# Unfrozen Skewed Turbulence for Wind Loading on Structures
###### Abstract  
The paper introduces an algorithm to generate a three-variate four-dimensional wind turbulence field suited for yawed wind dynamic load simulation. At large yaw angles, a relaxation of [PERSON]'s hypothesis of frozen turbulence becomes relevant as well as the flow phase lag in the along-wind direction, which modulates the real and imaginary parts of the coherence. To capture such a general wind action on a structure, a modified spectral representation method is used where the coherence of turbulence is described as a complex-valued function. The one-point and two-point co-spectra are implemented in the simulation setup using a square-root-free Cholesky decomposition of the spectral matrix. The numerical procedure is illustrated based on turbulence characteristics derived from data collected during storm Aina (2017) on the Norwegian coast by three-dimensional sonic anemometers. During this event, a remarkable 3-hour stationary time series with a mean wind speed of 24 m s\({}^{-1}\) at a height of 49 m above ground was recorded. Since no computational grid is needed, the velocity fluctuations with representative spatio-temporal characteristics can be directly simulated on structural elements of slender structures. Such an algorithm may be essential for the design of super-long span bridges in coastal areas. turbulence; bridge; yaw angle; [PERSON]'s hypothesis; synthetic turbulence generation Article  
Unfrozen Skewed Turbulence for Wind Loading on Structures  
[PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]. flow characteristics in the along-wind and cross-wind direction differ significantly [9; 10] such that their projection onto a bridge-based coordinate system will result in different fluctuating loadings, depending on the yaw angle. Although some authors (e.g., [11; 12]) discussed different approaches to model the buffeting load for a non-zero yaw angle, few studies describe the two-point turbulence characteristics in skewed flow conditions, especially for load computations in the time domain. Yet, such a description may become crucial to model the dynamic wind loading on slender structures such as long-span bridges, inclined stay cables [13; 14] or wind turbines, which are sensitive to yaw errors [15]. Whereas the wind direction along a bridge is largely governed by the topography, the variability of the wind direction due to earth rotation, i.e., the direction shear, concerns the design of tall vertical structures. Recent studies have highlighted the influence of the direction shear on wind turbine performances [16] but also on the wind loading on high-rise buildings [17; 18]. The coherence of turbulence is a correlation function in the frequency space that reflects the spatial distributions of eddies. At a given distance, larger eddies are more correlated than smaller ones. Therefore, the coherence decreases when the distance or the wavenumber increases. The coherence is at the heart of the dynamic wind load modelling for the design of structures. The so-called root-coherence is the normalized cross-spectral density of turbulence fluctuations and is a complex-valued function. In the frequency space, the turbulent characteristics of skewed flows are addressed in the ESDU 86010 standard [19], where the imaginary part of the root-coherence becomes non-negligible. This reflects the fact that eddies impinge on different parts of a structure at different times (Figure 1). [PERSON]'s hypothesis of frozen turbulence [20] assumes that the same eddy moving in the streamwise direction at a speed \(\overline{u}\) will be characterized by no loss of correlation except due to the time lag \(\Delta t=d_{x}/\overline{u}\), where \(d_{x}\) is the streamwise separation (Figure 1). Turbulence may no longer be considered frozen if the streamwise distance becomes large compared to a typical length scale of turbulence. ESDU [19] relaxes [PERSON]'s hypothesis by modelling the root-coherence in the along-wind direction, albeit with parameters that can deviate substantially from those found in the literature [10; 21; 22].