# Dynamic Stall Mechanisms of Pitching Airfoil: IDDES Study Across Different Mach Numbers
## 2 Numerical Methods
### Validation for CFD Method  
To validate the CFD method for simulating oscillating airfoils under deep stall conditions, experimental data of a pitching NACA 0012 airfoil [3] are utilized. The simulation parameters are set as follows: Mach number (\(Ma\)) = 0.283, angle of attack (\(\alpha\)) = 14.91\({}^{\circ}\) + 9.87\({}^{\circ}\) sin(\(\omega t\)), reduced frequency (\(k\)) = 0.151, and Reynolds number (\(Re\)) = 3.45 \(\times\) 10\({}^{6}\). Different time steps and grid resolutions are used to assess temporal and grid independence. Detailed grid information is provided in Table 1. In the calculation, the URANS method is first employed until convergence is achieved. The IDDES method is then used to compute for seven full cycles, with results from the last five cycles recorded and averaged.  
Figure 1: Sliding mesh surrounding the computational model: (**a**) computational domain; (**b**) sectional mesh and close-up view.  
Figure 2: Schematic diagram of boundary conditions.  
Figure 3 compares numerical results computed using three time steps (\(\Delta\)t denotes the non-dimensional time step) on the medium grid. Figure 4 compares results from three grid resolutions against experimental data. Notably, the CFD method overpredicts the peak values of drag and nose-down moment--a discrepancy also reported in prior studies [23; 36]--likely stemming from inherent limitations of CFD methods. Conversely, the attached flow stage and stall initiation are well simulated, along with the peak lift magnitude. Across different time steps and grid densities, results exhibit good agreement during the upstroke, whereas some discrepancies arise in the downstroke. This divergence is primarily attributed to turbulent fluctuations in the massively separated flow after stall. Table 2 presents the computational results of critical dynamic stall parameters for three grids, all of which exhibit good consistency--further validating grid convergence. Overall, numerical predictions agree favorably with experimental measurements, validating the efficacy of the CFD method for analyzing dynamic stall phenomena of pitching airfoils. Subsequent calculations in the study adopt the medium grid and a time step of \(\Delta\)t = 0.005.  
\begin{table}
\begin{tabular}{c c c} \hline
**Grid Resolution** & **Grid Points** & **Cell Size in the LES Region** \\ \hline Coarse & \(248\times 95\times 51\) & \(0.02c\) \\ Medium & \(306\times 112\times 67\) & \(0.015c\) \\ Fine & \(421\times 142\times 101\) & \(0.01c\) \\ \hline \end{tabular}
\end{table}
Table 1: Detailed information of grids.  
Figure 4: Comparisons of numerical results with different grids and experimental data for NACA 0012 airfoil at \(Ma\) = 0.283: (**a**) _Cl;_ (**b**) _Cd;_ (**c**) _Cm._  
Figure 3: Comparisons of numerical results with different time steps and experimental data: (**a**) _Cl;_ (**b**) _Cd;_ (**c**) _Cm._  
To validate the present CFD method at a high Mach number of 0.5, a modified VR-12 airfoil is employed. The airfoil features a 5%c trailing edge tab with a zero tab angle, and its coordinates are available in Reference [37]. The computation parameters are set as \(Ma=0.5\), \(\alpha=8^{\circ}+8^{\circ}\) sin(\(\omega\)t), \(k=0.05\), and \(Re=4.3\times 10^{6}\).  
Figure 5 shows a comparison among the experimental data [13], numerical results by [PERSON] et al. using the RANS method [23], and the current calculated results. The present CFD method accurately captures the angles of lift stall, drag divergence, and moment stall. The peak value of the lift coefficient is in good agreement with the experimental data. During the initial pitching down stage, both the RANS method used by [PERSON] et al. and the present IDDES method exhibit limitations due to fully turbulent separation. Nevertheless, the present IDDES method successfully captures the gradual lift and moment stall behavior and outperforms the RANS method, which predicts an abrupt lift and moment stall.