# Dynamic Stall Characteristics of the Bionic Airfoil with Different Waviness Ratios
## 3 The Numerical Calculation Verification
### Numerical Simulation Validation  
The results of the validation case are shown in Figure 5. The NACA0012 and NACA0015 airfoils were used for the numerical calculation verification and the results were compared with the corresponding experimental and numerical results. The primary flow conditions, Mach number, and other parameters of the calculation examples in this paper are consistent with the related wind tunnel tests.  
In the upstroke stage (Figure 5a), the lift coefficient was consistent with the experimental [29] and numerical results [30; 31], but in the downward (attack angle decreases) process, there was a deviation between the calculated results and the experimental values, which is related to the strong non-equilibrium of the separated turbulence. As the parameters in the turbulence model are often obtained by simple flow, however, in wind tunnel experiments, the measurement of the surface pressure at a high attenuation frequency will be very challenging due to the complex flow-field structure under deep stall ([PERSON] and [PERSON] [32]).  
Figure 5b depicts the variation of the lift coefficient of the NACA0015 airfoil with the angle of attack during the dynamic stall. Additionally, it is compared with the experimental results of literature [33] and the 2D numerical simulation results of literature [34] using RSM and SST k-w models. The 3D numerical simulation calculations were used in this article. As shown in Figure 5b, the general trend of the lift curve calculated was consistent with the experimentally measured trend during most of the pitch cycle. The difference between the numerical simulation and the experiment of the lift coefficient is due to many  
Figure 4: Computational domain and grids. (**a**) grid independence verification; (**b**) time step independence verification.  
factors. For example, the URANS method cannot faithfully resolve complex flows. The RSM and SST k-w methods produce almost the same results on the upstroke movement of the airfoil. The SST k-w model results in a smoother lift curve and more minor lift changes during the downstroke. Considering the complexity of dynamic stall, we judge that the results obtained using these two models are consistent.  
The dynamic stall angle of the 3D numerical calculation results was slightly different from the experimental results, but the lift coefficient during the airfoil ascent phase was closer to the experimental value. Moreover, the results of the 3D simulation were qualitatively similar to the experimental results. The main flow structure was captured. Therefore, our simulation accuracy was considered acceptable because dynamic stalls and their controls are notoriously difficult to predict. Our work focuses on helping readers qualify the physics and control mechanisms of dynamic stalls rather than accurately solving dynamic stalls themselves.  
To further verify the validity of the calculation results, the transient vorticity field from our simulation was compared with the phase-averaged flow field from the experiment [33]. The flow separation region and the vorticity of the airfoil suction surface can be seen in the flow visualizations in Figure 6, which were obtained from the numerical simulation and experimental observations. The close similarity between the experimental and simulated flow patterns can also be seen.  
However, there were some differences between the numerical results and experimental results. The numerical method used in this paper can simulate dynamic stalls and capture relevant flow-field structures as well as flow details. The SST k-w model was used in all the simulations below.  
Figure 5: Comparison of current simulation with other results. (**a**) NACA0012; (**b**) NACA0015.  
Figure 6: Flow patterns around the baseline NACA 0015 airfoil.