## 3 Results and Discussion
### 2D U-Turn Duct Flow  
It is well known that one of the serious weaknesses of conventional eddy viscosity models (EVMs) is that they are unable to capture the effects of strong streamline curvature. To ensure that the EVMs are able to do this, it is usually necessary to make corrections to the models. Several successful solutions have been published to date. For details, readers can refer to [PERSON] [33] and [PERSON] et al. [34]. The 2D U-turn duct \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Sep.** & **Reatt.** & **Len.** \\ \hline LES & 0.22\(H\) & 4.72\(H\) & 4.50\(H\) \\ \hline SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-SAS & 0.18\(H\) & 4.62\(H\) & 4.44\(H\) \\ \hline SST-SAS & 0.22\(H\) & 5.79\(H\) & 5.57\(H\) \\ \hline SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) & 0.24\(H\) & 7.54\(H\) & 7.30\(H\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Locations of separation (Sep.) and reattachment (Reatt.) points and the length of the main recirculation region (Len.) in the 2D periodic hills flow. Figure 8: Comparison of the mean skin friction coefficients distributed along the bottom wall. Figure 9: Visualization of the turbulent structures using iso-surface of \(Q=30~{}1/s^{2}\). flow is challenging for EVMs because of the strong streamline curvature, and it has been a benchmark for assessing the ability of turbulence models. The widely used reference data are the experimental measurements of [PERSON] et al. [35]. For example, [PERSON] et al. [36] used this flow to test their new eddy viscosity formulation. [PERSON] and [PERSON] [37] used this flow to validate the rotation and curvature correction in the SST \(k\)-\(\omega\) model. [PERSON] and [PERSON] [38] used this flow to demonstrate the ability of their three-equation variant of the SST \(k\)-\(\omega\) model to capture the effect of the streamline curvature. In contrast to the eddy-viscosity-based RANS models, the LES can predict the strong streamline curvature effects without modifications because the turbulence spectra are resolved. However, the increased computational cost limits its utilization. Hybrid RANS-LES models are worthy of attention as approaches for maintaining a balance between accuracy and efficiency. The 2D U-turn duct flow was used to test the performance of the proposed SAS model. Figure 10 illustrates the computational model used in this study. In accordance with the experiment of [PERSON] et al. [35], the Re based on the bulk velocity \(U_{b}\) at the inlet and the height of the inlet channel \(H\) was 100,000. The domain size in the lateral direction was 4\(H\). The periodic condition was specified in the lateral direction. At the inlet, the fully developed profiles, which were extracted from a preliminary computation in a periodic straight duct at the same Re with the relevant turbulence models, were used. The total number of computational meshes was 5.88 million. The time step size was 0.015\(H\)/\(U_{b}\). The statistics began at 10,000 steps and continued to 30,000 steps to ensure that the mean values of the variables remained unchanged with time. An arc coordinate (s) was constructed to facilitate the analysis of the results, as shown in Figure 10. A comparison of the mean stream-wise velocity profiles for different transverse sections is shown in Figure 11. It was clearly demonstrated that the two hybrid RANS-LES models (the proposed SAS and SST-SAS models) had a positive effect compared to the RANS model (the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model). At \(s/H=0\), all three models were able to predict the velocity profile well, and the difference was small. At \(s/H=\pi/2\), the two hybrid models showed improved results near the concave outer wall compared to the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model, which significantly underpredicted the velocity. At \(s/H=\pi\), the two hybrid models showed improved results for the entire section.