# Assessment of Two Streamline Curvature Correction Methods for an Elliptic Blending Turbulence Model
## 3 Results
### 2D Infinite Serpentine Passage Flow  
The 2D flow in an infinite serpentine passage is a severe test for conventional LEVMs, as it is characterized by a large streamline curvature. Therefore, it has been used by various researchers for assessing the performance of turbulence models [19; 20; 21]; in particular, the steady case in the DNS of [PERSON] and [PERSON] [22] has been widely used as a reference. The computational model, which was constructed according to the three-dimensional (3D) model used by [PERSON] and [PERSON] [22], is shown in Figure 1. The boundary conditions are also illustrated. The Reynolds number (Re) of the flow (based on the passage height \(H\) and the bulk velocity at inlet, \(U_{b}\)) was 5600. The flow is periodic, such that a periodic boundary condition was applied at the inlet and outlet. The mass flow rate was specified according to the requirement of Re. A grid-dependence test was performed using different resolution meshes, and, finally, the total number of mesh cells was selected as 330,000, which was fine enough to obtain grid-independent results. The velocity magnitude contours predicted by different turbulence models are plotted in Figure 2. It is clear that both the original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model and the STT \(k\)-\(\omega\)-CC model overpredicted the re-circulation region downstream of the bends. The velocity distributions predicted by the two new models proposed in the present paper were very close. To illustrate the velocity distribution more quantitatively, the velocity profiles along different transverse lines (\(l_{1}\), \(l_{2}\), \(l_{3}\), and \(l_{4}\), as shown in Figure 1) are shown in Figure 3. It can be observed that the two proposed models successfully predicted the velocity profiles at all locations. However, at \(l_{1}\) and \(l_{2}\), the original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model underpredicted the velocity near the outer concave wall, but strongly overpredicted the velocity near the inner  
Figure 1: Computational model of the 2D infinite serpentine passage flow. Figure 2: The velocity magnitude contours predicted by different turbulence models for the 2D infinite serpentine passage flow. convex wall. In contrast, at \(l_{3}\) and \(l_{4}\), the original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model overpredicted the velocity near the concave wall, but underpredicted the velocity near the convex wall. This model predicted higher re-circulation downstream of the bend (Figure 3c), and a significant flow recovery delay was observed (Figure 3d). The velocity predicted by the STT \(k\)-\(\omega\)-CC model was the worst compared to the other models. Figure 4 shows the friction coefficient (\(C_{f}=2\tau_{\text{tr}}/\left(\rho U_{b}^{2}\right)\)) along the inner and outer walls near bend 1. To facilitate the analysis, an arc coordinate (\(s\)) was used, as shown in Figure 1. In the first half of the inner wall (\(s/H<1.5\)), the \(C_{f}\) predicted by the four models were similar and agreed with the DNS results well. However, in the latter part (\(s/H>1.5\)), the \(C_{f}\) predicted by the four models differed dramatically. The predictions of the STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC1 and STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-CC2 models were similar and agreed well with the DNS data. The original STT \(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model and the STT \(k\)-\(\omega\)-CC model delayed the recovery of the \(C_{f}\) significantly. For \(C_{f}\) along the outer wall, the predictions of the four models displayed obvious differences, even in the first half of the wall. Therefore, the results obtained from the two proposed models were better. Figure 3: Comparison of velocity profiles along different transverse lines for the 2D infinite serpentine passage flow: (**a**) \(l_{1}\); (**b**) \(l_{2}\); (**c**) \(l_{3}\); and (**d**) \(l_{4}\).