## 3 Slipstream Velocities
### Effect of Yaw Angle  
This is a method widely used when researching the aerodynamics of a train under a crosswind, including the velocity of train-induced slipstream in the presence of a crosswind [41]. The relationship between train-induced slipstream and crosswind velocities is shown in Figure 7; \(V_{w}\) is the crosswind speed, \(V_{t}\) is the train speed vector, and \(U_{\text{res}}\) is the resultant wind with the resultant yaw angle \(\alpha\) (the relative angle between the velocities of the wind and the train). To investigate the effect of yaw angle flow on the train, the yaw angle was 21.8\({}^{\circ}\) depending on the train speed and wind speed, as listed in Table 1, the normalized slipstream velocity magnitude results from both sides of the train \(U\) is defined as  
\[U=\frac{\sqrt{u^{2}+v^{2}+w^{2}}}{u_{res}} \tag{1}\]  
where \(u\), \(v\), and \(w\) are longitudinal, lateral, and vertical velocity components measured by the Cobra probe, respectively. \(u_{res}\) is the relative speed between the train and wind speed. Ensemble-averaged \(U\) from samples at \(y\) = 1.68 m, \(y\) = 2.25 m, and \(y\) = 2.81 m are considered for the no-crosswind and 21.8\({}^{\circ}\) crosswind cases with various train speeds and wind speeds. These lateral positions correspond to 0.1 m, 0.6 m, and 1.11 m from the train. The flow variables in the slipstream will be considered at the height of 1.21 m above the surface of the bridge deck. The \(U\) on the windward side of the train for the no-crosswind and 21.8\({}^{\circ}\) crosswind cases at \(z\) = 1.21 m are shown in Figure 8. In the no-crosswind case, the highest peak velocities are \(U\) = 0.173, \(U\) = 0.148, and \(U\) = 0.08, respectively. The peak \(U\) in the no-crosswind case is higher than the crosswind in the 21.8\({}^{\circ}\) cases at \(y\) = 1.68 m and \(y\) = 2.25 m. At \(y\) = 2.81 from the train side, the velocity in the no-crosswind and 21.8\({}^{\circ}\) cases decreases in the nose region due to the train effect wane. Furthermore, the velocity in the no-crosswind case is lower than in the crosswind cases. Due to the typical aerodynamic features of a train under the crosswind, the main focus for the aerodynamics of a train has traditionally been the use of the relative yaw angle between the wind speed and train speed. To understand the effect of yaw angle on the slipstream of a train running on the bridge, two cases of \(Vt\) = 6 m/s, \(Vv\) = 15 m/s, and \(Vt\) = 8 m/s, \(Vv\) = 20 m/s have been used to compare the slipstream profiles obtained in the same yaw angle. As shown in Figure 8, the slipstream velocities show some similarities in the nose region, boundary layer region, and wake region; especially at \(y\) = 1.68 m, the curves of the two 21.8\({}^{\circ}\) cases almost overlap. With increasing distance from the train side, the difference between the nose and wake region's velocities becomes more pronounced. These observations indicate that the method using a relative yaw angle may be applicable to the case when the flow is near the train. Once the slipstream is further out of the train, the aerodynamic interference caused by the bridge has significant effects on the slipstream. Thus, the method of relative yaw angle is no longer applicable. Figure 7: Schematic of the yaw angle of the resultant wind experienced by a moving vehicle in a crosswind. Figure 9 shows the \(U\) on the leeward side of the train at \(z=1.21\) m. The nose region velocities of both crosswind cases at \(y=1.68\) m, \(y=2.25\), and \(y=2.81\) m are 15% larger than the no-crosswind peak. At \(y=2.25\) m, the nose peak of \(U\) in \(21.8^{\circ}\) crosswind cases leaped by more than double the no-crosswind case. With increasing distance from the train, the difference between the velocities in the nose region becomes more pronounced such that the no-crosswind velocity is observed to decrease fastest with distance from the train side, followed by the \(21.8^{\circ}\) crosswind cases.