## 3 Numerical Methods and [PERSON]'s \(\gamma-R_{\theta}\) Transition Model Formulation  
In the present work, the open source code OpenFOAM V.5.0 [31] was adopted for the deterministic flow solver. The steady incompressible Navier-Stokes governing equations were discretized spatially at the second order and the SIMPLE algorithm was adopted for continuity. [PERSON] proposed the \(\gamma-Re_{\theta}\) transitional model [21] based on the fully turbulent SST k-\(\omega\) model. It includes two transport equations for intermittency, \(\gamma\), and for the transported transition momentum thickness Reynolds number \(\widetilde{\text{Re}}_{\theta t}\). The dimensionless intermittency and Reynolds number based on the momentum thickness are formulated as follows. \[\rho\frac{D\gamma}{Dt}=\frac{\partial}{\partial x_{j}}\Bigg{[}\Big{[}\mu+ \frac{\mu_{t}}{\sigma_{t}}\Big{]}\frac{\partial\gamma}{\partial x_{j}}\Bigg{]} +F_{length}\epsilon_{\text{a1}}\rho S(\gamma F_{onset})^{0.5}(1-c_{\epsilon 1 }\gamma)+\epsilon_{\text{a2}}\rho\Omega\gamma F_{turb}(1-c_{\epsilon 2 }\gamma) \tag{16}\]  
\[\rho\frac{D\widetilde{\text{Re}}_{\theta t}}{Dt}=\frac{\partial}{\partial x_ {j}}\Bigg{[}\sigma_{\theta t}(\mu+\mu_{t})\frac{\partial\widetilde{\text{Re} }_{\theta t}}{\partial x_{j}}\Bigg{]}+c_{\theta t}\frac{(\rho U)^{2}}{500\mu}( \text{Re}_{\theta t}-\widetilde{\text{DR}}\epsilon_{\theta t})(1-F_{\theta t}) \tag{17}\]  
The modeled transport equations are controlled by the following functions:  
\[F_{onset}=\max(F_{onset2}-F_{onset3},0),F_{turb}=\exp[-(R_{T}/4)^{4}], \tag{18}\]  
\[F_{onset2}=\min[\max(F_{onset1},F_{onset1}^{4}),2],F_{onset1}=\frac{\text {Re}_{v}}{2.193\text{Re}_{\theta c}}, \tag{19}\]  
\[F_{onset3}=\max(1-(R_{T}/2.5)^{3},0), \tag{20}\]  
\[F_{\theta t}=\min\Bigg{\{}\max\Bigg{[}F_{wake.}\exp\Bigg{[}-\Bigg{[}\frac{U^{ 2}}{375\Omega\widetilde{v}Re_{\theta t}}\Bigg{]}^{4}\Bigg{]},1-\bigg{[}\frac{ c_{\text{a2}}\gamma-1}{c_{\text{a2}}-1}\bigg{]}^{2}\Bigg{]},1\Bigg{\}},F_{wake}=\exp \Bigg{[}-\bigg{[}\frac{\text{Re}_{\omega}}{10^{5}}\bigg{]}^{2}\Bigg{]}, \tag{21}\]  
with the following parameters:  
\[\text{Re}_{v}=\frac{\rho Sy^{2}}{\mu},R_{T}=\frac{\rho k}{\mu\omega},\text{ Re}_{\omega}=\frac{\rho\omega y^{2}}{\mu},\text{Re}_{v}=\frac{\rho Sy^{2}}{\mu},R_{T}= \frac{\rho k}{\mu\omega},\text{Re}_{\omega}=\frac{\rho\omega y^{2}}{\mu} \tag{22}\]  
The details of the intermittency equation are presented in a previous work [21]. Both \(F_{length}\) and \(R_{\theta c}\) in the above equation are empirical correlations and functions of the transition momentum-thickness Reynolds number \(R_{\theta t}\), which is also a function of the turbulence intensity \(Tu\) and the streamwise pressure gradient \(\lambda_{\theta}\) in the free flow. The model constants are \(\text{c}_{\text{a1}}=2.0\), \(\text{c}_{\text{e1}}=1.0\), \(\text{c}_{\text{a2}}=0.06\), \(\text{c}_{\text{e2}}=50.0\), \(c_{\text{a}}=0.5\), \(\sigma_{\gamma}=1.0\), \(\sigma_{\theta t}=2.0\) and \(c_{\theta t}=0.03\). For predicting separation induced transition, the following modification is utilized. \[\gamma_{sep}=\min(2.0\max(0,\frac{\text{Re}_{v}}{3.235R_{\theta c}}-1)F_{ reattach},2)F_{\theta t} \tag{23}\]\[F_{reattach}=e^{-(\frac{R_{T}}{2D})}{}^{4} \tag{24}\]  
The effective value of \(\gamma\) is thus obtained as follows. \[\gamma_{effective}=\max(\gamma,\gamma_{sep}) \tag{25}\]  
The empirical correlation used in this model is based on the pressure gradient parameter and turbulent intensity defined as follows. \[\rho\frac{D\widetilde{\text{Re}}_{\theta t}}{Dt}=\frac{\partial}{\partial x_ {j}}\Bigg{[}\sigma_{\theta t}(\mu+\mu_{t})\frac{\partial\widetilde{\text{Re} }_{\theta t}}{\partial x_{j}}\Bigg{]}+c_{\theta t}\frac{(\rho U)^{2}}{500\mu}( \text{Re}_{\theta t}-\widetilde{\text{DR}}\epsilon_{\theta t})(1-F_{\theta t}) \tag{17}\]0  
\(\mathrm{Re}_{\theta}\) is defined as follows.