## 4 Conclusions  
In this study, a computational analysis was carried out utilizing a flat plate model to investigate the phenomena of jet pulsation and periodic wake behavior in film cooling applications. Based on the numerical simulation results across three different blowing ratios (M = 0.3, 0.5, and 1.0) and various phase lags, it was confirmed that the coupling of periodic wake and pulsating jet significantly impacts film cooling efficiency. When the blowing ratio was low (M = 0.3), the film cooling efficiency could be improved by coupling the wake-affected surface with the low-blowing ratio stage of the pulsating jet. At higher blowing ratios (M = 1.0), the film cooling effect was optimized by integrating the pulsating low-pressure phase at the film hole with the jet pulsating high-blowing ratio. Moreover, it was found that the cooling efficiency of the pulsating film reached its peak across all blowing ratios when the phase difference was equal to zero, which indicates that the optimal coupling strategy depends on the blowing ratio and phase lag. Thus, the hypothesis was generally accepted. The primary conclusions drawn from the research are as follows:  
1. For a blowing ratio of M = 0.3, the variations in the blowing ratio are minimal, resulting in the predominance of the wake sweeping process. Therefore, it was essential to couple the surface affected by the wake with the low blowing ratio stage of the pulsating jet. Moreover, due to the low overall blowing ratio, the film structure was susceptible to being lifted and damaged by the wake, adversely affecting the cooling efficiency.
2. At a blowing ratio of M = 0.5, an increase in the average blowing ratio led to the predominance of the lift generated by the wake at the film hole. The jet lift induced by the wake was aligned with the high blowing ratio phase of the cycle. Consequently, when this lift coincided, the influence of the wake on jet pulsation was relatively diminished.
3. At a blowing ratio of M = 1.0, the elevated average blowing ratio caused significant jet lift. The wake's influence on the film's cooling efficiency was primarily concentrated on the leading and trailing edges of the lifted area. The interaction at these regions encouraged the higher jet to adhere closely to the wall, thereby enhancing the film-cooling effect.
4. In scenarios where the blowing ratio was low, coupling the wake-affected surface with the low-blowing ratio stage of the pulsating jet can achieve improved film cooling efficiency. Conversely, at higher blowing ratios, integrating the pulsating low-pressure phase at the film hole with the jet pulsating high-blowing ratio can optimize the film cooling effect. When the phase difference was equal to zero, the cooling efficiency of the pulsating film attains its peak across all blowing ratios, indicating a highly effective coupling strategy.  
Conceptualization, L.S.; methodology, L.S. and Q.P.; software, Q.P.; validation, Q.P.; formal analysis, Q.P. and L.S.; investigation, Q.P. and L.S.; resources, L.S.; data curation, Q.P.; writing--original draft preparation, Q.P.; writing--review and editing, Q.P. and L.S.; visualization, Q.P.; supervision, L.S.; project administration, L.S.; funding acquisition, L.S. All authors have read and agreed to the published version of the manuscript.  
This research is supported by the National Natural Science Foundation of China (Grant No. 12172227).  
Not applicable.  
Data Availability Statement: Some or all data and models used during this study are available from the corresponding author upon request.  
Conflicts of Interest: The authors declare no conflicts of interest.