## 2 Transition Modeling  
### Transport Equations for the Transition SST Model  
The transition shear stress transport (SST) model, known as the \(\gamma\)-\(Re_{\theta}\) model, is used in this study. This model is based on coupling of the SST \(k\)-\(\omega\) transport equations with two other transport equations for the intermittency, and for the transition momentum thickness Reynolds number (\(Re_{\theta}\)). The intermittency equation is used to control the production of turbulent kinetic energy in the boundary layer. The TO Reynolds number transport equation is developed in order to avoid the additional non-local operations introduced by quantities used in the experimental correlations. These correlations are typically based on the TI or the pressure gradient outside the boundary layer. This model is able to provide an accurate prediction of the TO point as it allows the intermittency to vary across the boundary layer. Instead of using the momentum-thickness Reynolds number, the \(\gamma\)-\(Re_{\theta}\) model is based on the vorticity Reynolds number \((Re_{y})\)[25], which is dependent solely on local variables, as:  
\[Re_{y}\ =\ \frac{\rho y^{2}}{\mu}\frac{\partial u}{\partial y}\ =\ \frac{\rho y^{2}}{\mu}\Omega \tag{1}\]  
where, \(\Omega\) is the vorticity (shear strain rate) and \(y\) is the wall normal distance. The maximum value of \(Re_{y}\) is dependent on the momentum thickness Reynolds number as follows:  
\[Re_{\theta}\ =\ \frac{\max\left(Re_{y}\right)}{2.193} \tag{2}\]  
The transport equation for the intermittency is formulated to trigger the transition process and is given by [25]:  
\[\frac{\partial(\rho\gamma)}{\partial t}\ +\ \frac{\partial\left(\rho U_{j} \gamma\right)}{\partial x_{j}}\ =\ P_{\gamma 1}\ -\ E_{\gamma 1}\ +\ P_{\gamma 2}\ -\ E_{\gamma 2}\ +\ \frac{ \partial}{\partial x_{j}}\left[\left(\mu\ +\ \frac{\mu_{t}}{\sigma_{\gamma}}\right)\frac{\partial\gamma}{ \partial x_{j}}\right] \tag{3}\]  
where the transition sources, \(P_{\gamma 1}\) and \(E_{\gamma 1}\) are defined as follows:  
\[P_{\gamma 1}\ =\ C_{a1}F_{length}\rho S\left[\gamma F_{onset}\right]^{C_{ \gamma 3}} \tag{4}\]  
\[E_{\gamma 1}\ =\ C_{e1}P_{\gamma 1}\gamma \tag{5}\]  
where, \(S\) is the strain rate magnitude, \(F_{length}\) is an empirical correlation that controls the length of the transition region, and \(C_{a1}\) and \(C_{e1}\) have constant values of 2 and 1, respectively.  
\(P_{\gamma 2}\) and \(E_{\gamma 2}\) in Equation (3) represent the destruction and relaminarization sources defined as:  
\[P_{\gamma 2}\ =\ C_{a2}\rho\Omega\gamma F_{turb} \tag{6}\]  
\[E_{\gamma 2}\ =\ C_{e2}P_{\gamma 2}\gamma \tag{7}\]  
The transport equation for the transition momentum thickness Reynolds number \(R\overline{e}_{\theta t}\) is defined as follows:  
\[\frac{\partial(\rho R\overline{e}_{\theta t}\ )}{\partial t}\ +\ \frac{ \partial\left(\rho U_{j}R\overline{e}_{\theta t}\ \right)}{\partial x_{j}}\ =\ P_{\theta t}\ +\ \frac{ \partial}{\partial x_{j}}\left[\sigma_{\theta t}\ (\mu\ +\ \mu_{t})\ \frac{\partial R \overline{e}_{\theta t}}{\partial x_{j}}\right] \tag{8}\]  
Outside the boundary layer, the source term \(P_{\theta t}\) is employed to force the transported scalar \(R\overline{e}_{\theta t}\) to match the local value of \(R\overline{e}_{\theta t}\) calculated from an empirical correlation. The source term is defined as  
\[P_{\theta t}\ =\ c_{\theta t}\frac{\rho}{t}(Re_{\theta t}\ -\ R\overline{e}_{ \theta t})\ (1.0\ -\ F_{\theta t}) \tag{9}\]  
\[t\ =\ \frac{500\mu}{\rho U^{2}} \tag{10}\]  
The studied fan where \(t\) is a time scale and the blending function \(F_{\theta t}\) is used to turn off the source term in the boundary layer and allow the transported scalar \(R\overline{e}_{\theta t}\) to diffuse in from the freestream. In the free stream, \(F_{\theta t}\) is 0, and in the boundary layer it equals 1.