## 3 Results and Discussion
### Flow around the SD7003 Airfoil  
The SD7003 airfoil, designed by Selig [PERSON], is a well-known airfoil optimized for low Reynolds number applications, typically in the range of \(10^{4}\) to \(10^{5}\). This airfoil is widely used in small-scale aircraft, unmanned aerial vehicles (UAVs), and model aircraft due to its favorable aerodynamic performance at low speeds. The SD7003 airfoil performs well in low Reynolds number regimes, making it an ideal choice for applications where  
Figure 5: SD7003 computational mesh.  
Figure 6: Detailed view of the boundary layer mesh around the SD7003 airfoil.  
low-speed aerodynamics are critical. A key characteristic of the SD7003 is its tendency to form laminar separation bubbles. These bubbles occur when laminar flow separates from the suction side of the airfoil, then it becomes turbulent and subsequently reattaches. The presence of these laminar separation bubbles significantly impacts the airfoil's performance, particularly in terms of lift and drag characteristics.  
In this study, we utilized the SD7003 airfoil to analyze separation behavior using the Spalart-Allmaras turbulence model. The simulations were conducted at a Reynolds number of 60,000 and angles of attack of 4\({}^{\circ}\) and 8\({}^{\circ}\). These conditions were chosen to investigate the flow separation and transition phenomena that are critical for low Reynolds number aerodynamics.  
At an angle of attack of 4\({}^{\circ}\), the airfoil is expected to exhibit moderate flow separation, primarily near the trailing edge. At a higher angle of attack of 8\({}^{\circ}\), more pronounced separation and the formation of laminar separation bubbles are expected.  
It is well known that the SA model has limitations in accurately predicting separation, especially at low Reynolds numbers where complex laminar-to-turbulent transition processes occur. This makes the SD7003 airfoil a suitable test case for applying field inversion and machine learning approaches to verify whether they can improve the baseline model.  
By using field inversion, we can identify regions where the SA model deviates from high-fidelity data, allowing us to modify the turbulence model to better capture the separation phenomena.  
The results of the original SA model are shown in Figure 7. The turbulent viscosity and velocity fields exhibit similar behavior at both angles of attack. However, we should expect a clear difference, as the separation is anticipated to start near the trailing edge at \(\alpha=4^{\circ}\) and extend towards the leading edge at \(\alpha=8^{\circ}\).  
More exhaustive results are presented in Table 1, where CL and CD are compared to the high-fidelity results from a LES simulation [54]. There is not a significant difference in terms of CL for both angles of attack, even though the correct separation is not observed. In the next section, we will also show results in terms of CP and CF. For CD at \(\alpha=8^{\circ}\), we  
Figure 7: Results of the Spalart–Allmaras model for the SD7003 test case. (a) Velocity magnitude and (b) turbulent viscosity at \(\alpha=4^{\circ}\). (c) Velocity magnitude and (d) turbulent viscosity at \(\alpha=8^{\circ}\).  
observed good agreement with the experimental results, while a relative error of 52% is present for \(\alpha=4^{\circ}\).