# Unfrozen Skewed Turbulence for Wind Loading on Structures
## 5 Discussion and Conclusions
### Conclusions  
The present paper introduces an algorithm to generate a three-variate four-dimensional correlated stationary Gaussian wind field which accounts for non-zero yaw angles and relaxes [PERSON]'s hypothesis of frozen turbulence. This approach is suitable to model the wind load on slender structures, such as future super-long span bridges crossing Norwegian fjords, which can span from one to five-kilometre distances.  
For a structure located mainly in the atmospheric surface layer, the flow field is modelled using surface layer scaling. Turbulence is here governed by the roughness length, the reference wind speed at a reference height, and the one-point and two-point auto- and cross-spectral densities of the velocity fluctuations. Due to the non-zero values of the single-point cross-spectrum for the along-wind and vertical turbulence components, the spectral matrix is not necessarily positive definite. Therefore, the traditional Cholesky decomposition is replaced with the more general LDL factorization. To model non-frozen turbulence, a 3D root-coherence function that accounts for longitudinal separations in addition to the lateral and vertical ones was used.  
To illustrate the relevancy of the turbulence parameters selected, measurement data from three 3D sonic anemometers mounted on a meteorological mast on the west coast of Norway were used. The full-scale records were taken from storm Aina (2017), which was characterized by a 3-h mean wind speed of \(24\,\mathrm{m}\,\mathrm{s}^{-1}\) at \(49\,\mathrm{m}\) above ground. The exceptionally long duration and stationarity of the records allowed an in-depth characterization of the spectral characteristics of turbulence.  
The capability of the algorithm to simulate skewed flow conditions is illustrated for a diamond geometry. The time lag was modelled using a complex exponential in the frequency space, which is reflected by the co-coherence and quad-coherence estimates. Although the quad-coherence may not directly participate in the linearised wind load calculation in the frequency domain, it is required for time-domain simulations.  
For a 5-km floating pontoon bridge, the proximity of the deck to the sea surface means that appropriate root-coherence and one-point power spectral densities should be used in the wind load modelling. This information could be provided by the algorithm presented herein.  
Conceptualization, E.C.; methodology, E.C., N.D.; software, E.C.; validation, E.C., N.D.; formal analysis, E.C.; investigation, E.C., N.D., J.B.J., J.S. and J.W.; resources, J.W.; data curation, E.C.; writing--original draft preparation, E.C.; writing--review and editing, All co-authors; visualization, E.C. All authors have read and agreed to the published version of the manuscript. This research received no external funding. This research received no external funding. The algorithm to generate the skewed turbulence field with the relaxation of [PERSON]'s hypothesis of frozen turbulence is written in Matlab and is available under a BSD-3-Clause open source license ([[https://doi.org/10.5281/zenodo.7075723](https://doi.org/10.5281/zenodo.7075723)]([https://doi.org/10.5281/zenodo.7075723](https://doi.org/10.5281/zenodo.7075723)) (accessed on 16 September 2022)). The authors declare no conflict of interest.