## 4 Results and Discussion
### Effect of Turbulence
#### 4.2.4 The Power Spectral Density or Spectrum  
The power spectral density or spectrum \(E_{u}\) is defined as foll  
\[E_{u}(n)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-int}R_{uu}(\tau)d\tau \tag{4}\]  
where \(R_{uu}(\tau)=\overline{u(t)u(t+\tau)}\), and \(\tau\) is the time lag. There is little literature on spanwise shedding of train leeward vortices in turbulent environments to the authors' knowledge. Previous scholars [40; 41; 42] studied the tail-vehicle droop vibration of the [PERSON] vehicle model. With the increase in turbulence intensity, the vibration of the train body in a crosswind environment can no longer be neglected. Figures 17 and 18 show the results of the S3 plane measured by the cobra probe. The measurement points are arranged as shown in Figure 17a. The u, v and w three-way main frequencies are shown to be broadband, and the main frequency range is 0.1-0.3. P5 is the location of the vortex nucleus on the S3 surface, its v-directional vibration is the largest, and the main vibration frequency is 0.18, which is consistent with the location of the vortex nucleus in Figure 10b. \(Z\)-directional vibration at P6 is the largest, and the main frequency is 0.23. Figure 17: Spectrum analysis of horizontal measurement points near the S3 surface vortex core in wind tunnel: (**a**) test point arrangement; (**b**) v; (**c**) w; (**d**) u; (**e**) total pressure. The most dangerous conditions of the train occur at the moment of peak aerodynamic force, and the lift and side force of the head car and tail car are affected by the transient flow field, so the following studies the structure of the leeward vortex field with different nose lengths. \[Q=-\frac{1}{2}\Big{(}\|S\|^{2}-\|\varOmega\|^{2}\Big{)}=-\frac{1}{2}\Bigg{(} \bigg{(}\frac{\partial\overline{u}_{i}}{\partial x_{i}}\bigg{)}^{2}-\frac{ \partial\overline{u}_{i}\partial\overline{u}_{j}}{\partial x_{j}\partial x_{j }}\bigg{)}=\frac{1}{2}\frac{\partial\overline{u}_{i}\partial\overline{u}_{j}}{ \partial x_{i}\partial x_{j}} \tag{5}\]  
where \(S\) is the rate-strain tensor, and \(\varOmega\) is the rate-of-rotation tensor of the velocity variable. The iso-surfaces of positive values of \(Q\) reveal the locations in the flow where rotation dominates strain and indicate the vertical structures in the flow [35]. Figure 19 shows the transient flow field structure of the surrounding flow field for different train headway lengths at a \(15^{\circ}\) yaw angle, where the equivalent surface \(Q=\) 60,000, rendered using the instantaneous flow velocity. There is a short vortex V1 at the side of the head car, which starts from the first bogie and rolls upward. The intensity of the vortex can only maintain its rotation until the first bogie of the middle car, and the angle with the flow direction becomes smaller slowly as the length of the streamlined head car increases. At the same time, the flow separation occurs when the incoming flow crosses the top of the front end and flows to the leeward side, resulting in a longitudinal vortex V2, as shown in Figure 19b,c. V2 is stronger and is maintained from the front end to the rear end, and the angle with the incoming flow is similar to V1. The longer the nose length, the smaller the angle. When the nose length is short enough, V2 beside the head car basically blends in with the leeward side and results in the breakdown of large vortices into small ones, as shown in Figure 19a. After the incoming flow passes through the tail car bogie and the gap between the body and the ground, the flow also separates on the leeward side of the tail car, forming a vortex V3, which rotates in the opposite direction of V2 and moves upward after forming at the bottom of the leeward side, and then mixes with V2 in the middle of the body. Although the V3 vortex is not as strong as V2 because it is closer to the body, it can have a stronger influence on the aerodynamic performance of the tail car leeward side.