# The Spatial Structure of Passively Simulated Atmospheric Boundary Layer Turbulence
## 4 Results and Discussion
### Lateral Spatial Structure  
For lateral separation distances, the wind data were measured in the simulated turbulent boundary layer flows for nine cases. The time domain ensemble average shown in Equation (1) was used to calculate the correlation coefficients of different turbulent components, as shown in Figure 4.  
For the simulated boundary layer turbulence, the lateral correlation of the three turbulent fluctuation components is \(R_{v}>R_{u}>R_{w}\). According to the theoretical analysis of isotropic turbulence in the previous section, it can be seen that the turbulence satisfies \(180^{\circ}\) symmetry on the horizontal plane at the same height. For the horizontal and lateral distances, the \(v\) component is equivalent to the along-wind correlation function, and the \(u\) component and the \(w\) component are the lateral correlation functions. The experimental data show that the along-wind turbulent component correlation is stronger than the vertical fluctuating wind component, indicating that the isotropic turbulence theory can describe the lateral correlation distribution of the stimulated turbulence. However, there are differences between the two results, and this difference becomes more signif  
Figure 4: The correlation coefficients of different turbulent fluctuation components: (**a**) BL1 flow; (**b**) BL3 flow. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Wind Field**} & \multicolumn{3}{c}{**Integral Length Scale**} & \multicolumn{3}{c}{**Turbulence Intensity**} \\ \cline{2-7}  & \(L_{u}\) **(m)** & \(L_{v}\) **(m)** & \(L_{w}\) **(m)** & \(L_{u}\) **(\%)** & \(L_{v}\) **(\%)** & \(I_{w}\) **(\%)** \\ \hline BL1 & 1.216 & 0.376 & 0.301 & 12.8 & 10.7 & 8.4 \\ BL2 & 1.158 & 0.351 & 0.268 & 20.8 & 18.2 & 16.7 \\ BL3 & 1.027 & 0.324 & 0.232 & 27.1 & 22.1 & 18.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistical parameters of turbulence field. icant as the turbulence intensity increases, indicating the influence of the nonisotropic turbulence component. For transverse slender structures, such as long-span bridges, the spatial structure of the along-wind fluctuating component \(u\) and vertical fluctuating component \(w\) is generally more concerned. By fitting the cross-correlation coefficients (\(g(r)\)) measured in the BL1 wind field using the theoretical correlation function model [23; 29; 31], the parameters \(n\) and \(L\) were obtained, as shown in Figure 5.  
In Figure 5a, the lateral correlation of the along-wind fluctuating wind component can be sufficiently described by homogeneous isotropic turbulence theory, and the fitting result is \(n=0.614\) and \(L=0.797\). The fitting result of the along-wind fluctuating wind spectrum deviates from the theoretical model, but the error is within acceptable limits, as illustrated in Figure 5b. The results show that homogeneous isotropic turbulence theory works well with a horizontal lateral spacing. It can be used to accurately describe the energy distribution and spatial structure of the fluctuating velocity components in the frequency domain. Using the same method, the parameters \(n\) and \(L\) in the lateral correlation function of the vertical fluctuation component were fitted, as shown in Figure 6. The comparison results show that homogeneous isotropic turbulence theory can also be used to describe the lateral correlation of the \(w\) component of the simulated atmospheric turbulence. The fitting result is \(n=0.412\) and \(L=0.523\). The fitting parameter \(n\) is relatively close to the von Karman spectrum model (\(n=1/3\)), and the length scale is the same as the result in Table 3, \(L\approx 2L_{u}\), but there is a gap from the natural atmospheric turbulence. This result shows that the vertical component roughly satisfies the homogeneous isotropic assumption, and the fluctuating wind spectrum agrees well with the von Karman spectrum model. Although the lateral correlation function differs from natural atmospheric turbulence, the error is within engineering practice's acceptable range.