## 4 Detailed Design Method and CFD Model  
The two fundamental objectives of detailed design were (1) to get spanwise and stagewise distribution of aerodynamic, geometric, and thermodynamic parameter constraints of preliminary design results, and (2) to get a more detailed meridional flow path. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Upper Bound** & **Lower Bound** \\ \hline Solidity & 1.60 & 1.10 \\ Reaction & 0.60 & 0.40 \\ Outlet Mach number & 0.40 & 0.28 \\ Inlet radius ratio & 0.50 & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The bounds of four design parameters. The full radial equilibrium equation is expressed as the following equation [15]:  
\[ \begin{array}{l}\frac{1}{2}\frac{\partial}{\partial q}C_{m}^{2}=\frac{\partial H }{\partial q}-\frac{\partial S}{\partial q}+C_{m}\frac{\partial C_{m}}{\partial m }\sin(\phi+\gamma)+\frac{C_{m}^{2}}{r_{m}}\cos(\phi+\gamma)\\ -\frac{1}{2r^{2}}\frac{\partial}{\partial q}\left(r^{2}C_{\theta}^{2}\right)+ \frac{C_{m}}{r}\frac{\partial}{\partial m}(rC_{\theta})\tan\varepsilon\end{array}  \tag{15}\]  
where the subscript \(m\) denotes meridional component; \(r\) is the radial line; \(q\) is quasi orthogonal line; \(r_{m}\) is Meridional streamline curvature; \(S\) is specific entropy, and \(H\) is specific enthalpy. The \(\gamma\), \(\phi\), and \(\varepsilon\) are illustrated in Figure 2.  
The spanwise distribution of \(C_{\theta}\) is pre-set by:  
\[C_{1,\theta}=Qr+\frac{\mathrm{R}}{r}, \tag{16}\]  
\[C_{2,\theta}=Vr+\frac{\mathrm{Y}}{r}, \tag{17}\]  
V and Q are set as nonzero constants for the entire compressor (all stages with identical swirl). Y and R are determined by Equation (1), Equations (16) and (17), and midspan reaction. The \(C_{m}\) at the midspan obtained from the preliminary design results was set as the definite condition to sufficiently obtain the unique solution of Equation (15), which is numerically and iteratively solved by a finite differencing method. In the iteration process, 13 nodes are set at each calculation station. These nodes are initialized by being evenly distributed along the spanwise direction. The one-dimensional solution of \(C_{m}\) is equal to midspan \(C_{m}\). After each iteration, the nodes (except the casing node) are simultaneously updated, and the iteration process was considered to be completed when the node radii are no longer changed. From the assumptions of axisymmetric, the principle of streamline curvature method can be obtained, i.e., the flow can be restricted to the meridional plane. Hence the \(\varepsilon\) is settled to equal zero in this work. The value of \(\phi\) and \(r_{m}\) are calculated by performing cubic spline interpolation on the calculation stations along the meridional streamline direction. The \(\gamma\) is obtained by calculating the included angle between the axial gap centerline and the radial direction. The spanwise distribution of \(H\) is obtained by the definition of the Euler equation and kept unchanged along the spanwise in this work. The viscous effects as well as the spanwise distribution of \(S\) are reflected in flow loss, flow deviation, and aerodynamic blockage. From the solution of Equation (15), the flow angles and the other thermodynamic and aerodynamic parameters can be obtained. The blade metal angles were determined by flow angles, incidence angle, and deviation angle. Subsequently, the resultant three-dimensional geometry of the blade was attainable by radially stacking the two-dimensional airfoils. The shroud curve and hub curve were parameterized by two cubic spline interpolation curves. A constant spanwise chord of each blade was adopted in this work. To validate the numerical model used in the present research, CFD simulation of NASA Stage 35 [34] was first carried out. The energy equations and three-dimensional  
Figure 2: Coordinate graph of the streamline curvature method. steady Reynolds-averaged Navier-Stokes equations were solved with EURANUS/TURBO. The turbulence was closed by an S-A model.