Flight of the honeybee

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Abstract
Drag forces and lift forces acting on honeybee trunks were measured by using specially built sensitive mechanical balances. Measurements were made on prepared bodies in ‘good’ and in ‘bad’ flight position, with and without legs, at velocities between 0.5 and 5m·s-1 (Reynolds numbers between 4·102 and 4·103) and at angles of attack between-20° and +20°. From the forces drag coefficients and lift coefficients were calculated. The drag coefficient measured with a zero angle of attack was 0.45 at 3≤v≤5m·s-1, 0.6 at 2m·s-1, 0.9 at 1m·s-1 and 1.35 at 0.5m·s-1, thus demonstrating a pronounced effect of Reynolds number on drag. These values are about 2 times lower (better) than those of a “drag disc” with the same diameter and attacked at the same velocity. The drag coefficient (related to constant minimal frontal area) was minimal at zero angle of attack, rising symmetrically to larger (+) and smaller (-) angles of attack in a non-linear fashion. The absolute value is higher and the rise is steeper at lower speeds or Reynolds numbers, but the incremental factors are independent of Reynolds number. For example, the drag coefficient is 1.44±0.05 times higher at an angle of attack of 20° than at one of 0°. On a double-logarithmic scale the slope of the drag versus Reynolds number plot was 1.5: with decreasing Reynolds number the relationship between drag and velocity changes from quadratic (Newton's law) to linear (viscous flow). Trunk drag was not systematically increased by the legs at any velocity or Reynolds number or any angle of attack. The legs appear to shape the trunk “aerodynamically”, to form a relatively low-drag trunk-leg system. The body is able to generate dynamic lift. Highly significant positive linear correlations between lift coefficient and angle of attack were determined for the trunk-leg system in the typical flight position. Lift coefficient was +0.05 at zero angle of attack (possibly attained during very fast flight), +0.1 at 5° (attained during fast flight), +0.25 at +20° (attained during slow flight) and +0.55 at 45° (attained whilst changing over to hovering). Average slope ΔcLΔα was 0.66±0.07, and average profile efficiency was 0.10. Non-wing lift contribution due to body form and banking only accounts for a few percent of body weight during fast flight. A non-wing lift contribution due to the legs has been demonstrated. The legs increase trunk lift by 23–24%. Reynolds number lift effects are present but of no biological significance. Force and power calculations do not support maximum flight speeds substantially higher than approximately 7m · s-1 relative to the ambient air. At this speed body drag attains 35% and body lift 8.4% of the body weight, and parasite power is 5% of the maximum metabolic power.

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