# Assessment of Two Streamline Curvature Correction Methods for an Elliptic Blending Turbulence Model
## 3 Results
### 2D U-Turn Duct Flow  
The 2D U-turn duct flow is another good benchmark for assessing the performance of conventional LEVMs under strong streamline curvature, which has been used by many \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Sep.** & **Reatt.** & **Len.** \\ \hline DNS & 2.43 \(H\) & 3.83 \(H\) & 1.40 \(H\) \\ SST \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) & 1.95 \(H\) & 5.17 \(H\) & 3.67 \(H\) \\ SST \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC1 & 2.26 \(H\) & 4.26 \(H\) & 2.00 \(H\) \\ SST \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC2 & 2.18 \(H\) & 4.42 \(H\) & 2.24 \(H\) \\ SST \(k\)-\(\omega\)-CC & 1.65 \(H\) & 5.84 \(H\) & 4.19 \(H\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The separation point (Sep.), re-attachment point (Reatt. ), and the re-circulation length (Len. ), in the 2D infinite serpentine passage flow. Figure 4: Comparison of friction coefficient distribution along the inner (**a**) and outer (**b**) walls. researchers [2; 3; 5; 10; 16]. The experimental measurements of [PERSON] et al. [23] have been widely used as reference. The computational model was the same as [PERSON] and [PERSON] [3], as shown in Figure 5, in which the boundary conditions are also provided. At the inlet, we specified fully developed quantity profiles (\(u_{i}\), \(k\), \(\omega\), \(\varphi\) and \(\alpha\)), which were obtained from preliminary computations of the fully developed turbulent flow in a plane channel at the same Re using the relevant turbulence models. At the outlet, the static pressure was set to be equal to zero. The Re (based on the channel height \(H\) and the bulk velocity at the inlet \(U_{b}\)) was 100,000. The total number of mesh cells was 345,000. By doubling the number of nodes in each direction and finding that the difference in the quantity was negligibly small (i.e., less than 1%), we found that this number of meshes was sufficient for obtaining a grid-independent solution. Figure 6 shows the velocity magnitude contours and streamlines predicted by the different turbulence models. Intuitively, the main difference occurred at the flow separation region behind the bend. The separation region predicted by the STT \(k\)-\(\omega\)-CC model was the largest. Meanwhile, the separation regions predicted by the two proposed models were similar, with both being smaller than that predicted by the original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model. Figure 7 depicts the streamwise velocity profiles along the transverse lines near the bend. To facilitate the analysis, an arc coordinate (s), defined along the center-line of the duct, was used (as shown in Figure 5). At the beginning of the bend (\(s/H=0\), Figure 7a), the velocity profiles predicted by all turbulence models agreed well with the experimental data. At \(s/H=\pi/2\), as shown in Figure 7b, in the region near the outer concave wall, the original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model underpredicted the results. The three other models still underestimated the velocity relative to the experimental results but showed improvements  
Figure 5: Computational model of the 2D U-turn duct flow. Figure 6: The velocity magnitude contours predicted by different turbulence models for the 2D U-turn duct flow. compared to the original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model. At \(s/H=\pi\), as shown in Figure 7c, the results of all models were not very different. Among them, the velocity predicted by the STT \(k\)-\(\omega\)-CC model was closer to the experimental data. At \(s/H=\pi+2\), as shown in Figure 7d, the STT \(k\)-\(\omega\)-CC model yielded the worst velocity profile. Moreover, the results predicted by the two proposed models were better than those of the original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model. Figure 8 shows the turbulent kinetic energy profiles along the transverse lines near the bend.