## 4 Results
#### 4.1.1 SD7003 Airfoil  
The Selig-Donovan (SD) 7003 airfoil was investigated at \(\mathrm{Re}=6\times 10^{4}\). A C-topology grid with 768 (96 in the wake) cells in the stream-wise direction and 176 cells in the normal-to-the-wall direction were used. This grid was generated at \"Centro Italiano Ricerche Aerospaziali\" (CIRA)research center and extensively tested in the computation of this kind of flow field [25]. It was made available to the authors as participants of the previously mentioned GARTEUR project, AG59, which started in 2019.  
Here we compare the \(\gamma\)-\(\widetilde{\mathrm{Re}}_{\theta,I}\)-SA and \(\gamma\)-\(\widetilde{\mathrm{Re}}_{\theta,I}\)-SA20 transition models with the LES and \(k\)-\(\omega\) results of [PERSON] and [PERSON] [25, 26]. At \(\alpha=4^{\circ}\) the pressure coefficient, \(c_{p}=2(p-p_{\infty})/\rho u_{\infty}^{2}\), the obtained distributions put in evidence a good agreement between our RANS results and the literature ones; see Figure 1a. A negligible impact of the Spalart and Garbaruk correction can be noted. The skin friction coefficient behavior, \(c_{f}=2\tau_{w}/\rho u_{\infty}^{2}\), at the same angle of attack is shown in Figure 1b. It is quite evident that the RANS separation and reattachment points are in good agreement with the LES ones. The \(C_{f}\) peak is sufficiently reproduced in the downstream (transitional part) of the bubble. Evident discrepancies remain past the turbulent reattachment. In this case, the Spalart and Garbaruk correction slightly improves the overall model behavior.  
In Figure 2 pressure and skin friction coefficient distributions at \(\alpha=6^{\circ}\) are reported. At this angle of attack, it is possible to note a situation similar to the previous case. Results for \(\alpha=8^{\circ}\) are depicted in Figure 3; for the skin friction coefficient, we obtained a similar behavior in comparison with the other angles of attack presented above. On the other hand, the \(c_{p}\) coefficient prediction gets clear benefits from the \(c_{w2 LRe}\) expression. Indeed, it is very easy to note that the pressure wiggles, which appear near the transition point, are completely suppressed; see Figure 3a. Nevertheless, \(\gamma\)-\(\overline{\text{Re}_{\theta,t}}\)-SA-20 agrees rather well with the \(k\)-\(\omega\) solutions published by [PERSON] and [PERSON] (2018), rather than the literature LES data. At \(\alpha=10^{\circ}\), the impact of the Spalart and Garbaruk correction is even more marked, see Figure 4. The \(\gamma\)-\(\overline{\text{Re}_{\theta,t}}\)-SA computation produces the pressure distribution of a stalled airfoil, whereas the \(\gamma\)-\(\overline{\text{Re}_{\theta,t}}\)-SA20 model results are in satisfactory agreement with the \(k\)-\(\omega\) data.  
The force coefficients are compared, in Figure 5, to three sets of experimental data and to the numerical results obtained by [PERSON] and [PERSON] (2018). The lift coefficient, \(C_{L}=2L^{\prime}/\rho u_{\infty}^{2}c\), and the drag coefficient, \(C_{D}=2D^{\prime}/\rho u_{\infty}^{2}c\), are considered for comparison. The measurements depicted in Figure 5 were obtained from [PERSON] et al. (2019) at the University of Princeton in 1989, from [PERSON] et al. (2019) at the University of Illinois in 1996, and from [PERSON] et al. (2019) at the Horizontal Fee-Surface Wind Tunnel (HFWT) of the Air Force Research Laboratory in 2005. At low-medium angles of attack, the computed force coefficients are well in accordance with the experimental data. Airfoil stall is differently predicted by the models presented here. In particular, \(\gamma-\widetilde{\text{Re}_{\theta,t}}-\text{SA}\) shows the stall at \(\alpha=10^{\circ}\). On the contrary, \(\gamma-\widetilde{\text{Re}_{\theta,t}}-\text{SA}20\) seems to approach \(k-\omega\) of [PERSON] and [PERSON] [25], but unsteadiness appears in the solution for \(\alpha>10^{\circ}\).