# Numerical Investigation of Jet Angle Effect on Airfoil Stall Control
## 2 Methods
### Numerical Methods  
The compressible Reynolds-averaged Navier-Stokes (RANS) equations are computed numerically using the unstructured flow solver SU2, which was developed in Stanford university [39]. The RANS equations are obtained using the Favre average. Here, the Favre average is given by \(\tilde{\phi}=\frac{\partial\tilde{\phi}}{\tilde{\rho}}\) where \(\phi\) is an instantaneous variable and the overbar indicates the conventional Reynolds-averaged \(\phi\). The compressible RANS equations are given by  
\[\frac{\partial\tilde{\rho}}{\partial t}+\frac{\partial}{\partial x_{i}}( \tilde{\rho}\tilde{u_{i}})=0 \tag{3}\]  
\[\frac{\partial}{\partial t}(\tilde{\rho}\tilde{u_{i}})+\frac{\partial}{ \partial x_{j}}(\tilde{\rho}\tilde{u_{i}}\tilde{u_{j}})+\frac{\partial}{ \partial x_{j}}[\tilde{\rho}\tilde{g_{ij}}-\tau_{ij}^{turb}]=0 \tag{4}\]  
\[C_{v}\frac{\partial}{\partial t}(\tilde{\rho}\tilde{T})+\frac{\partial}{ \partial x_{j}}[(C_{v}\tilde{\rho}\tilde{u_{j}}\tilde{T})+\tilde{p}\tilde{u_{ j}}+\tilde{u_{i}}\tilde{\tau_{turb}}-\lambda\tilde{T}+C_{v}q_{j}^{turb}]=0 \tag{5}\]  
where \(\tau_{ij}^{turb}=2\mu_{t}\widetilde{S_{ij}^{*}}-\frac{2}{3}\tilde{\rho}k \delta j_{ij}\). The equation of state for a calorically perfect gas is used here. Unless notified otherwise, the following scales are used for non-dimensionalization : \(U_{\infty},c,P_{\infty},T_{\infty}\) and \(\rho_{\infty}\). The standard Spalart-Allmaras (SA) model is used to model the turbulent viscosity \(\mu_{t}\)[40]. The SA model is given by  
\[\frac{D\hat{\psi}}{Dt}=c_{b1}\tilde{S}\hat{\psi}+\frac{1}{\sigma}[\
abla(( \
u+\hat{\
u})\
abla\hat{\
u}+c_{b2}(\
abla\hat{\
u})^{2}]-[c_{w1}f_{w}]| \frac{\hat{\psi}}{d}]^{2} \tag{6}\]  
where \(\
u_{t}\) is defined as \(\
u_{t}=\frac{\mu_{t}}{\rho}=\hat{\psi}f_{v1}\). The working variable \(\hat{\
u}\) is obtained from the SA transport Equation (6). In addition, the following model coefficients are used: \(c_{b1}=0.1355,c_{b2}=0.622,\kappa=0.41,\sigma=\frac{2}{3},c_{w1}=\frac{c_{b1 }}{\kappa^{2}}+\frac{1+c_{b2}}{\sigma},c_{w2}=0.3,c_{w3}=2,c_{v1}=7.1\). The turbulent heat flux \(q_{j}^{turb}\) is modeled using the temperature gradient \(q_{j}^{turb}\equiv C_{p}\frac{\mu_{t}}{P\tau_{t}}\frac{\partial\tilde{T}}{ \partial x_{j}}\) where the turbulent Prandtl number is \(P\tau_{t}=0.9\) and the heat capacity at constant pressure \(C_{p}\) for the air is used. The numerical analysis is conducted with the assumption of fully turbulent flow. It should be noted that typical RANS approaches are limited to flow separation prediction [41]. The accuracy of current fully turbulent SA RANS computations will be discussed with comparison to relevant experimental data. The second-order Roe scheme is used for the Euler fluxes [42]. The second-order central difference is used for the viscous fluxes. The second-order Venkatakrishnan limiter is used for preserving the monotonicity of the flow [43]. For temporal discretization, the dual-time stepping method is used with the second-order accuracy [44]. The coordinates of OA209 airfoil are listed in Tables 1 and 2 (see Appendix A). It should be mentioned that the current OA209 airfoil, which was used in wind tunnel experiments [16; 17; 18; 36; 38], is slightly different to the version available from the web airofiltools.com [45]. O-type grids are generated around the current OA209 airfoil. According to the previous study [46], the leading edge separation occurs on the OA209 airfoil. Thus, dense grids are generated near the leading edge. Structured grids for the current simulation are created to capture the boundary layer on the OA209 airfoil. The first wall-normal grid point is located at \(y^{+}\lesssim 1.5\) in wall units. Grids for the clean airfoil are generated first, then a slot for the jet actuation is introduced on the clean airfoil grid. The current grid of \(420\times 150\) is used for the clean airfoil. The streamwise grid size at the leading edge is \(\Delta_{LE}/c=1.0\times 10^{-4}\).