# Estimation of Aircraft-Dependent Bumpiness Severity in Turbulent Flight
## 2 Methodology  
### Turbulence Generation Based on In-Situ EDR Value  
With the isotropy and large Reynolds number assumption, there exists an inertial subrange of wave numbers in the turbulence field. The turbulence energy is neither generated nor dissipated. Instead, it is just transferred inertially from larger to smaller eddies. The energy spectrum of turbulence \(E(k)\) only depends on \(\varepsilon\), the Eddy Dissipation Rate (EDR), by  
\[E(k)=A\varepsilon^{\frac{2}{3}}k^{-\frac{5}{3}} \tag{1}\]  
where \(A=1.6\) and \(k\) is the spatial frequency of turbulence [3]. The von [PERSON] and Dryden turbulence model allows researchers to describe the random characteristics of isotropic and small-scale turbulence in high-altitude. The energy spectrum of von Karman model has a roll-off rate of -5/3 in high-frequency section, while the roll-rate of Dryden model is -2 because Dryden model is an approximation of von Karman model. Furthermore, faster-moving aircraft is also responsive to larger scales, which are typically outside the inertial sub-range. The [PERSON] turbulence model, representing both the inertial sub-range and the larger scales beyond it, has been widely used in both aerodynamics and meteorological field [3]. For large frequencies in the inertial subrange, the energy spectrum of the von Karman model is consistent with that of the Kolmogorov model [30]. The von Karman energy spectrum is described as  
\[E(k)=\frac{55}{9}\frac{\Gamma\big{(}\frac{5}{6}\big{)}}{\Gamma\big{(}\frac{1}{ 3}\big{)}\sqrt{\pi}}\frac{\sigma_{i}^{2}}{k_{0}}\bigg{(}\frac{k}{k_{0}} \bigg{)}^{-\frac{5}{3}} \tag{2}\]  
where \(\Gamma(\cdot)\) is the gamma function, \(\sigma_{i}^{2}\) refers to the intensity of three turbulence components. By equating Equations (1) and (2), the relationship between \(\sigma_{i}^{2}\) and \(\varepsilon\) is  
\[\sigma_{i}^{2}=\varepsilon^{\frac{2}{3}}A\pi^{\frac{1}{2}}L_{0}^{\frac{2}{3}} \frac{9}{55}\frac{\Gamma\big{(}\frac{1}{3}\big{)}}{\Gamma\big{(}\frac{5}{6} \big{)}} \tag{3}\]  
Based on the assumption of isotropy, the intensities of three turbulence components are identical, which can be described as \(\sigma_{Wx}=\sigma_{Wy}=\sigma_{Wz}\). The length scale in three directions is nominally set by \(L_{0}=669\) m [31]. The EDR indicator is generally defined as \(\varepsilon^{\frac{1}{3}}\), the cubic root of eddy dissipation rate. According to Equation (3), the turbulence intensity is obtained from in-situ EDR indicator. With the provided intensity and length scale, the longitudinal and vertical temporal spectra of von [PERSON] model are described by  
\[\left\{ \begin{array}{l}\Phi_{Wx}(\omega)=\sigma_{Wx}^{2}\frac{L_{Wx}}{\pi V_{T} }\frac{1}{[1+(aL_{Wx}\frac{\omega}{V_{T}})^{2}]^{\frac{5}{6}}}\\ \\ \Phi_{Wz}(\omega)=\sigma_{Wz}^{2}\frac{L_{Wz}}{\pi V_{T}}\frac{1+\frac{5}{3}(2 aL _{Wz}\frac{\omega}{V_{T}})^{2}}{[1+(2 aL_{Wz}\frac{\omega}{V_{T}})^{2}]^{\frac{ 1}{6}}}\end{array} \right. \tag{4}\]  
where \(\omega\) is the temporal frequency of turbulence. \(L_{Wx}\) and \(L_{Wz}\) stand for the longitudinal and vertical length scale of turbulence, respectively. In order to obtain the vertical acceleration response of aircraft, a sample of turbulence components needs to be generated beforehand by feeding unit-intensity white noise into forming filters. By decomposing Equation (4), the transfer functions of forming filter are described by  
\[\left\{ \begin{array}{l}G_{Wx}(s)=\frac{K_{Wx}}{T_{Wx}s+1},K_{Wx}=\sigma_{Wx} \sqrt{\frac{L_{Wx}}{\pi V_{T}}},T_{Wx}=(\frac{aL_{Wx}}{V_{T}})^{\frac{5}{6}}\\ \\ G_{Wz}(s)=\frac{K_{Wz}}{T_{Wz}s+1},K_{Wz}=\sigma_{Wz}\sqrt{\frac{L_{Wz}}{\pi V _{T}}},T_{Wz}=\frac{(\frac{2 aL_{Wz}}{V_{T}})^{\frac{11}{6}}}{\sqrt{\frac{8}{3 }}\frac{2 aL_{Wz}}{V_{T}}}\end{array} \right. \tag{5}\]  
By reducing to the one-order for rational approximation and further discretizing with the first-order backward differential method, the forming filters in Equation (5) are transformed into the following differential equations as [32]  
\[\left\{