## 3 Results
### Hysteresis Loops at Different Reduced Frequencies
In the moderate reduced frequency case (Case 2), the linear segment ends up at a larger angle of attack and the segment ab disappears, due to the greater pitching speed. Under  
Figure 4: Lift coefficient hysteresis loops of two airfoils in four cases in a clean air flow: (**a**) NACA0012 airfoil; (**b**) S809 airfoil. Figure 5: Pressure coefficients of four points for case 1.  
the effect of light stall, the lift coefficient in segment ac increases at a smaller speed. Then, in the segment cd, the leading-edge vortex sheds, and the lift coefficient drops sharply, which is defined as deep stall (or full stall). In the following segment, the lift coefficient slowly decreases to the initial value of the up stroke. In the high reduced frequency case (Case 3), the range of linear segment further increases and the segment ac becomes smaller. There are only two segments during the whole up stroke, and the deep stall also disappears, due to the high pitching speed. That is because the virtual camber effect, which was discussed by [PERSON] et al. (2016), starts to dominate the flow field. In addition, the fluctuations in case 2 and 3 are caused by the small number of the inner iteration per step, which is described in Figure 2a. In the following discussion, the fluctuation will become larger, due to the hysteresis effect, especially in the loop of the relative increment between clean air and air-particle cases. Therefore, the mean value is more accurate. Next, the cases in the air-particle flow are added to the comparison. For the three cases above, increment loops of the lift coefficient of two airfoils, caused by particles, are presented. Figures 6 and 7 display the loops for Case 1 at different particle mass concentrations and the relative increments between concentrations of 0% and 1%, respectively. For NACA0012 airfoil, particles have little effect on the shape of the lift coefficient loop but a decrease for its inclined angle. The flat loop of the relative increment indicates the small loss in aerodynamic performance throughout and its little difference between up and down strokes. The peak value of \(-2.016\%\) is obtained at the maximum angle of attack. For S809 airfoil, the relative increment loop is also divided into several segments, as well as the discussion in Figure 4b. During the up stroke, particles make the linear segment end up at a larger angle of attack, which increases with the particle concentration. The segment ab is obtained at larger angles of attack; its relative increment decreases first and then increases. In these two segments, the relative increments are both small enough to be out of consideration. Due to the large difference between increments at Point b and Point b\({}_{1}\) by particles, the change of segment bb\({}_{1}\) strongly depends on the particle concentration. It means that the angle of attack, at which the maximum lift coefficient is obtained, has two possibilities. As shown in Figure 6b, the lift coefficient of segment bb\({}_{1}\) increases with the angle of attack at the particle concentration of 0%, and a decrease is presented at the concentration of 2%. The concentration of 1% is the critical value. Then, the maximum lift coefficients at concentrations of 0% and 2% are 1.127 and 1.087, which are obtained at angles of attack of 10.5\({}^{\circ}\) (Point b\({}_{1}\)) and 9.25\({}^{\circ}\) (Point b), respectively. In this segment, the relative increment rapidly increases. In addition, particles make the Point b\({}_{1}\) move to a smaller angle of attack and the larger the concentration, the larger the movement distance and smaller the range of segment bb\({}_{1}\). In the following three segments, the lift coefficient in the air-particle flow is approximately parallel to that in the clean air flow. The relative increment continues to increase, and the maximum value of \(-6.55\%\) is obtained at the angle of attack of 11.3\({}^{\circ}\) in segment b\({}_{1}\)b\({}_{2}\).