## 2 Numerical Simulation
### Numerical Method  
Detached eddy simulation (DES) is a hybrid method of large eddy simulation (LES) and Reynolds average Navier-Stokes (RANS), RANS is employed to simulate near-wall flows, while LES is used to analyze flow separation in other regions. However, DES has very strict requirement on the quality of grids, and when the near-wall grids are not properly arranged, the boundary layers will be solved by RANS instead of LES, which would cause the modeled stress depletion (MSD) [20]. DDES is an improvement of DES that can improve the calculation efficiency and avoid the appearance of MSD.  
The incompressible N-S equation is expressed as:  
\[\frac{\partial(u_{i})}{\partial t}+\frac{\partial(u_{i}u_{j})}{\partial x_{j}} =-\frac{\partial p}{\rho\partial x_{i}}+v\frac{\partial}{\partial x_{j}}( \frac{\partial u_{i}}{\partial x_{j}})+f_{i}(i,j=1,2,3) \tag{1}\]  
Figure 1: Scale models: (**a**) dimensions of the bridge and train models; (**b**) schematic of the curved wind barrier; (**c**) three-dimensional view of the curved wind barriers with four curvatures (unit: mm).  
\[\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{2}\]  
where \(u_{i}\) and \(u_{j}\) are the components of velocity, \(t\) is the time, \(p\) is the pressure, \(\rho\) is the density, \(v\) is the kinematic viscosity, and \(f_{i}\) is the body force of the \(i\) direction.  
The SST \(k-\omega\) model can accurately simulate complex separation flows and is one of the most widely used turbulence models in engineering [21]. SST model formulations are as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho u_{j}k)}{\partial x_{ j}}=\rho P-\beta^{*}\rho\omega k+\frac{\partial}{\partial x_{j}}\Bigg{[}(\mu+ \sigma_{k}u_{t})\frac{\partial k}{\partial x_{j}}\Bigg{]} \tag{3}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho u_{j}\omega)}{ \partial x_{j}}=\frac{\gamma}{\
u_{t}}P-\beta\rho\omega^{2}+\frac{\partial}{ \partial x_{j}}\Bigg{[}(\mu+\sigma_{\omega}u_{t})\frac{\partial\omega}{ \partial x_{j}}\Bigg{]}+2(1-F_{1})\frac{\rho\sigma\omega^{2}}{\omega}\bullet \frac{\partial}{\partial x_{j}}\bullet\frac{\partial\omega}{\partial x_{j}} \tag{4}\]  
\[F_{1}=\tanh\Bigg{\{}\left\{\min\left[\max\left(\frac{\sqrt{k}}{\beta^{*} \omega y^{*}},\frac{500v}{y^{2}\omega}\right)\frac{4\rho\sigma\omega^{2}k}{ CD_{k\omega}y^{2}}\right]\right\}^{4}\Bigg{\}} \tag{5}\]  
where \(CD_{k\omega}=\max\left(2\rho\sigma\omega^{2}\frac{18\delta\omega}{\omega \delta x_{j}\delta x_{j}},10^{-10}\right)\) and \(y\) is the distance nearest to the wall.  
The turbulent eddy viscosity \(\
u_{t}\) is defined as:  
\[\
u_{t}=\frac{a_{1}k}{\max(a_{1}\omega,SF_{2})} \tag{6}\]  
where \(S\) is an invariant measure of the strain rate and \(F_{2}\) is the second blending function defined by:  
\[F_{2}=\tanh\left\{\left[\max\left(\frac{2\sqrt{k}}{\beta^{*}\omega y^{*}}, \frac{500v}{y^{2}\omega}\right)\right]^{2}\right\} \tag{7}\]  
where \(\rho\) is the density, \(u_{j}(j=1,2,3)\) represents the velocity components, \(\mu\) is the laminar viscosity, and \(\beta\), \(\beta^{*}\), \(\gamma\), \(\sigma_{k}\), \(\sigma_{\omega}\), \(\sigma_{\omega^{2}}\), and \(a_{1}\) are constants.  
In order to simulate the flow field of unsteady and large separated flow around the train-bridge system with a curved wind barrier, the delayed detached eddy simulation (DDES) based on the shear stress transport (SST) \(k-\omega\) turbulence model was adopted in this paper. The discretizations of the computational domain and the governing equations adopted the finite volume method. The commercial CFD software ANSYS FLUENT 19.2 with a pressure-based solver was used to perform all simulations of this study.