## 2 Calculation Model and Numerical Method
### Turbulence and Transition Model  
Because the Reynolds number of the monitoring device was medium, to accurately evaluate the aerodynamic resistance, the transition impact needed to be considered. Additionally, a large range of separated flows exists when operating high-speed trains. To increase the accuracy of the numerical simulation, we used a coupling \(\gamma\)-Re\({}_{\theta}\) transition model and IDDES method. The \(\gamma\)-Re\({}_{\theta}\) transition model not only includes the intermission factor transport equation [19; 20; 21; 22], but it also uses the momentum thickness Reynolds number as a transport scalar and establishes the transport equation for it; this allows the momentum thickness Reynolds number of the incoming flow to be used for the boundary layer due to convection, which thus allowed us to obtain the local transition initial momentum thickness Reynolds number. Then, we could judge the beginning of the transition by comparing the Reynolds number of the vortex and momentum thickness. The intermittent factor (\(\gamma\)) and the trans  
Figure 2: Computing grid; (**a**) Grid near the wall; positive operation: condition 1; (**b**) Grid far field; positive operation: condition 1; (**c**) Grid near the wall; negative operation: condition 2; (**d**) Grid far-field; negative operation: condition 2.  
port equation of the Reynolds number of the transition initial momentum thickness (\(\overline{\text{Re}_{\theta t}}\)) was as follows:  
\[\frac{\partial(\rho\gamma)}{\partial t}+\frac{\partial(\rho Uj\gamma)}{\partial x _{j}}=P_{\gamma}-E_{\gamma}+\frac{\partial}{\partial x_{j}}[(\mu+\frac{\mu_{t}} {\sigma_{\gamma}})\frac{\partial\gamma}{\partial x_{j}}] \tag{2}\]  
\[\frac{\partial(\rho\overline{\text{Re}_{\theta t}})}{\partial t}+\frac{ \partial(\rho Uj\overline{\text{Re}_{\theta t}})}{\partial x_{j}}=P_{\theta t} +\frac{\partial}{\partial x_{j}}[\sigma_{\theta t}(\mu+\frac{\mu_{t}}{\sigma_{ \gamma}})\frac{\partial\gamma}{\partial x_{j}}] \tag{3}\]  
\[P_{\gamma}=F_{length}C_{41}\rho S[\gamma F_{onset}]^{0.5}(1-C_{41}\gamma) \tag{4}\]  
\[E_{\gamma}\text{= C}_{a2}\rho\Omega\gamma F_{turb}(C_{a2}\gamma-1) \tag{5}\]  
\[F_{onset}=\frac{\text{Re}_{\text{v}}}{2.193\text{Re}_{\theta c}} \tag{6}\]  
\[P_{\theta t}\text{= C}_{\theta t}\frac{\rho}{t}(\text{Re}_{\theta t}-\overline{ \text{Re}_{\theta t}})(1.0-F_{\theta t}) \tag{7}\]  
\[\text{Re}_{\theta t}\text{= F}(\text{Tu})\text{F}(\lambda_{\theta}) \tag{8}\]  
\[F_{length}\text{= f}_{1}(\overline{\text{Re}_{\theta t}}) \tag{9}\]  
\[\text{Re}_{\theta c}\text{= f}_{2}(\overline{\text{Re}_{\theta t}}) \tag{10}\]  
\[\frac{\partial(\rho\gamma)}{\partial t}+\frac{\partial(\rho Uj\gamma)}{\partial x _{j}}=P_{\gamma}-E_{\gamma}+\frac{\partial}{\partial x_{j}}[(\mu+\frac{\mu_{t}} {\sigma_{\gamma}})\frac{\partial\gamma}{\partial x_{j}}] \tag{2}\]0  
\[\frac{\partial(\rho\gamma)}{\partial t}+\frac{\partial(\rho Uj\gamma)}{\partial x _{j}}=P_{\gamma}-E_{\gamma}+\frac{\partial}{\partial x_{j}}[(\mu+\frac{\mu_{t}} {\sigma_{\gamma}})\frac{\partial\gamma}{\partial x_{j}}] \tag{2}\]1  
where \(P_{\gamma}\) and \(E_{\gamma}\) are the generation and destruction terms of the \(\gamma\) equation, respectively; \(P_{\theta t}\) is the generation terms of the \(\overline{\text{Re}_{\theta t}}\) transport equation; \(\text{Re}_{\theta t}\) is the Reynolds number of the momentum thickness outside the boundary layer, which is related to the turbulence intensity and pressure gradient of the incoming flow; and \(\lambda_{\theta}\) is the parameter representing the pressure gradient. Two key factors influence the term \(P_{\gamma}\) and determine the threshold function (\(F_{onset}\)) at the beginning of the transition and the parameters of the transition region length (\(F_{length}\)). In the upstream laminar flow region, the value \(F_{onset}\) is 0.