# Partial Oscillation Flow Control on Airfoil at Low Reynolds Numbers
## 2 Research Methods  
### Numerical Simulation Method  
In the present study, the Reynolds-averaged Navier-Stokes (RANS) equations have been rigorously solved, with the utilization of an unsteady numerical methodology to emulate the dynamics of the flow field. The inviscid terms have been discretized utilizing a second-order upwind scheme, while the viscous terms have been handled with a second-order central difference scheme. In the temporal advancement process, we have adopted the dual-time stepping method [17]. Given that the Reynolds number of the airfoil under scrutiny in this study falls within the range of \(10^{5}\), categorizing it as a low Reynolds number regime, the transition from laminar to turbulent boundary layer must be meticulously accounted for in the CFD [18]. The turbulence modeling approach employed in the numerical simulations of this study is the \(\gamma\)-\(Re_{\theta}\) transitional model, which has been developed by [PERSON] and [PERSON] [19]. This particular model integrates the empirical relationships governing transition phenomena with the Shear Stress Transport (SST) \(k\)-\(\omega\) turbulence model, thereby combining the strengths of both approaches. As a result, it has emerged as a widely adopted transitional modeling solution in contemporary practice. In comparison to the SST turbulence model, the \(\gamma\)-\(Re_{\theta}\) transition model incorporates two supplementary transport equations: one for the local transition Reynolds number \(\overline{Re}_{\theta t}\) and another for the intermittency factor \(\gamma\). These equations extend the model's capacity to capture transitional flow phenomena. \(\overline{Re}_{\theta t}\) serves as a predictive criterion for determining the onset location of transition, whereas the intermittency factor \(\gamma\) is employed to simulate the flow dynamics within the transition zone. The governing transport equation for the intermittency factor \(\gamma\) is expressed as follows [19]:  
\[\frac{\partial(\rho\gamma)}{\partial t}+\frac{\partial\big{(}\rho U_{j}\gamma \big{)}}{\partial x_{j}}=P_{\gamma 1}-E_{\gamma 1}+P_{\gamma 2}-E_{\gamma 2}+\frac{ \partial}{\partial x_{j}}\Bigg{[}\Big{(}\mu+\frac{\mu_{t}}{\sigma_{\gamma}} \Big{)}\frac{\partial\gamma}{\partial x_{j}}\Bigg{]} \tag{1}\]  
The equation of the local transition Reynolds number \(\overline{Re}_{\theta t}\) is as follows:  
\[\frac{\partial\big{(}\rho\overline{Re}_{\theta t}\big{)}}{\partial t}+\frac{ \partial\big{(}\rho u_{j}\overline{Re}_{\theta t}\big{)}}{\partial x_{j}}=P_ {\theta t}+\frac{\partial}{\partial x_{j}}\Bigg{[}\sigma_{\theta t}(\mu+\mu_{ t})\frac{\partial\overline{Re}_{\theta t}}{\partial x_{j}}\Bigg{]} \tag{2}\]  
In Equations (1) and (2), \(P_{\gamma 1}\) and \(E_{\gamma 1}\) are source terms of transition; \(P_{\gamma 2}\) and \(E_{\gamma 2}\) are source terms of the relaminarization. [PERSON] et al. [20] emphasized that to ensure the precise simulation of both laminar and transitional boundary layers, it is essential to maintain the y+ value closely proximate to 1 during computations under conditions of low Reynolds numbers. The H-type 2D structured grid is used in this study. The chord length of the airfoil is denoted by \(c\). The computational domain extends its boundaries to be 20 times the chord length (c) upstream of the airfoil's leading edge and 25 times the chord length downstream of the trailing edge. Moreover, the top and bottom boundaries of this domain are set at a uniform distance of 15 times the chord length from the airfoil's chord line. The grid for the numerical simulation is shown in Figure 1.  
To validate the accuracy of the aerodynamic properties of an airfoil at low Reynolds numbers using the grid configuration and unsteady CFD approach employed in this research, a numerical simulation has been performed on the NACA 0009 airfoil and its outcomes are juxtaposed against the results from wind tunnel experiments.