## 3 Results and Discussion
### Blowing Ratio \(M=1.0\)  
Figure 13 illustrates the temperature contours during a single period for varying phase lags between the incoming wake and the pulsating jet in the central plane \(Z=0\). The velocity of the pulsating jet at the orifice varies over one period, reaching maximum velocity at \(1/4\) T and minimum velocity at \(3/4\) T. In Figure 13a, it is evident that the cooling jet, upon ejection from the orifice, ascends from the bottom wall due to its high velocity. However, upon interaction with the incoming wake, the elevated cooling jet is compelled to reattach to the wall at x/D = 5-15, affording cooling protection for the bottom wall. Subsequently, as the jet velocity diminishes and continues to interplay with the pulsating jet, the adhesion of the cooling jet to the wall is enhanced, resulting in an expanded cooling coverage. The combined impact of the wake and jet is subject to variation based on diverse phase lags between the incoming wake and the pulsating jet. Figure 13b demonstrates superior cooling jet coverage compared to Figure 13a at the equivalent instant within a single period. In Figure 13c, with a phase lag of 0.5, at \(1/4\) T, the cooling jet, upon ejection from the orifice, demonstrates an initial upward movement analogous to the previous cases. However, due to the specific phase of the pulsating jet and the influence of the wake, the cooling jet begins to be affected by the encroaching wake. As time progresses to \(3/4\) T, the velocity of the jet notably decreases, and the effect of the wake becomes increasingly pronounced. The wake constrains the cooling jet to reattach to the wall at shorter distances when compared to Figure 13a,b, leading to a cooling coverage that is more concentrated in proximity to the film hole. In Figure 13d, where the phase lag is 0.75, the cooling jet similarly shows an upward trend at \(1/4\) T. However, as the wake traverses the plate at \(3/4\) T, its influence on the jet becomes substantial. The wake not only compels the jet to reattach to the wall but also modifies the shape and extent of the film coverage. In comparison to other phase-lag cases, the film coverage illustrated in Figure 13d exhibits a distinct distribution pattern. It reveals a more dispersed cooling area behind the film hole at both \(3/4\) T and T, which may be attributable to the combined effects of variations in the jet velocity and the robust perturbation induced by the wake. These distinctions between Figure 13c,d underscore how the phase lag between the incoming wake and the pulsating jet influences the performance of the cooling jet and the resulting film cooling efficiency at a blowing ratio of M = 1.0. Figure 10: Contours of vorticity in one period for different phase lags in \(Z=0\) at M = 0.5. Figure 11: _Cont._  
The interaction between the incoming wake and the pulsating jet is illustrated in Figure 14, where vorticity contours are presented. It is evident that as the passing wake approaches the ejected cooling flow, it induces a bending effect on the cooling jet towards the bottom wall. At t = 1/4 T, illustrated in Figure 13a, the preceding wake has just passed the bottom wall while the subsequent wake is still distant, exerting minimal influence on the cooling jet. Consequently, the incoming wake attaches to the bottom wall anterior to the cooling jet, as depicted in Figure 14b,c, thereby mitigating the upward movement of the cooling jet and ensuring effective coverage of the cooling protection. Consequently, the conjoined effect of the incoming wake and the pulsating jet significantly impacts the coverage of the cooling jet. The conjoined effect of the wake and the pulsating jet can be discerned from the temperature contours in Figure 15.  
Figure 11: Contours of temperature on the bottom wall in one period for different phase lags at M = 0.5. Figure 12: Streamwise distribution of time-mean span averaged film cooling efficiency at M = 0.5. Figure 13: Cont.