# A Spectrum Correction Method Based on Optimizing Turbulence Intensity
## 2 Materials and Methods
### Power Spectrum Correction Method  
In the practical application of simulating a uniform pulsating wind field, since the dimensionless power density spectrum of one-dimensional space accords with Gaussian distribution (Figure 1 shows the longitudinal dimensionless wind speed power density spectrum curve of several classical spectra), the relationship between the simulating pulsating wind speed variances \(\sigma^{2}\) and the power spectrum density function \(S(f)\) is defined as below [18,19]:  
\[\sigma^{2}=\int_{0}^{\infty}S(f)df \tag{12}\]  
Therefore, when the analog frequency range \([F_{\max},\,F_{min}]\) is a segment interval and is cut out from the spectral density function originally defined interval \((0,\infty)\), theoretically, the variance of the simulated fluctuating wind speed \(\sigma_{theory}\)[2] is defined as:  
\[\sigma_{theory}^{2}=\int_{F_{min}}^{F_{max}}S(f)df \tag{13}\]  
The defined standard deviation of the fluctuating wind speed \(\sigma\) can be determined by an empirical formula in the simulation algorithm (e.g., Equation (20) in Section 3.1), according to the calculation formula of turbulence intensity \(I_{T}=\sigma/v(t)\), choosing the segment interval \([F_{\max},\,F_{min}]\) will cause the simulated results to produce truncated standard deviation, which will inevitably affect the numerical error of the simulated turbulence intensity due to the existence of the deviation of \(\sigma\) and \(\sigma_{theory}\). The deviation between the two can be calculated as:  
\[\varepsilon=\left|\frac{\sigma_{theory}-\sigma}{\sigma}\right| \tag{14}\]  
The standard deviation's deviation \(\varepsilon\) affects the simulated turbulence intensity to a certain extent, and the smaller the deviation is, the smaller the influence is. It can be seen from Equations (12)-(14) that when increasing the proportion of the interval length in the simulated frequency interval, the theoretically calculated standard deviation's deviation will decrease. However, it can be seen from the simulation algorithm introduced in Section 2.2 that the frequency step has a significant impact on the simulation results. On the other hand, according to the formula \(df=\frac{F_{max}-F_{min}}{N}\), after determining the basic range of the simulation frequency interval, a slight change caused by the minor increase in cutoff frequency relative to the magnitude of \(N\) is almost negligible. When the ratio of the turbulence integral length to the average wind speed \(X_{k}=L_{k}/\overline{U}\) is constant, the upper limit of the interval expands to the right indefinitely, and the deviation will not significantly decrease [10]. This is also consistent with the simulation conclusion in this paper: under the same simulation conditions, when the initial frequency changes by one step unit, the impact on the standard deviation's deviation \(\varepsilon\) is significantly greater than that when the cutoff frequency changes by one step unit (see Section 3.2 for details). In addition, the simulation results also show that the theoretical truncation bias \(\varepsilon\) caused by the truncation of the simulation frequency range will affect the numerical value of the simulated object in the simulation process, which leads to the simulation bias \(\hat{\varepsilon}\). Simulation deviation is not only related to truncation deviation, but also to the setting of other simulation parameters and the application of interpolation methods. The simulated deviation is a manifestation of the error generated by simulating the turbulence intensity, which can be expressed by the following calculation formula:  
\[\hat{\varepsilon}=\left|\frac{\hat{\sigma}-\sigma}{\sigma}\right| \tag{15}\]  
where, \(\hat{\sigma}\) denotes the standard deviation of actual simulated wind speed.