## 3 Results and Discussion
### 3D Asymmetric Diffuser Flow
It is worth noting that the flow patterns from the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model were very similar to those obtained by [PERSON] et al. [22] using the SST \(k\)-\(\omega\) and BL - \(\overline{v^{2}}/k\) models. This demonstrated that 3D diffuser flow is a challenge for many linear eddy viscosity models. Figure 3 showed a more quantitative comparison of the mean stream-wise velocity profiles, which were extracted from a spanwise midplane (\(y/H=-1.665\)). It can be seen that both the proposed SAS and the SST-SAS models reasonably predicted the upward movement of the velocity peak, but the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model did not. Overall, the predictions of the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model were very poor. In the upstream sections (\(x/H=4\) and \(x/H=8\)), the predictions of the SST-SAS model were slightly better than those of the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-SAS model. In contrast, the predictions of the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-SAS model were better than those of the SST-SAS model in the downstream sections (\(x/H=12\), \(x/H=15.5\), and \(x/H=18.5\)). The pressure coefficients, \(C_{p}=2\left(p-p_{ref}\right)/\left(\rho U_{b}^{2}\right)\), where \(p_{ref}\) is the reference pressure at \(x=0\), distributed along the intersection line of the spanwise midplane (\(y/H=-1.665\)) and the bottom wall, were compared with those from the DNS in Figure 4. As can be seen, the prediction by the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-SAS model was in good agreement with that of the DNS. The SST-SAS model underpredicted the increase in pressure, whereas the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model was not able to predict the increasing trend of the pressure. Figure 1: Schematic of the geometry and relevant boundary conditions of the 3D diffuser flow. Figure 4: The pressure coefficients distributed along the intersection line of the spanwise midplane and the bottom wall. Figure 3: The normalized mean stream-wise velocity profiles on the spanwise midplane. Figure 2: The contours of the normalized mean stream-wise velocity at different cross-sections. The attractive characteristic of the SAS is that the formation of the turbulent spectrum is allowed by adjusting the length scale to match the resolved structures. The turbulent structures can be illustrated using the \(Q\)-criterion (\(Q=(S^{2}-\Omega^{2})/2\)). Figure 5 shows the turbulent structures predicted by the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-SAS and SST-SAS models using an iso-surface of \(Q=100\ 1/s^{2}\). The color gradient in the figures represents the mean viscosity ratio (the ratio of eddy viscosity to molecular viscosity). It was clear that, in the diffuser, the turbulent structures predicted by the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-SAS model were smaller than those predicted by the SST-SAS model. This explained why the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-SAS model was better than the SST-SAS model overall. The difference between the results of the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-SAS and SST-SAS models reveals the importance of the underlying RANS model in the SAS method, as this is the major difference between these two models. The SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-SAS model can yield more accurate near-wall turbulence [14], further improving the entire flow.