# The Refined Simulation Methods of Fluctuating Wind and Aerodynamic Forces Relative to Moving Trains
## 2 The Wind Spectrum of Moving Trains Under Different Wind Environments  
### Inhomogeneous and Non-Stationary Wind Environments  
In inhomogeneous and non-stationary wind environments, the train runs in the direction of \(\varphi_{0}\) at the speed of \(V_{tr}\), as shown in Figure 1. The wind direction is along the positive direction of the \(X\)-axis, and the mean wind speed \(U(y,t)\) is inhomogeneously distributed along the \(Y\)-axis and changes with time. In other words, the mean wind speeds for different spatial locations are unequal at the same moment, and the mean wind speedsfor the same spatial location are also unequal at different times. Points \(P\) and \(P^{\prime}\) are located on the moving train, and points \(P\) and \(P^{\prime}\) are located at spatial positions \(j\) and \(k\) at time \(t\), respectively. At this time, the fluctuating wind speeds at points \(P\) and \(P^{\prime}\) on the moving train are equal to the fluctuating wind speed at spatial positions \(j\) and \(k\). Points \(P\) and \(P^{\prime}\) move to the spatial positions \(j^{\prime}\) and \(k^{\prime}\) with the train at time \(t+\tau\). At this time, the fluctuating wind speeds acting on the train points \(P\) and \(P^{\prime}\) are equal to the fluctuating wind speeds at the spatial positions \(j^{\prime}\) and \(k^{\prime}\). According to the relationship of wind speeds relative to the moving train at different times illustrated in Figure 1, taking the along-wind fluctuating wind as an example, the correlation function of the fluctuating wind on the moving train at points \(P\) and \(P^{\prime}\) can be expressed as follows:  
\[R_{ \begin{subarray}{c}\sim,\\ u\end{subarray} }(\Delta\eta,\tau,t)=E\Big{[}\widetilde{u}(x_{P},y_{P},t) \widetilde{u}(x_{P^{\prime}},y_{P^{\prime}},t+\tau)\Big{]}=E\Big{[}\widetilde{ u}\big{(}x_{j},y_{j},t\big{)}\widetilde{u}(x_{k^{\prime}},y_{k^{\prime}},t+ \tau)\Big{]} \tag{1}\]  
where \(R_{ \begin{subarray}{c}\sim,\\ u\end{subarray} }(\Delta\eta,\tau,t)\) is the cross-correlation function of the along-wind fluctuating wind for the moving train at time \(t\); \(\Delta\eta\) is the distance between points \(P\) and \(P^{\prime}\); \(\tau\) is the time interval; \(\widetilde{u}(x_{P},y_{P},t)\), \(\widetilde{u}(x_{P^{\prime}},y_{P^{\prime}},t+\tau)\) are the along-wind fluctuating wind acting on the train points \(P\) and \(P^{\prime}\) at time \(t\) and \(t+\tau\), respectively; and \(\widetilde{u}\big{(}x_{j},y_{j},t\big{)}\), \(\widetilde{u}(x_{k^{\prime}},y_{k^{\prime}},t+\tau)\) are the along-wind fluctuating wind of the spatial positions \(j\) and \(k^{\prime}\) at time \(t\) and \(t+\tau\), respectively. The fluctuating wind could be regarded as a superposition of a series of vortices of various sizes, moving under the transport of the mean wind. As shown in Figure 1b, based on [PERSON]'s frozen turbulence hypothesis [19], the vortex of the fluctuating wind at the spatial position \(k^{\prime}\) at time \(t+\tau\) reaches the spatial position \(m\) at time \(\tau^{\prime}\) under the transport of the mean wind \(U(x_{k^{\prime}},y_{k^{\prime}},t+\tau)\) before it has time to evolve. Thus, the fluctuating wind at the spatial position \(k^{\prime}\) at time \(t+\tau\) is the same as that of the spatial position \(m\) at time \(t+\tau+\tau^{\prime}\). The equation is as follows:  
\[\widetilde{u}(x_{k^{\prime}},y_{k^{\prime}},t+\tau)=\widetilde{u}\big{(}x_{m },y_{m},t+\tau+\tau^{\prime}\big{)} \tag{2}\]  
Figure 1: Wind turbulence relative to moving trains: (**a**) stereo view; (**b**) plan view. \[\tau^{\prime}=\frac{\Delta x}{U(x_{k^{\prime}},y_{k^{\prime}},t+\tau)} \tag{3}\]  
where \(U(x_{k^{\prime}},y_{k^{\prime}},t+\tau)\) is the mean wind at the spatial position \(k^{\prime}\) at time \(t+\tau\); \(\widetilde{u}(x_{m},y_{m},t+\tau+\tau^{\prime})\) is the along-wind fluctuating wind at spatial position \(m\) at time \(t+\tau+\tau^{\prime}\).