## 2 Model Formulation and Implementation  
The basis of the proposed SAS model is the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model, which belongs to the elliptic blending RANS model category [14]. The SAS model consists of four model equations for the turbulent kinetic energy \(k\), specific dissipation rate \(\omega\), wall-normal turbulent anisotropy \(\varphi\), and elliptic variable \(\alpha\). The equations of the model are as follows:  
\[\frac{\partial\rho k}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i} k)=G_{k}-\rho f_{k}\beta^{*}k\omega+\frac{\partial}{\partial x_{j}}\Bigg{[} \Big{(}\frac{\mu}{2}+\sigma_{k}\mu_{t}\Big{)}\frac{\partial k}{\partial x_{j} }\Bigg{]}, \tag{1}\]  
\[\frac{\partial\rho\omega}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u _{i}\omega)=f_{\omega}\gamma\frac{\omega}{k}G_{k}-\rho\beta\omega^{2}+C_{D}+ \frac{\partial}{\partial x_{j}}\Bigg{[}\Big{(}\frac{\mu}{2}+\sigma_{\omega} \mu_{t}\Big{)}\frac{\partial\omega}{\partial x_{j}}\Bigg{]}+Q_{SAS}, \tag{2}\]  
\[\frac{\partial\rho\phi}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{ i}\phi)=(1-\alpha^{p})\rho f_{wall}+\alpha^{p}\rho f_{hom}-\frac{\phi}{k}G_{k}+ \frac{2}{k}\mu_{t}\sigma_{k}\frac{\partial\phi}{\partial x_{j}}\frac{\partial k }{\partial x_{j}}+\frac{\partial}{\partial x_{j}}\Bigg{[}\Big{(}\frac{\mu}{2} +\sigma_{\phi}\mu_{t}\Big{)}\frac{\partial\phi}{\partial x_{j}}\Bigg{]}, \tag{3}\]  
\[0=\frac{1-\alpha}{L^{2}}+\frac{\partial}{\partial x_{j}}\Bigg{(}\frac{\partial \alpha}{\partial x_{j}}\Bigg{)}. \tag{4}\]  
In addition, an additional source term \(Q_{SAS}\) is introduced into the \(\omega\)-equation, and the remaining terms are the same as the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model. For the meaning of each term and the value of each coefficient, please refer to [PERSON] et al. [14]. The source term \(Q_{SAS}\) is the same as the SST \(k\)-\(\omega\) model-based SAS model (denoted as the SST-SAS model) [5]. It is defined as follows:  
\[Q_{SAS}=\max\Bigg{[}\rho\eta_{2}\kappa S^{2}\bigg{(}\frac{L_{t}}{L_{nk}} \bigg{)}^{2}-C_{SAS}\frac{2\rho k}{\sigma_{q}}\max\Bigg{(}\frac{1}{k^{2}}\frac {\partial k}{\partial x_{j}}\frac{\partial k}{\partial x_{j}},\frac{1}{\omega ^{2}}\frac{\partial\omega}{\partial x_{j}}\frac{\partial\omega}{\partial x_{j} }\Bigg{)},0\Bigg{]}, \tag{5}\]  
with \(\eta_{2}=3.51\), \(\kappa=0.41\), \(C_{SAS}=2\), and \(\sigma_{q}=2/3\). The modeled turbulence length scale and the von Karman length scale are  
\[L_{t}=\frac{\sqrt{k}}{c_{\mu}^{2}\omega}, \tag{6}\]  
and  
\[L_{nk}=\kappa\bigg{|}\frac{U^{\prime}}{U^{\prime\prime}}\bigg{|}, \tag{7}\]  
respectively. Here, \(c_{\mu}=0.09\), \(U^{\prime}=S\) is the magnitude of the shear stress, and \(U^{\prime\prime}\) is the magnitude of the velocity Laplacian, which is given by  
\[U^{\prime\prime}=\sqrt{\frac{\partial^{2}U_{i}}{\partial x_{j}^{2}}\frac{ \partial^{2}U_{i}}{\partial x_{k}^{2}}}. \tag{8}\]  
For the purpose of comparison, all test cases were computed using three models: the proposed SAS model (denoted as the SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\)-SAS model), SST-SAS model, and SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) model. The FLUENT computational fluid dynamics (CFD) code was used as the computing platform. The SST-SAS model is built into the software itself. The proposed SAS and SST-\(k\)-\(\omega\)-\(\varphi\)-\(\alpha\) models were implemented using the user-defined function (UDF) functionality. A pressure-based coupled algorithm was used to solve the governing equation system. The bounded second-order implicit scheme was used for the integration of the transient terms. The bounded central differencing scheme and the second-order upwind scheme were used to discretize the convection terms in the momentum equation and in the turbulence equations, respectively. The gradients and derivatives were evaluated by the least-squares cell-based method.