# Unfrozen Skewed Turbulence for Wind Loading on Structures
## 2 Materials and Methods
### One-Point Velocity Spectra  
The spectral characteristics of turbulence are described herein for a neutral atmosphere, which is dominating under strong wind conditions (see e.g., ([PERSON] et al., 2018)). The one-point power spectral densities of the velocity fluctuations \(S_{i}(f,x,y,z)\), \((i)=\{u,v,w\}\) are functions of both the frequency \(f\) and the location \((x,y,z)\). For the sake of brevity, only the dependency on \(f\) is explicitly shown herein. The target power spectral densities for the horizontal and vertical wind components are modelled using the so-called \"blunt\" and \"pointed\" models, respectively ([PERSON] et al., 2018; [PERSON] et al., 2018)  
\[\frac{fS_{u}(f)}{u_{*}^{2}} =\frac{a_{u}f_{r}}{\left(1+b_{u}f_{r}\right)^{5/3}} \tag{1}\] \[\frac{fS_{v}(f)}{u_{*}^{2}} =\frac{a_{v}f_{r}}{\left(1+b_{v}f_{r}\right)^{5/3}}\] (2) \[\frac{fS_{w}(f)}{u_{*}^{2}} =\frac{a_{w}f_{r}}{1+b_{w}f_{r}^{5/3}} \tag{3}\]  
where \(f_{r}=fz/\overline{u}\) is a reduced frequency; \(a_{i}\) and \(b_{i}\), with \(i=\{u,v,w\}\), are coefficients empirically estimated. For a neutral atmosphere, the hypothesis of local isotropy in the inertial subrange ([PERSON] et al., 2018) can be used to reduce the number of parameters in Equations (1) to (3). The assumption, already exploited by [PERSON] et al. (2018), means that \(fS_{u}(f)/u_{*}^{2}\simeq 0.3f_{r}^{-2/3}\), \(fS_{v}(f)/u_{*}^{2}\simeq 0.4f_{r}^{-2/3}\) and \(fS_{w}(f)/u_{*}^{2}\simeq 0.4f_{r}^{-2/3}\) at \(f_{r}\gg 1\). This implies that \(b_{u}=(a_{u}/0.3)^{3/5}\), \(b_{v}=(a_{v}/0.4)^{3/5}\) and \(b_{w}=a_{w}/0.4\). As a result, the following relation is obtained ([PERSON] et al., 2018):  
\[\frac{S_{w}}{S_{u}}\simeq\frac{S_{v}}{S_{u}}\sim\frac{4}{3}\text{ at }f_{r}\gg 1 \tag{4}\]  
The above relationship is generally observed for the neutral atmosphere and is a necessary condition for realistic 3D flow simulation in the near-neutral atmospheric surface layer ([PERSON] et al., 2018; [PERSON] et al., 2018; [PERSON] et al., 2018). Another interesting property of Equation (1) is that the maximal value of the normalized spectrum \(S_{u}\) is reached at a frequency \(f_{k}\) such that  
\[\frac{f_{k}S_{u}(f_{k})}{u_{*}^{2}}\approx 1 \tag{5}\]  
To account properly for the covariance between the \(u\) and \(w\) components, the co-spectrum between the longitudinal and vertical velocity components needs to be modelled. The co-spectrum, which is the real part of the cross-spectrum, is denoted \(Co_{uw}\). Defining the friction velocity as  
\[u_{*}=\sqrt{-u^{\prime}w^{\prime}}, \tag{6}\]  
the co-spectrum can be modelled using a similar expression as that used by [PERSON] et al. (2018):  
\[\frac{fCo_{uw}(f)}{u_{*}^{2}}=-\frac{a_{uw}f_{r}}{\left(1+b_{uw}f_{r}\right)^ {7/3}} \tag{7}\]  
where \(a_{uw}\) and \(b_{uw}\) are two constants to be determined empirically. The above definition of the friction velocity implies that \(\overline{v^{\prime}w^{\prime}}\approx 0\) and thus \(S_{vw}\approx 0\). Although this may not always be observed in full-scale ([PERSON] et al., 2018), it is a reasonable approximation for modelling purposes. Similarly, it is generally assumed that the co-spectral estimate \(S_{uv}\) has negligible valuessuch that turbulent momentum fluxes are driven by the vertical wind component only. If the friction velocity is defined as in Equation (6), the integration of Equation (7) over the frequencies should be equal to \(u_{*}^{2}\), which implies that \(b_{uw}=3/4a_{uw}\). [PERSON] et al. [29] studied the friction velocity based on the work of [PERSON] et al. [38], which defined \(u_{*}\) as in Equation (6). However, they obtained \(a_{uw}=14.0\) and \(b_{uw}=9.6\), i.e., \(b_{uw}\approx 0.69a_{uw}\). Therefore, to simulate correlated turbulent wind histories with consistent values between the target and computed friction velocity, the co-spectrum proposed by [PERSON] et al. [29] was not used directly. Instead a modified co-spectrum formulation with \(b_{uw}=3/4a_{uw}\) is chosen.