# Shock Reduction through Opposing Jets--Aerodynamic Performance and Flight Stability Perspectives
## 2 Problem Formulation
### Computational Setup  
In this research, \(NACA\) 0012 airfoil is selected as a reference aerodynamic surface. The airfoil has a blunt leading edge and moderate thickness to chord ratio. Moreover, it is one of the most generic airfoil shapes studied in aerodynamic literature. The airfoil has 1 m chord with 0.004 m jet orifice at leading edge (Figure 1(a)). Figure 2: Numerical grid distribution with boundary conditions. A fully structured two-dimensional mesh was generated. The boundary layer resolution was made such that \(Y^{+}\) is less than one on the surface. Pressure inlet (jet), pressure outlet (static pressure at outlet) and the pressure far field (to model free-stream compressible flow up to an optimum distance) boundary conditions were selected for numerical analysis. Isentropic relations, as in Equation (1), were used to calculate the static pressure and temperature (\(P_{\infty}\), \(T_{\infty}\)) at sea level stagnation conditions (\(P_{\infty}\), \(T_{\infty}\)) [12]. \[\frac{P_{\infty}}{P_{\infty}}=[1+\frac{\gamma-1}{\gamma}M^{2}]\frac{\gamma}{ \gamma-1}\qquad;\qquad\frac{T_{\infty}}{T_{\infty}}=[1+\frac{\gamma-1}{\gamma }M^{2}], \tag{1}\]  
where \(\gamma\) is the specific heat ratio. The discretized domain used for all simulations is shown in Figure 2c,d. The compressible flow around the airfoil is simulated by using ANSYS(r) FLUENT. Two dimensional Unsteady Reynolds Averaged Navier Stokes (\(URANS\)) and turbulence model equations were solved using a density based implicit solver, as it works effectively for supersonic flows. The ideal gas law was used to model the flow properties in the far field. However, the viscosity of air was defined by [PERSON]'s viscosity law to capture the complex flow features like shock wave and re-circulation regions accurately. The \(k\)-\(\omega\)\(SST\) turbulence model was used to calculate the complex flow field which is expressed as  
\[\frac{\partial(\rho k)}{\partial t}+div(\rho kV)= div(\mu+\frac{\mu_{t}}{\sigma_{k}}grad(k))+P_{k}-\beta^{*} \tag{2}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+div(\rho\omega V)= div(\mu+\frac{\mu_{t}}{\sigma_{\omega_{1}}}grad(\omega))+\gamma_{2}(2\mu_{t}S_{ij }.S_{ij}-\frac{2}{3}\rho k\frac{\partial U_{i}}{\partial x_{j}}\delta_{ij})- \beta_{1}\rho\omega^{2}+A_{1}, \tag{3}\]  
however, \(\rho\) is density, \(\omega\) is specific turbulent dissipation rate and \(k\) is the turbulence kinetic energy. Where \(A_{1}\) in Equation (3) represents the blending function \(F_{1}\) which ensures the smooth transition between the \(k\)-\(\omega\) and \(k\)-\(\epsilon\). This transition can be expressed in Equation (4) as  
\[A_{1} =2(1-F_{1})\rho\sigma_{\omega_{2}}\frac{1}{\omega}\frac{\partial k}{ \partial x_{j}}\frac{\partial\omega}{\partial x_{j}}. \tag{4}\]  
The value of blending function (\(F_{1}\)) becomes one near wall but at far position its value become zero. The building of turbulence in stagnation regions is prevented by the implementation of production limiter (\(P_{k}\)). So, the generation of turbulent viscosity can be expressed as  
\[\mu_{t}=\frac{\alpha_{1}\rho k}{max(\alpha_{1}\omega SF_{2})}, \tag{5}\]  
where \(F_{2}\) is a second blending function with strain rate, \(S\). This model was proved to be superior as it was a combined version of \(k\)-\(\omega\) (for the inner boundary layer) and \(k\)-\(\epsilon\) (for the outer region and outside of boundary layer). The models switch for prediction of far field flow characteristic.