## 2 Methods
### Calculation Strategy  
DPM is employed to study the dynamics of multiphase flows involving combinations of gas-liquid, gas-solid, liquid-solid, and gas-liquid-solid interactions. It carries out Lagrangian trajectory calculations for the dispersed phases such as particles, droplets, or bubbles, along with their interaction with the continuous phase. Multiphase flow scenarios encompass a variety of applications like channel flows, sprays, sedimentation processes,separation techniques, and cavitation phenomena. For the purpose of evaluating and comparing the computational efficacy of three different turbulence modeling approaches in analyzing the aerodynamic response of a sedan moving through a dune section within a wind-sand flow environment, this study utilizes the DPM to model the wind-sand flow dynamics. In this method, sand particles are injected into air, and the Eulerian-Lagrangian approximation is used to calculate the air-sand interaction. Equation (10) presents the equilibrium equations of a sand particle in the air stream.  
\[m_{p}\frac{\mathrm{d}\overrightarrow{u}_{p}}{\mathrm{d}t}=F_{g}+F_{d}+F_{l}+F_ {m}+F_{b} \tag{10}\]  
where \(F_{g}\), \(F_{d}\), \(F_{l}\), \(F_{m}\), and \(F_{b}\) are gravity, drag, Saffman lift, Magnus, and Brownian forces, respectively; \(m_{p}\) is the mass of a sand particle; and \(\overrightarrow{u}_{p}\) is the velocity of the particle.  
The aerodynamic loads (aerodynamic drag \(F_{x}\), aerodynamic lift \(F_{y}\), lateral force \(F_{z}\), rolling moment \(M_{x}\), yaw moment \(M_{y}\), and pitching moment \(M\)) of the sedan are obtained by integrating the forces or moments subjected to several calculation segments. The corresponding calculation strategies and formulas can be found in the literature [25]. Among these loads, aerodynamic loads that lead to the yaw, sideslip, and roll behaviors of the sedan are the \(F_{z}\), \(M_{x}\), and \(M_{y}\)[13]. To capture the aerodynamic forces acting on the sedan as it moves, the surface mesh cells of the sedan serve as the computational domain's boundary faces. The transient pressures exerted on these surface cells are integrated and computed in real-time through the implementation of a tailored UDF program.  
To compare and analyze the sedan's aerodynamic loads in different cases, the lateral force coefficient \(C_{z}\) is obtained by dimensionalizing the lateral force \(F_{z}\) using \(F^{\prime}\). The rolling moment \(M_{x}\) and yaw moment \(M_{y}\) are dimensionless, using \(M^{\prime}\) to obtain the rolling moment coefficient \(C_{mx}\) and yaw moment coefficient \(C_{my}\). The calculation equations for \(F^{\prime}\) and \(M^{\prime}\) are shown in Equations (11) and (12). Equation (13) is used to add dimension to the pressure on the sedan surface.  
\[F^{\prime}=U_{s}^{2}S\rho/2 \tag{11}\]  
\[M^{\prime}=U_{s}^{2}SH\rho/2 \tag{12}\]  
\[C_{p}=\frac{2(p_{j}-p_{0})}{\rho U_{c}^{2}} \tag{13}\]  
where \(U_{s}\) and \(\rho\) denote the sedan's running speed and the atmospheric density, respectively; and \(S\) and \(H\) denote the side area and height of the sedan, respectively. \(U_{c}\) and \(C_{p}\) represent the velocity of the crosswind and the pressure coefficient on the surface of the car, respectively.