Table 1.
Comparison of mass (), wing length (
), total wing area (
), aspect ratio and wing loading (
) for different insects.
Figure 1.
Key variables and validation results.
(A) Schematic defining the variables used to parameterize the kinematics and deformation of the wing in the current study. () and (
) are lab and wing-attached coordinated frames respectively. The wing shape and position at the middle of downstroke (gold) and upstroke (aqua) are shown. The red dot identifies the leading-edge and
is the cambered shape associated with the wing surface at one spanwise location in both wing positions. The local camber is defined as the ratio of maximum camber (
in the figure) to the local chord length (
, which is the straight line joining the leading and trailing edge of
). The angle-of-incidence (AoI) is defined as the angle between
and the
plane. (B) Simulation result for the observed butterfly wing (OBW) in forward flight showing streaks mimicking smoke traces and vortical structures at early downstroke. (C) Vanessa cardui with its brightly colored wings. Center-of-masses (CoMs) for different parts of the butterfly are marked. Point 1: abdomen; Point 2: whole body; Point 3: head and thorax; Point 4: hind wing and Point 5: fore wing. (D) Comparison of the lift force predicted by the simulation and the experimentally estimated value. The error-bars indicate the uncertainties in the experimental estimate.
Figure 2.
Description of key features in the generation of flat-plate wing (FPW) models.
Each FPW model is aligned to a unique set of three selected points on the OBW as shown: wing root, wing tip and hindwing tip for FPW 1; wing root, wing tip and forewing notch for FPW 2; wing root, point at the leading-edge at 2/3 wing-span, and 1/3 wing-chord at this spanwise location for FPW 3.
Figure 3.
Local camber and twist of the butterfly wing.
(A) Time variation of local maximum camber normalized by local chord length , through the wingbeat at different span-wise position. Note that
, where
is time period for one wingbeat. A positive value means the leading and trailing edges of the wing bends downward and vice verse. (B) variation of local AoI across the wing (
, where
is the wing length) which is a measure of wing twist. (C) variation of local AoI through the wingbeat. In Fig. (A) and (C), the red shaded region shows the spanwise variation of the AoI and the black line denotes the mean value.
Figure 4.
Time-variation of AoI and spanwise variation of AoA.
(A) Time-variation of the angle-of-incidence (AoI). Note that the AoI of the TOW along the span exactly matches that of the OBW since they have identical wing twist. (B) Spanwise variation of the angle-of-attack (AoA) during mid-downstroke and mid-upstroke for different models sampled at three locations along the wing.
Figure 5.
Comparison of instantaneous (A) lift, (B) thrust and (C) power during one flapping cycle for different models.
Table 2.
Comparison of mean lift (), total force(
), power
,
and
during one flapping cycle obtained from our simulations.
Figure 6.
Vortex structures and associated pressure for various cases at selected time-instances.
(A–D) OBW model at = 0.20, 0.33, 0.80 and 0.93 respectively. (E–F) FPW 1 model at
= 0.33 and 0.80 respectively. (G) FPW 3 model at
= 0.93. For all the cases the vortices are visualized by plotting one isosurface of the swirl strength (corresponding to a non-dimensional value of 20.5) and pressure is visualized by plotting color contours on the isosurfaces. Salient vortical features are labeled in each panel for ease of discussion.
Figure 7.
Comparison of aerodynamic traction vectors at 2/3 of the wing span during one wingbeat for different models.