## 2 Numerical Models  
### Numerical Method  
Detached-eddy simulation (DES) model was first carried out by [PERSON] [28] to compensate the lack of Reynolds-Averaged Navier-Stokes (RANS) in unsteady turbulence prediction and avoid the heavy demand of grid and computation of Large-Eddy Simulation (LES). Then, for the purpose of solving the modeled stress depletion (MSD), delayed-detached eddy simulation (DDES) was carried out by [PERSON] [29]. DDES introduces the second length scale of turbulence model in its length scale instead of filter scale of LES. [PERSON] et al. [30] introduced an improved delayed-detached eddy simulation (IDDES), which combines DDES and LES for wall modeling. In the IDDES method, the model stress loss issue from the original DES technique is removed using the DDES length scale, and the boundary layers are predicted using WMLES. In the recent years, this hybrid approach has been widely utilized to examine separated flows which are unstable and geometrically associated [31]. [PERSON] et al. [32] used the \(sst\)\(k-\omega\) IDDES method to perform the reverse flow past an NACA0012 airfoil and had satisfactory results compared to experimental measurements. Moreover, in the field of high-speed train aerodynamics simulations which have complex shapes and usually high Reynolds numbers, the \(sst\)\(k-\omega\) IIDDES method is quite popular among scholars due to its capacity of capturing the train wake flow and vortex structures [31; 33; 34; 35]. Hence, \(sst\)\(k-\omega\) IIDDES is adopted in the present study. The key point of IIDDES is to modify the dissipation component in the governing equation for the turbulent kinetic energy equation. The governing equations for the kinetic energy \(k\) are written as  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho u_{i}k)}{\partial x_{ i}}=\tau_{ij}S_{ij}-\frac{\rho k^{1.5}}{l_{k-\omega}}+\frac{\partial}{\partial x_{ i}}\bigg{[}(\mu_{l}+\sigma_{k}\mu_{t})\frac{\partial k}{\partial x_{i}}\bigg{]} \tag{1}\]  
where \(l_{k-\omega}=\sqrt{k}/(\beta^{*}\omega)\); \(\rho\) is the fluid density; t is the time; \(u_{i}\) is the velocity; \(x_{i}\) is the position; \(\tau_{ij}\) is the Reynolds stress tensor; \(S_{ij}\) is the mean strain rate tensor; \(\mu_{l}\) is the laminar viscosity coefficient; \(\sigma_{k}\) is model coefficient; \(\mu_{t}\) is the turbulent eddy viscosity. The IIDDES model defines the length scale as  
\[l_{IDDES}=\tilde{f}_{d}(1+f_{e})l_{k-\omega}+\big{(}1-\tilde{f}_{d}\big{)}C_{ DES}\Delta \tag{2}\]  
where \(\Delta=\min[C_{w}\max(\mathrm{d},\Delta),\Delta]\); \(\mathrm{d}\) is the distance to the closest wall boundary; \(\Delta=\max(\Delta\mathrm{x},\Delta\mathrm{y},\Delta\mathrm{z})\) represents the maximum of the local grid scales; \(C_{w}=0.15\), \(C_{DES}\) is a constant calculated by the blending function \(F_{1}\)  
\[C_{DES}=(1-F_{1})C_{DES}^{k-\varepsilon}+F_{1}C_{DES}^{k-\omega} \tag{3}\]  
with \(C_{DES}^{k-\varepsilon}\) is 0.61 and \(C_{DES}^{k-\omega}\) is 0.78; \(\tilde{f}_{d}\) is a blending function defined as  
\[\tilde{f}_{d}=\max[(1-f_{dt}),f_{b}] \tag{4}\]  
with  
\[f_{dt}=1-\tanh\Big{[}(C_{dt}r_{dt})^{3}\Big{]} \tag{5}\]  
\[r_{dt}=\frac{\mu_{t}}{\kappa^{2}d^{2}[(S^{2}+\Omega^{2})/2]^{0.5}} \tag{6}\]  
\[f_{b}=\min\Big{[}2\exp\Big{(}-9\alpha^{2}\Big{)},1.0\Big{]} \tag{7}\]  
\[\alpha=0.25-d/\Delta \tag{8}\]  
where \(C_{dt}\) is 20 and \(\kappa\) is the von [PERSON] constant; \(f_{e}\) is elevating function defined as  
\[f_{e}=f_{2}\cdot\max[(f_{1}-1.0),0.0] \tag{9}\]  
with  
\[f_{1}=\left\{ \begin{array}{ll}2\exp\big{(}-9\alpha^{2}\big{)},&\alpha<0\\ 2\exp\big{(}-11.09\alpha^{2}\big{)},&\alpha\geq 0\end{array} \right. \tag{10}\]  
\[f_{2}=1.0-\max(f_{t},f_{l}) \tag{11}\]  
\[f_{t}=\tanh\bigg{[}\Big{(}C_{t}^{2}r_{dt}\Big{)}^{3}\bigg{]} \tag{12}\]  
\[f_{l}=\tanh\bigg{[}\Big{(}C_{t}^{2}r_{dt}\Big{)}^{10}\bigg{]} \tag{13}\]  
\[r_{dl}=\frac{\mu_{l}}{\kappa^{2}d^{2}[(S^{2}+\Omega^{2})/2]^{0.5}} \tag{14}\]  
Please consult [PERSON] et al. [30] for more information.