Fig 1.
Schematic drawing of the morphological model.
(a) Wing contour. Coordinates are normalized to the wing length R. (b) The body is generated by circular sections of variable radius, which changes depending on the position along the centre line (dash-dotted arc). The body axis Obxb is the thorax-abdomen principal axis, approximately. The coordinates of the wing pivot points in the body frame of reference Obxbybzb are (−0.07R, ±0.18R, 0.115R). Note that, since β(t) is prescribed in the simulations and only (xc, zc) are dynamically calculated, the position of the body point of reference with respect to the body contour is chosen arbitrarily. (c) The insect’s position with respect to the ground is described by the body point of reference coordinates (xc, zc) and the position angle β. (d) Definition of the wing’s angles with respect to the stroke plane frame of reference Ospxspystzsp. The origin Osp is the wing pivot point.
Table 1.
Kinematics and leg model parameters of the takeoffs considered in the present study.
Table 2.
Numerical parameters of the takeoffs considered in the present study.
Fig 2.
Time evolution of the angular position of the body and of the wings during the voluntary takeoff.
Gray shaded regions correspond to downstrokes. η − β is the angle between the horizontal plane and the stroke plane, i.e., the global stroke plane angle [26].
Fig 3.
(a) Visualization of the wings, body and ground surface, and the wake at 4 subsequent time instants. Blue semi-transparent iso-surfaces show the Q-criterion, Q/f2 = 15. (b) Vertical and horizontal displacement. To obtain distance zc from the ground for the IGE case, add 1.08 mm. The black dash-dotted line indicates zc = R. (c) Components of the leg force. (d) horizontal and (e) vertical components of the aerodynamic force and (f) the aerodynamic power. The black dash-dotted line in panel e indicates the weight. Solid circles connected by dotted lines show wingbeat cycle averages. The results for OGE and IGE are shown, but the curves in panels b to f overlap because the difference is negligible.
Fig 4.
(a) Vertical and horizontal displacement. Dash-dotted line indicates zc = R. (b) Components of the leg force. (c) horizontal and (d) vertical components of the aerodynamic force. The black dash-dotted line in panel d indicates the weight. Solid circles connected by dotted lines show wingbeat cycle averages. Note that, in panels b and c, the lines for IGE and OGE almost coincide.
Fig 5.
The difference between the cases IGE and OGE, in terms of the wingbeat cycle averaged aerodynamic force normalized by the body weight (a) horizontal and (b) vertical components. The forces acting on the wings and the body are shown separately. The total force, which is their sum, is also shown.
Fig 6.
Aerodynamic power ratio IGE/OGE.
Fig 7.
(a) Data obtained by periodization of the last wingbeat shown in [15]. It is used in section Takeoffs with simplified kinematics. Also, in section Ground effect in hovering flight, this kinematics is referred to as ‘P1’ (‘P’ for ‘Periodic’). The cycle begins from the upstroke. (b) Data adapted from [6]. In section Ground effect in hovering flight, it is referred to as ‘P2’. The cycle begins from the downstroke.
Fig 8.
Takeoffs with simplified kinematics.
(a) Horizontal and (b) vertical difference between the wingbeat-averaged force IGE and OGE, normalized to the insect weight. (c) Vertical velocity of the body point of reference (rate of climb) versus time. (d) Maximum normalized force difference versus takeoff rate of climb at the moment when the legs lose contact with the ground.
Fig 9.
Flow visualization at the end of the 5th wingbeat (t = 23.78 ms) for the simplified kinematics cases.
Iso-surfaces Q/f2 = 15 are shown for 4 different takeoffs IGE: (a) N/m; (b)
N/m; (c)
N/m; (d)
N/m.
Fig 10.
Visualization of the leading-edge vortex during the 6th wingbeat for the simplified kinematics cases, IGE.
Left column (a-e) shows the dimensionless pressure distribution over the surface of the model and over a semisphere of radius 0.9R around the body point of reference. Middle column (f-j) shows the pressure iso-contours for two different takeoffs: N/m and 0.0430 N/m. Right column (k-o) compares iso-contours of the dimensionless vorticity magnitude for the same two takeoffs. Time instants are t = 24.73, 25.68, 26.63, 27.58 and 28.53 ms (tf = 5.2, 5.4 5.6, 5.8 and 6, where f is the wingbeat frequency).
Fig 11.
(a) The ratio of wingbeat averaged vertical force IGE to OGE, Fz,ave,IGE/Fz,ave,OGE. (b) The ratio of wingbeat averaged power IGE to OGE, Pave,IGE/Pave,OGE.
Fig 12.
Time evolution of the wake during hovering.
Iso-surfaces of the Q-criterion, Q/f2 = 15, are shown at the end of the downstroke. ‘P1’ kinematics with 100% wingbeat amplitude.