# Wind Characteristics in Mountainous Valleys Obtained through Field Measurement
## 3 Results
### Turbulence Characteristics
#### 3.2.5 Turbulence Integral Scale  
The turbulence integral scale is a measure of the average eddy size of turbulence. Corresponding to the three directions related to the longitudinal, horizontal, and vertical fluctuating velocity components \(u\), \(v\), and \(w\), respectively, a total of nine turbulent integral scales exist. According to the principle of a smooth random process, the \(x\)-direction of the coordinate axis is defined as follows:  
\[\mathrm{L}_{a}^{x}=\frac{1}{\sigma_{a}^{2}}\int_{0}^{\infty}C_{a_{1}a_{2}}(x) \mathrm{d}x \tag{11}\]  
where \(a\) is the direction of the pulsating wind (\(u\), \(v\), or \(w\)), \(C_{a_{1}a_{2}}(x)\) is the cross-correlation function of the pulsating component between the two points, and \(\sigma_{a}^{2}\) is the variance of the  
Figure 28: Fitting error of the northeast side.  
Figure 29: Fitting error of the southwest side.  
pulsating wind speed component. The above equation is a theoretical definition. The cross-correlation function of the pulsating components can be rewritten as an autocorrelation function, thus transforming the spatial correlation of two points in the integral scale formula into a temporal correlation of the same point.  
The autocorrelation function integral was used to calculate the turbulence integral scale along the downwind pulsating wind speed along the downwind x-axis during the observation period at the bridge site. Figure 30 presents the probability distribution of the integral scale \(\mathrm{L}_{u}^{x}\) based on a large number of wind speed samples during strong wind periods. The figure reveals a wide distribution of integration scales, but mainly concentrated within 100 m. From the statistical results, the value of \(\mathrm{L}_{u}^{x}\) on the northeast side is between 0 m and 650 m. The highest probability is found in the range of 50-100 m, accounting for about 40% of the total, with a mean value of 87.64 m. The \(\mathrm{L}_{u}^{x}\) values on the southwest side ranged from 0 m to 800 m. The 50-100 m range showed the highest probability and accounted for about 40% of these values, with a mean value of 94.13 m.  
The histograms of the integral scale frequency statistics for the two measurement points in Figure 30 are similar. To compare the two histograms (\(H_{1}\) and \(H_{2}\)), it was necessary to define a measure of the concept of similarity, called the degree of similarity,  
\[\mathrm{d}(H_{1},H_{2})=\frac{\frac{\sum\limits_{i}^{n}(H_{1}(i)-\overline{H_{ 1}})(H_{2}(i)-\overline{H_{2}})}{\sqrt{\frac{\sum\limits_{i}^{n}(H_{1}(i)- \overline{H_{1}})}^{2}\frac{n}{i}(H_{2}(i)-\overline{H_{2}})^{2}}}} \tag{12}\]  
where \(\overline{H_{k}}=\frac{1}{n}\frac{n}{i}H_{k}(i)\), \(n\) is the number of statistical intervals in the histogram, and \(k\) takes values of 1 and 2.  
Equation (12) was used to calculate the similarity coefficients, and the distribution frequencies for each integral scale interval are shown in Table 3. The similarity was calculated as 99.53%. The integral scale of the measuring points on the northeast and southwest sides has highly consistent statistical characteristics in the case of large samples. Figure 31 shows the correlation between the integral scale and the mean wind speed. The correlation between the integral scale and the mean wind speed is weak, and the integral scale is highly discrete.  
Figure 30: Frequency statistical histogram of the integral scale: (**a**) \(\mathrm{L}_{u}^{x}\) on the northeast side; (**b**) \(\mathrm{L}_{u}^{x}\) on the southwest side.