### vehicle models

The numerical analysis of the current research is carried out with the ANSYS. The input dimensions for the considered sample, LR discovery, are \(4.835\mathrm{m}\), \(1.887\,\mathrm{m}\), \(1.915\,\mathrm{m}\)and \(2.51\,\mathrm{m}\) for length, height, width without mirrors, and wheelbase, respectively, as seen in Figure 1, which shows 3-D of the sample under consideration.

This study proposed different models to improve and reduce downforce \({CD}\). The ventilation duct, one of the considered modifications along the LR vehicle, is shown in blue color in Figure 2. This modified model has the same frontal area as the basic model, namely 3.144 m^{2}. The benefits of this modification lead to a reduction \({CD}\), reduces the turbulence behind the vehicle and cools the engine. The channel design has been carefully done to avoid the capacitance effect; Because of this, the dimensions of the channel are changed along with the car. The size of the channel is \(0.228 \,{\mathrm{m}}^{2}\) (1.11 m × 0.205 m) at the inlet then decreases \(0.099 \,{\mathrm{m}}^{2}\) (0.492 m × 0.203 m) at center and approximately \(0.06 \,{\mathrm{m}}^{2}\) (1.11 m × 0.054 m) at the end of the carriage.

The second modification tested in this work is shown in blue color in Fig. 3, this technique is important for the stability of the vehicle by increasing the pressure on the roof to improve the aerodynamic behavior. The dimensions of the roof modification were chosen based on the optimal conditions of the vehicle design and the expected aerodynamic behavior. The width of the roof modification at the beginning of the main roof is \(0.832\mathrm{m}\) then sinks up \(0.5\mathrm{m}\) and will \(1.183\mathrm{m}\) at the end of the roof. The depth of this modification is \(50\mathrm{mm}\).

The trench on the roof model has a frontal area that is around less than the base model \(0.025 {\mathrm{m}}^{2}\) (\(500\mathrm{mm}\times 50\mathrm{mm}\)), which means 1/125.76 as is the face of the base model \(3.011 {\mathrm{m}}^{2}\) while it is for the trench model \(2.986 {\mathrm{m}}^{2}\). Therefore, we can neglect the area reduction to compare the base model with the ditch model. The following images show the front views of these two models.

Another modification is a combination of the two previous modifications. These modifications could potentially produce lower drag and higher downforce than the base model.

### computing domain

The dimensions of the calculation area have been carefully chosen to avoid the possible effects of the walls and the dimensions are \(44.835\mathrm{m}\), \(9\mathrm{m}\)and \(13.915\mathrm{m}\) for the length, height and width. Furthermore, the dimensions of care \(4.835\mathrm{m}\), \(1.915\mathrm{m}\)and \(1.887\mathrm{m}\) for the length, height and width; Consequently, the dimensions of the compression area are larger enough to eliminate the effect of the walls and capture the important physics, especially behind the car where the vortices occupy a large area, as can be seen in Fig. 4.

### Numeric grid

The accuracy of the simulation depends on the quality of the mesh. The calculation domain has been divided into three layers according to the physics variation: The area near the car containing a large variation of physics must be smooth and fine enough to capture the physics. As we move away from the car, the physical variation decreases, so the level 2 grid cells get a little larger. As we move further, the physical variation becomes even smaller and the mesh required to capture the physics in this area becomes coarser than seen in fig. 5, 6, 7 and 8. Meshing is an extremely important step in design and analysis and this step was performed with Ansys.

It is important to check the mesh dependency before the analysis in order to choose a smaller number of cells where the results become independent of the cell change. This technique is very important to get accurate results in less time. An extensive study to select the appropriate number of elements was accomplished, which was 13 × 10^{6} Elements. In addition, the maximum skewness of the standard mesh was 0.897066 and the minimum orthogonal quality was 0.012722. j^{+} Each Discovery car configuration has a size range based on location. Figure 9 illustrates y^{+} for the modified roof model of the Discovery car. Using a turbulence model with a wall function is the best choice with respect to this number of y^{+}.

### boundary conditions

In this work, uniform inflow velocities of \(28\), \(34\)and \(40\mathrm{m}/\mathrm{s}\) were used. The air density (\(\rho\)) is \(1.225\mathrm{kg}/{\mathrm{m}}^{3}\). Reynolds numbers (\(Regarding\)) of the system were 9 × 10^{6}11×10^{6}and 13×10^{6} for the above speeds. Experimental tests for the base model of the Discovery car were conducted in a MIRA wind tunnel in the UK^{26.27}. The Discovery car and all its tires were in the MIRA wind tunnel. The static road under the vehicle is used in the numerical simulation as well as the experimental test in the MIRA wind tunnel. The inlet airflow velocity was about 28 m/s at the inlet section of a wind tunnel. In the experimental test, the turbulence intensity was 2.65% at the inlet of the MIRA wind tunnel. The same turbulence intensity is used in all numerical simulations. Two types of wall boundary conditions for the side walls and the top wall of the calculation area could have been: (i) non-slip and (ii) stationary walls. As a result, there was no negative impact on the analyses, as the outer walls of the domain were identical. The tires of the car have all stopped, analogous to the wind tunnel model. To optimize the design and mesh size, the underbody surface has been made flat.

### Discretization and numerical structure

The second-order downwind approach was used for momentum, turbulent kinetic energy, and turbulent dissipation rate. Also in terms of spatial discretization, it was used for printing. The relaxation factor was set to 0.25. The four turbulence models adopted for this research were viable *k*–\(\varepsilon\)default *k*–\(\Omega\)shear stress transport *k*–\(\Omega\) (SST) and a Reynolds stress model (RSM). Most previous researchers have used these models to study automotive performance of vehicles and they would result in acceptable processing time. In this study, the two formulas given below are used to calculate the drag and lift coefficients:

$${C}_{D}=\frac{{F}_{D}}{\left(\rho {V}^{2} A\right)/2}$$

$${C}_{L}=\frac{{F}_{L}}{\left(\rho {V}^{2} A\right)/2}$$

\({F}_{D}\) represents the resistance (\(\mathrm{N}\)), \({F}_{L}\) represents the buoyancy force (\(\mathrm{N}\)), \(\rho\) is the air density (\(\mathrm{kg}/{\mathrm{m}}^{3}\)), \(V\) represents the initial airspeed (\(\mathrm{m}/\mathrm{s}\)), and A represents the front cross-sectional area of the car (\({\mathrm{m}}^{2}\)).

### Setup of the car model

ANSYS 16.0 was used to simulate the full-scale model mentioned above. The solution domain created in ANSYS Meshing (version 16.0) consisted of the global domain and three VCBs around the car. Irregular tetrahedral cells were used throughout the domain. Velocity profiles around vehicle surfaces were accurately estimated using five prismatic cell inflation layers. The real model of the automobile and the model used in the simulation have different variations. Side mirrors, spinning wheels and a number of intricate geometric pieces under the car are all present in a real car. The tires were completely fixed in the simulation model as in the wind tunnel model. To reduce both geometry and mesh, a flat surface was used for the simulation model. Assumptions for real, experimental and numerical models are listed in Table 1.