## 4 Numerical Results of the Proposed Reconstruction Method  
### Presentation of Reconstruction Results of Velocity Field and Pressure Field  
In what follows, five stacking planes with heights of 0.22, 0.25, 0.30, 0.35 and 0.4 m and dimensions of \(1\times 1\) m are selected to represent the 3D wind field. The 3D velocity field and relative pressure field corresponding to the boundary condition of 90, 15 m/s is selected to validate the performance of the reconstruction algorithm. A series of parameters, such as the number of basis vectors, the number of sensors and the measurement noise need to be considered. In the simulation, the measurement noise \(\gamma\) is defined by Gaussian noise:  
\[\gamma\sim N\Big{(}0,\sigma^{2}\Big{)} \tag{15}\] \begin{table}
\begin{tabular}{c c} \hline
**Mesh Number** & **Relative Error (\%)** \\ \hline
986,523 & 0.12 \\
415,243 & 0.22 \\
195,489 & 0.58 \\ \hline \end{tabular}
\end{table} 
Table 1: Mesh independence test. Figure 6: Hexahedral structure grid and boundary layer grid around wind tunnel model. Here, \(\sigma\) is the standard deviation and \(N\) is the normal distribution function related to \(\sigma^{2}\); the noise level is defined by \(\sigma\):  
\[Noise\_level=\sigma\times 100\% \tag{16}\]  
However it can be seen from Equation (10) that the selection of the number of basis vectors is crucial for reconstruction calculation. Before reconstruction, the joint effect of the number of basis vectors and the measurement noise on the reconstruction error will be explored to determine the number of basis vectors. First, the number of sensors and the noise level are determined to be 100 (only 1% of total data) and 10%, respectively, and then the number of basis vectors is increased from 1 to 20 to obtain the reconstruction error curve. To ensure accurate results, 1000 calculations for each reconstruction are performed to obtain the average value. The calculation results are shown in Figure 7.  
Figure 7 is a double \(y\)-axis diagram, from which it could be found that the reconstruction error of relative pressure is nearly ten times higher than that of velocity. This is because the distribution of relative pressure is characterized by the fact that most of the pressure values are very small (close to 0), and the pressure gradient is very large. In addition, the reconstruction error of the velocity field is not monotonically reduced, whereas the reconstruction error of the pressure field shows a steady downward trend. This reflects the reconstruction characteristics of velocity and pressure fields: First, the feature extraction of the velocity field is more accurate, which allows better reconstruction results with fewer basis vectors. However, this may lead to an increase in the sensitivity of reconstruction accuracy to noise level, i.e., when the number of basis vectors continues to increase, the reconstruction accuracy will decrease instead. Correspondingly, when feature extraction is more accurate, an excessive increase of the number of basis vectors will result in reconstruction errors being more sensitive to noise level. To verify the accuracy of PCA feature extraction, an energy proportion \(t\in(0,1]\) is specified to represent the importance of eigenvectors. If \(n\) eigenvalues are \(\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{n}\), then \(t\) is defined as follows:  
\[t_{i}=\frac{\lambda_{i}}{\sum\limits_{i=1}^{n}\lambda_{i}} \tag{17}\]  
Figure 8 shows the \"energy proportion\" of each eigenvalue in the velocity database and the pressure database. It can be seen from Figure 8 that the energy of the eigenvalue  
Figure 7: Relative reconstruction error (%) corresponding to different number of basis vectors. decreases rapidly in both databases; the first six and eight basis eigenvalues account for 99.9% \"energy\" of the total system in the velocity database and the pressure database, respectively, which is sufficient to represent the basis vectors.