## 3 Results and Discussion
### Analysis of the CFD Calculation Results
The correlation function is defined as follows:  
\[R_{ij}(r,x)=\left\langle u_{i}^{\prime}(x,t)u_{j}^{\prime}(x+r,t)\right\rangle, \tag{7}\]  
where \(u_{i}\) and \(u_{j}\) are components of the pulsating velocity, and \(x\) and \(r\) are vectors. When \(i=j\), this expresses the autocorrelation of a certain pulsation, and when \(i\
eq j\), it means that the two pulsations are correlated. The autocorrelation diagram is shown in Figure 13.  
The above formula can be transformed into dimensionless correlation coefficients as follows:  
\[R_{ij}=\frac{\left\langle u_{i}^{\prime}(x,t)u_{j}^{\prime}(x+r,t)\right\rangle }{\sqrt{\left\langle u_{i}^{\prime 2}(x,t)u_{j}^{\prime 2}(x+r,t)\right\rangle}}. \tag{8}\]  
\[Q=\frac{1}{2}(\left\|\Omega\right\|^{2}-\left\|\mathrm{E}\right\|^{2}) \tag{6}\]0 Relationship between the vortex core radius and vortex core spacing with flow direction. \[Q=\frac{1}{2}(\left\|\Omega\right\|^{2}-\left\|\mathrm{E}\right\|^{2}) \tag{6}\]1 Autocorrelation of pulse velocity \(u\). The correlation coefficient magnitude represents the statistical similarity between two random variables or the probability that the fluctuation takes the same sign at different positions in the flow field. \(R^{\prime}_{ij}=\pm 1\) denotes that two random variables are completely correlated, while \(R^{\prime}_{ij}=0\) denotes that they are completely unrelated. For uniform turbulence, the correlation coefficient is independent of the coordinate position, and related only to the correlation distance. In order to investigate the correlation of the turbulent momentum downstream of the VGs, the correlation characteristics of the turbulent momentum \(v^{\prime}\) along the flow direction at certain points at different heights on a pair of centrosymmetric planes of the VGs were compared. Figure 14 illustrates the autocorrelation coefficient curve \(R^{\prime}_{v^{\prime}v^{\prime}}(r_{1})\) for the calculation of the fluctuating velocity \(v^{\prime}\) at different positions on the central symmetry plane of the domain. Five reference points at different heights and different points along the flow direction were selected for coherence analysis (Figure 14a). The minimum correlation distance was 0.5 mm, the maximum correlation distance was 675 mm, and the VGs were placed at the 0 position. According to Figure 14b, the coherence coefficient at the reference point is \(1\), indicating that the reference point was correlated with itself. In addition, the larger the correlation coefficient near the reference point, the larger the correlation coefficient amplitude, and the higher the correlation between the fluctuation and the reference point. A comparison between the following four cases demonstrated that when \(\lambda/H=3\), the magnitude of the correlation coefficient was the largest at most locations, which indicated that at this VG spacing, the structure of turbulence in the downstream domain preserved a high organization. When \(\lambda/H=7\) and 9, the magnitude of the correlation coefficient was larger at most locations, while that of \(\lambda/H=5\) was closer to \(R^{\prime}_{v^{\prime}v^{\prime}}(r_{1})=0\) at most locations. The correlation between the turbulent vortices downstream of the VGs and the reference point was poor, indicating that this VG spacing (\(\lambda/H=5\)) was more suitable for suppressing flow separation. The width between the correlation coefficient curve and the two intersections reflected the scale of the coherent structure in the downstream turbulent flow field. It can be observed that the scale of the vortices increased gradually as the flow developed downstream. Figure 15 shows the average velocity distribution at different positions along the downstream direction. \(<\)\(u\)\(>\)/\(U_{\infty}\) represents the ratio of the time-averaged velocity to the mainstream velocity and \(y\)/\(H\) is the ratio of the normal wall height to the VG height.