## 6 Conclusions and Further Works  
Aerodynamic forces are significantly essential to the dynamic response analysis and overturning risk assessment of trains in crosswinds. Starting with the simulation of aerodynamic characteristics using the CFD method, an analytical model for predicting unsteady aerodynamic forces of a moving train subjected to a crosswind was set out in this study. Compared to the conventional approach widely used in previous investigations, the newly proposed model can capture the effects of both three-dimensional wind turbulence and varying flow direction of natural wind. The following conclusions can be drawn:  
1. The flow pattern of a moving train exposed to a crosswind is characterized by three-dimensional structures on account of the aerodynamic coupling effect of the train-induced wind and crosswind, and it is mainly determined by the resultant wind speed \(V_{R}\) and yaw angle \(\beta\), related to the train speed \(V\) and wind speed \(U\) with a flow angle of \(\alpha\).
2. A generalized sine expression can be used to predict the aerodynamic coefficient of a moving train varying with the resultant wind yaw angle, and based on wind tunnel test data, it is proved that the prediction formula is possible to be applied to most types of trains.
3. The quasi-steady and weighting function methods are developed through a mathematical process to calculate unsteady aerodynamic forces of a moving vehicle in a three-dimensional stochastic wind field, which shows good agreements with previous models. Moreover, the filtering and time lag effects are found in the weighting function approach.
4. When the complete wind turbulence is considered, greater force fluctuation and peak can be found, and also, variations in the wind direction are crucial to unsteady aerodynamic forces of moving trains, which can lead to maximum mean force at approximately 90\({}^{\circ}\) wind direction. However, when the flow direction of crosswind  
Figure 22: Side force peak value/mean ratio at different turbulence combinations: (**a**) train speed effect (\(U=20\) m/s, \(\alpha=90^{\circ}\)); (**b**) crosswind angle effect (\(U=20\) m/s, \(V=300\) km/h).  
deviates from 90\({}^{\circ}\), consideration of only a portion of the turbulence components may underestimate the dynamic response of trains.  
Our next work is the use of the simulated results to calculate the dynamic responses of trains in crosswinds with considerations of the track irregularity, track defects, foundation settlement, etc., which can also give some significant suggestions for the optimization and design of trains. Moreover, the simulated forces are useful in being applied to determine the characteristic wind curves of trains through a dynamic multi-body simulation. However, we acknowledge that some fundamental issues in the simulation remain to be explored and studied in further research, for example, the determination of aerodynamic admittance function.  
Conceptualization, Z.Y. and N.Z.; Methodology, Z.Y. and N.Z.; Software, Z.Y., X.L. and Z.L.; Formal analysis, Z.Y. and X.L.; Investigation, X.L. and Z.L.; Data curation, Z.Y., X.L. and Z.L.; Writing--original draft, Z.Y.; Writing--review & editing, N.Z., X.L. and Z.L.; Supervision, N.Z.; Funding acquisition, Z.Y. and N.Z. All authors have read and agreed to the published version of the manuscript.  
This work was accomplished by the supports of the China Postdoctoral Science Foundation (Grant No. 2022M/10870) and the National Natural Science Foundation of China (Grant No. 51708465).  
Not applicable.  
The authors declare no conflict of interest.