# Numerical Study of Endwall Modification with Micro-Scale Ribs in a Turbine Cascade
## 3 Results and Discussions  
### Film Cooling Characteristic  
In reality, due to the existence of the micro-scale rib, the coolant could flow along the rib geometry. In this section, four types of endwalls (original, horizontal rib, tilted rib, and vertical tib) were numerically studied. Boundary conditions are shown in Table 3. The Reynolds number (Re) of the mainstream was 326,000, which corresponds to the \(U_{\rm m}\) value, 40 m/s. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mesh** & **Total Number of Cells** & **Area-Averaged \(\eta\)** & **Deviation/\%** & **GCI/\%** \\ \hline Coarse & 23,989,236 & 0.16841 & 8.08 & — \\ Moderate & 35,531,716 & 0.152809 & 1.93 & 3.17 \\ Fine & 66,834,042 & 0.155591 & 0.14 & 0.18 \\ Richardson & — & 0.155815 & — & — \\ extrapolation & — & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Grid independence details obtained using the GCI method. Figure 5: Detail of mesh topology. \begin{table}
\begin{tabular}{c c} \hline \hline \(U_{\rm m}\) (m/s) & **40** \\ \hline \(T_{\rm m,in}\) (K) & 298.15 \\ \(T_{\rm c,in}\) (K) & 288.15 \\ \(MFR\) (\%) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Boundary conditions. The definition of \(\eta\) can be expressed as Equation (4). \[\eta=\frac{T_{m,in}-T_{aw}}{T_{m,in}-T_{c,in}} \tag{4}\]  
where \(T_{m,in}\) represents the temperature of the mainstream inlet, \(T_{aw}\) is the adiabatic wall temperature, and \(T_{c,in}\) is the temperature of the coolant. The numerical results of the cooling characteristics of the horizontal rib are presented in Figure 6. With a fixed end position of the ribs (0.57 \(C_{\mathrm{ax}}\)) and rib spacing, the initial rib position of the horizontal ribs (with a rib spacing value of 8 mm) is located at the outlet of the upstream slot. So, the coolant first flows through the rib surface and then through the endwall surface. As a result, there is a significant resistance on the endwall cooling, and the cooling magnitude and the coverage area rapidly diminish. The coolant dissipates at about 0.4 \(x/C_{\mathrm{ax}}\). For the horizontal ribs with a rib spacing of 10 mm, there is a short distance from the outlet of the upstream slot to the initial rib position, so it exhibits high cooling effectiveness in the circumferential direction. After that, \(\eta\) decreases rapidly, and the coverage of the coolant in the axial direction is not as good as that of the case of 8 mm horizontal ribs. The direction of horizontal ribs is similar to the flow direction of the PV in general, and the coolant coverage resembles it on the smooth endwall. However, in the axial direction, due to the restriction by ribs, the circumferential coverage of the coolant is apparently reduced. The numerical results of the cooling characteristics of the tilted rib are presented in Figure 7. The tilted ribs guide the coolant toward the PS, and since the direction of the endwall's secondary flow is roughly opposite to the direction of the ribs in this case, the \(\eta\) transition is relatively abrupt at 0.3-0.4 \(x/C_{\mathrm{ax}}\) (red circle in Figure 7). Also, the guiding effect of the ribs improves the cooling effectiveness of the leading edge near the SS, but inhibits the cooling effectiveness in the corner of the SS and blade. In addition, there is no significant difference in the endwall cooling effectiveness contour for tilted ribs with different rib spacings. The coolant coverage area on the endwall is mainly influenced by PV. The cooling characteristics of the endwall with different rib types are different mainly because the rib direction produces different inhibition effects on the PV. The numerical results of the cooling characteristics of the vertical rib are presented in Figure 8. The vertical rib has the largest interference degree to the secondary flow near the endwall, so the coolant that is sucked to the suction surface is reduced, and the cooling coverage on the endwall circumferential direction is more extensive.