# Numerical Study on Cooling Performance of a Steam-Cooled Blade Based on Response Surface Method
## 2 Numerical Methods
### Numerical Verification  
Figure 4 shows the mesh model of the steam cooled blade channel. As shown in Figure 4, both the fluid and solid domain meshes of the steam cooled blade channel are divided using structured meshing. Among them, Figure 4a is the fluid domain mesh of the cascade channel and neither is given because the meshes of the steam flow channel of the steam cooled blade and its inlet and outlet steady flow sections are simpler. To match the SST \(k\)-\(\omega\) turbulence model, the mesh of the near-wall region of the fluid domain is refined, and the grid size of the first layer of the near-wall surface is set to 0.001 mm, the mesh growth ratio is 1.2, and the maximum grid size is 1.5 mm. Figure 4b shows the solid domain mesh of the steam cooled blade, and since the solid domain mesh does not need to be encoded, the minimum grid size is set to 0.5 mm. In addition, the mesh nodes at the intersection of the solid and fluid domains are matched to reduce the information transfer error at the fluid-solid interface. To ensure the economic viability and reliability of the method of numerical simulation, mesh independence verification was conducted on the mesh of steam-cooled blade channel. Figure 4: Mesh model of the steam-cooled blade channel: (**a**) Fluid domain mesh; (**b**) Solid Domain Mesh. A total of five sets of meshes were divided, and Table 1 shows the average cooling efficiency and average dimensionless temperature of the steam-cooled blade when the total number of meshes is 1.22, 1.83, 2.33, 2.9, and 3.6 million, respectively. From Table 1, It is clear that the average cooling efficiency and the average dimensionless temperature of the steam-cooled blade surface increase as the mesh number increases, in which the fluid domain mesh increases by about 500,000 each time and the solid domain mesh increases by about 100,000 each time. When the total number of meshes is increased from 2.9 million to 3.6 million, the differences of the average cooling efficiency and the average dimensionless temperature of the steam-cooled blade surface are less than 1%, which means that the mesh meets the requirements of mesh independence. The experimental and numerically calculated values of the local cooling efficiency distribution and the average cooling efficiency on the surface of the steam cooled blade are given in Figure 5 for comparison. Figure 5a shows the local cooling efficiency distribution of the steam-cooled blade surface when the mainstream inlet temperature is 695 K, the mainstream outlet pressure is 140 kPa, mainstream inlet to outlet pressure ratio is 1.4, temperature ratio of steam to mainstream is 0.65, and the flow ratio of steam to mainstream multiplied by 100 is 5.5. As can be seen from Figure 5a, the distribution of the local cooling efficiency of the steam-cooled blade obtained from the numerical calculation and the experiment is in good agreement, and the trend of the two curves is basically the same, with a maximum deviation of 4.6%. Figure 5b shows the comparison between the experimental and numerical calculated values of the average cooling efficiency of the steam-cooled blade when the flow ratio of steam to mainstream multiplied 100 (\(M_{\rm r}\)) increases from 3 to 8, with the remaining working parameters kept constant. From Figure 5b, it can be seen that the change trend of the average cooling efficiency of steam-cooled blade obtained from numerical calculation and experiment is basically the same, that is, the average cooling efficiency increases with the increase of \(M_{\rm r}\). At the same time, the average cooling efficiency of the numerically calculated blade is close to the experimental value. The maximum error appears when \(M_{\rm r}\) is 4, the error is about 7.4%, and the average error is 4.9%.