# Dynamic Stall Characteristics of the Bionic Airfoil with Different Waviness Ratios
## 4 Results and Analysis
### Formation of Streamwise Vorticity  
The main flow mechanism of the streamwise vortex formation can be explained from two different perspectives. From the first perspective, the spanwise vorticity varies along the airfoil's spanwise direction according to [PERSON]'s law due to the cyclic distribution in the spanwise direction caused by tubercles. This means that the function of the tubercles produces ripples in the spanwise vortex of the airfoil. Similarly, each tubercle can be seen as a separated finite airfoil section. Considering that this finite airfoil produces a counter-rotating tip vortex in the direction of the flow, the periodic change of the spanwise circulation leads to the generation of a counter-rotating flow vortex. The second perspective has two main mechanisms for generating flow vortices: the so-called \"skew induction\" and\"stress induction.\" The skew induction mechanism is also called the first type of Prandtl secondary flow. The stress-induced mechanism is also known as the second type of Prandtl secondary flow, which arises from the anisotropy of the turbulent flow [35].  
To illustrate the formation and development mechanism of streamwise vortices from the second perspective, we took the airfoil (R = 1.0) as an example. The tilt-induced vorticity would be generated because the vorticity (such as in the boundary layer) rotates in the flow direction or rotates laterally. This quasi-inviscid phenomenon (the first kind of Prandtl secondary flow) exists in both laminar and turbulent flows (Figures 9 and 10).  
The flow deflection is distinguished by the changes in the curvature of the streamlines for an airfoil with a wavy leading edge. The streamlines in Figure 8 were selected to determine the structure of the significant vortex characteristics in the airflow near the airfoil surface. The sudden change of the streamline direction indicates that the vorticity reorganized near the leading edge  
We further studied the flow structure of an airfoil with a wavy leading edge by examining the shear stress lines on the airfoil surface. Figure 10a,b show that a pair of rotating focal points, namely F1 and F2, appeared near the wavy leading edge of the airfoil with R = 1.0, which was reflected by the 3D streamlines. The flow field structure was consistent. A critical point appeared downstream of this area, specifically the saddle point S, which specifies the boundary of the separation bubble. There were two other focal points, namely F3 and F4, at the rear edge of saddle point S, suggesting that the airflow develops on the upper surface of the airfoil in the form of a tornado-like vortex.  
In Figure 10c, the streamwise vorticity of the airfoil with R = 1.0 at an angle of attack of 18\({}^{\circ}\) is shown on the planes perpendicular to the streamwise direction. It can be seen that the circulation behind the trailing edge of the airfoil had a distribution in the form of a  
Figure 8: Iso-surfaces of vorticity (Q = 200000 s\({}^{-2}\)) with velocity. (**a**) 14\({}^{\circ}\uparrow\) (**b**) 18\({}^{\circ}\uparrow\) (**c**) 20\({}^{\circ}\uparrow\) (**d**) 17\({}^{\circ}\downarrow\) (**e**) 15\({}^{\circ}\downarrow\) (**f**) 13\({}^{\circ}\downarrow\).  
Figure 9: Three-dimensional streamlines of an airfoil (R = 1.0).  
vortex pair. The vorticity at the sections 0.85 \(C\), 1.175 \(C\), and 1.5 \(C\) away from the trailing edge of the airfoil showed a gradual dissipation and attenuation trend.