## 2 Materials and Methods
### Computational Fluid Dynamics Simulation
#### 2.3.2 Computational Setup for CFD Simulation  
To clarify the influence of butterfly kinematics parameters on free flight, the CFD simulations are conducted in this paper. The direction of the mainstream is taken as the positive \(X\) direction, the vertically upward direction is taken as the positive \(Y\) direction, and the horizontal direction is taken as the \(Z\) direction. The directions are expressed by the Cartesian unit vectors \(\mathbf{e}_{x}\), \(\mathbf{e}_{y}\), and \(\mathbf{e}_{z}\). The equations governing the flow field include an equation for the three-dimensional, transient, and incompressible continuity and the Navier-Stokes equation,  
\[\
abla\cdot\mathbf{u}=0 \tag{5}\]  
\[\frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot\
abla)\mathbf{u}=-\frac{1}{\rho_{ air}}\
abla p+v_{air}\
abla^{2}\mathbf{u}. \tag{6}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Description** & **Value** \\ \hline \(S\) (m) & Wingspan & \(8.5\times 10^{-2}\) \\ \(\Delta l\) (m) & Wing-root offset & \(1.5\times 10^{-3}\) \\ \(c\) (m) & Chord length & \(3.6\times 10^{-2}\) \\ \(L_{t}\) (m) & Thoracic length & \(1.0\times 10^{-2}\) \\ \(L_{tt}\) (m) & Abdominal length & \(1.5\times 10^{-2}\) \\ \(h_{tw}\) (m) & Wing thickness & \(2.0\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters defining the simulation model of the butterfly. where \(\mathbf{u}\) is the velocity vector of the flow in the Cartesian coordinate system, defined as \(\mathbf{u}=u_{0}\mathbf{e}_{x}\). \(p\) is the pressure. The fluid is considered to be 20 \({}^{\circ}\)C. The values of the air properties are adopted from those at 20 \({}^{\circ}\)C: the density of air \(\rho_{air}=1.205\) kg/m\({}^{3}\) and the kinematic viscosity \(v_{air}=1.5\times 10^{-5}\)m\({}^{2}\)/s. It should be noted that the gravitational term does not appear in Equation (6). The velocity is constant on the inflow boundary plane: \(V_{\infty}\), incoming flow velocity. The streamwise length of the computational domain is \(L_{x}=0.8\)m, which is approximately 20 times as long as the aerodynamic chord length. The boundary conditions of the flows are shown in Figure 2a. The periodic boundary condition is employed for the transverse boundaries, and, on the other sides, parallel to the longitudinal \(X\)-axis, the no-slip condition is used. The transverse length of the computational domain is \(L_{y}=0.4\) m, \(L_{z}=0.5\) m, which is approximately five times as long as the wing-tip length. The results are not affected by the values of \(L_{x}\), \(L_{y}\), and \(L_{z}\). The average wing-tip velocity \(\bar{V}_{tip}\) equals the reference velocity defined as \(V_{ref}=2A_{f}\pi rf/180^{\circ}\). \(A_{f}\) is the stroke amplitude, with the value about \(120\pm 10^{\circ}\), \(f\) is the flapping frequency, \(r\) is the single wing length which equals \(L_{tip}/2\). The governing parameter of the above equations is the Reynolds number \((\text{Re})\) given by  
\[\text{Re}=\frac{V_{ref}L_{ref}}{v_{air}}. \tag{7}\]  
where \(L_{ref}\) is the reference length which is equal to \(\bar{c}\). \(\bar{c}\) is the mid-chord length (\(2.5\times 10^{2}\) m). It should be noted that the definition of the Re is the same as that used in [2; 31]. In this simulation case, the Re during butterfly forward flight is about 3000. Two key parameters of the flapping wing propulsion are the Strouhal number [32] and the advanced ratio, which are defined as \(St=2L_{ref}f/V_{\infty}\) and \(J=V_{\infty}/V_{ref}\), respectively. Set \(V_{\infty}\) to 1 m/s, \(f=10\) Hz, \(V_{ref}=1.75\) m/s, \(St\) is about 1.75, and \(J\) is about 0.57. Three-dimensional flow fields were characterized with the isosurface of the vorticity \(\omega\), so that the vortex structures of the butterfly are identified.