## 3 Prediction Model of Aerodynamic Coefficients
### Aerodynamic Coefficients  
Figure 11 shows the simulated aerodynamic coefficients of a moving train at the resultant yaw angle ranging from \(0^{\circ}\) to \(90^{\circ}\). It is apparent that the aerodynamic coefficient of the head car is greater than that of the middle and tail cars. It is known from previous analysis that the flow structure of a moving train is featured by a three-dimensional characteristic due to the aerodynamic interaction between the train-induced wind and natural wind, which can be mapped onto the surface pressure distribution of the train along the length, leading to different forces on cars. Also, it can be found that the aerodynamic coefficient of the head and middle cars increases at low yaw angles but decreases at higher yaw angles. A similar trend also can be observed on the leading vehicle in terms of previous investigations [14; 34]. However, it should be noted that there is a steady increase in the aerodynamic coefficient of the tail car as the yaw angle increases. In addition, the trend of the rolling coefficient in the resultant yaw angle is noticeably similar to that of the side coefficient, indicating that the rolling moment mainly results from the side force. As discussed above, the resultant yaw angle is essential to the aerodynamic coefficient of a moving train. In general, it is a fact that when the train speed and wind speed with an incoming direction are given, we can obtain the corresponding aerodynamic coefficient of the train by calculating the resultant yaw angle. Thus, it is fundamental to explore the explicit expression between the resultant yaw angle and aerodynamic coefficient. The experimental study carried out by [PERSON] and [PERSON] [32] found that, the aerodynamic coefficient at large yaw angles follows the curve \(C_{F}(\beta)=C_{F}(90^{\circ})\cdot\sin^{2}(\beta)\), where \"\(F\)\" means \"\(S\)\", \"\(L\)\" or \"\(R\)\", and \(C_{F}(90^{\circ})\) is the force coefficient at \(90^{\circ}\) yaw angle. In the simulation of unsteady aerodynamic cross wind forces on trains, [PERSON] [34] gave a crude approximation of the side and lift coefficient \(C_{F}(\beta)=C_{F}(90^{\circ})\cdot\sin(\beta)\). Despite the fact that these expressions given by [PERSON] and [PERSON] [32] and [PERSON] [34] are quite simple and easy, they are not robust for a variety of vehicles. To solve this problem, this study attempts to propose a generalized prediction formula of the aerodynamic coefficient, expressed as follows:  
\[C_{F}(\beta)=a+b\sin(c\beta+d) \tag{8}\]  
Figure 11: Aerodynamic coefficients of the train at various resultant wind angles: (**a**) side coefficient; (**b**) lift coefficient; (**c**) Rolling coefficient. where \(a\), \(b\), \(c\), and \(d\) are the fitted parameters based on experimental or numerical data, respectively. With the use of the aerodynamic coefficient of the head car, Figure 12 plots the comparison results by means of different prediction formulas and Table 1 lists the fitted coefficients. It is visible that Equation (8) can give a good agreement for the simulated side and lift coefficients when compared to the works by [PERSON] and [PERSON] [32] and [PERSON] [34], and the coefficient of determination R-squared, usually used to evaluate the fitting accuracy, is close to 1, implying that the prediction results are significantly reliable. A simple explanation for Equation (8) is that when the resultant wind speed is assumed to be a fixed value, the velocity component perpendicular to the car body and mainly determining the aerodynamic force on the train can be written by the sine function between \(V_{R}\) and \(\beta\). Based on this, there may be a sinusoidal expression between the aerodynamic force coefficient and resultant angle. In fact, the proposed prediction formula in Equation (8) is not only applicable to the aerodynamic coefficients of the train in this study, but also useful for most railway vehicles.