# Unfrozen Skewed Turbulence for Wind Loading on Structures
## 2 Materials and Methods
### Coherence Modelling  
The root-coherence between two velocity components \(i\) and \(j\), where \(i,j=\{u,v,w\}\) and between two points located at coordinates \((x_{1},y_{1},z_{1})\) and \((x_{2},y_{2},z_{2})\) is defined as  
\[\mathrm{coh}_{ij}(f,x_{1},y_{1},z_{1},x_{2},y_{2},z_{2})=\frac{S_{ij}(f,x_{1}, y_{1},z_{1},x_{2},y_{2},z_{2})}{\sqrt{S_{i}(f,x_{1},y_{1},z_{1})S_{j}(f,x_{2},y_{2},z _{2})}}, \tag{10}\]  
where \(S_{ij}(f,x_{1},y_{1},z_{1},x_{2},y_{2},z_{2})\) is the two-point cross-spectral density between the \(i\) and \(j\) components; \(S_{i}(f,x_{1},y_{1},z_{1})\) and \(S_{j}(f,x_{2},y_{2},z_{2})\) are the one-point spectra estimated at coordinates \((x_{1},y_{1},z_{1})\) and \((x_{2},y_{2},z_{2})\), respectively. For the sake of simplicity, the root-coherence is assumed to be a function of the spatial separations rather than the measurement locations, i.e.,  
\[\mathrm{coh}_{ij}(f,x_{1},y_{1},z_{1},x_{2},y_{2},z_{2})\approx\mathrm{coh}_{ ij}(f,d_{x},d_{y},d_{z}), \tag{11}\]  
where \(d_{x}=|x_{1}-x_{2}|\), \(d_{y}=|y_{1}-y_{2}|\), and \(d_{z}=|z_{1}-z_{2}|\). Although this assumption is widely used in structural design, its validity is questionable at large and/or vertical separations [22; 58; 59]. Equations (10) and (11) require a fairly homogeneous turbulent field, which is compatible with the turbulence generation considered herein but such a condition is not always achievable in full-scale. Small inhomogeneities due to a wind direction shear or a vertical mean wind shear are known to be handled properly by turbulence generators, which is the reason why they are widely applied for the simulation of surface-layer turbulence in flat and uniform terrain. However, abrupt changes in the wind direction or the mean wind speed due to local topographic effects may not be adequately captured by the coherence of turbulence. Similarly, flow separation and recirculation areas downstream of hills are not meant to be described in detail by turbulence generators. In such a situation, it is recommended to study the wind loading using, if possible, wind tunnel tests, computational flow dynamic simulations and site-specific measurements. The root-coherence is a complex-valued function, the real part of which is often named co-coherence and denoted \(\gamma_{ij}\). Its imaginary part, called quad-coherence, is denoted \(\rho_{ij}\)  
\[\mathrm{coh}_{ij}(f,d_{x},d_{y},d_{z})=\gamma_{ij}(f,d_{x},d_{y},d_{z})+i\rho_ {ij}(f,d_{x},d_{y},d_{z}) \tag{12}\]  
In the present study, the root-coherence of each velocity component is inspired by the Davenport model, which is a positive real-valued exponential function [60] and extended to three spatial dimensions. This model was introduced in the 1960s to model the turbulent wind loading on structures. The Davenport model was later generalized to any velocity component and any direction (e.g., [52; 61]). Although the Davenport model is purely empirical, it is widely used in wind engineering, wind energy and micro-meteorology for its simplicity and ability to characterize the coherence of turbulence at relatively small separation, in general under 50 m. When the separation distance is no longer negligible compared to a typical turbulence length scale, the root-coherence becomes lower than unity at zero frequency [9]. In this case, an additional parameter can be introduced in the Davenport model, leading to  
\[\mathrm{coh}_{ij}(f,d_{x},d_{y},d_{z}) \approx\exp\left\{-\frac{1}{\overline{u}}\sqrt{C_{x}+C_{y}+C_{z}}\right\} \tag{13}\] \[C_{x} =(c_{x1}^{j}fd_{x})^{2}\] (14) \[C_{y} =(c_{y1}^{j}fd_{y})^{2}+\left(c_{y2}^{j}d_{y}\right)^{2}\] (15) \[C_{z} =(c_{z1}^{j}fd_{z})^{2}+\left(c_{z2}^{j}d_{z}\right)^{2} \tag{16}\]  
where \(j=\{u,v,w\}\); \(d_{x}\), \(d_{y}\) and \(d_{z}\) are the longitudinal, lateral and vertical spatial separation, respectively. The coefficients \(C_{x}\), \(C_{y}\) and \(C_{z}\) are decay parameters. The coefficients \(c_{y2}^{j}\), \(c_{z2}^{j}\) are proportional to the inverse of a time scale of turbulence [22; 62].