## 2 Governing Equations
### log \(\gamma\)-SA Transition Model  
As already introduced in Section 1, we also consider a second strategy to include transitional effects in the Spalart-Allmaras turbulence model. Specifically, in our approach, only one equation for \(\tilde{\gamma}=\log\gamma\), Equation (27), is solved to account for the laminar-to-turbulent transition. This method, formerly introduced by [PERSON] and [PERSON] [20], is very attractive because it allows guaranteeing the positivity of the intermittency. Moreover, the logarithmic distribution of a variable ensures a much smoother behavior with respect to the use of the primitive variable itself. This is a focal point of our approach, because the baseline model formulation exhibits a blow-up of the computations. The transport equation for \(\tilde{\gamma}\) is obtained by changing \(\gamma=\exp(\tilde{\gamma})\) in the intermittency equation reported in [PERSON] et al. [13]:  
\[\frac{\partial\tilde{\gamma}}{\partial t}+\mathbf{\
abla}\cdot(\mathbf{u}\tilde{ \gamma})=\text{P}_{\hat{\gamma}}-\text{D}_{\hat{\gamma}}+\mathbf{\
abla}\cdot[( \
u+\
u_{t})\mathbf{\
abla}\tilde{\gamma}]+(\
u+\
u_{t})|\mathbf{\
abla}\tilde{\gamma} |^{2}. \tag{27}\]  
Production and destruction terms for \(\tilde{\gamma}\) equation have the following expressions:  
\[\text{P}_{\hat{\gamma}}=\text{F}_{\text{length}}S\big{(}1-e^{\tilde{\gamma}} \big{)}\text{F}_{\text{onset}}\qquad\text{D}_{\hat{\gamma}}=c_{a2}\Omega\text {F}_{\text{turb}}\big{(}c_{a2}e^{\tilde{\gamma}}-1\big{)}\text{F}_{\text{onset}}. \tag{28}\]The F\({}_{\text{onset}}\) term in Equation (28) is defined as in the \(\gamma\)-\(\widetilde{\text{Re}_{\theta,t}}\) framework. Differently, F\({}_{\text{onset,2}}\) and F\({}_{\text{onset,3}}\) are re-calculated as follows:  
\[ \begin{split}\text{F}_{\text{onset,2}}&=\text{min}( \text{F}_{\text{onset,1}},2.0),\\ \text{F}_{\text{onset,3}}&=\text{max}\Bigg{(}1- \bigg{(}\frac{\text{R}_{\text{T}}}{3.5}\bigg{)}^{3},0\Bigg{)}.\end{split}  \tag{29}\]  
As regards F\({}_{\text{turb}}\), the following relation is adopted for this LCTM model:  
\[\text{F}_{\text{turb}}=\exp\left(-\frac{\text{R}_{\text{T}}}{2} \right)^{4}. \tag{30}\]  
We want to also remark that the constants \(c_{a2}\) and \(c_{e2}\) assume the same values reported in Equation (15). The parameter F\({}_{\text{length}}\) is fixed at 0.5 as in [PERSON] et al. (2019) in order to achieve a smooth growth of the intermittency. On the other hand, Re\({}_{\theta,c}\) is defined here as:  
\[\text{Re}_{\theta,c}=f_{local}(Tu_{l})f_{far}(Tu_{\infty}), \tag{31}\]  
where:  
\[ \begin{split} f_{local}(Tu_{l})&=803.73(Tu_{l}+0.6 067)^{-1.027},\\ f_{far}(Tu_{\infty})&=-3.162 Tu_{\infty}^{2}-0.4565 Tu_{\infty}+1.7.\end{split}  \tag{32}\]  
Looking at Equations (31) and (32), it is very easy to note that both the local turbulence intensity and the far-field turbulence information are involved. This is an interesting feature of the \(\log\gamma\)-SA correlation based transition model. Indeed, a local turbulence intensity controlling term, \(f_{local}\), is adopted to reflect the effect of local turbulence intensity variation on Re\({}_{\theta,c}\). The formulation of [PERSON] (2017) is used for the local turbulence intensity, \(Tu_{l}\). The integration of the \(\log\gamma\) model and the Spalart-Allmaras turbulence model is handled using a scheme similar to the \(\gamma\)-\(\widetilde{\text{Re}_{\theta,t}}\) one. The production term of the modified turbulent viscosity, P\({}_{\tilde{v}}\), is treated exactly as in Equation (22). However, for the coefficient \(\gamma_{sep}\), we use a correlation similar to the one suggested by [PERSON] et al. (2019):  
\[\gamma_{sep}=\text{min}\bigg{(}8.0\max\bigg{[}0.0,\bigg{(}\frac{\text{Re}_{v} }{2\text{Re}_{\theta,c}}\bigg{)}-1\bigg{]},\gamma_{lim}\bigg{)} \tag{33}\]  
In the above presented equation, \(\gamma_{lim}\) is a user-defined parameter imposed equal to 2.5. As regards the destruction term appearing in the turbulence equation, D\({}_{\tilde{v}}\), we use the same approach proposed by [PERSON] et al.