## 2 Governing Equations
### \(\gamma\)-\(\widetilde{\mathrm{Re}_{\theta,t}}\)-\(S\) Transition Model  
In this approach, we use two transport equations to model the transition:  
\[\frac{\partial\gamma}{\partial t}+\mathbf{\
abla}\cdot(\mathbf{u} \gamma)=\mathrm{P}_{\gamma}-\mathrm{D}_{\gamma}+\mathbf{\
abla}\cdot\Bigg{[} \Bigg{(}\
u+\frac{\
u_{t}}{\sigma_{f}}\Bigg{)}\mathbf{\
abla}\gamma\Bigg{]}, \tag{7}\] \[\frac{\partial\widetilde{\mathrm{Re}_{\theta,t}}}{\partial t}+ \mathbf{\
abla}\cdot\Big{(}\widetilde{\mathrm{u}\mathrm{Re}_{\theta,t}}\Big{)}= \mathrm{P}_{\theta,t}+\mathbf{\
abla}\cdot\Big{[}\sigma_{\theta,t}(\
u+\
u_{t}) \mathbf{\
abla}\widetilde{\mathrm{Re}_{\theta,t}}\Big{]}.\]  
The source terms in the \(\gamma\) equation are defined as:  
\[\mathrm{P}_{\gamma} =c_{a1}S[\gamma\mathrm{F}_{\mathrm{onset}}]^{0.5}(1-c_{e1}\gamma )\mathrm{F}_{\mathrm{length}}, \tag{8}\] \[\mathrm{D}_{\gamma} =c_{a2}\Omega\gamma\mathrm{F}_{\mathrm{turb}}(c_{e2}\gamma-1),\]  
and in \(\mathrm{P}_{\gamma}\), the term \(\mathrm{F}_{\mathrm{onset}}\) is computed as:  
\[\mathrm{F}_{\mathrm{onset}}=\max(\mathrm{F}_{\mathrm{onset},2}-\mathrm{F}_{ \mathrm{onset},3},0) \tag{9}\]  
with:  
\[\mathrm{F}_{\mathrm{onset},2} =\min\Big{(}\max\Big{(}\mathrm{F}_{\mathrm{onset},1},\mathrm{F}_ {\mathrm{onset},1}\Big{)},4\Big{)}, \tag{10}\] \[\mathrm{F}_{\mathrm{onset},3} =\max\Bigg{(}2-\Big{(}\frac{\mathrm{R}_{\mathrm{T}}}{2.5}\Big{)} ^{3},0\Bigg{)},\] \[\mathrm{F}_{\mathrm{onset},1} =\frac{\mathrm{Re}_{v}}{2.193\mathrm{Re}_{\theta,c}}.\]  
In Equation (10), the terms \(\mathrm{Re}_{v}\) and \(\mathrm{R}_{\mathrm{T}}\) are obtained as follows:  
\[\mathrm{Re}_{v}=\frac{Sd^{2}}{\
u},\qquad\mathrm{R}_{\mathrm{T}}=\frac{\
u_{t} }{\
u}. \tag{11}\]The aspects concerning the terms \(\mathrm{F_{length}}\) and \(\mathrm{Re}_{\theta,c}\) are described in Section 2.1.1. As regards \(\mathrm{D_{\gamma}}\), the coefficient \(\mathrm{F_{turb}}\) is defined as:  
\[\mathrm{F_{turb}}=\exp\biggl{(}-\frac{\mathrm{R_{T}}}{4}\biggr{)}^{4}. \tag{12}\]  
For the source terms in the transport equation for \(\widetilde{\mathrm{Re}_{\theta,t}}\), \(\mathrm{P_{\theta,t}}\), the following equation is adopted:  
\[\mathrm{P_{\theta,t}}=\frac{c_{\theta,t}}{\mathrm{T}}\left(\mathrm{Re}_{ \theta,t}-\widetilde{\mathrm{Re}_{\theta,t}}\right)(1-\mathrm{F_{\theta,t}}). \tag{13}\]  
In Equation (13), the last term \(\mathrm{F_{\theta,t}}\) is defined as:  
\[\mathrm{F_{\theta,t}}=\min\left(\max\left(\exp\left(-\frac{\left|\mathbf{u} \right|^{2}}{375\Omega\
u\widetilde{\mathrm{Re}_{\theta,t}}}\right)^{4},1- \left(\frac{\gamma-1/c_{\ell 2}}{1-1/c_{\ell 2}}\right)^{2}\right),1.0\right). \tag{14}\]  
The term \(\mathrm{T}\), which appears in \(\mathrm{P_{\theta,t}}\), is defined as follows: \(500\
u/\left|\mathbf{u}\right|^{2}\). Finally, the handling of \(\mathrm{Re}_{\theta,t}\) in Equation (13) is further discussed, together with the \(\mathrm{F_{length}}\) coefficient, in the next subsection.  
For the transition model, the following closure constants are adopted:  
\[c_{a1}=2.0,\quad c_{a2}=0.06,\quad c_{e1}=1.0, \tag{15}\] \[c_{e2}=50,\quad c_{\theta,t}=0.03,\quad\sigma_{f}=1.0,\] (16) \[\sigma_{\theta,t}=2.0. \tag{17}\]