## 3 Delay of Buffet Onset  
Under the same free-stream conditions of the validation case plotted in Figure 2, steady numerical simulations were conducted for the NACA0012 airfoil at different incidence angles. The calculated lift coefficients at different incidence angles are plotted, as shown in Figure 4. By referring to the work conducted by [PERSON] and [PERSON] [57] and [PERSON] [26,27], the maximum curvature method proposed by [PERSON] and [PERSON] [26,27] is briefly introduced as follows. By comparing the fitting precisions of polynomial aerodynamic curves, including lift and moment coefficient curves, it was found that a quartic polynomial can give the best fitting precision. It is written as follows:  
\[\mathrm{y}=\mathrm{ax}^{4}+b\mathrm{x}^{3}+c\mathrm{x}^{2}+dx+e \tag{1}\]  
where a, \(b\), \(c\), \(d\), and \(e\) are fitting constants obtained by least-squares fitting, \(x\) denotes the incidence angle, and \(y\) denotes either the lift or moment coefficient. In the present study, the lift coefficient was used. The calculated curvature of \(\mathrm{y}\) is as follows:  
\[\mathrm{K(x)}=\frac{\left|12a\mathrm{x}^{2}+6 bx+2c\right|}{\left(1+\left(4a \mathrm{x}^{3}+3b\mathrm{x}^{2}+3c\mathrm{x}+d\right)^{2}\right)^{3/2}} \tag{2}\]  
Figure 4: Lift coefficients vs. incidence angle. Figure 3: Comparison and validation of unsteady numerical results. when the minimum curvature radius is achieved, the first-order derivative of K(x) equals zero:  
\[\text{K}^{\prime(\text{x})}=0 \tag{3}\]  
By solving Equation (3), the buffet onset incidence angle is obtained. The accuracy of this method is valid for NACA0012 airfoil buffet onset predictions, compared to wind tunnel tests, as validated in reference [26; 27]. By using the above method, the curve plotted in Figure 4 was fitted by a quartic polynomial, and the maximum curvature was then obtained to be 0.8, corresponding to a transonic buffet onset margin of 3.9\({}^{\circ}\). By inspecting the surface pressure distribution curve plotted in Figure 2, it can be found that the shockwave on the upper wing's surface resides in the range of 41%\(\sim\)51% chord length; thus, 41% chord length, 46% chord length, and 51% chord length are denoted as \"41% x/c\", \"46% x/c\", and \"51% x/c\", respectively. The tangential blowing position on the upper surface is first selected to be the \"46% x/c\" case. The geometric shape of the tangential blowing nozzle and the structured grid is plotted in Figure 5. This tangential blowing nozzle is a Venturi nozzle with its contraction ratio equaling 5.0. Its mass flow flux is governed and can be adjusted by setting the total pressure value at the inlet boundary of the Venturi nozzle. With the blowing total pressure equaling the freestream total pressure, the lift coefficients at different incidence angles were compared with those of a clean airfoil, as plotted in Figure 6. Based on the maximum curvature method, the transonic buffet onset margin with \"46% x/c\" blowing was found to be 4.5\({}^{\circ}\), and its corresponding curvature was 0.04. It can be seen from Figure 6 that, at Mach 0.751 when the incidence angle is below 2.5\({}^{\circ}\), the airfoil lift with tangential blowing was a little lower. When the incidence angle is greater than 3.0\({}^{\circ}\), and especially when the lift slope of the airfoil without blowing is lower for incidence angles beyond 4.5\({}^{\circ}\), the airfoil lift with tangential blowing was higher. Thus, tangential blowing can not only postpone the transonic buffet onset margin, but the airfoil lift can also be increased. Figure 5: Grid of a tangential blowing slot. To investigate the control effort, pressure coefficient distributions, pressure contours, and streamlines will be presented. At incidence angles of 4\({}^{\circ}\) and 5\({}^{\circ}\), the surface pressure coefficients of the airfoil with and without tangential slot blowing were plotted together, as shown in Figure 7. It is seen that the shock location is shifted downstream by tangential slot blowing.