## 3 Results  
### Hysteresis Loops at Different Reduced Frequencies  
In this section, hysteresis loops of the lift coefficient and their difference at three reduced frequencies, listed in Table 2, in clean air and air-particle flows are discussed. All cases are computed on a 12-core workstation with a CPU frequency of 3.4 GHz. Figure 4 shows the hysteresis loops of two airfoils, in four cases, in a clean air flow. For NACA0012 airfoil, the lift coefficient has a linear change with the angle of attack in the steady case, and the values during the up stroke are approximate to those during the down stroke in every pitching case. Three loops also have little difference, except for the inclined angle. However, for S809 airfoil, according to the analysis of the steady case by [PERSON] et al. [34], the lift coefficient stops linearly increasing and decreases slightly near the angle of attack of \(8^{\circ}\), due to the forward movement of the boundary-layer separation point from the trailing edge, which is called the light stall. During the following stroke, the lift coefficient increases again and drops into deep stall at an angle greater than \(16^{\circ}\). In three pitching cases, the reduced frequency has large effect on the hysteresis loop. The lift \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{Re\times 10^{6}}\) & \(\mathbf{\omega_{m}}\) (\({}^{\circ}\)) & \(\mathbf{\omega_{A}}\) (\({}^{\circ}\)) & \(\boldsymbol{f}\) (Hz) & \(\boldsymbol{\omega_{\text{red}}}\) \\ \hline
1.01 & 8 & 5.5 & 1.85 & 0.077 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Correlation pitching parameters of the validation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \begin{tabular}{c} **Case** \\ **Number** \\ \end{tabular}  & \begin{tabular}{c} **Stokes** \\ **Number (St)** \\ \end{tabular}  & \(\mathbf{Re\times 10^{6}}\) & \(\mathbf{\omega_{m}}\) (\({}^{\circ}\)) & \(\mathbf{\omega_{A}}\) (\({}^{\circ}\)) & \(\boldsymbol{f}\) (Hz) & \(\boldsymbol{\omega_{\text{red}}}\) & \ \begin{tabular}{c} **Mean Computation Time of a** \\ **Pitching Period in the** \\ **Air–Particle Flow (hour)** \\ \end{tabular}  \\ \hline \begin{tabular}{c} Case 1 \\ \end{tabular}  & \multirow{4}{*}{0.053} & \multirow{4}{*}{1.01} & \multirow{4}{*}{8} & 0.37 & 0.0154 & \multirow{4}{*}{60} \\ \cline{1-1} \cline{5-5  
coefficients during the up stroke are greatly different from those during the down stroke, which will be discussed as follows, and the mechanisms are also revealed. In the low reduced frequency case (Case 1), the loop during the up stroke has similar trend with the steady results. As shown in the Figure 4, the loops are divided into several segments for comparison purposes. Due to the hysteresis effect, a larger range of the linear segment is obtained, and the segment ab has a smaller range than that of the steady case. In segment bb\({}_{1}\), the lift coefficient increases again because of the higher suction peak, which is shown in Figure 5. With the angle of attack increasing, the leading-edge vortex and suction peak barely change, the boundary-layer separation point goes forward rapidly. At the same time, the continuously decreasing pitching speed also cause the lift coefficient to approach the value of the steady case. They both lead to the decrease of the lift coefficient in segment b\({}_{1}\)b\({}_{2}\). Then, a similar change to the steady case is presented, until the angle of attack reaches to its maximum value. In the segment de, the lift coefficient keeps close to the steady result because of the slow pitching speed. Notably, the light stall is still effective in this segment. With the further decrease of the angle of attack in segment ef, the airfoil no longer suffers the effect of light stall, the boundary-layer reattaches, and the lift coefficient increases. Finally, the linear change reappears at angles of attack smaller than 8.1\({}^{\circ}\), and the lift coefficients are larger than those during the up stroke.