# Multi-Objective Optimization of Cascade Blade Profile Based on Reinforcement Learning
## 3 Numerical Method  
### Governing Equations, Turbulence, and Transition Model  
The governing equations of general Reynolds-averaged Navier--Stokes (RANS) equations are as follows:  
\[\frac{\partial}{\partial t}\int_{\Omega}\mathbf{U}d\Omega+\int_{S}\mathbf{F}_{I} \cdot d\vec{S}+\int_{S}\mathbf{F}_{V}\cdot d\vec{S}=\int_{\Omega}\mathbf{S}d\Omega, \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{x_{i}}\) & \(\mathbf{\mu_{\mathbf{x_{i}}}}\) & \(\mathbf{\sigma_{\mathbf{x_{i}}}}\) \\ \hline
0.2 (SS) & \(-0.12\) & \(0.046\) \\
0.3 (SS) & \(0\) & \(0.046\) \\
0.4 (SS) & \(0.06\) & \(0.046\) \\
0.5 (SS) & \(0.04\) & \(0.046\) \\
0.6 (SS) & \(0\) & \(0.046\) \\
0.2 (PS) & \(-0.04\) & \(0.046\) \\
0.3 (PS) & \(-0.04\) & \(0.046\) \\
0.4 (PS) & \(-0.04\) & \(0.046\) \\
0.5 (PS) & \(-0.04\) & \(0.046\) \\
0.6 (PS) & \(-0.04\) & \(0.046\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The range of bump function coefficients \(\alpha_{i}\). Figure 3: Distribution of blade profiles perturbed by shape functions. Figure 2: Shape functions of profile perturbation. where \(\mathbf{\Omega}\) is the volume, and \(\mathbf{S}\) is the surface of the control volume. The conservative variables \(\mathbf{U}\), the inviscid flux \(\mathbf{\bar{F}}_{I}\), and the viscid flux \(\mathbf{\bar{F}}_{V}\) in the time-averaged form can be written as:  
\[\mathbf{U}= \begin{bmatrix}\overline{\rho}\epsilon_{1}\\ \overline{\rho}\epsilon_{2}\\ \overline{\rho}\epsilon_{3}\\ \overline{\rho}\epsilon_{\bar{E}}\end{bmatrix} \mathbf{\bar{F}}_{I}= \begin{bmatrix} \overline{\rho}\tilde{v}_{i}\tilde{v}_{i}+\overline{v}^{*}\delta_{1i}\\ \overline{\rho}\tilde{v}_{2}\tilde{v}_{i}+\overline{p}^{*}\delta_{2i}\\ \overline{\rho}\tilde{v}_{3}\tilde{v}_{i}+\overline{v}^{*}\delta_{3i}\\ (\overline{\rho}\tilde{E}+\overline{p}^{*})\tilde{v}\end{bmatrix} -\mathbf{ \bar{F}}_{V}= \begin{bmatrix}0\\ \overline{\tau}_{i1}\\ \overline{\tau}_{i2}\\ \overline{\tau}_{i3}\\ \overline{\tau}_{i}+\tilde{v}_{i}\overline{\tau}_{ij}\end{bmatrix} . \tag{4}\]  
Based on [PERSON]'s assumption, the Reynolds stress can be calculated with turbulent eddy viscosity \(\mu_{t}\), which is obtained by solving turbulence models. For turbulence calculation, Spalart-Allmaras turbulence model (SA) coupled with Abu-Ghannam and Shaw transition model (AGS) is used in this paper. SA turbulence model is a one-equation turbulence model proposed and developed by [PERSON] and [PERSON] in 1992 [36]. SA model is often used in the calculation of the internal flow field of turbomachinery ([37; 38; 39; 40], [etc.]) because of its robustness and its ability to treat complex flows [41; 42]. The principle of SA model is aimed at solving an additional transport equation for kinematic eddy viscosity. In a non-conservative manner, the equation contains an advective, a diffusive, and a source term. The transport equation is as follows:  
\[\frac{\partial\tilde{v}}{\partial t}+\vec{v}\cdot\
abla\tilde{v}=\frac{1}{ \sigma}\{\
abla\cdot[(\
u+(1+c_{b2})\tilde{v})\
abla\tilde{v}]-c_{b2}\tilde{v }\Delta\tilde{v}\}+S_{T}. \tag{5}\]  
The turbulent viscosity is given by  
\[\
u_{t}=\tilde{v}f_{v1} \tag{6}\]  
and  
\[\left\{ \begin{array}{c}f_{v1}=\frac{\chi^{3}}{\chi^{3}+c_{3}^{3}},\chi=\frac {\tilde{v}}{\
u}\\ S_{T}=\tilde{v}P(\tilde{v})-\tilde{v}D(\tilde{v}),\tilde{v}P(\tilde{v})=c_{b1 }\tilde{S}\tilde{v},\tilde{v}D(\tilde{v})=c_{w1}f_{w}(\frac{\tilde{v}}{d})^{ 2}\\ \tilde{S}=Sf_{v3}+\frac{\tilde{v}}{\kappa^{2}d^{2}}f_{v2},f_{v2}=\frac{1}{(1+ \chi/c_{v2})^{3}},f_{v3}=\frac{(1+\chi/c_{v1})(1-f_{v2})}{\chi}\\ f_{w}=g(\frac{1+c_{b3}^{6}}{g^{6}+c_{w3}^{3}})^{\frac{1}{6}},g=r+c_{w2}(r^{6}- r),r=\frac{v}{g^{6}\kappa^{2}d^{2}}\end{array} \right., \tag{7}\]  
where \(\tilde{v}\) is the turbulent working variable, \(\
u\) is the molecular viscosity, \(d\) is the distance to the closest wall, and \(S\) denotes the magnitude of vorticity. The constants arising in the model are referred to Reference [42].