# Multi-Objective Optimization of Cascade Blade Profile Based on Reinforcement Learning
## 3 Numerical Method
### Cascade Performance Calculation  
The three-dimensional compressible Navier-Stokes equation is solved by using a finite-volume based RANS solver NUMECA FineTurbo. Multigrid technique is used during the solving process in order to accelerate the convergence. To test the accuracy of the transition position prediction, the VKI turbine blade is chosen as the validation case. The VKI turbine blade, as is shown in Figure 4, has been investigated a lot, both numerically and experimentally, and the experimental data on its transition position can be referred to in Ubaldi [51]. The blade-to-blade grid used to validate the calculation is shown in Figure 5.  
Figure 6 shows the experimental and the calculated data of the transition position of VKI turbine. The experiment was carried out by [PERSON] [51]. Details about experimental measurement can be referred to in Appendix A.1. With Laser Doppler Velocimetry (LDV), the boundary layer integral parameters distribution is measured, and the start of transition (ST), transition position (TP), and end of transition (ET), marked in Figure 6,  
Figure 4: VKI turbine blade profile. Figure 5: Computation grid of VKI cascade. were concluded by [PERSON], according to the boundary layer integral parameter distributions from the experiment [51]. The intermittency factor is used to trigger the transition locally, which is computed on each point of the blade surface and at each iteration via the solver. The intermittency factor is 0 before transition and 1 after the transition onset. According to the calculated intermittency factor distribution in Figure 6, the computational transition position fits well with the experimental transition position (magenta dashed line in Figure 6), which validates the accuracy of the calculation of transition position. The geometry of the initial cascade blade profile to be optimized in this paper is shown in Figure 7, with the basic parameters listed in Table 2. The blade-to-blade grid is generated as a structured grid, with the topology shown in Figure 8. The length of the inlet domain is as long as the chord line, and the outlet domain is twice as long as the chord line. Figure 6: The validation of transition position prediction. The scatters represent the \(Re_{\theta}\) distribution obtained from [PERSON]’s experiment [51]. The solid line represents the calculated intermittency factor distribution by computational fluid dynamics in this paper. Figure 7: Geometric shapes of initial cascade blades. A three-dimensional grid is used as the cascade grid, with five points across the span of the cascade, generated via NUMECA AutoGrid. The boundary condition at the hub and shroud is set to the mirror boundary condition, to eliminate the three dimensional effect. The grid points distribution is listed in Table 3. The total number of grid points is 407,465. The height of the cells at the walls is set to \(3\times 10^{-6}\) m to ensure that \(y^{+}\) is below 10. The 2D cross-sectional view of the computational grid is shown in Figure 9. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** \\ \hline Chord length \(c\) (mm) & 109.5 \\ Solidity & 1.792 \\ Blade geometric inlet angle \(\beta_{1k}\) (\({}^{\circ}\)) & 50.0 \\ Blade geometric outlet angle \(\beta_{2k}\) (\({}^{\circ}\)) & 11.4 \\ Blade camber angle (\({}^{\circ}\)) & 38.6 \\ Blade stagger angle (\({}^{\circ}\)) & 32.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometric parameters of cascade. Figure 8: An illustration of blade-to-blade topology. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Block Name** & **Type** & **Points (\(x,\theta,r\))** & **Total Points** \\ \hline Inlet & H & \(193\times 41\times 5\) & 39,565 \\ Up blade region & H & \(721\times 29\times 5\) & 104,545 \\ Blade passage & O & \(673\times 33\times 5\) & 111,045 \\ Down blade region & H & \(721\times 29\times 5\) & 104,545 \\ Outlet & H & \(233\times 41\times 5\) & 47,765 \\ Total & & & 407,465 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Grid parameters.