## 4 Conclusions and Future Work  
In the current study, various geometrical parameters of shaped cooling holes were investigated using LES to determine their effects on film cooling efficiency. Numerical simulations were performed for all cases and validated by comparison with experimental results obtained by other researchers. The predicted effects of the different design variables on film cooling performance to LES showed similar trends to those observed in experiments. Based on these results, the following conclusions can be drawn:  
1. The results show that the shaped cooling holes used in this study have larger lateral expansion angles compared to the circular reference hole, resulting in better propagation of the coolant in the transverse direction. In addition, due to two upstream wave structures along the flow direction, the cooling jets interact with the main hot flow at an earlier point near each hole outlet, which contributes significantly to improved film cooling performance. These factors together improve the lateral averaged cooling effectiveness compared to the circular reference hole. 2. All formed film holes have lower pressure drop, and the velocity fields show that their cooling jets have a smaller angle to the wall surface, indicating better wall attachment. 3. Among all design cases, Case 3 (3-lobe reversed) shows superior overall cooling performance with an increase in average cooling effectiveness near the outlet of approximately 27% compared to the reference hole. In contrast, Case 2 (3-lobe) shows even lower cooling efficiency than the reference hole, highlighting that a well-designed shape is critical for optimal performance. 4. Analysis of vortex structures for all cases shows that more complex coherent vortices form behind all shaped film holes than behind circular ones, with hairpin vortices dominating. Vortex structures near holes are located closer to the wall regions around X/D < 6, where jet separation phenomena can be avoided, while they extend significantly downstream beyond X/D > 6 in both vertical and spanwise directions, contributing to further enhancement of their respective performance. In summary, these results suggest that shaped cooling holes are a promising method for improving gas turbine engine efficiency through enhanced film cooling. This study can be applied to liner cooling in combustors of advanced gas turbines. Compared with the circular film cooling holes, the lobe-shaped film cooling holes can withstand more heat load, which enables the gas turbine to have a higher output power at the same coolant ratio. There are limitations, however, as this study only examined single-hole flat plate configurations using LES without considering realistic combustor geometries or the interaction of multiple cooling holes. In the future, a more detailed analysis of vortex structures should also be conducted. Future studies should investigate such scenarios experimentally and numerically under realistic combustor conditions to further validate the potential of shaped cooling holes to improve the overall performance of gas turbine engines. At the same time, to further improve the understanding of numerical simulation details and the need for personalized customization, future research can use open-source software such as OpenFOAM. More and more scholars have begun to use OpenFOAM [33; 34]. **Author Contributions:** Conceptualization, K.W. and F.L. ; methodology, K.W. and Y.A. ; software, K.W. and T.Z. ; validation, K.W. and T.Z. ; formal analysis, K.W. ; investigation, F.L. and Y.A. ; resources, K.Z. and F.L. ; data curation, K.Z. ; writing--original draft preparation, K.W. ; writing--review and editing, K.W. ; visualization, K.W. and Y.A. ; funding acquisition, F.L. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was supported by the National Science and Technology Major Project (2017-III-0006-0031, J2017-III-0008-0034 and J2019-III-0002-0045). **Institutional Review Board Statement:** Not applicable.