# Large Eddy Simulation of Film Cooling with Bulk Flow Pulsation: Comparative Study of LES and RANS
## 4 Results and Discussion
### Streamwise Velocity and Fluctuation Profiles at x/D = 5 and z/D = 0  
Figure 10 shows the time-averaged dimensionless streamwise velocity profiles at \(x/D=5\) and z/D = 0 for 0 and 32 Hz. The experimentally measured streamwise velocity profile at 0 Hz [15] has a local peak between \(y/D=0\) and 0.5 because of coolant jet penetration into the boundary layer. The streamwise velocity profile obtained using LES predicted this peak well, whereas the various RANS models failed to predict the peak exactly, as shown in Figure 10a. In Figure 10b, the time-averaged dimensionless streamwise velocity profiles at 32 Hz obtained by LES match well with the experimental results [15], whereas the RANS realizable k-\(\varepsilon\) model did not predict the velocity profile between \(y/D=0\) and 1 exactly, and the velocity gradient predicted by RANS was rather different from those in the experimental and LES results. Figure 9: Q-criterion contours colored by y-vorticity in each phase with iso-surfaces of level 1000 for three dimensions under 32 Hz pulsation: (**a**) LES (Smagorinsky–Lilly) and (**b**) unsteady RANS (realizable k-\(\varepsilon\)). Figure 11 shows the time-averaged dimensionless U\({}_{\rm rms}\) profiles at \(x/D=5\) and \(z/D=0\) for 0 and 32 Hz. The almost-zero U\({}_{\rm rms}\) values obtained using the RANS realizable k-\(\varepsilon\) model are considerably much smaller than those obtained experimentally, as illustrated in the figure. The film cooling efficiency predicted using the realizable k-\(\varepsilon\) model was the closest to the experimental value, but the model yielded the weakest prediction of velocity fluctuation. As shown in Figure 11a, the time-averaged U\({}_{\rm rms}\) values at 0 Hz obtained by LES matched well with the experimental result. A peak of U\({}_{\rm rms}\) values at around \(y/D=0.9\) was predicted well by LES. The Reynolds stress model or the k-\(\omega\) model predicted the velocity fluctuation better than the realizable k-\(\varepsilon\), but the predicted value was still less than half the experimental value. Figure 11b illustrates that the time-averaged U\({}_{\rm rms}\) values at \(x/D=5\) and \(z/D=0\) under 32 Hz pulsation obtained by LES were slightly smaller than the experimental values, even though the shape of the plot is similar to the experimental plot. Figure 11 shows that the LES approach is superior to the URANS approach in predicting the U\({}_{\rm rms}\) values of pulsating flows. Compared with the case of no pulsation, the turbulence intensity increased below \(y/D=0.5\) near the wall, and LES predicted this increase appropriately. Figure 12 illustrates the phase-averaged dimensionless U\({}_{\rm rms}\) profiles at \(x/D=5\) and \(z/D=0\) for 32 Hz pulsation in three phases during a period as obtained by LES and URANS. Figure 11: Time-averaged dimensionless \(U_{rms}\) profiles obtained using RANS and LES models at \(x/D=5\) and \(z/D=0\): (**a**) 0 Hz and (**b**) 32 Hz. Figure 10: Time-averaged streamwise velocity profiles at \(x/D=5\) and \(z/D=0\): (**a**) 0 Hz and (**b**) 32 Hz. The time-averaged dimensionless U\({}_{\rm rms}\) profiles at \(x/D=5\) and \(z/D=0\) as obtained by the LES Smagorinsky-Lilly model and the URANS realizable k-\(\varepsilon\) model under 32 Hz pulsation are also shown in the figure. As shown in Figure 12a, as t/\(\tau\) decreases from 0 to 0.4, the \(y/D\) values corresponding to the maximum values of U\({}_{\rm rms}\) decrease because the heights of the large vortex at \(x/D=5\) decrease, as illustrated in Figure 8a. When t/\(\tau\) increases from 0.4 to 0.8, the \(y/D\) values corresponding to the maximum U\({}_{\rm rms}\) values increase because the height of the large vortex at \(x/D=5\) increases. Therefore, the \(y/D\) values corresponding to the maximum U\({}_{\rm rms}\) values are periodic because of the outline of the large coolant vortices. This outline is attributed to the periodic blowing ratios expressed by Equation (2), even though the averaged blowing ratio is 0.5.