## 2 Numerical Approach
### \(\gamma-\widetilde{\mathrm{Re}}_{\theta t}\) Transition Model  
The \(\gamma-\widetilde{\mathrm{Re}}_{\theta t}\) transition model is based on the local variables of flow field proposed by [PERSON] and [PERSON], in which the intermittence factor \(\gamma\) and the transition momentum thickness Reynolds number \(\widetilde{\mathrm{Re}}_{\theta t}\) are added to construct the transport equations. The \(\gamma\) is used to depict the flow characteristics of the transition region, which represents the probability of turbulence at a point in flow field, and the value range is \(0\leq\gamma\leq 1\). \(\widetilde{\mathrm{Re}}_{\theta t}\) is used to control the generation of intermittence factors and to form a criterion for predicting the starting position of transition.  
Transport equation of \(\gamma\) is:  
\[\frac{\partial(\rho\gamma)}{\partial t}+\frac{\partial\big{(}\rho U_{j} \gamma\big{)}}{\partial x_{j}}=P_{\gamma 1}-E_{\gamma 1}+P_{\gamma 2}-E_{ \gamma 2}+\frac{\partial}{\partial x_{j}}\bigg{[}\bigg{(}\mu+\frac{\mu_{j} }{\sigma_{j}}\bigg{)}\frac{\partial\gamma}{\partial x_{j}}\bigg{]} \tag{4}\]  
Transition source term and fracture/laminar fluidization source term are defined as follows:  
\[P_{\gamma 1}=C_{a1}F_{length}\rho S[\gamma F_{onset}]^{C\gamma 3} \tag{5}\]  
\[E_{\gamma 1}=C_{e1}P_{\gamma 1}\gamma \tag{6}\]  
\[P_{\gamma 2}=C_{a2}\rho\Omega\gamma F_{turb} \tag{7}\]  
\[E_{\gamma 2}=C_{e2}P_{\gamma 2}\gamma \tag{8}\]  
where \(S\) is strain rate, \(F_{length}\) is the empirical correlation for controlling the length of transition region, \(\Omega\) is vorticity, the constant terms are \(C_{a1}=2\), \(C_{e1}=1\), \(C_{a2}=0.06\), \(C_{e2}=50\), \(C_{\gamma 3}=0.5\).  
The vorticity Reynolds number is defined as:  
\[\mathrm{Re}_{v}=\frac{\rho v^{2}S}{\mu} \tag{9}\]  
The other parameters are as below:  
\[R_{T}=\frac{\rho k}{\mu\omega} \tag{10}\]  
\[F_{onset1}=\frac{\mathrm{Re}_{v}}{2193\mathrm{Re}_{\theta c}} \tag{11}\]  
\[F_{onset2}=\min\Bigl{(}\max\Bigl{(}F_{onset1},F_{onset}^{4}\Bigr{)},2.0 \Bigr{)} \tag{12}\]  
\[F_{onset3}=\max\biggl{(}1-\Bigl{(}\frac{R_{T}}{2.5}\Bigr{)}^{3},0\Bigr{)} \tag{13}\]  
\[F_{onset}=\max\bigl{(}F_{onset2}-F_{onset3},0\bigr{)} \tag{14}\]  
\[F_{turb}=e^{-\bigl{(}\frac{R_{T}}{4}\bigr{)}^{4}} \tag{15}\]The transport equation of \(\gamma-\widetilde{\text{Re}}_{\theta t}\) is:  
\[\frac{\partial\Big{(}\rho\widetilde{\text{Re}}_{\theta t}\Big{)}}{\partial t}+ \frac{\partial\Big{(}\rho U_{j}\widetilde{\text{Re}}_{\theta t}\Big{)}}{ \partial x_{j}}=P_{\theta t}+\frac{\partial}{\partial x_{j}}\Bigg{[}\sigma_{ \theta t}(\mu+\mu_{t})\frac{\partial\widetilde{\text{Re}}_{\theta t}}{ \partial x_{j}}\Bigg{]} \tag{16}\]  
The source term is defined as:  
\[P_{\theta t}=c_{\theta t}\frac{\rho}{t}\Big{(}\text{Re}_{\theta t}-\widetilde{ \text{Re}}_{\theta t}\Big{)}(1.0-F_{\theta t}) \tag{17}\]  
The coupled equation with the shear stress transport (SST) flow model is:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{j}}\Big{(} \rho u_{j}k\Big{)}=\frac{\partial}{\partial x_{j}}\Bigg{[}\Gamma_{k}\frac{ \partial k}{\partial x_{j}}\Bigg{]}+G^{*}_{k}-Y^{*}_{k}+S_{k} \tag{18}\]  
\[G^{*}_{k}=\gamma_{eff}\widetilde{G}_{k} \tag{19}\]  
\[Y^{*}_{k}=\min\Big{(}\max\Big{(}\gamma_{eff},0.1\Big{)},1.0\Big{)}Y_{k} \tag{20}\]  
Other parameters are defined by reference [30; 31; 32].