# Numerical Simulation of Flapping Airfoil Aerodynamic Characteristics
## 3 Results and Analysis
### Effects of Motion Parameters on the Aerodynamic Characteristics of Flapping Airfoils
#### 3.1.1 Research on Flapping Airfoil Lift Mechanism  
The flapping airfoil selected for the research of lift mechanism is elliptical, with a maximum thickness of 0.12 h. The parameter values in the flapping airfoil motion function are \(\alpha_{0}=45^{\circ}\), \(h_{a}=\) h, \(\alpha_{a}=90^{\circ}\), \(\phi=90^{\circ}\), \(f=100\) Hz, \(Re\approx 1075\), \(St\approx 0.16\), \(J=0\). The left flapping stroke and the right flapping stroke are included in a single flapping motion. The left flapping stroke and the right flapping stroke are symmetrical movements, and the flapping stroke can be regarded as the minimal unit of motion for flapping airfoil periodic motion. The lift coefficient curve of flapping airfoils is demonstrated in Figure 6. In the absence of inflow, a flapping stroke is divided into 5 stages: \(\varphi\): Rotating loop flow lift enhancement [28; 29; 30; 31]. \(\kappa\): Trailing vortex capture lift enhancement. \(\lambda\): Leading edge vortex shedding due to the rotation mechanism. \(\mu\): Leading edge vortex stall delay mechanism. \(\
u\): Translational motion-induced leading edge vortex shedding. Among them, \(\varphi\), \(\kappa\), \(\lambda\), \(\mu\), and \(\
u\) correspond to stages 1, 2, 3, 4, and 8 in Figure 6, respectively. \(\varphi\): Rotating loop flow lift enhancement. Flapping airfoils begin to rotate rapidly as they are about to move to a certain end. As time progresses, for the upper and lower airfoil surfaces of the airfoil flip, the Magnus effect is created, the lift is obtained by flapping the airfoil, and the lift coefficient is continuously improved during this process. \(\kappa\): Trailing vortex capture lift enhancement. During the flapping airfoil rotation procedure, new leading edge vortices were generated, after the completion of the previous stage \(\varphi\), the flapping airfoil continued to flutter towards the other end. During this period,  
Figure 5: Mesh sensitivity check (45\({}^{\circ}\) diagonal trajectory). Figure 6: Flapping airfoil lift coefficient curve. the trailing vortex that had just fallen off was captured, and the kinetic energy of the trailing vortex was absorbed by the flapping airfoil. The leading edge vortex and the low-pressure area on the upper airfoil surface were enhanced, and the lift coefficient of the flapping airfoil was rapidly upgraded during this stage. \(\lambda\): Leading edge vortex shedding due to the rotation mechanism. The flapping airfoil continues to flutter towards the other end, the rotation speed slows down, the translation speed increases, and at the same time the leading edge vortex produced by the rotation is detached by the upper airfoil surface, and the low pressure area of the upper airfoil surface is weakened, and the lift begins to decline until a new leading edge vortex is created. \(\mu\): Leading edge vortex stall delay mechanism. After the \(\lambda\) phase the flapping airfoil continues to flutter towards the other end, a new leading edge vortex is developed, a process dominated by translational motion, and the leading edge vortex is provided with kinetic energy. The upper airfoil surface is connected by the newly generated leading edge vortex, and a low pressure area is formed, so that the flapping airfoil lift is raised. \(\
u\): Translational motion-induced leading edge vortex shedding. When the flapping airfoil quickly moves to the other end, the translation speed is decreased, the rotation speed is raised, the leading and trailing edge vortices of the flapping airfoil are detached, the low-pressure area on the upper airfoil surface is weakened, and the lift continues to decrease with the strength of the leading edge vortex until the next lift mechanism begins.