# Numerical Study of Endwall Modification with Micro-Scale Ribs in a Turbine Cascade
## 3 Results and Discussions
### Heat Transfer Characteristic  
The general consensus indicates that the rib will inevitably cause heat transfer enhancement, which has an adverse influence on the turbine cascade. However, the last section indicates that the ribbed enwall possesses the advantage of film cooling whether for the endwall surface or the SS of the blade. Therefore, this section mainly focuses on the heat transfer of the interaction between the rib and mainstream for the uncooping case. Meanwhile, the original endwall was used for comparison. The mainstream was the same as that in the last section, which was 40 m/s. The heat transfer coefficient (\(h\)) is defined as Equation (6):  
\[h=\frac{\Delta q}{T_{w}-T_{aw}}=\frac{q_{w}}{T_{w}-T_{aw}} \tag{6}\]  
It is necessary for the \(h\) calculation to be calculated twice for acquiring the adiabatic wall-temperature (\(T_{aw}\)) and constant wall-temperature (\(T_{w}\)), respectively. The \(q_{w}\) represents the heat flux on the endwall surface when the wall was set as the constant-temperature condition. Therefore, through subtraction of the two calculation results, the \(h\) of the endwall was obtained. The comparison of the original endwall and horizontal case is shown in Figure 12. For the original endwall, there were mainly three regions of high \(h\). The first region pointed at the LE of the blade (SS and PS). Due to the stagnation of the mainstream here and the horseshoe vortex, the first region presented an obvious trajectory of high-\(h\) distribution. The second region pointed at the leading edge, which was caused by the wake. These regions presented a high value of \(h\). The last high-\(h\) region pointed at the passage, which was caused by the PV. Although the magnitude of high \(h\) was lower than those of the others, its range coincided with the PV, which is extensive. These three regions of high \(h\) were the basic model for turbine cascade. For the horizontal case, the high-h region was still in the vicinity of the LE. However, the region was separated by the rib channel, showing the discontinuity. The streamline was curved due to the rib and thus the high-\(h\) sub-region was  
Figure 11: Streamwise vorticity of the plane \(x/C_{\mathrm{ax}}\) = 0.5. (**a**) Original; (**b**) Horizontal s = 8 mm; (**c**) Horizontal s = 10 mm; (**d**) Tilted s = 8 mm; (**e**) Tilted s = 10 mm; (**f**) Vertical s = 8 mm; (**g**) Vertical s = 8 mm. partially shifted along the rib in each rib channel, where \(h\) was enhanced. Second, in the vicinity of the horseshoe vortex (SS leg, \(y\)/\(P\approx 0.5\)), a low-\(h\) region was presented. This is because the intensity of the area around the horseshoe vortex decreased. It is speculated that this is a comprehensive result of the interaction between the horseshoe vortex and the rib-like vortex. In the vicinity of the PS corner, there was another high-h region, which was caused by the crossflow and rib-like vortex. As stated in the last paragraph, the high h was more obvious in the PV coverage region. It indicated that the rib enhanced the disturbance around the endwall, which is dominated by the rib-like vortex. Therefore, two high-\(h\) sub-regions were distributed on both sides of each rib channel in the vicinity of the throat, which were caused by PV and cross-flow, respectively. In addition, the effect of rib spacing did not show too much a difference for the horizontal case, indicating that the \(h\) distribution was nearly the same. The \(h\) of the tilted case is shown in Figure 13. Note that the rib orientation was pointed straight to the LE of the PS. The flow in the rib channel was versus the horseshoe vortex and thus the stagnation line, presenting a line of low \(h\), in front of the LE. The flow in the rib channel presented a tendency along the rib geometry, meaning that the rib-like vortex offset the secondary flow to some extent. Therefore, it is unclear whether the horseshoe or the PV would offset the original position, and the \(h\) presented a unique distribution.