# Assessment of Two Streamline Curvature Correction Methods for an Elliptic Blending Turbulence Model
## 3 Results
### 2D Periodic Hill Flow  
The 2D periodic hill flow is also a popular benchmark for assessing turbulence models [24; 25; 26; 27]. The LES results of [PERSON] et al. [28] have been widely used as a reference. The computational model was constructed based on the LES mentioned above and is illustrated in Figure 10. The boundary conditions are also shown. The Re (based on the hill height and the bulk velocity at inlet) was 10,595. The total number of mesh cells was 60,800, which was large enough to obtain a grid-independent solution. At the inlet and outlet, a periodic boundary condition was applied. The mass flow rate was specified according to the requirement of Re. The velocity magnitude contours and streamlines predicted by the different turbulence models are presented in Figure 11, while the x-direction velocity profiles at different transverse lines are shown in Figure 12. It is easy to see that both the original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model and the STT \(k\)-\(\omega\)-CC model predicted excessively large re-circulation zones, while the two proposed models significantly reduced the re-circulation zones. Notably, the STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC2 model yielded the smallest re-circulation zone. The velocity profiles obtained with the two proposed models agreed excellently with the LES at all locations. The original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model and the STT \(k\)-\(\omega\)-CC model predicted higher peak velocities due to the larger re-circulation zone. Figure 13 depicts the friction coefficient (\(C_{f}\)) along the bottom wall, and Table 2 lists the locations of the separation point, re-attachment point, and the length of the re-circulation zone. It is evident that the \(C_{f}\) predicted by the STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC2 model was in excellent agreement with the LES data. The STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC1 model also yielded good \(C_{f}\). The original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model and the STT \(k\)-\(\omega\)-CC model predicted similar \(C_{f}\), which significantly deviated from the LES result. The separation point was predicted by the original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model slightly later, but the re-attachment point was predicted much later, leading to a much larger re-circulation region. Although the two proposed models  
Figure 11: Velocity magnitude contours predicted by different turbulence models. Figure 10: Computational model of the 2D periodic hill flow. also predicted the separation point slightly later, they improved the re-attachment point location strongly, leading to an accurately predicted length of the re-circulation region. The separation point yielded by the STT \(k\)-\(\omega\)-CC model was very close to LES, but the re-attachment point was strongly delayed. In summary, the STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC2 model gave the best results for the 2D periodic hill flow, though the STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC1 also predicted good results. The original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model failed to predict the accurate re-circulation zone behind the hill, demonstrating that \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Sep.** & **Reatt.** & **Len.** \\ \hline LES & 0.22 \(H\) & 4.54 \(H\) & 4.32 \(H\) \\ SST \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) & 0.26 \(H\) & 7.57 \(H\) & 7.31 \(H\) \\ SST \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC1 & 0.26 \(H\) & 5.02 \(H\) & 4.76 \(H\) \\ SST \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC2 & 0.32 \(H\) & 4.54 \(H\) & 4.22 \(H\) \\ SST \(k\)-\(\omega\)-CC & 0.23 \(H\) & 7.73 \(H\) & 7.50 \(H\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The separation point (Sep.), re-attachment point (Reatt. ), and the re-circulation length (Len. ), in the 2D periodic hill flow. Figure 12: Comparison of velocity profiles at different transverse lines. Figure 13: Comparison of friction coefficients along the bottom wall.