## 3 Computational Methods
### Mesh Sensitivity Study  
To ensure an appropriate set of meshes was used, a meshing sensitivity study was conducted to determine how the mesh element sizing and density impact the mean flow. The general process for mesh sensitivity studies is to create multiple meshes with different densities, ranging from coarse to fine. The boundary conditions and initial conditions are held constant for each mesh to ensure a proper comparison. Based on the results using a meaningful quantity of interest, the mesh with sufficient density to accurately resolve the behavior is selected. Generally, mesh comparisons are performed using a baseline comparison against a significantly finer mesh than the rest as a stand-in for an analytical solution. This study used a total of four meshing models, ranging from the coarsest to the finest, labeled as M1, M2, M3, and M4, respectively. Detailed meshing parameters for these four meshing models are summarized in Table 1. To compare these four meshing models, a timed cooled injection scenario was used, involving two injections with a time delay. The overall transient lasted 10 s with an injection flow velocity of 0.545 m/s at 285 K (\(\sim\)53 F). The mean flow rate for the study was varied to cover three groups spanning from 39.52 to 178.30 GPM at a set temperature of 308 K (\(\sim\)95 F). To compare these four meshing models, the area-weighted average temperature and velocity traces at five different cross-sections and the temperature and velocity traces at two local positions were selected, as shown in Figure 8. Figure 8 shows the five cross-sections (indicated by the blue solid regions) and two local points (indicated by the blue stars). Both the temperature and velocity are plotted at different locations, including five locations equally distributed along the flow direction and at the two sensor locations. Figure 9 shows the area-weighted average temperature traces at the three different cross-sections shown in Figure 8. Figure 10 shows the two local temperature traces at the two star-marked locations shown in Figure 8.  
Figure 9 shows the area-weighted average temperature of the cross-sections evenly distributed along the flow direction. The three average temperature comparisons exhibit almost identical trends, and the two local temperature comparisons are at 50.8 mm (2 in) and 152.4 mm (6 in) from the injection point. In Figure 10, some variation is observed during the injection period, with a temperature difference of around 0.2-0.4 K. This amount is insignificant in total magnitude when compared with the experimental temperature sensor uncertainty in the experimental flow loop. Comparing the different meshes, the mesh cell count does not significantly affect the temperature distribution along the mean flow. This is confirmed by the investigation of the velocity profiles across the four meshes, which helped in selecting the suitable mesh. Figure 7: Detailed meshing around the injection pipe. For the velocity comparison, the cross-sections and two local points were also selected, as shown in Figures 11 and 12, respectively. Figures 11 and 12 show that M1 and M2 generated a higher velocity than M3 and M4 do, and the average velocity data from M1 and M2 were not sufficiently resolved when compared with the M3 and M4 meshes. Excluding the injection period, the meshes with a higher cell count were able to better resolve the momentum behavior. Figure 12 shows that the overall local velocity is greater than the average value. The reason for this is that the velocity distribution was not uniform across the cross-section.