## 3 Results Analysis and Discussion  
### Cooling Effectiveness  
First, Figure 6 shows the contour maps of wall temperature distribution for four types of cooling holes. From the figure, it can be seen that a wider low-temperature zone is formed behind the shaped cooling holes due to the larger outlet area and the larger transverse expansion angle. In addition, an extremely low-temperature zone may also form at the troughs due to the lobed structure in Case 2 and Case 4. In Case 1 of circular holes, it can be observed that a blow-off phenomenon occurs at the outlet with this blow-off ratio. As a result, hot mainstream air enters behind a circular hole on both sides and pushes out the coolant, resulting in a high-temperature zone behind the hole. After the blowdown phenomenon occurs, cooling air reattachment occurs in Case 1 at X/D = 0.5, resulting in a decrease in wall temperature downstream along the flow direction. However, this problem does not occur with formed holes (Case 2-4). Figure 5: Comparison of LES and experimental results [32]. The comparison of Case 2 and Case 3 shows that even if the same geometric structure is the same, the direction has a significant influence on the cooling effect. When the expansion direction is along the flow (forward), the low-temperature zone at the hole outlet is lower and can reach up to 2 K than when it is against the flow (backward). However, both the local low-temperature zones at the cooling hole outlets and the overall low-temperature regions downstream are narrower for Case 2 than for Case 3, indicating that the lateral expansion of the coolant is worse in Case 2, resulting in significantly higher temperatures at side surfaces. If you compare the temperature distribution maps of Case 3 and Case 4, you can see that the range below 357 K for the four-lobed structure is much smaller, but the total range below 360 K is larger. This indicates that although the cooling effectiveness in the region after the hole outlet is not as good as the three-lobed structure of Case 3, the overall cooling effectiveness in Case 4 is better. This also indicates that the structure of Case 4 allows better mixing between the cooling air and the high-temperature main stream gas, resulting in a more effective film in lateral expansion. The laterally averaged wall temperatures for the four structures are 361.5063, 361.5962, 360.9564, and 361.2596 K, respectively. Figure 7 shows the distributions of the side-averaged cooling effectiveness profiles. From the figure, it can be seen that, except for the shaped hole in Case 2, both Case 3 and Case 4 perform better than the circular hole in terms of overall cooling effectiveness. Along the flow direction, the increase and subsequent decrease in cooling effectiveness in Case 1 also indicates some degree of blowdown phenomenon. The cooling effectiveness of Case 2 decreases continuously and is finally the lowest among all cases studied here. A comparison of Case 3 with Case 1 shows that the cooling effectiveness at the outlet of the hole in Case 3 also increases and then decreases, but different from Case 1. This is due to the significant lateral expansion caused by the lobe-like structure. A comparison of Case 3 with Case 4 shows that the overall cooling performance of Case 3 is much better than that of Case 4, especially near the hole outlet, where the values for the cooling performances of Case 3 are significantly higher than those of Case 4. Figure 6: Comparison of wall time-averaged temperature distribution contour maps. The laterally averaged cooling effectiveness of Case 3 is approximately 50% higher at the X/D = 1 cross-section and approximately 27% higher near the outlet (X/D = 18) compared to Case 1. Similarly, the laterally averaged cooling effectiveness for Case 4 increases by approximately 24% near the outlet (X/D = 18). For Case 2, on the other hand, the side-averaged cooling effectiveness decreases by approximately 8% near the outlet. In the case of a single hole, Case 3 (three-lobed backward) has the best cooling effect.