# Wind Speed Spectrum of a Moving Vehicle under Turbulent Crosswinds
## 5 Coherence Function  
The cross-correlation function of the wind speed at points \(P\) and \(P\)'in Figure 3 can be written as follows:  
\[\rho_{uu}(\Delta\eta,\tau)=E[u(0,\eta,t)u(0,\eta^{\prime},t+\tau)] \tag{24}\]  
where \(u(0,\eta,t)\) and \(u(0,\eta^{\prime},t+\tau)\) are, respectively, the longitudinal wind speeds at points P and P' at time \(t\) and \(t+\tau\).  
The time delay \(\tau\) can be transformed into an equivalent spatial interval \(\tau\overline{V}\) according to [PERSON]'s freezing hypothesis in Figure 2. Equation (24) can be presented as the correlation characteristics of points \(P\) and \(P_{e}^{\prime}\):  
\[\rho_{uu}(\Delta\eta,\tau)=E[u(0,\eta,t)u(0,\Delta y_{e}+\eta,t+\tau+\tau^{ \prime})]=R_{uu}(\Delta y_{n},\tau+\tau^{\prime}) \tag{25}\]  
\[\Delta y_{n}=(\eta^{\prime}-\eta-V_{T}\tau)\sin\phi \tag{26}\]  
\[\rho_{uu}(\Delta\eta,\tau)=\rho_{uu}(\Delta y_{n},\tau+\tau^{\prime})=\int \limits_{0}^{\infty}S_{uu}(\Delta y_{n},f)\cos[2\pi f(\tau+\tau^{\prime})]df \tag{27}\]  
\[S_{uu}(\Delta y_{n},f)=C_{oh}(\Delta y_{n},f)S_{u}(f) \tag{28}\]  
where \(S_{uu}(\Delta\eta,f)\) is the cross-power spectral density of two points on a moving vehicle with a distance \(\Delta\eta\). \(S_{uu}(f)\) is the power spectral density of a single point.  
By substituting Equation (27) into Equation (28), Equation (27) can be written as follows:  
\[\rho_{uu}(\Delta\eta,\tau)=\rho_{uu}(\Delta y_{n},\tau+\tau^{\prime})=\int \limits_{0}^{\infty}S_{u}(f)C_{oh}(\Delta y_{n},f)\cos[2\pi f(\tau+\tau^{ \prime})]df \tag{29}\]  
\[C_{oh}(\Delta y_{n},f)=\frac{S_{uu}(\Delta y_{n},f)}{S_{u}(f)} \tag{30}\]  
\(C_{oh}(\Delta y_{n},f)\) could be presented as follows:  
\[C_{oh}(\Delta y_{n},f)=\exp(-\frac{fC_{u}|\Delta y_{n}|}{U}) \tag{31}\]  
In Equation (31), \(Cu\) is the correlation coefficient which could not evaluate the influence of the turbulence integral scale on the correlation characteristics.  
[PERSON] [24] proposed an empirical model with appropriate corrections as follows:  
\[\begin{split}& C_{oh}(\Delta y_{n},f)=\exp(-A_{I}\Delta y_{n})\\ & A_{I}=\left(c_{2}^{2}+\left(c_{3}f/2\pi\
u L\right)^{2}\right)^{ 0.5\mathrm{c}_{1}}\end{split} \tag{32}\]  
where \(c_{1}\), \(c_{2}\) and \(c_{3}\) are the parameters to be fitted. The parameters are dimensioned, and Equation (32) can be corrected as follows:  
\[\begin{split}& C_{oh}(\Delta y_{n},f)=\exp(-A_{I_{\
u}}\Delta y_{n })\\ & A_{J_{\
u}}=\left(c_{2}^{2}+\left(c_{3}fL_{\
u}/2\pi U\right)^{ 2}\right)^{0.5\mathrm{c}_{1}}/L_{\
u}\end{split} \tag{33}\]  
The correlation coefficients can then be obtained from Equation (33). However, the most important problem is that the expression for a correlation function cannot be obtained easily. [PERSON] [18] obtained the change law of the correlation function using an integral approach. Considering the wind speed spectrum peak, when the \(V_{\mathrm{ratio}}\) value is larger than 1, the increasing rate could reach 8%. After comprehensive consideration, when the vehicle's moving speed is larger than the wind speed, the wind speed spectrum and correlation characteristics should be considered.  
In Figure 11, some of the coherence function points on the vehicle at different moving speeds are shown. It could be known that the coherence function increases with the increasing speed ratio of the vehicle's moving speed to the wind speed. The correlation characteristics of the moving vehicles, which are of great influence, could be investigated by wind tunnel tests.