Fig 1.
(a) Typical non-uniform Cartesian mesh employed in the simulations. (b) Schematic of the control volume (not to scale) employed for the simulations and FPM. (c) Kinematics of the wing of the hovering hawkmoth and (d) fruit fly. In these plots, the trajectory of the leading edge of the wings at 2/3 span is identified by a thick line which is blue during downstroke and pink during upstroke. The chordlines at 2/3 span are also identified by black lines with circular “heads”. Time series of three characteristic angles (see inset in (f)) that define the wing kinematics for the (e) hawkmoth and (f) the fruit fly. For the hawkmoth, the instantaneous 3D wing shape and kinematics were quantified via high-speed stereo videogrammetry from recordings of the animal hovering steadily in front of an artificial flower [9]. For the fruit fly, a flat-plate wing was constructed from a high-resolution image of a fruit fly wing. Flapping kinematics consisting of three angular degrees of freedom were then extracted via high-speed stereo videogrammetry of a fruit fly, hovering shortly after takeoff, and imposed on the wing, resulting in rigid wing flapping kinematics. Fruit fly wings exhibit little deformation and the use of rigid wing kinematics is typical of mechanical [10] and recent computational [11–13] models of their flight.
Fig 2.
Components of the instantaneous lift coefficient over one flapping cycle for the (a) hovering hawkmoth at Re = 1000, and the (b) fruit fly at Re = 100. The stroke is divided into two phases: downstroke (D) and upstroke (U). Lift coefficient is defined as CF = F/(ρAβ2 f2 L2) where F is force, ρ is the fluid density, A and L the wing area and wing length respectively, and β and f the stroke amplitude and frequency respectively. The vortex-induced lift (VIL) Fω exhibits a large and distinct peak near mid-downstroke for both wings. In upstroke both wings generate positive VIL, but the magnitudes are significant only for the hawkmoth.
Table 1.
Stroke-averaged lift coefficients for the hawkmoth (Re = 1000) and fruit fly (Re = 100) wings. Values in parentheses denote the percentage of the total attributed to the given component. Overall, the vorticity in the flow contributes 82.5% of the total lift for the hawkmoth and ∼ 114% of the net lift over one stroke of the fly wing. This greater than 100% contribution results from the fruit fly’s operation at a lower Reynolds number, where viscous force induces a sizeable (25% of the net) negative lift over the cycle. The balance of the positive lift for both insects comes from the kinematic component Fκ; 17.3% for the hawkmoth and 9.5% for the fruit fly.
Fig 3.
Vortex structures and local contribution of vortices to lift production.
Vortex structures are identified by plotting an isosurface of the imaginary part of the complex eigenvalue of the local deformation tensor [31]. Isosurfaces are shaded by the lift coefficient per unit volume (cm3 for the hawkmoth and mm3 for the fruit fly) contributed by local vorticity. Panels (a-c) and (d-e) show the three phases in the flapping of the hawkmoth and fruit fly respectively. The hawkmoth wing, which is operating at a higher Reynolds number, generates a number of distinct vortices including the LEV, a tip-vortex and a root-vortex. (a) the downstroke peak in VIL for the hawkmoth wing, the bulk of the lift-producing vorticity patches are associated with the LEV. (b) at t/T = 1.5, the end downstroke, there is a spiralling LEV and a strong tip-vortex but neither of them generates much lift. (c) at t/T = 1.9, at the peak in VIL during late upstroke, the LEV on the ventral surface of the wing is weak but the wing-tip vortex provides a noticeable contribution to lift. For the fruit fly wing, which operates at a lower Reynolds number, the LEV is the most dominant vortical structure in downstroke (d) and upstroke (f); vorticity at the end of downstroke (e) is negligible.
Fig 4.
Time variation of the components of added-mass lift for the (a) hawkmoth and (b) fruit fly over one flapping cycle. Surface contours of added-mass lift
at instants corresponding to peak values in the flapping cycle for the (c) hawkmoth and (d) fruit fly. Chord-lines at a few instances with black and green vectors corresponding to
and
respectively where ΔΦ(2) is the difference in scalar Φ(2) on the two sides of the wing, are also shown. Note the supination and downward movement of the wing in early upstroke. During this time, the vectors corresponding to
and
are pointing in similar directions so that their dot-product generates a positive (upward) lift force, creating a net positive centripetal acceleration reaction contribution during upstroke. Scaling of various components of lift with Reynolds numbers for the (e) hovering hawkmoth and (f) fruit fly. The black, green, red and cyan lines represent total, vortex-induced, added-mass and viscous components of lift, respectively.