# Numerical Simulation of Flapping Airfoil Aerodynamic Characteristics
## 2 Control Equations, Numerical Methods, and Model Validation
### Control Equation  
Flapping airfoil motion can be simplified as a combination of translational and rotational motion, and the flapping airfoil motion function is as Equation (1):  
\[\left\{\begin{array}{l}h(t)=h_{a}\sin(2\pi ft)\\ \alpha(t)=\alpha_{0}-\alpha_{a}\sin(2\pi ft+\phi)\end{array}\right. \tag{1}\]  
Among them, \(\alpha\) is the rotation angle of the airfoil around the center of gravity, \(h\) is the horizontal displacement of the airfoil center of gravity, \(\phi\) is the rotation angle phase, \(\alpha_{0}\) is the initial angle, \(h_{\alpha}\) is the displacement amplitude, \(\alpha_{\alpha}\) is the rotation angle amplitude, and \(f\) is the flapping frequency.  
The schematic diagram of flapping airfoil movement is demonstrated in Figure 1.  
Because the Reynolds number of the flutter flow field is small, a new shear layer is generated at the airfoil surface when the airfoil is fluttered, and the flow field can be simplified to laminar flow, and thus the control equations of the flutter flow field can be streamlined to the incompressible Navier-Stokes equations, as Equations (2) and (3):  
\[\
abla\cdot u=0 \tag{2}\]  
\[\frac{\partial u}{\partial t}+u\cdot\
abla u=-\frac{1}{\rho}\
abla p+\
u\
abla^ {2}u \tag{3}\]  
where \(u\) is the fluid velocity vector, \(p\) is the flow field pressure, and \(\
u\) is the kinematic viscosity coefficient.  
The lift coefficient \(C_{L}\) and the periodic average lift coefficient \(\overline{C}_{L(T)}\) are defined as Equations (4) and (5):  
\[C_{L}=\frac{F_{lift}}{\frac{1}{2}\rho u^{2}S} \tag{4}\]  
\[\overline{C}_{L(T)}=\frac{\int_{nT}^{(n+m)T}C_{L}dt}{mT} \tag{5}\]  
Among them, \(\rho\) is the fluid density, \(S\) is the characteristic area, and \(F_{lift}\) is the lifting force on the object. In this article, the lifting force is positive along the \(Y\)-axis, and \(u\) is the characteristic velocity. When the inflow velocity is 0, the maximum flapping velocity is taken as the characteristic velocity. \(m\), \(n\) are positive integers.  
The resistance coefficient \(C_{D}\) and the periodic average resistance coefficient \(\overline{C}_{D(T)}\) are defined as Equations (6) and (7):  
\[C_{D}=\frac{F_{drag}}{\frac{1}{2}\rho u^{2}S} \tag{6}\]  
\[\overline{C}_{D(T)}=\frac{\int_{nT}^{(n+m)T}C_{D}dt}{mT} \tag{7}\]  
where \(F_{drag}\) is the resistance, the resistance is positive along the \(x\)-axis.  
Reynolds number \(Re\) is used to characterize relative size of fluid inertia force and viscous force in flow field. which is defined as follows:  
\[Re=\frac{U_{ref}L}{\
u} \tag{8}\]  
where \(L\) is the characteristic length, \(U_{ref}\) is characteristic velocity and \(v\) is kinematic viscosity of the fluid.  
The power coefficient consumed by the flapping motion is as follows:  
\[C_{p}(t)=\mathrm{F}(t)\dot{h}(t)/(\frac{1}{2}\rho S\mathrm{u}^{3})+M(t)\dot{ \alpha}(t)/(\frac{1}{2}\rho S\mathrm{u}^{3}) \tag{9}\]  
Figure 1: Schematic diagram of flapping airfoil motion.  
where F(\(t\)) is the force applied to the flap and \(M(t)\) is the torque applied to the flap.  
The average power coefficient over one cycle of the flapping airfoil is as follows:  
\[\bar{C}_{P}=\frac{1}{T}\int_{t}^{t+T}C_{P}(t)dt \tag{10}\]  
This gives the lifting efficiency of the flapping airfoil as follows:  
\[\eta=\bar{C}_{L}/\bar{C}_{P} \tag{11}\]