# Partial Oscillation Flow Control on Airfoil at Low Reynolds Numbers
## 3 Calculation Model  
This study focuses on a particular type of high-lift laminar flow airfoil as its subject of investigation. This airfoil boasts a relative thickness of 15.3%, with its maximum thickness occurring at 30.7% of the chord length. Moreover, it features a relative camber of 7.0%. The calculation model depicting partial oscillations on an airfoil is illustrated in Figure 5. The chord length of the airfoil is denoted as \(c\), and the span of the flexible surface measures 0.1\(c\). The distance between the center of the partial flexible surface and the leading edge of the airfoil is 0.1\(c\), 0.2\(c\)...0.7\(c\).  
The displacement of the oscillating surface can be articulated as follows:  
\[\Delta y=A\sin(2\pi ft)\sin(\pi d_{c}/l) \tag{6}\]  
Figure 4: Water tunnel test system and the model of airfoil. (**a**) Water tunnel test system; (**b**) test model of the airfoil.  
Figure 5: Calculation model of the partial oscillation on airfoil.  
Figure 3: Schematic diagram of the water tunnel test system.  
where \(\Delta y\) is the oscillation displacement perpendicular to chord direction; \(l\) is the length of the oscillation surface; \(d_{c}\) is the chord distance between the oscillation point and the starting position of oscillation; \(A\) is the amplitude; \(f\) is the frequency of oscillation; and \(t\) is the time.  
Both the fixed and oscillating boundaries of the airfoil surface employ no-slip wall boundary conditions, wherein the fixed boundary remains stationary throughout the computation. The nodes on the oscillating boundary, adhering to Equation (6), undergo a periodic up-and-down motion within the two-dimensional plane, with grid deformation managed by the spring smoothing method described in Section 2.2. Should the oscillation cycle be denoted as \(T\), Figure 6 presents the varying shapes of the flexible surface on the airfoil across distinct time points within a single cycle.  
During the deformation process of the airfoil surface, variations in grid density may occur, potentially impacting the computation results. To further investigate the impact of grid distribution variations on computational outcomes, considering the deformation at position P1 as an example, under the condition of amplitude \(A=0.001c\) where the local skin deformation reaches its lowest point, the grid becomes notably sparse. To address this, the refinement of the grid in the direction perpendicular to the wall surface was performed at this instance, ensuring that the post-refinement grid distribution matches that prior to deformation. The comparison of the grid distribution near the oscillation surface before and after refinement is depicted in Figure 7.  
Figure 6: Deformation of the airfoil in one cycle. (**a**) \(t=0.2\)\(T\); (**b**) \(t=0.4\)\(T\); (**c**) \(t=0.6\)\(T\); (**d**) \(t=0.8\)\(T\).  
Figure 7: Grid distribution near oscillation surface. (**a**) Initial grid; (**b**) refined grid.  
Calculations were performed under the conditions of \(Re=3\times 10^{5}\), an angle of attack of \(3^{\circ}\) and with the surface held stationary, using both refined and initial unrefined grids. The results are summarized in Table 1.  
The computational results presented in Table 1 reveal that the aerodynamic coefficient errors obtained from the initial, unrefined grid are minimal, all falling within a 1% margin, compared to those derived from the refined grid. Consequently, it is inferred that during the oscillation of the airfoil surface, due to the small amplitude of oscillation, the variation in grid distribution is negligible. Thus, the computational errors introduced by the grid distribution changes can be disregarded.