# Aerodynamic Efficiency Improvement on a NACA-8412 Airfoil via Active Flow Control Implementation
## 2 Numerical Method  
### Governing Equations and Turbulence Model  
Due to the unsteadiness of the problem, three possible turbulence models arose: Direct Numerical Simulation (DNS), Large Eddy Simulations (LES) and Unsteady Reynolds Averaged Navier-Stokes (URANS). Due to the large computational power required to perform both DNS and LES simulations, URANS was selected as the turbulence model used. More specifically, the Spalart-Allmaras model was chosen to perform all the simulations involved in the present study because of its suitability for low Reynolds cases and its easy convergence when the Courant-Friedrichs-Levy number \(CFL<1\). The Navier-Stokes equations for incompressible flow, take the form:  
\[\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{\partial x_{j}}= -\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\
u\frac{\partial^{2}u_{i}}{ \partial x_{j}\partial x_{j}} \tag{2}\]  
Substituting each variable \(\phi\) by the average \(\bar{\phi}\) and fluctuation \(\phi^{\prime}\)terms.  
\[\phi=\bar{\phi}+\phi^{\prime} \tag{3}\]  
Being \(\phi(x,t)\) a generic flow variable, it can be written as the sum of its mean \(\bar{\phi}(x,t)\) and fluctuating components \(\phi^{\prime}(x,t)\), being \(\phi(x,t)=\bar{\phi}(x,t)+\phi^{\prime}(x,t)\). By substituting it in the Navier-Stokes equations and taking the time average, it results in the following Reynolds-averaged Navier-Stokes equations,  
\[\frac{\partial\bar{u}_{i}}{\partial x_{i}}=0 \tag{4}\]  
\[\frac{\partial\bar{u}_{i}}{\partial t}+\frac{\partial\bar{u}_{i}\bar{u}_{j}}{ \partial x_{j}}=-\frac{1}{\rho}\frac{\partial\bar{p}}{\partial x_{i}}+\
u \frac{\partial^{2}\bar{u}_{i}}{\partial x_{j}\partial x_{j}}-\frac{\partial \overline{u^{\prime}_{i}u^{\prime}_{j}}}{\partial x_{j}} \tag{5}\]  
where the term \(\overline{u^{\prime}_{i}u^{\prime}_{j}}\) is the Reynolds stress tensor denoted by \(R_{ij}\) and was approximated using Boussinesq hypothesis. The deviatoric part of the tensor is given as follows,  
\[R_{ij}-\frac{1}{3}R_{kk}\delta_{ij}=-2\
u_{i}S_{ij} \tag{6}\]  
where \(\
u_{t}\) is the kinematic eddy viscosity that needs to be modeled using one of the RANS models.  
As it has been previously mentioned, the turbulence model selected was the Spalart-Allmaras (S-A). The model proposed by [36] solves a single transport equation for the modified form of the turbulent kinetic energy called \(\hat{v}\). This new parameter is identical to \(v_{t}\) except in the viscous-affected region nearby the wall.  
\[v_{t}=\hat{v}\Bigg{(}\frac{\chi^{3}}{\chi^{3}+C_{v1}^{3}}\Bigg{)} \tag{7}\]  
The transport equation of \(\hat{v}\) is given as:  
\[\frac{\partial\hat{v}}{\partial t}+u_{j}\frac{\partial\hat{v}}{\partial x_{j} }=\frac{1}{\sigma}\Bigg{[}\frac{\partial}{\partial x_{j}}\Bigg{(}(\
u+\hat{v} )\frac{\partial\hat{v}}{\partial x_{j}}\Bigg{)}+C_{b2}\frac{\partial\hat{v}}{ \partial x_{i}}\frac{\partial\hat{v}}{\partial x_{i}}\Bigg{]}+C_{b1}(1-f_{t2}) \hat{S}\hat{v}-\bigg{[}C_{w1}f_{w}-\frac{C_{b1}}{\kappa^{2}}f_{t2}\bigg{]} \bigg{(}\frac{\hat{v}}{d}\bigg{)}^{2} \tag{8}\]  
where:  
\[\begin{array}{ll}f_{\omega}=g\bigg{[}\frac{1+C_{v3}^{6}}{\hat{v}^{6}+C_{v3} ^{6}}\bigg{]}^{1/6},&g=r+C_{\omega 2}(r^{6}-r),&r=\min\Big{[}\frac{\hat{v}}{ 5\kappa^{2}d^{2}},10\Big{]}\\ \\ \tilde{S}=\Omega+\frac{\hat{v}}{\kappa^{2}d^{2}}fv_{2},&f_{t2}=C_{t3} exp(-C_{t4}\chi^{2}),&f_{t2}=1-\frac{\chi}{1+\chi f_{v1}}\\ \\ f_{v1}=\frac{\chi^{3}}{\chi^{3}+C_{v1}^{3}},&\chi=\frac{\hat{v}}{ \
u}\end{array} \tag{9}\]  
Being \(d\) the distance from a given point to the nearest wall, and \(\Omega\) the magnitude of the vorticity. The model constants have the following default values:  
\[\begin{array}{ll}C_{b1}=0.1355,&C_{b2}=0.622,&\sigma=\frac{2}{3},&C_{\omega 1 }=\frac{C_{v4}}{\kappa^{2}}+\frac{(1+C_{b2})}{\sigma}\\ C_{\omega 2}=0.3,&C_{\omega 3}=2.0,&C_{v1}=7.1,&\kappa=0.4187,&C_{t3}=1.2\\ C_{t4}=0.5\end{array} \tag{10}\]