FIGURE 2 (a) The computational domain of AP1 model; (b) The enlarge and near wall boundary layer of the of AP1 model’s leading edge; (c) Comparison between the numerical and the experimental lift coefficient of Kesel’s flat wing model (Kesel et al., 2000).
The flow regime of a flapping insect wing is regarded to be unsteady (Flint et al., 2017) which asks for a time-resolved simulation approach. Therefore, a 3D finite volume method and a pressure-based solver were used to solve the time-dependent incompressible Navier-Stokes equations (Ansys FLUENT 19.2). The k-\(\omega\) SST low R emodel was selected, because this relatively viscous model had sufficient ability to predict and capture the unsteady flow characteristics and the consequent aerodynamic forces, as discussed in previous studies (Fairuz et al., 2016). The velocity inlet boundary condition was set to free flow from infinity with the value of \(\ v_{\infty}=1m/s\). The UDF (User-Defined Function) was used to define the motion equation for wing flapping. The wing models rotated around the Z axis, and the motion equation was defined as:
\(\varnothing(t)=\frac{1}{2}\times\frac{\text{θπ}}{180}\sin\left(2\pi ft+\frac{7\pi}{8}\right)+\frac{1}{2}\times\frac{\text{Aπ}}{180}\)(1)
Where, Φ(t) denotes the flapping angle over time; θdenotes the flapping amplitude, the value is 90 degrees (Jeffries et al., 2013); f denotes the flapping frequency; A denotes the phase of the flapping angle, the value is 30 degrees.
Then, the change of angular velocity φ(t) with time is,
\(\varphi\left(t\right)=49.35\cos\left(2\pi ft+2.75\right)\) (2)
The lift coefficient (Cl ) and the drag coefficient (Cd ) were monitored and recorded as the key quantities in this paper, which were defined as:
\(C_{l}=\frac{F}{0.5\rho{v_{r}}^{2}S_{r}}\) (3)
\(C_{d}=\frac{D}{0.5\rho{v_{r}}^{2}S_{r}}\) (4)
Where F , D denotes the lift and drag, respectively;ρ denotes the air density;vr denotes the reference velocity (the value is 1 m/s, which is the average speed of ladybird (Jeffries et al., 2013)), and Sr denotes the reference area.
The wing parameters and aerodynamic characteristics of three models are shown in the Table 1. In order to ensure the accuracy of the simulation results, each model should be simulated at least 50 flapping cycles, and the time period and time step of the transient simulations depend on the flapping period, that is, flapping frequency. When the monitoring value such as Cl and Cd , tend to be stable in each cycle, the convergence can be guaranteed.
Table 1 Parameters of Airfoil models