# Dynamic Stall Mechanisms of Pitching Airfoil: IDDES Study Across Different Mach Numbers
## 4 Aerodynamics at a Mach Number of 0.5
### Effect of Reduced Frequency at Ma = 0.5  
This section discusses the effect of reduced frequency on airfoil dynamic stall characteristics at \(Ma=0.5\). The airfoil pitching frequency maintains consistent with the \(Ma=0.283\) case, corresponding to reduced frequencies of \(k=0.057\), \(0.085\), and \(0.113\). Figure 25 presents  
Figure 24: Cycle-averaged and time-resolved aerodynamic damping for the pitching airfoil at \(Ma=0.5\).  
the airfoil aerodynamic coefficients across varying reduced frequencies. Analysis shows that increasing the reduced frequency delays lift stall, drag divergence, and moment stall, while enhancing the hysteresis of aerodynamic forces. Notably, compared to the \(Ma=0.283\) case, the influence of reduced frequency is less pronounced at the current Mach number. After stall onset, the airfoil aerodynamic loads under all reduced frequencies exhibit a gradual variation trend.  
Figure 26 depicts pressure coefficients at the airfoil leading edge for \(k=0.085\) across varying AoAs, while Figure 26b,c illustrate the same at fixed AoAs with different reduced frequencies. Analysis of Figure 26a reveals a leading-edge shock wave that strengthens and moves downstream with increasing AoA, until shock-induced boundary layer separation triggers upstream shock migration.  
Figure 25: Calculated forces of the pitching airfoil for different reduced frequencies at \(Ma=0.5\): (**a**) _CI;_ (**b**) _Cd;_ (**c**) _Cm._  
Observations from Figure 26b,c show that at identical AoAs, increasing \(k\) shifts the shock wave closer to the leading edge with reduced intensity. Shock-induced separation first occurs at \(k=0.057\) when AoA reaches \(11.56^{\circ}\). These results indicate that higher reduced frequency delays aerodynamic response, thereby postponing shock-induced separation and stall onset. However, this delaying effect is less pronounced compared to low-Mach cases, highlighting compressibility as the dominant factor at higher Mach numbers.  
Figure 27 depicts the cycle-averaged and time-resolved aerodynamic damping of airfoil pitching motion across different reduced frequencies. Analysis shows that increasing the reduced frequency enhances the cycle-averaged damping, indicating a lower tendency for stall flutter. At high Mach numbers, the moment coefficient vs. AoA curves do not exhibit crossover (as shown in Figure 25c), and increasing the reduced frequency further amplifies the hysteresis of aerodynamic forces, thereby enhancing the stability of airfoil oscillatory motion. Examination of the time-resolved damping curves reveals that, due to the relatively weak delaying effect of increased reduced frequency on dynamic stall at high Mach numbers, the discrepancies in minimum transient damping coefficients across reduced frequencies remain negligible.  
Figure 26: Pressure coefficients near the airfoil leading-edge for different reduced frequencies at \(Ma=0.5\): (**a**) \(k=0.085\); (**b**) \(\alpha=9.10^{\circ}\uparrow\); (**c**) \(\alpha=10.15^{\circ}\uparrow\); (**d**) \(\alpha=11.56^{\circ}\uparrow\).