## 2 Methods
### RNG k-\(\epsilon\), LES, and IDDES Theory  
The turbulence modeling approach is a closed set of equations describing the mean values of turbulence on the basis of the Reynolds average equation of motion and the pulsating equation of motion, which rely on the combination of theory and experience and introduce a series of model assumptions. The RNG _k-\(\epsilon\)_, LES, and IDDES modeling theory are proposed based on the Reynolds-averaged Navier-Stokes equations (Equations (3) and (4)). \[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{i})}{\partial x_{i}}=0 \tag{3}\]  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial(\rho u_{i}u_{j})}{ \partial x_{j}}=-\frac{\partial p}{\partial x_{i}}+\frac{\partial\sigma_{ij}} {\partial x_{j}}+\frac{\partial\Big{(}-\rho u_{i}^{\prime}u_{j}^{\prime}\Big{)} }{\partial x_{j}} \tag{4}\]  
where \(u_{i}\) and \(u_{j}\) denote the Reynolds-averaged velocity component with the averaging sign omitted; \(\rho\) and \(p\) are the fluid density and pressure, respectively; \(u_{i}^{\prime}\) is the pulsation velocity; and \(\sigma_{ij}\) is the stress tensor component. The _k-\(\epsilon\)_ model is the most widely used turbulence modeling approach in flow computation, including three forms (Standard _k-\(\epsilon\)_, RNG _k-\(\epsilon\)_, and Realizable _k-\(\epsilon\)_), and their governing equations can be found in the literature [31]. Compared to the standard _k-\(\epsilon\)_ model, the RNG _k-\(\epsilon\)_ models developed based on reformed group theory add an extra factor (Equations (5) and (6)) to the \(\epsilon\) equation and consider the influence of eddies on the turbulence. Thus, the RNG _k-\(\epsilon\)_ model demonstrates superior performance compared to the standard _k-\(\epsilon\)_ model in terms of computational accuracy for high-velocity flows, vortex dynamics, and regions close to walls [32; 33; 34]. \[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_ {i})=\frac{\partial}{\partial x_{j}}\left(\alpha_{k}\mu_{eff}\frac{\partial k }{\partial x_{j}}\right)+G_{k}+G_{b}-\rho\epsilon-Y_{M}+S_{k} \tag{5}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Turbulence Modeling** & **The Condition of the** & **The Speed of the** & **The Driving Speed** & **Calculation Time (h)** \\  & **Approaches** & **Flow Field** & **Crosswind (m/s)** & **of the Sedan (m/s)** & **Calculation Time (h)** \\ \hline
1 & RNG _k-\(\epsilon\)_ & Crosswind & 20 & 27.78 & 127 \\
2 & LES & Crosswind & 20 & 27.78 & 203 \\
3 & IDDES & Crosswind & 20 & 27.78 & 141 \\
4 & IDDES & Crosswind–sand & 20 & 27.78 & 165 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters for calculation of the cases. \[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}}(\rho eu _{i})=\frac{\partial}{\partial x_{j}}\Bigg{(}\alpha_{\varepsilon}\mu_{eff}\frac{ \partial\varepsilon}{\partial x_{j}}\Bigg{)}+C_{1\varepsilon}\frac{ \varepsilon}{k}(G_{k}+C_{3\varepsilon}G_{b})-C_{2\varepsilon}\rho\frac{ \varepsilon^{2}}{k}-R_{\varepsilon}+S_{\varepsilon} \tag{6}\]  
where \(u\) and \(\mu_{eff}\) denote the flow velocity and the effective turbulent viscosity, respectively; \(\rho\), \(\varepsilon\), \(k\), and \(R_{\varepsilon}\) are the fluid density, the dissipation rate, the turbulent kinetic energy, and the extra factor, respectively; \(G_{k}\) is the turbulent kinetic energy generation term caused by the average velocity gradient; \(G_{b}\) is the turbulent kinetic energy generation term caused by the buoyancy force; \(T_{M}\) denotes the influence of the turbulent pulsating expansion on the total dissipation rate; \(S_{k}\) and \(S_{\varepsilon}\) are user-defined source terms; \(C_{1\varepsilon}\), \(C_{2\varepsilon}\), and \(C_{3\varepsilon}\) are empirical coefficients; \(\alpha_{k}\) and \(\alpha_{\varepsilon}\) are the reciprocals of \(k\) and \(\varepsilon\) effective Prandtl numbers, respectively.