FIGURE 1 (a) The diagram of the R e withl /L . The l is defined as the distance between each cross section and the wing base, and the L is defined as the length of the ladybird right hindwing. (b) The H. axyridishindwing and sampling positions of corrugation, in which from positions P1 to P3 indicate 0.2L, 0.4L and 0.6L corresponding to 20%, 40% and 60% of wing length. In hindwing, C, costa; R, radius; Cu, cubitus; AP, anal posterior. MP is media posterior. WM1, WM2 are the wrinkled membrane. (c) The cross-sections detail of P1-P3 by VHX-6000; (d) Sketches of the cross-sections of the corrugated hindwing, airfoil 1 to 3, are obtained from P1 to P3, respectively. (e) Geometrical models AP1 to AP3. Sketches of veins are enlarged for clarity. So, they are not drawn to scale, whereas the corrugations average height, h and the chord length, c of the cross-sections are drawn to scale.

2.3 | The veins of corrugated airfoil

After observing the cross section of hindwing by the 3D microscope system with super wide depth of field (VHX-6000, Keyence, Japan), three cross sections from P1 to P3 were fixed with resin and the microstructure of the first groove on the leading edge was observed by the laser scanning confocal microscope (LSCM Olympusols3000, Zeiss, Japan).

2.4 | Parameters setting

Based on the cross section morphologies P1, P2 and P3 of the ladybeetle hindwing (see Figure 1c), 2D corrugated airfoil models were derived by Ansys Workbench 19.2: CA models AP1, AP2 and AP3 (see Figure 1e). Hereby the thickness of the wing membrane as well as the average vein diameter of AP1 as an example were measured to be 2.00 ± 0.01 μm and 35.00 ± 0.41 μm respectively, using Scanning Electron Microscope images (SEM microscope). Then AP1, AP2 and AP3 were meshed (see Figure 2a for AP1 as an example) for aerodynamic simulations. To test for the most suitable mesh resolution, three series of simulations with different meshes were set up with the same model (AP1). The surface mesh size of the first series was 0.2mm; the mesh size of the second series was 0.1mm; the mesh size of the third series was 0.05mm. Further simulation setting was: pressure-based, steady calculation, k-e standard viscous model, angle of attack is 5° and air speed is 1m/s. The calculation results show that the relative error in the lift coefficient obtained by using the first and second mesh series is 13.9%, and when using the second and third grids is 4.8%. After comparing the calculation precision and calculation time, the mesh size of the second series was selected finally and applied to all three models AP1, AP2 and AP3. The computational size of three models were relatively 1682232 elements and 841787 nodes (AP1), 1750487 elements and 875980 nodes (AP2), 1734086 elements and 867762 nodes (AP3), and the y + value (grid height of the first layer) of all models was set at 0.0005 mm, the geometry of the near wall boundary layer elements can be observed in Fig 2(b). For verification, the lift coefficient (Cl ) of a 2D flat airfoil (Kesel et al., 2000) at Re = 10000 at angles of attack is between 0 and 10° (interval 2°) is calculated using the second series mesh size. The calculated result was compared with Kesel’s experimental results (Kesel et al., 2000) and shown in Figure 2(c). It can be seen that the calculated numerical value of the lift coefficient of the flat wing is basically similar to the experimental value, indicating that the second series mesh guarantees sufficient accuracy.