## 2 Materials and Methods
### Finite Element Method  
The mathematical model of air in the radial interval between the interacting surfaces of the BGD and the OM is based on the Navier-Stokes (Reynolds-averaged Navier-Stokes equations) (RANS) equation's average, according to Reynolds [84; 85]. The SST model of turbulence [86; 87; 88] and the \(\gamma\)-model of laminar and turbulent transition [89] are used for modeling. The main equations for describing these models are presented below. The gamma model of laminar and turbulent transition is described by one differential equation for the intermittency coefficient \(\gamma\):  
\[\frac{\partial(\rho\gamma)}{\partial t}+\frac{\partial(\rho V_{j}\gamma)}{ \partial x_{j}}=P_{\gamma}-E_{\gamma}+\frac{\partial}{\partial x_{j}}\Bigg{[} \bigg{(}\mu+\frac{\mu_{t}}{\sigma_{\gamma}}\bigg{)}\frac{\partial\gamma}{ \partial x_{j}}\Bigg{]}, \tag{4}\]  
where \(\rho\) is the air density; \(t\) is the time; \(x\) is the coordinate; \(V\) is the vector of air velocity; \(P_{\gamma}\) and \(F_{\gamma}\) are, respectively, the generative and dissipation members of the managing directors of laminar and turbulent transition; \(\mu\) is the molecular dynamic viscosity of gas; \(\mu_{t}\) is the turbulent dynamic viscosity of gas; and \(\sigma_{\gamma}\) = 1.0 is the model constant. The coupling between the transition model and the SST turbulence model is the same as for the \(\gamma\)-Re\({}_{\theta}\) model [90], and it is accomplished by modifying the equations of the original SST model as follows:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{j}}(\rho V_{ j}k)=\widetilde{P}_{k}+P_{k}^{\text{lim}}-\widetilde{D}_{k}+\frac{\partial}{ \partial x_{j}}\Bigg{(}(\mu+\sigma_{k}\mu_{t})\frac{\partial k}{\partial x_{ j}}\Bigg{)}, \tag{5}\]  
\[\frac{\partial}{\partial t}(\rho\omega)+\frac{\partial}{\partial x_{j}}(\rho V _{j}\omega)=\alpha\frac{P_{k}}{v_{t}}-D_{\omega}+Cd_{\omega}+\frac{\partial}{ \partial x_{j}}\Bigg{(}(\mu+\sigma_{\omega}\mu_{t})\frac{\partial\omega}{ \partial x_{j}}\Bigg{)}, \tag{6}\]  
\[\widetilde{P}_{k}=\gamma P_{k}, \tag{7}\]  
\[\widetilde{D}_{k}=\max(\gamma,0.1)\cdot D_{k}, \tag{8}\]  
\[\mu_{t}=p\frac{a_{1}\cdot k}{\max(a_{1}\cdot\omega,F_{2}\cdot S)}, \tag{9}\]  
\[S_{ij}=\frac{1}{2}\Bigg{(}\frac{\partial V_{i}}{\partial x_{j}}+\frac{ \partial V_{j}}{\partial x_{i}}\Bigg{)};\,S^{2}=2S_{ij}S_{ij}, \tag{10}\]  
where \(k\) is the kinetic turbulent energy; \(\omega\) is the specific speed of dissipation of the kinetic energy of turbulence; \(P_{k}\) and \(D_{k}\), are, respectively, the original generation and dissipation of the SST model; \(P_{k}^{\text{lim}}\) is the additional part, which provides the correct gain of turbulent viscosity in the transitional area at a very low level of turbulent viscosity of the running stream; \(v_{t}\) is the turbulent kinematic viscosity of gas; and \(\sigma_{k}\), \(\alpha\), and \(\alpha_{1}\) are, respectively, the empirical constants of the model. The generative member in Equation (5) appears as follows:  
\[P_{\gamma}=F_{length}\rho S\gamma(1-\gamma)F_{onset}, \tag{11}\]  
where \(F_{length}\) is the empirical correlation which controls the length of the transitional area (with \(F_{length}\) = 100) and \(F_{onset}\) is the function controlling the provision of the beginning of the transition. The destruction/relaminarization source is identical to the one used in the \(\gamma\)-Re\({}_{\theta}\) model [83] and is defined as follows:  
\[E_{\gamma}=c_{a2}\rho\Omega\gamma F_{turb}(c_{e2}\gamma-1), \tag{12}\]where \(c_{a2}=0.06\) and \(c_{e2}=50\) are the empirical constants, \(\Omega=\sqrt{2\Omega_{ij}\Omega_{ij}}\) is the invariant of the tensor of vorticity, and \(F_{turb}=e^{-(\frac{k_{T}}{2})^{4}}\); \(R_{T}=\frac{\rho k}{\mu\omega}\). The formulation of the function \(F_{onset}\) is similar to that used in the \(\gamma\)-Re\({}_{\theta}\) model. It is used to start the intermittency production (i.e., to activate the source value (11)).