## 2 Theoretical Analysis of Reflection Error Correction for Ambient Radiation of Turbine Blade
### Three-Dimensional Dynamic Reflection Model of Turbine Blades  
By establishing a coordinate transformation relationship of the turbine blade surface at different positions, the coordinates of each point on the blade surface at different positions can be described. Using the actual shape of an engine blade, a three-dimensional geometric model of a turbine blade was established. As shown in Figure 3, the \(x\)-axis is the engine axis, the rotor blades rotate around this axis, and the \(y\)-axis is parallel to the initial state rotor blade axis line \(r\). The \(z\)-axis direction is determined by the right-hand rule. When the rotor blade rotates, line \(r\) rotates onto line \(r^{\prime}\), and the angle \(\varphi\) between the two straight lines represents the position angle of the rotor blade. A three-dimensional model of the rotor blade at \(\varphi=0\) was established, and the transformation relationship between the coordinates of each point on the rotor blade and \(\varphi\) was analyzed. First, we intercepted the blades with a set of planes parallel to the plane formed by the \(x\)-\(z\) axis with a certain step length and fit these intercept lines to obtain a set of equations:  
\[z_{0}=f_{j}(x)\ \ \ \ \ j=1,2,3\ldots\text{N} \tag{4}\]  
Figure 3: Position changes of rotor blades at different angles. Figure 2: Schematic diagram of the reflection relationship of discretized blade panel. Then, we analyzed the coordinate change of any point on the rotor blade section line \(z_{0}=f_{j}(x)\) in Figure 3 when it rotates. When the rotor blade rotates to the position angle \(\varphi=\varphi_{0}\), the \(y\)-axis coordinate of the point becomes  
\[ \begin{array}{c}y^{\prime}=y\cos\varphi_{0}+f_{j}(x)\sin\varphi_{0}\\ j=1,2,3,\ \ldots,\ N\end{array}  \tag{5}\]  
The \(x\)-axis coordinate of the point will not change during the rotation, and the \(z\)-axis coordinate will become  
\[ \begin{array}{c}z^{\prime}=\left[f_{j}(x)+y^{\prime}\sin\varphi_{0}\right] /\cos\varphi_{0}\\ j=1,2,3\ldots N\end{array}  \tag{6}\]  
By obtaining the position coordinates of each point on the cross-section of the rotor blade, a geometric model of the rotor blade at any position angle \(\varphi\) can be established. The method of modeling the guide vane is the same as that of the rotor blade. The established blade model was divided into panels, and the radiation angle factor of each panel was calculated. In order to ensure calculation accuracy, we divided the suction surface and pressure surface of all blades for analysis. Considering the calculation accuracy and speed comprehensively, we divided the suction surface and pressure surface of each blade into 9720 panels. The three-dimensional geometric model of the turbine blades is shown in Figure 4. According to the relative position of the turbine blades, the suction surface of the target blade, and the suction surface and pressure surface of the upper rotor blade, three adjacent guide vanes were modeled and discretized. A guide vane near the target blade was selected to analyze the radiation angle factor between each panel on the guide vane and the center point \(P_{0}\) of the panel on the target blade. When the segmentation is sufficiently fine, each panel can be approximated as a rectangle, and the three-dimensional coordinates of all panel endpoints on the guide vane obtained by the geometric model can be expressed as matrices \(\mathbf{X_{1},Y_{1},Z_{1}}\). As shown in Figure 5, the position vector \(\mathbf{p_{1_{k,l}}}\) and the length vector \(\mathbf{a_{1_{k,l}}}\) and width vector \(\mathbf{b_{1_{k,l}}}\) of the end point (for the \(k,l\) element in the matrix) of a panel on the guide vane can be expressed as  
\[\mathbf{p_{1_{k,l}}}=\left[ \begin{array}{c}1/2(\mathbf{X_{1}}(k-1,l)+\mathbf{X_{1}}(k,l+ 1))\\ 1/2(\mathbf{X_{1}}(k-1,l)+\mathbf{X_{1}}(k,l+1))\\ 1/2(\mathbf{X_{1}}(k-1,l)+\mathbf{X_{1}}(k,l+1))\end{array} \right] \tag{7}\]  
Figure 4: The three-dimensional geometric model of the turbine blades. \[\mathbf{a_{1k,l}}=\left[