## 3 Results and Discussion
### Flow around the NACA0021 Airfoil  
We also evaluated the FIML approach using the NACA0021 airfoil. This airfoil is a notable test case frequently used to assess the performance of transitional RANS models. For this analysis, we referenced the work of [PERSON] et al. [53]. The mentioned study investigated the flow around a two-dimensional NACA0021 airfoil at low Reynolds numbers and turbulence intensities to examine long separation bubbles and their impact on performance. Two transition models were evaluated: the correlation-based \(\gamma-Re_{\theta}\) intermittency model and laminar-kinetic-energy-based \(k-k_{L}-\omega\) model. The authors compared experimental and computed lift and drag coefficients for the NACA0021 airfoil at a Reynolds number of 120,000. The \(\gamma-Re_{\theta}\) model under-predicted both lift and drag coefficients, with increased divergence at higher angles. It also predicted a gradual stall behavior typical of thick airfoils at high Reynolds numbers. Conversely, the \(k-k_{L}-\omega\) model showed better agreement with experimental lift coefficients in the attached-flow regime and accurately predicted drag coefficients and stall behavior, capturing the abrupt stall due to the separation bubble bursting at low Reynolds numbers. However, a deviation from the experimental lift coefficient is still present at a higher angle of attack. In this work, our objective is to understand the ability of the FIML paradigm to improve the SA model in order to predict flow separation and achieve good results in terms of lift and drag coefficients at higher angles of attack. For this test case, we had experimental CL and CD for \(\alpha\in[0^{\circ},20^{\circ}]\). We used six angles of attack, specifically \(\alpha=8^{\circ},10^{\circ},12^{\circ},14^{\circ},16^{\circ}\) and \(20^{\circ}\). In Figure 17, we show the velocity and turbulent viscosity fields of the original SA model for \(\alpha=8^{\circ},12^{\circ},\text{and}16^{\circ}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **CL** & **CD** \\ \hline \(\alpha=4^{\circ}\) & LES [54] & 0.59 & 0.021 \\  & SA & 0.551 & 0.0099 \\  & SA (FI\(-\)CL) & 0.586 & 0.0147 \\  & SA (U-Net\(-\)CL) & 0.584 & 0.0147 \\ \hline \(\alpha=8^{\circ}\) & LES [54] & 0.92 & 0.043 \\  & SA & 0.895 & 0.0435 \\  & SA (FI\(-\)CL) & 0.917 & 0.0436 \\  & SA (U-Net\(-\)CL) & 0.917 & 0.0494 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of lift and drag coefficients for SD7003 airfoil. Figure 16: Results of the U-Net-augmented Spalart–Allmaras model for the SD7003 test case. (**a**) Velocity magnitude and (**b**) turbulent viscosity at \(\alpha=6^{\circ}\). The velocity field shows a growing separation starting from the trailing edge of the airfoil, which leads to an increase in turbulent viscosity in this region. However, this is insufficient for predicting accurate CL and CD values. In Figure 18, we show the results of running the original SA model for various angles of attack, covering both pre-stall and post-stall conditions. The CL behavior indicates that as the angle of attack increases, the original model failed to accurately capture the stall and subsequent separation. We begin by performing an optimization based on the experimental values of CL. In Figure 19, we present the velocity and turbulent viscosity fields after the optimization has been completed for angles of attack \(\alpha=8^{\circ},12^{\circ},\text{and}16^{\circ}\). Notably, there is a significant difference, particularly for \(\alpha=16^{\circ}\). At this angle of attack, a marked separation is evident in the velocity field, and the shape of the turbulent viscosity distribution is also substantially altered. Figure 20 illustrates the comparison between experimental data, the original SA model, and the augmented SA model. The augmented model incorporates the correction field obtained after the optimization using field inversion. An improvement in CL is observed with the augmented SA model compared to the original SA model.