## 2 Numerical Methods
### Aerodynamics  
Incompressible, unsteady Navier-Stokes and continuity equations were used for the solution of turbulent flow around the vehicle. Several approaches are available for the solution of the governing equations of turbulent flows. These approaches include directly \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Vehicle’s Parameter** & **Symbol** & **Data** & **Unit** \\ \hline Length & \(L\) & 12.2 & [m] \\ Height & \(h\) & 3.6 & [m] \\ Width & \(w\) & 2.6 & [m] \\ Track width & \(T\) & 2.25 & [m] \\ Length of the wheelbase & \(L_{ub}\) & 5.9 & [m] \\ Distance from the front axle to CG & \(f\) & 3.7 & [m] \\ Distance from the rear axle to CG & \(b\) & 2.2 & [m] \\ Mass & \(m\) & 13,650 & [kg] \\ Cornering stiffness for the front tyre & \(C_{yl12}\) & 250 & [kh/rad] \\ Cornering stiffness for the rear tyre & \(C_{yl94}\) & 450 & [kh/rad] \\ Roll damping of the suspension & \(C_{\phi}\) & 100 & [kh/rad] \\ Roll stiffness of the suspension & \(K_{\phi}\) & 1000 & [khN/rad] \\ Yaw moment of inertia & \(I_{zz}\) & 200,000 & [kgm\({}^{2}\)/rad] \\ Roll moment of inertia & \(I_{xx}\) & 30,000 & [kgm\({}^{2}\)/rad] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Features of the vehicle employed in the study. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\mathbf{k_{y}}\) & \(\mathbf{k_{\Psi}}\) & \(\mathbf{k_{l}}\) \\ \hline Driver 1 & 1.2 & 30 & 0.9 \\ Driver 2 & 1.6 & 40 & 1.2 \\ Driver 3 & 2.0 & 50 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Sets of proportional control gain parameters indicate three different driver’s attitudes. solving all flow scales, e.g., direct numerical solution (DNS), modelling all or certain scales of flow, e.g., Reynolds-averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES), or approaches using hybrid RANS-LES methods, e.g., detached eddy simulation (DES). The search for less computationally expensive yet sufficiently accurate methods for solving turbulent flows around vehicles remains a critical need [23; 24; 25]. For that purpose, hybrid RANS-LES methods, such as DES, were proposed, in which elements of both RANS and LES were combined to provide a balance between computational efficiency and accuracy in simulating turbulent flows. In DES, the simulation dynamically switches between RANS and LES in different regions of the flow, allowing for a more accurate representation of turbulent phenomena. Specifically, RANS is utilized in regions where the flow is predominantly attached and steady, while LES is employed in areas with unsteady and separated flow. However, due to some disadvantages of DES, such as its grid sensitivity in boundary layers and log-layer mismatch, new methods named delayed detached eddy simulation (DDES) and improved delayed detached eddy simulation (IDDES) were proposed [26]. IDDES integrates DDES with an enhanced RANS-LES hybrid model, specifically designed to address wall modelling in LES by adapting to grid resolution [27]. Additionally, [PERSON] et al. [7] stated that studies employing two-way coupling between aerodynamics and vehicle dynamics in the previous literature commonly utilized scale-resolving turbulence models such as LES and IDDES in aerodynamic simulations, known for their accuracy in turbulent flow solutions compared to RANS equations. Therefore, in the present study, IDDES was used to solve the turbulent flow around the vehicle model. The Ground Transportation System (GTS), a 1/8-scaled simplified heavy vehicle model, was utilized [22]. Numerous studies [21; 28; 29; 30; 28; 22; 31] confirmed the applicability of GTS in aerodynamic studies of heavy vehicles. Figure 2 provides the features of the GTS, including the coordinate system, along with qualitative descriptions of aerodynamic force and moments. The computations utilized a deterministic crosswind velocity profile characterized by a cosine inlet and exit profile with a flat top, as depicted in Figure 3. The mathematical formulation of the crosswind's velocity profile is defined in Equation (8). \[\mathrm{w}_{\mathrm{cw}}(x)=\left\{