## 2 Computational Setup
### Computational Solver  
The Large Eddy Simulation was used to resolve the flow field around the square-back [PERSON] model with and without control. The governing equations in the LES method are the incompressible Navier-Stokes equation and the continuity equation, which are expressed as follows:  
\[\frac{\partial(\rho\overline{\mathrm{u}_{\mathrm{i}}})}{\partial\mathrm{t}}+ \frac{\partial\big{(}\rho\overline{\mathrm{u}_{\mathrm{i}}u_{\mathrm{j}}}\big{)} }{\partial\mathrm{x}_{\mathrm{j}}}=\frac{\partial\overline{\mathrm{p}}}{ \partial\mathrm{x}_{\mathrm{i}}}+\frac{\partial}{\partial\mathrm{x}_{\mathrm{j} }}\left(\upmu\frac{\partial\overline{\mathrm{u}_{\mathrm{i}}}}{\partial \mathrm{x}_{\mathrm{j}}}\right)-\frac{\partial\tau_{\mathrm{ij}}}{\partial \mathrm{x}_{\mathrm{j}}}\, \tag{1}\]  
\[\frac{\partial\rho}{\partial\mathrm{t}}+\frac{\partial\big{(}\rho\overline{ \mathrm{u}}_{\mathrm{i}}\big{)}}{\partial\mathrm{x}_{\mathrm{i}}}=0, \tag{2}\]  
where \(\overline{\mathrm{u}}\) and \(\overline{\mathrm{p}}\) stand for spatial filtered fields, \(\rho\) is the density of the air, and \(\upmu\) denotes the dynamic viscosity.  
\(\tau\) represents the subgrid-scale (SGS) turbulent stresses:  
\[\tau_{\mathrm{ij}}-\frac{1}{3}\tau_{\mathrm{kk}}\delta_{\mathrm{ij}}=-2\upmu_{ \mathrm{t}}\overline{\mathrm{S}}_{\mathrm{ij}}, \tag{3}\]  
where \(\upmu_{\mathrm{t}}\) is the SGS turbulent viscosity. \(\overline{\mathrm{S}}_{\mathrm{ij}}\) represents the filtered rate-of-strain tensor:  
\[\overline{\mathrm{S}}_{\mathrm{ij}}=\frac{1}{2}\Bigg{(}\frac{\partial\overline{ \mathrm{u}}_{\mathrm{i}}}{\partial\mathrm{x}_{\mathrm{j}}}+\frac{\partial \overline{\mathrm{u}}_{\mathrm{j}}}{\partial\mathrm{x}_{\mathrm{i}}}\Bigg{)} \tag{4}\]  
Although the rapid development of high-performance computing systems and advanced algorithms in recent years, the application of LES in the academic community is still limited, especially in industrial research. One of the restrictions when applying LES in automotive aerodynamics is the requirement of a vast computational resource to resolve the flow field in the near-wall region. The flow in this region significantly influences the development of the near-wake structure and the aerodynamic forces on the model. At high Reynolds numbers Re, the size of vortices decreases rapidly as the flow approaches the solid boundaries. Thus, a fine mesh and a very small time step are required at the near-wall region, and LES turns unaffordable. One promising approach to overcome this limitation of the LES is the use of Wall-Modeled LES (WMLES) [26]. In WMLES, large eddies of flow far from the solid boundary are resolved, while the flow in the near-wall region is modeled. In this study, WMLES was used to calculate the flow fields around the square-back Ahmed model.  
The finite volume based ANSYS Fluent 2021 solver was used to solve the WMLES equations. The SIMPLEC algorithm was applied for pressure-velocity coupling, and Least Square Cell Based approach was used to discretize the gradient terms. Furthermore, the momentum terms were calculated with the Bounded Central Differencing method. Finally, the Bounded Second Order Implicit scheme was imposed for the time discretization. A time step of \(\Delta\)t = 1 \(\times\) 10\({}^{-5}\) s was specified to numerically solve for the flows around the model in the unactuated and actuated cases, at various operating conditions. In all simulated cases, the CFL \(<\) 1.5 condition was satisfied, as shown in Figure 6. 30,000 time steps, corresponding to 0.3 s of iteration time were run, and solution convergence was obtained within the first 12,000 time steps. The last 18,000 time steps were used to determine the flow topology, aerodynamic forces on the body, and to perform the spectral analysis in the following section.