## 5 Unsteady Aerodynamic Forces Prediction with a Complete Turbulence  
### Prediction Model  
When a complete turbulence field with a longitudinal component \(u(t)\), lateral component \(v(t)\), and vertical component \(w(t)\), is considered, the transient speed vectors on a moving train are drawn in Figure 18. It should be noted that the resultant wind speed \(V_{R}\) and yaw angle \(\beta\) are time-varying as a result of the fluctuating turbulence, which can be expressed by:  
\[V_{R}=\overline{V_{R}}+{V_{R}}^{\prime} \tag{12}\]  
\[\beta=\overline{\beta}+\beta^{\prime}+\beta^{\prime\prime} \tag{13}\]  
\[\overline{V_{R}}=\sqrt{\left(V+U\cos\alpha\right)^{2}+\left(U\sin\alpha\right) ^{2}} \tag{14}\]  
\[\overline{\beta}=\arctan\left(\frac{U\sin\alpha}{V+U\cos\alpha}\right) \tag{15}\]  
where \(\overline{V_{R}}\) and \({V_{R}}^{\prime}\) are the mean component and the fluctuating component of the resultant wind speed, respectively; \(\overline{\beta}\) is the mean component of the resultant wind yaw angle; \(\beta^{\prime}\) and \(\beta^{\prime\prime}\) are the fluctuating components resulting from the longitudinal turbulence \(u(t)\) and lateral turbulence \(v(t)\), respectively. Noting that the resultant wind speed also can be given in terms of the geometrical relations shown in Figure 18:  
\[{V_{R}}^{2}=\left[V+\left(U+u\right)\cos\alpha+v\sin\alpha\right]^{2}+\left[ \left(U+u\right)\sin\alpha-v\cos\alpha\right]^{2}+w^{2} \tag{16}\]  
It is assumed that the train speed \(V\) and wind speed \(U\) are significantly greater than the fluctuating components. Neglecting the high-order fluctuations of wind turbulence, an approximate expression from Equations (12), (14) and (16) can be obtained:  
\[{V_{R}}^{2}\approx\overline{V_{R}}^{2}+2\overline{V_{R}}{V_{R}}^{\prime} \approx\overline{V_{R}}^{2}+2u(U+V\cos\alpha)+2 Vv\sin\alpha \tag{17}\]  
Thus, the fluctuation turbulence on a moving train can be written as  
\[{V_{R}}^{\prime}\approx\frac{\left(U+V\cos\alpha\right)u+Vv\sin\alpha}{ \overline{V_{R}}} \tag{18}\]  
Figure 18: Transient speed vectors on a moving train in a competed turbulence field. It can be seen from Equation (18) that the resultant wind turbulence of a moving train depends on the longitudinal and lateral turbulence. It should be noted that the following geometrical relations are possible to be observed from \[V_{R}=\overline{V_{R}}+{V_{R}}^{\prime} \tag{12}\]0  
\[\left\{ \begin{array}{l}\cos\left(\alpha-\overline{\beta}\right)=\frac{U\sin \alpha}{V_{R}}\\ \sin\left(\alpha-\overline{\beta}\right)=\frac{V\sin\alpha}{V_{R}}\end{array} \right. \tag{19}\]  
Hence, an equivalent expression of Equation (17) can be obtained  
\[{V_{R}}^{\prime}\approx u\cos\left(\alpha-\overline{\beta}\right)+v\sin \left(\alpha-\overline{\beta}\right) \tag{20}\]  
Based on the quasi-steady hypothesis, the unsteady aerodynamic forces \(F\) of a moving vehicle, consisting of a mean \(\overline{F}\) and a fluctuation \(F^{\prime}\), can be expressed by a mean resultant speed \(\overline{V_{R}}\) and a fluctuating resultant speed \(V_{R}\)\({}^{\prime}\):  
\[F=\overline{F}+F^{\prime}=0.5\rho AC_{F}(\beta,\theta)\left(\overline{V_{R}}+{ V_{R}}^{\prime}\right)^{2} \tag{21}\]  
Through Taylor series expansion at \(\theta=0\) and \(\beta=\overline{\beta}\), the aerodynamic coefficient can be written as an approximate expression:  
\[C_{F}(\theta,\beta)\approx C_{F}\left(\overline{\beta},0\right)+\left.\frac{ \partial C_{F}}{\partial\beta}\right|_{ \begin{array}{l}\theta=0\\ \beta=\overline{\beta}\end{array} }\left(\beta^{\prime}+\beta^{\prime\prime} \right)+\left.\frac{\partial C_{F}}{\partial\theta}\right|_{ \begin{array}{l} \theta=0\\ \beta=\overline{\beta}\end{array}