## 4 Development of a Performance-Based Design Method Using Airfoil Cascades
### Numerical Results  
As a result of the optimization, a quadratic equation that gives the relationship between the design parameters and the objective function is obtained in the following form:  
\[f= \begin{bmatrix}0.95&0.036&-0.015&-0.003&-0.01&-0.015\\ 1.22&0.024&0.001&0.001&-0.015&-0.006\\ 1.52&0.083&-0.053&0.037&0.027&-0.002\end{bmatrix}\begin{bmatrix}1\\ x_{1}\\ x_{2}\\ x_{1}^{2}\\ x_{2}^{2}\\ x_{1}x_{2}\end{bmatrix}  \tag{6}\]  
Figure 7: (**a**) Schematic view of a geometrical process for extracting the cascade of airfoils using the cylindrical section, (**b**) computational domain of the cascaded airfoils designed for the two-dimensional simulation, and (**c**) distribution of the angle of attack along the spanwise direction obtained from the three-dimensional analysis and used as an inlet boundary condition in the two-dimensional analysis. The response equation for the radial strip for r/R = 0.45, 0.65, and 0.8, respectively, is shown in matrix form by row in Equation (6). In all the selected radial strips, the results indicate that the camber inlet angle significantly affects the L/D more than the camber outlet angle. The regression equations of Equation (6) are shown visually in Figure 8 as two- and three-dimensional contour plots and three-dimensional graphs, respectively, showing the response surfaces. The contour plots indicate all the relationships between the camber inlet angle, camber outlet angle, and L/D. Plotting the response surface to the design variable and L/D in three dimensions reveals a gradient trend comparable at all the current design points and a propensity to decline in a particular direction. Additionally, the surface has a significant curvature, particularly at the design point when r/R = 0.65 occurs. The optimal point of the design parameter can be represented in two-dimensional contour plots, where the dark area indicates the significant value of the objective function within the range of the design parameter. The lower right corner of each target's contour plot displays the darkest area. This indicates that a higher camber inlet angle corresponds to a lower camber exit angle, which raises the L/D value. Furthermore, Figure 9 illustrates the association between each design parameter and the objective function through XY graphs. When the camber inlet angle grows and the camber outlet angle drops compared to those of the existing airfoils, the XY plots verify that the value of the design variable attains the maximum value of the L/D. Figure 8: Iso-contour plots of the response surface at the target radial strip with a three-dimensional (left) and two-dimensional view (right): (**a**) r/R = 0.45, (**b**) r/R = 0.65, and (**c**) r/R = 0.8. Figure 10a shows the airfoil shapes obtained by optimizing the camber inlet and outlet angles. The figure illustrates the optimized airfoil form with a red line, contrasting it with the original airfoil design represented by a black line. Remarkably, a tendency is found in all three design radial positions; the maximum camber positions rise in the direction of the leading edge. The constraint of the design process is to maintain a fixed airfoil thickness throughout. The camber angles are adjusted to ensure that the thickness remains constant. This approach allows us to preserve the critical aerodynamic characteristics while optimizing other aspects of the airfoil geometry. A comprehensive two-dimensional numerical analysis is conducted to validate the efficacy of these optimized airfoils. Figure 10b  
Figure 9: XY plots of the response surface at the target radial strip: (**a**) r/R = 0.45, (**b**) r/R = 0.65, and (c) r/R = 0.8.  
displays the numerically predicted L/D values obtained by comparing the optimized and original airfoils. Figure 10b demonstrates a significant enhancement in the L/D ratio for the improved cascaded airfoils compared to the original ones across all the design radial strips. The important performance metric grew by about 20% on average.