## 3 Results and Discussion
### 2D Periodic Hills Flow  
The 2D periodic hills flow is a popular benchmark for assessing turbulence models. The case with Re = 10,595 (based on the hill height and the bulk velocity above the hill crest) was comprehensively investigated using a highly resolved LES by [PERSON] and [PERSON] [28] and later by [PERSON] et al. [29]. Using the same geometry, [PERSON] et al. [30] performed experimental studies and numerical simulations. They not only reproduced the results of [PERSON] et al. [29], but also extended the research to a wide range of Re. The LES data have usually been used as references to assess other turbulence models. A few examples are described. [PERSON] et al. [31] compared the abilities of many RANS models (linear, nonlinear, and second-moment closure) for this flow. [PERSON] and [PERSON] [8] compared the abilities of three RANS models (the \(\mathrm{BL}-\overline{v^{2}}/k\) model, the \(\overline{v^{2}}-f\) model of [PERSON] and [PERSON] [32], and the SST \(k\)-\(\omega\) model). Among these RANS models, only the nonlinear \(k\)-\(\omega\) and the \(\overline{v^{2}}-f\) models could predict reasonable flow features. The linear \(k\)-\(\epsilon\) model yielded premature reattachment. On the contrary, the SST \(k\)-\(\omega\) model delayed the reattachment and yielded excessively long recirculation. [PERSON] and [PERSON] [3] used this case to evaluate the abilities of many hybrid RANS-LES models, including the DES, delayed DES (DDES), layering RANS-LES, RANS-limited LES, and SAS. [PERSON] and [PERSON] [5] and [PERSON] and [PERSON] [27] used this case to investigate the performance of SAS. The performance of the proposed SAS model was also evaluated using this flow. A schematic of the model geometry and the relevant boundary conditions is shown in Figure 6. It should be noted that, in all the simulations mentioned above, the computational domain was the region between two adjacent hill crests, and two periodic conditions were applied in the stream-wise and lateral directions. However, in the FLUENT CFD code used in this study, only one periodic condition was permitted in one case. Therefore, the region between the first and sixth hill crests was chosen as the computational domain, accompanied by a periodic condition in the lateral direction. In the streamwise direction, the inlet and outlet conditions were specified. As shown in Figure 6, the computational domain was divided into five sections, namely, S1, S2, S3, S4, and S5. The Re based on the hill height  
Figure 5: Visualization of the turbulent structures using iso-surface of \(Q=100\ 1/s^{2}\). \(H\) and bulk velocity above the crest \(U_{b}\) was Re = 10,595. The domain size in the lateral direction was 4\(H\). To avoid the influence of the inlet and outlet on the results, the results in S4 were selected for subsequent analysis. In the simulations of the two SAS models, the inlet conditions were extracted from precursor simulations using the corresponding RANS models. For example, the steady flow was obtained using the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model, and the velocity components and turbulent quantities, which were extracted from the intersection surfaces of S4 and S5, were applied as the inlet condition in the simulation of the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-SAS model. Similarly, for the simulation of the SST-SAS model, the inlet condition was extracted from a precursor simulation using the SST \(k\)-\(\omega\) model. The total number of computational meshes was 4.76 million. The time step size was \(0.005H/U_{b}\). The statistics began at 10,000 steps and continued to 20,000 steps to ensure that the mean values of the variables remained unchanged with time. The LES data of [PERSON] [28] were used as references. A comparison of the mean stream-wise velocity profiles for different transverse sections is shown in Figure 7. It was evident that the velocity profiles predicted by the proposed SAS model were in good agreement with those predicted by the LES.