & 52 & 60 \\ Number of divisions in temperature collection region in the axial direction & 114 & 150 & 200 & 250 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mesh sensitivity study. Figure 8: Temperature and velocity locations for the mesh sensitivity study. Figure 9: Area-weighted average temperature traces at (**a**) 0.1 m, (**b**) 0.3 m, and (**c**) 0.5 m from the inlet. Figure 11: Area-weighted average velocity traces at (**a**) 0.1 m, (**b**) 0.3 m, and (**c**) 0.5 m from the inlet. Figure 10: Two local temperatures at the star-marked locations shown in Figure 8.  
One example of the 3D velocity distribution of the fully developed turbulent flow is shown in Figure 13. Figure 13 shows that the velocity is not uniformly distributed across the cross-section. The endpoint of the temperature sensor location is close to the wall, where the local velocity is relatively smaller than the average velocity. Figure 12 supports this conclusion, indicating that the model with a larger mesh count or degrees of freedom exhibits lower velocity, just like the previous average velocity plots. To further examine how the M1, M2, and M3 meshes resolved the velocity compared with the M4 mesh, which has the highest density mesh, the absolute percentage error for both average and local velocities between the M4 mesh and the other three meshes was calculated, as shown in Figures 14 and 15. Both Figures 14 and 15 show differences between M1, M2, and M4, while the absolute percentage error between M3 and M4 is close to zero. This shows that M3 is sufficient to be used due to the minimal velocity differences between M3 and M4. Therefore, M3 was chosen as the base meshing model. Figure 12: Two local velocities at the star-marked locations shown in Figure 8.  
Figure 13: One example of 3D velocity distribution of fully developed turbulent pipe flow. M3 in Table 1 was used as the base mesh to ensure high-quality temperature and velocity simulations, with additional refinement of the wall region mesh. The \(u^{+}\) vs. \(y^{+}\) plot was used as a reference to judge whether the wall region was simulated correctly. After several simulations, it was found that no single meshing model was sufficient for all the flow-rate cases with suitable \(y^{+}\) values. The main reason for this is that the range of wall region sizes is not sufficient for the wall region. Figure 8:0 Absolute percentage errors between the meshing (area-weighted average velocity) at (a) 0.1 m, (b) 0.3 m, and (c) 0.5 m from the inlet. Figure 8:1 Absolute percentage errors of two local velocities. of the mean flow was quite large, from around 40 GPM up to 170 GPM (approximately 2.5 \(\times\) 10\({}^{-3}\) m\({}^{3}\)/s to 1.1 \(\times\) 10\({}^{-2}\) m\({}^{3}\)/s), with Reynolds numbers ranging from 5.7 \(\times\) 10\({}^{4}\) to 2.5 \(\times\) 10\({}^{5}\). A single meshing schema might better resolve the flow at lower flow rates but would not work well for higher flow rates. To solve the problem caused by the wide flow-rate range and achieve better accuracy, the flow-rate cases were divided into three subgroups: low flow rate, medium flow rate, and high flow rate. Each group used a unique mesh designed for a specific range of flow rates.