# Spectral Characteristics of Fluctuating Aerodynamic Forces Acting on Rectangular Prisms
## 3 Results and Discussion
### Normalized Across-Wind Fluctuating Load Spectrum Model  
As one of the primary tasks of wind-induced structural response computation, selecting an appropriate and accurate spectrum model for fluctuating wind load is needed to accurately assess various wind effects. Based on the quasi-definite theory and strip assumption, the generalized aerodynamic spectrum for the fluctuating along-wind load is generated by converting the along-wind turbulence component. However, the spectral properties of the fluctuating across-wind load and torque are more complex, and the quasi-definite theory is thus no longer applicable. Therefore, various empirical models are created by fitting the experimentally observed spectrum. Previous analyses have found that although a number of factors can affect the fluctuating across-wind load spectrum, its relationship with the side ratio of the rectangular prism from the flow around state is particularly strong. The fluctuating wind load spectrum is a narrow-band phenomenon with a spectral peak, and it is primarily governed by vortex shedding at small side ratios (\(D/B<2\)). When the side ratio is sufficiently large (\(D/B\geq 2\)), the separated shear flow reattachment causes the aerodynamic force spectral energy to be redistributed to a wider bandwidth, leading to a broadband process with a petite spectral peak in the fluctuating wind load spectrum. Based on this feature concerning the fluctuating across-wind load spectral function, and with reference to existing spectral models [3; 6; 7; 8; 9; 16; 18; 19; 20; 21; 22; 24; 25; 37; 38], this study provides an empirical formula based on the side ratio of rectangular prisms. These expressions are as follows:  
\[\frac{fS_{L}(f)}{\sigma_{L}^{2}}=\sum_{i=1}^{N}\frac{\beta_{i}\cdot(f/f_{si}) ^{2}}{\left[1-\left((f/f_{si})^{2}\right)\right]^{2}+4\beta_{i}^{2}\cdot(f/ f_{si})^{2}}+\eta\cdot\exp\Big{\{}-[\alpha_{1}-\alpha_{2}\cdot(f/f_{si})^{2}] \Big{\}} \tag{1}\]  
where, \(N\) is the parameter related to the side ratio when \(D/B<2\), \(N=1\) and when \(D/B\geq 2\), \(N=2\); \(\beta_{i}\) is the spectral bandwidth correction factor; \(\eta\) is the spectral peak adjustment factor; \(f_{si}\) is the vortex shedding frequency; \(\alpha_{1}\) and \(\alpha_{2}\) are the low-band and high-band fluctuation correction parameters, respectively. The nonlinear least squares method was used to fit the experimental data to obtain the determined parameters, as follows:  
\[\beta_{1}=-0.13\cdot(D\Big{/}_{B})^{3}+0.927\cdot(D\Big{/}_{B})^{2}-1.707\cdot D \Big{/}_{B}+1.051 \tag{2}\]  
\[\beta_{2}=0.383\cdot D\Big{/}_{B}-0.05 \tag{3}\]\[f_{s1}=1.488\cdot\left( \begin{array}{c}D\Big{/}_{B}\end{array} \right)^{3}-10.581 \cdot\left( \begin{array}{c}D\Big{/}_{B}\end{array} \right)^{2}+18.996\cdot D \Big{/}_{B}-0.257 \tag{4}\]  
\[f_{s2}=-1.395\cdot D\Big{/}_{B}+9.142 \tag{5}\]  
\[\eta=-0.063\cdot\left( \begin{array}{c}D\Big{/}_{B}\end{array} \right)^{3}+0.55 5\cdot\left( \begin{array}{c}D\Big{/}_{B}\end{array} \right)^{2}-1.52\cdot D \Big{/}_{B}+1.146 \tag{6}\]  
\[\alpha_{1}=-0.685\cdot\left( \begin{array}{c}D\Big{/}_{B}\end{array} \right)^{3 }+5.706\cdot\left( \begin{array}{c}D\Big{/}_{B}\end{array} \right)^{2}+13.808 \cdot D\Big{/}_{B}+9.812 \tag{7}\]  
\[\alpha_{2}=-1.051\cdot\left( \begin{array}{c}D\Big{/}_{B}\end{array} \right)^{3 }+8.61\cdot\left( \begin{array}{c}D\Big{/}_{B}\end{array} \right)^{2}-21.222 \cdot D\Big{/}_{B}+14.783 \tag{8}\]  
According to the aforementioned empirical model, the results of the measured spectral functions are displayed in Figure 13. They demonstrate that the fit is good and can generally explain the spectrum characteristics of various side ratios. It is evident that the fitting accuracy is high because the proposed empirical formula considers the effects of the flow-induced high-frequency fluctuations and spectral bandwidth changes caused by the incoming turbulence, side ratio, vortex shedding, and separation reattachment.