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.