## 3 Results
### Hysteresis Loops at Different Mean Angles of Attack  
In the previous section, the comparison is presented at a mean angle of attack of 8\({}^{\circ}\). Another two mean angles of attack are adopted to further reveal their difference. The default particle mass concentration is also 1% and the more details are listed in Table 3. Then, the computed results of these three cases are discussed, respectively. In Figure 12a, for Case 4, the loop of NACA0012 airfoil is hardly changed by particles, due to its insensitivity during both up and down strokes. However, in Figure 12b, the loops of S809 airfoil during the beginning of the down stroke have apparently smaller lift coefficients than those during the end of the up stroke, which is caused by the light stall. Although particles have an obvious change in this region, it is still too small to be paid  
Figure 11: Relative increment loops of two airfoils after loading particles with a concentration of 1%. Figure 10: Lift coefficient hysteresis loops of two airfoils for Case 3: (**a**) NACA0012 airfoil; (**b**) S809 airfoil. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case Number** & **Stokes Number (St)** & **Re \(\times\) 10\({}^{6}\)** & **\(\mathbf{\omega_{m}}\) (\({}^{\circ}\))** & **\(\mathbf{\kappa_{A}}\) (\({}^{\circ}\))** & **\(\mathbf{f}\) (Hz)** & **\(\mathbf{\omega_{red}}\)** \\ \hline Case 4 & & & 4 & & 1.85 & 0.0770 \\ \hline Case 2 & 0.053 & 1.01 & 8 & 5.5 & 1.85 & 0.0770 \\ \hline Case 5 & & & 11 & & 1.85 & 0.0770 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The pitching parameters of three cases. attention to. Therefore, the effect of particles on the lift coefficient is slight and negligible at the default particle mass concentration in the whole loop. For two airfoils, the angle of attack, at which the maximum lift coefficient is obtained, is unchanged by particles. In order to ascertain this phenomenon, two larger particle mass concentrations of 3% and 5% are adopted, and the results are listed in Table 4. It is found that particles do not change the angle at which the maximum lift coefficient is obtained, even though it is under the effect of light stall at moderate and high reduced frequencies. The comparison for Case 2 was presented in Figures 8 and 9. In Table 5, for NACA0012 airfoil, all angles of attack, at which the maximum lift coefficient is obtained, were 13.5\({}^{\circ}\), while for S809 airfoil, the value decreases with the mass concentration. This may illustrate that the angle of attack, at which the maximum lift coefficient angle is obtained, can be changed by particles in the region, which suffers the effect of deep stall. Finally, as shown in Figure 13 for Case 5, two airfoils both suffer the effect of deep stall and have different degradations of the lift coefficient near the maximum angle of attack. In \begin{table}
\begin{tabular}{c c c} \hline \multirow{2}{*}{**Particle Mass Concentration**} & \multicolumn{2}{c}{**Maximum Lift Coefficient Angle (\({}^{\circ}\))**} \\ \cline{2-3}  & **NACA0012** & **S809** \\ \hline
0\% & & \\ \hline
1\% & 9.5 & 9.5 \\ \hline
3\% & & \\ \hline
5\% & & \\ \hline \end{tabular}
\end{table} 
Table 4: Angles of attack, at which the maximum lift coefficient is obtained, for Case 4. Figure 12: Hysteresis loops of two airfoils for Case 4: (**a**) NACA0012 airfoil; (**b**) S809 airfoil. \begin{table}
\begin{tabular}{c c c} \hline \multirow{2}{*}{**Particle Mass Concentration**} & \multicolumn{2}{c}{**Maximum Lift Coefficient Angle (\({}^{\circ}\))**} \\ \cline{2-3}  & **NACA0012** & **S809** \\ \hline
0\% & & 12.8 \\ \hline
1\% & 13.5 & 12.6 \\ \hline
3\% & & 12.3 \\ \hline
5\% & & 12 \\ \hline \end{tabular}
\end{table} 
Table 5: Angles of attack, at which the maximum lift coefficient is obtained, for Case 2. Figure 14, during the up stroke, the relative increment of NACA0012 airfoil is smaller than \(-1.25\%\) at angles of attack less than \(15^{\circ}\) and increases to \(-3.01\%\) at \(16.5^{\circ}\). With the angle of attack decreasing, it increases again at a greater speed rate.