## 5 Unsteady Aerodynamic Forces Prediction with a Complete Turbulence
### Case Study  
With the use of aerodynamic parameters described above and taking the head car as an example, Figure 19 plots the simulated unsteady aerodynamic forces time histories of a moving train about 100 s by means of the quasi-steady calculation and the weighting function approach. It is obvious that the fluctuating parts of unsteady side and lift forces always oscillate up and down around the mean value, which is to be expected as the aerodynamic force is derived from the fluctuating time series swinging around the mean wind velocity. According to the definition of unsteady force in Equation (32), it can be found that the significant difference between the quasi-steady approach and the weighting function approach is how fluctuation in the turbulence leads to variation in the force. It is clear from Figure 19 that the side and lift force curves through the weighting function approach are smoother for the reason that some high frequency turbulences are filtered. In addition, the unsteady force in the weighting function exhibits a slight time lag in comparison with the quasi-steady result, which follows the turbulence fluctuation exactly. Note that the following discussion of unsteady force will be based on the result obtained from the weighting function method. When the train speed and wind speed are given, the varying crosswind direction can lead to impacts on the aerodynamic coefficient and resultant wind speed to affect the unsteady loads of a moving train. Figure 20 shows the effect of crosswind direction on the mean side force. It clearly illustrates that the increasing crosswind angle brings out an increase in the mean component of unsteady force firstly and then a decrease, and the corresponding maximum appears around 80\({}^{\circ}\)-90\({}^{\circ}\) incoming flow direction of crosswind that is approximately perpendicular to the travel line, which is possible to be unfavorable to the train's stability. Figure 21 shows the simulated side force time histories at three typical crosswind directions (30\({}^{\circ}\), 90\({}^{\circ}\), 150\({}^{\circ}\)) to explore different wind turbulence effects on the unsteady force. Observing the results at 90\({}^{\circ}\) wind direction, we can see that the unsteady force time series resulting from both longitudinal and lateral turbulences are similar to the results only including the longitudinal turbulence, indicating there is no visible effect of lateral turbulence on the force fluctuation. However, at conditions of 30\({}^{\circ}\) and 150\({}^{\circ}\) crosswind angles, it is clearly visible that the longitudinal, lateral and vertical turbulence can considerably contribute to the unsteady force curves. One explanation might simply be that when the crosswind direction swings around 90\({}^{\circ}\), lateral turbulence is approximatelyparallel to the moving direction of the vehicle, which can only contribute to less effective turbulent components normal to the car-body to produce small fluctuation in the force. Besides, it also can be found that the wind turbulence in the vertical direction has a great impact on the unsteady force throughout the crosswind direction because it is always normal to the moving train. The extreme value of unsteady force, in general, related to the extreme index used to assess the running safety and ride quality of railway vehicles, is of great importance. Considering different wind turbulences and using simulated 100s realizations of the unsteady force time series, Figure 22 shows the ratio of the maximum value to the mean. We  
Figure 21: Simulated unsteady side force time histories at different crosswind angles (\(U=20\) m/s, \(V=300\) km/h): (**a**) \(\alpha=30^{\circ}\), \(V_{R}=101.1\) m/s, \(\beta=5.7^{\circ}\); (**b**) \(\alpha=90^{\circ}\), \(V_{R}=85.7\) m/s, \(\beta=13.5^{\circ}\); (**c**) \(\alpha=150^{\circ}\), \(V_{R}=66.7\) m/s, \(\beta=8.6^{\circ}\). Figure 20: Mean side force at a variety of crosswind angles (\(U=20\) m/s, \(V=300\) km/h).