# Correction of Aero-Optical Effect with Blow-Suction Control for Hypersonic Vehicles
## 3 Methods and Validation
### Numerical Simulation Method  
The RANS method exhibits high computational efficiency but low-fidelity for turbulence simulation in practical engineering, while the LES method offers high-fidelity at the expense of computational efficiency. In order to accurately capture the flow structure of the outflow field of the optical window in the proposed seeker model with appropriate computational efficiency, the high-fidelity Detached Eddy Simulation (DES) method is used to numerically simulate the flow field around the optical window.  
Figure 6: Schematic diagram of aero-optical effects.  
Figure 7: The ray passing through the nonuniform flow field forms the distorted wavefront, \(<\)\(OPL\)\(>\), and \(OPD\).  
The SST-DES method is a DES hybrid method proposed by [PERSON] [25] in 2001 on the basis of the two-equation SST model, and its governing equations are as follows:  
\[\frac{\partial k}{\partial t}+U_{j}\frac{\partial k}{\partial x_{j}}=\frac{1}{ \rho}P_{k}-\beta^{*}k\omega F_{\text{DES}}+\frac{1}{\rho}\cdot\frac{\partial}{ \partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{\text{t}}}{\sigma_{k}}\bigg{)} \frac{\partial k}{\partial x_{j}}\Bigg{]} \tag{6}\]  
\[\frac{\partial\omega}{\partial t}+U_{j}\frac{\partial\omega}{\partial x_{j}}= \frac{1}{\rho}P_{\omega}-\beta\omega^{2}+\frac{1}{\rho}\cdot\frac{\partial}{ \partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{\text{t}}}{\sigma}\bigg{)}\frac{ \partial\omega}{\partial x_{j}}\Bigg{]}+2(1-F_{1})\frac{1}{\sigma_{\omega_{2} }}\cdot\frac{1}{\omega}\cdot\frac{\partial k}{\partial x_{j}}\cdot\frac{ \partial\omega}{\partial x_{j}} \tag{7}\]  
\[S_{D,k}=-\beta^{*}k\omega F_{\text{DES}} \tag{8}\]  
where \(k\) and \(\omega\) are the turbulent kinetic energy and specific dissipation rate; \(\mu_{t}\) is the dynamic turbulent eddy viscosity coefficient. \(P_{k}\) and \(P_{\omega}\) are the generated terms of Equations (4) and (5), respectively. \(\beta^{*}\), \(\sigma_{k}\), \(\sigma_{\omega_{2}}\) are the SST model's constants. \(\rho\) is the density; \(F_{\text{DES}}\) is the switching function of the SST-DES method, and its expression is as follows:  
\[F_{\text{DES}}=\max\left[\frac{L_{\text{RANS}}}{C_{\text{DES}}\Delta},1\right] \tag{9}\]  
The length scale of the RANS model, LRANS, is defined as \(L_{\text{RANS}}=/(\beta^{*}\omega)\). The value of \(F_{\text{DES}}\) is 1 in proximity to the wall and exceeds 1 further away from it. For additional details regarding other parameters and expressions in Equations (4)-(6), please refer to [26].  
The SST-DDES method was proposed by [PERSON] et al. [27]. The switching function of the SST-DES method was modified to reduce the dependence of the method on the grid. The switch function expression after its change is as follows:  
\[F_{\text{DDES}}=\max\left[(1-F_{\text{SST}})\frac{L_{\text{RANS}}}{C_{\text{DES }}\Delta},1\right] \tag{10}\]  
In the formula, \(F_{\text{SST}}\) is equal to \(F_{1}\) or \(F_{2}\), and \(F_{1}\) and \(F_{2}\) are the boundary layer identification functions inherent to the SST model, whose value range is similar to \(r_{\text{d}}\), and are the delay factors of the SST-DDES method.  
\[r_{\text{d}}=\frac{v_{t}+v}{\sqrt{U_{ij}U_{ij}\kappa^{2}{d_{w}}^{2}}} \tag{11}\]  
where \(v_{t}\) is the moving eddy viscosity coefficient; \(v\) is the viscosity coefficient of moving molecules; \(\kappa\) is the Karman constant, with a value of 0.41. \(r_{\text{d}}\) is approximately equal to 1 in the logarithmic layer, and then gradually decreases to 0 in the viscous top layer. This simulation employs the Roe scheme for spatial dispersion and the LU-SGS scheme for time advancement. The initial flow field is obtained through steady calculations, followed by unsteady calculations consisting of 20,000 steps, with a physical time step of \(10^{-7}\) s.