# Numerical Study on Cooling Performance of a Steam-Cooled Blade Based on Response Surface Method
## 2 Numerical Methods
### Numerical Methods  
The numerical model of the steam cooled blade passage is shown in Figure 3. For the convenience of the study, the periodic passage with three flow channels in the test section is simplified to a single passage, and the flow channel on both sides of the blade in the numerical model is set to periodic boundary condition. For the accuracy of the numerical simulation, the inlet and outlet of the steam channel are extended by 100 mm each, which can make the numerical calculation better. The ANSYS CFX software (V18, ANSYS Inc., Pittsburgh, PA, USA) is used for the flow-solid conjugate heat transfer calculation. Since the flow-solid conjugate heat transfer method considers the influence brought by the solid  
Figure 1: Experimental schematic diagram of the steam-cooled blade. Figure 2: Cascade channel and the experimental blade: (**a**) Cascade channel; (**b**) The experimental blade and its cooling structure. heat conduction, it is widely used in the research of turbine blade cooling technology. The finite difference method based on finite elements is used to discretize the control equations and solve the three-dimensional compressible Reynolds time-averaged \(N\)-\(S\) equation, and the diffusion, source and convection terms in the equation are separated using a high-precision discrete format, and the residual targets for continuity, momentum and energy are set to \(10^{-6}\). The boundary conditions of the numerical calculation are basically the same as the experimental conditions, and the mainstream inlet is set as the total temperature and total pressure; the steam inlet is set as the static temperature and static pressure, and the turbulence level is 5%; the mainstream outlet is set as the average static pressure, and its value is the average value of the experimental obtained static pressure, and its value is allowed to vary within 5% of the average value; the steam outlet is set as the mass flow rate, and the magnitude is the experimental value. The fluid-solid contact surface is set as a flow-solid interface with the same temperature and heat flux distribution; the rest of the solid surface is set as the adiabatic wall surface. Figure 3 also presents the mainstream inlet, mainstream outlet, steam inlet, and steam outlet of the steam-cooled blade channel. According to the research on turbine blade cooling performance by our research group [24; 25], it is observed that the SST \(k\)-\(\omega\) turbulence model has a high accuracy in the prediction of turbine blade cooling performance and is in the best agreement with the experimental data. To guarantee the accuracy and comparability of numerical simulation results, the SST \(k\)-\(\omega\) turbulence model is also used in this paper for the numerical simulation of the steam-cooled blade channel. [PERSON] et al. [26] proposed the SST \(k\)-\(\omega\) model based on the \(k\)-\(\omega\) model, using the \(k\)-\(\omega\) model in the wall boundary layer and the k-\(\varepsilon\) model in the wall boundary layer, which can be used to improve the accuracy of the prediction of flow separation phenomena. The SST \(k\)-\(\omega\) model equation is as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_{ i}}=\frac{\partial}{\partial x_{j}}\left(\Gamma_{k}\frac{\partial k}{\partial x_{j}} \right)+G_{k}+Y_{k}+S_{k} \tag{1}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho\omega u_{i})}{ \partial x_{i}}=\frac{\partial}{\partial x_{j}}\left(\Gamma_{\omega}\frac{ \partial\omega}{\partial x_{j}}\right)+G_{\omega}+Y_{\omega}+S_{\omega}+D_{\omega} \tag{2}\]  
The laws of conservation of mass, momentum and energy are used to solve the fluid domain, where the variables (\(u\), \(v\), \(w\), \(T\)) all obey a universal conservation principle.