# Unsteady Aerodynamic Design of a Flapping Wing Combined with a Bionic Wavy Leading Edge
## 2 Aerodynamic Characteristics Analysis of Baseline Flapping Wing
### Calculation Method and Verification  
Flapping wing has flight characteristics of low speed and low Reynolds number. The numerical calculation in this paper is based on the three-dimensional unsteady Reynolds-averaged N-S Equation, which based on the finite volume method. The second-order up-wind scheme is used in the convection direction, the central difference is used in the diffusion term, and the Coupled algorithm is used to solve [36].  
The two-Equation shear stress (\(k-\omega\) SST) turbulence model [37] modified by the low Reynolds number is used for calculation. The turbulence model simulates the transition process with the damping function in the low Reynolds number eddy viscosity model, which has wide applicability. The \(k-\omega\) SST turbulence model Equation can be expressed as the following Equation (2):  
\[\begin{array}{l}\mu_{t}\ =\ \frac{\rho a_{1}k}{\max\left(u,u\Delta t \right)T_{2}}\\ \frac{\partial(\rho k)}{\partial t}+u_{i}\frac{\partial(\rho k)}{\partial x_{i }}\ =\ P_{k}-\beta_{k}\rho k\omega+\frac{\partial}{\partial x_{i}}\Big{[} \Big{(}\mu_{t}+\frac{\mu_{t}}{\alpha_{t}}\Big{)}\frac{\partial k}{\partial x_ {i}}\Big{]}\\ \frac{\partial(\rho\omega)}{\partial t}+u_{i}\frac{\partial(\rho\omega)}{ \partial x_{i}}\ =\ C_{\omega}P_{\omega}-\beta_{\omega}\rho\omega^{2}+\frac{ \partial}{\partial x_{i}}\Big{[}\Big{(}\mu_{t}+\frac{\mu_{t}}{\omega_{\omega} }\Big{)}\frac{\partial\omega}{\partial x_{i}}\Big{]}+2\rho(1-F_{1})\frac{1}{ \omega_{\omega 2}}\frac{1}{\omega}\frac{\partial k}{\partial x_{i}}\frac{ \partial\omega}{\partial x_{i}}\end{array} \tag{2}\]  
The low Reynolds number-corrected \(k-\omega\) SST turbulence model constructs the mode coefficient \(\alpha^{*}\) in the eddy viscosity coefficient \(\mu_{t}\ =\ \alpha^{*}\rho\kappa/\omega\) of the \(k-\omega\) Equation as Equation (3) [38], where \(\text{Re}_{t}=\rho\kappa/\mu\omega\) represents the turbulent Reynolds number:  
\[\alpha^{*}\ =\ \alpha^{*}_{\infty}\Bigg{[}\frac{\alpha^{*}_{0}+\frac{R_{t}}{ R_{k}}}{1+\frac{R_{t}}{R_{k}}}\Bigg{]} \tag{3}\]  
In order to verify the reliability of the numerical simulation, the rectangular straight wing of NACA0014 airfoil with aspect ratio of \(AR=8\) is selected. The mode of motion is sinusoidal oscillation, as shown in Equation (4). \(z(t)\) is the wing displacement in the vertical direction at time \(t\), amplitude \(h=0.4c\), reduction frequency \(k=0.2\), and Reynolds number \(Re\ =\ 1\times 10^{6}\)[34].  
\[z(t)\ =\ h\cos(kt) \tag{4}\]  
The boundary conditions of the calculation model are referred to [34]. Since the overset grid method is used in the subsequent calculation, the surrounding and far field of the wing are divided into two parts during grid division. Figures 3 and 4 show the comparison of the average lift coefficient and the average thrust coefficient of the flapping wing in one period calculated by using the overset grid method, and the results calculated by [PERSON] [34]. The two have good consistency, indicating that the numerical simulation method used in this paper is reliable.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Reynolds number \(Re\) & \(1.2\times 10^{5}\) & Plunging amplitude \(\psi_{1}(^{\circ})\) & 15 \\ \hline Wing chord length \(c(\text{m})\) & 0.12 & Plunging motion equation \((^{\circ})\) & \(\psi(t)\ =\ 15\cos(16\pi t)\) \\ \hline Wing aspect ratio \(AR\) & 8 & Pitching amplitude \(\alpha_{1}(^{\circ})\) & 15 \\ \hline Approaching velocity \(U_{\infty}(\text{m/s})\) & 15 & Setting angle of wing \(\alpha_{0}(^{\circ})\) & 0 \\ \hline Mach number \(Ma\) & 0.0435 & Pitching axis location \((m)\) & 0.25c \\ \hline Flapping frequency \(f(\text{Hz})\) & 8 & Phase difference \(\phi(^{\circ})\) & 90 \\ \hline Reduced frequency \(k\) & 0.2 & Pitching motion equation \((^{\circ})\) & \(\alpha(t)\ =\ 15\cos(16\pi t+90)\) \\ \hline \end{tabular}
\end{table}
Table 2: Relevant parameters of flapping wing.