# Unfrozen Skewed Turbulence for Wind Loading on Structures
## 5 Discussion and Conclusions  
### Skewed Turbulence Generation on a Diamond Geometry  
To illustrate the performances of the turbulence generator, ten realizations of a wind field with the same statistical characteristics as for storm Aina are computed. Their statistical properties are estimated for each time series and ensemble-averaged to reduce the random error. The flow field is computed in the middle of the elements of a diamond geometry (Figure 7), denoted \(e_{1}\), \(e_{2}\), \(e_{3}\) and \(e_{4}\), which has the advantage of including both along-wind and cross-wind separations. The diamond geometry is located in a horizontal plane and does not intend to represent a real structure. Instead, it is used to demonstrate that the turbulent field is properly generated for both along-wind and cross-wind distances. The wind field is generated with a sampling frequency of 4 Hz and 21 data  
Figure 6: Co-coherence and quad-coherence of the three velocity components estimated for a wind direction of 331\({}^{\circ}\), \(\overline{u}=18\) m s\({}^{-1}\) at \(z=49\) m above ground and a yaw angle of 69\({}^{\circ}\) (\(d_{x}=7.5\) m, \(d_{y}=2.9\) m, and \(d_{z}=16\) m). The data were recorded on 8 December 2017 from 20:00 to 23:00. The solid lines correspond to the best fit of Equation (32) for the co-coherence and Equation (33) for the quad-coherence. points, leading to time series with a duration of ca. 1 h. The reference mean wind speed is set as 24 m s\({}^{-1}\) at a reference height of 49 m above ground with a roughness length \(z_{0}\) of 0.05 m which is within the range of values found in fjord-like topographies [36; 85]. In the following example, the time series are generated at the same height, so there is no need to prescribe a mean wind speed profile. The target velocity spectra are computed based on Equations (1)-(3) and (7) with the parameters estimated in Table 3. Similarly, the target co-coherence model is from Equation (13) with the parameters estimated in Table 4.  
In Figure 7, the axes \(X\) and \(Y\) correspond to the longitude and latitude of the vertices, such that the \((X,Y)\) plane is horizontal. The line joining \(e_{1}\) and \(e_{2}\) is aligned with the wind direction. Therefore, the along-wind and cross-wind distance between \(e_{2}\) and \(e_{1}\) are \(d_{x}=20\) m and \(d_{y}=0\) m, respectively. On the other hand, the line joining \(e_{1}\) and \(e_{4}\) is perpendicular to the flow, such that \(d_{x}=0\) m and \(d_{y}=20\) m. Finally, the line joining \(e_{1}\) and \(e_{3}\) combines both cross-wind and along-wind separations, with \(d_{x}=20\) m and \(d_{y}=20\) m. Figure 8 displays the time series of the \(u\) velocity component on \(e_{1}\) and \(e_{2}\). The normalized cross-correlation function \(R_{u}(e_{1},e_{2})\) in the bottom panel of Figure 8 reaches its maximum near 0.80 s, which correspond to the advection time between \(e_{1}\) and \(e_{2}\). The largest value of \(R_{u}(e_{1},e_{2})\) is below 1, which underlines the loss of correlation in the along-wind direction due to the along-wind root-coherence introduced to relax [PERSON]'s hypothesis of frozen turbulence. Figure 7: Diamond geometry used to validate the skewed turbulence generation algorithm. The flow is simulated in the middle of elements 1 to 4 with a northern wind direction. For each structural element \(e_{i}\) (\(i=1,2,3,4\)), the axial and normal velocity components are denoted \(v_{a}\) and \(v_{n}\), respectively. The co-coherence estimates between the different elements are shown as a scatter plot in Figure 9. In this figure, the solid lines correspond to Equation (32) for the co-coherence and to Equation (17) for the cross-coherence. We remind that Equation (17) is only used here for comparison with the simulated cross-coherence, which was numerically implemented using Equation (25). The excellent agreement between target and estimated co-coherence values indicates that the method adopted to generate a yawed flow field with a lower-than-unity co-coherence in the along-wind direction is appropriate.