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Figures
- Fig. 1.
Boundary and initial conditions of the Navier–Stokes (NS) equations of air around a fly. The fly flies with respect to an inertial coordinate system (X,Y,Z) fixed to earth in a volume bounded by the outer surface Sob where the air remains at rest (outer boundary condition). The fly takes off at t=0 when the fly and fluid are at rest (initial condition). The air at the surface of the fly Sfb adopts the same velocity as the fly's body surface (inner boundary condition). Gravity is indicated with g.
- Fig. 2.
The velocity profile in the boundary layer around an airplane wing depends on the observer; airplane spotter (inertial observer) versus pilot (local observer). In engineering problems it is customary to attach the reference frame (analogous to the observer) to the wing (like the pilot is). This transformation simplifies mathematical analysis, numerical simulations and experiments, e.g. through the use of wind tunnels. The main advantage is that the surface of interest remains stationary with respect to the reference frame. A similar approach can be used for propellers; in this case the reference frame (or unfortunate observer) is fixed to the spinning and forward moving propeller, which again simplifies analysis. Here we propose to use a similar approach for studying the aerodynamics of the even more complicated motion of flapping fly wings.
- Fig. 3.
The kinematics and morphology of a forward flying fly. (A)The fly flies with velocity ubody while flapping its wings with angular velocity Ωwing. The cross-product of the wing angular velocity vector and the local radius r induces three velocity gradients along the wing; a velocity gradient due to deviation u
, stroke uϕ and angle of
attack variation uα. There are two reference frames,
the inertial reference frame (X,Y,Z) attached to earth and the local
reference frame (x,y,z) attached to the fly's wing at the joint.
(B)Wing morphology. Wing radius is the radial distance between the wing root
and tip, R. The wing radius of gyration Rg can be
calculated using a blade element method and is roughly equal to half the wing
radius (Ellington, 1984). The
average chord length of the wing c can be calculated by dividing the
single-wing area S by the single-wing span bs. We
define the single-wing aspect ratio as R/c. (C)The definition of wing
deviation , stroke ϕ and angle of attack α, which
depend on time t (Sane and
Dickinson, 2001). - Fig. 4.
Simplified forward flight model of a fly. (A) We first assume that the side slip angle of the fly, with respect to its body velocity, is zero. (B) Definition of the stroke plane angle β (Ellington, 1984), flight path angle ξ (Ellington, 1984) and the angle between the normal vector of the stroke plane and the body velocity γ. (C) A hovering fly induces three rotational accelerations in the flow due to stroke (downstroke shown). Shown are the angular aang, centripetal acen and Coriolis aCor accelerations that are induced in the fluid near the wing and result from the wing stroke, its propeller-like swing. The angular velocity of the wing stroke is Ωstroke and its angular acceleration is Ωstroke. The local velocity of fluid with respect to the local reference frame (x,y,z) is uloc. We color coded the stroke plane of the wing orange and the `outer shell' blue, which features the white wingtip path. (Note that the depicted direction of angular acceleration was chosen to prevent image clutter.)
- Fig. 5.
Flapping wing in forward flight. Graphical representation of the dimensionless numbers in the NS equation that describe the wingtip kinematics; A* the dimensionless stroke amplitude andλ * the dimensionless wavelength. The body and wingtip speed are indicated with blue and orange vectors, respectively. The dashed line is the wingtip path for γ=0° (fast forward or climbing flight). This figure holds for arbitrary flight direction with respect to gravity.
- Fig. 6.
The absolute average of the wingtip velocity (Uave) can be approximated well (Uapprox) within the same order of magnitude using Eqn 34 for any combination of γ and J (advance ratio).
- Fig. 7.
Graphical representation of A* and λ* for γ=0°; fast forward or climbing flight. Under these conditions arctan(1/J) is the average induced angle of attackα ind of the flapping wing. The maximum induced angle of attack amplitude at midstroke can be calculated using arctan(π/2J) (Lentink et al., 2008). The effective angle of attack of the wing αeff is equal to the induced angle of attack αind minus the geometric angle of attack αgeo.
- Fig. 8.
Propeller in forward flight. Graphical representation of the dimensionless numbers in the NS equation that describe the wingtip kinematics; D* and s*. Note that the induced angle of attack is again arctan(1/J).
- Fig. 9.
Is there an analogy between the influence of rotational accelerations on the stall characteristics of wind turbines and flapping fly wings? (A) Slender wings, such as wind turbine blades, have been found to possess distinctly different stall mechanisms near the root, for r/c<3 where they stall locally `3D', and near the tip, where they stall locally `2D'. Fly wings are much less slender than wind turbine blades, therefore their whole wing is `submerged' in the region of `3D stall'. Clearly the circular path (in the stroke plane) of outward wing sections becomes progressively less curved compared with chord length. We illustrated how the local `rotational component' of the radial speed component Ω·Δr becomes less and less important towards the tip compared with the `linear component' of radial speed Ωr. By taking Δr=c we find that the ratio of `linear component' over `rotational component' of radial speed is exactly r/c. (B) Based on the observations in A we designed flapping wing kinematics that allow us to change the relative importance of rotational accelerations R/c for a fly wing, which has a constant geometric slenderness. We achieved such a gradual transition from a linear velocity distribution to a constant velocity over the wing by gradually sticking the fly wing further out on an extension rod between the wing and its joint. In this we keep the stroke amplitude `actuator disc' areas swept by the wing in the stroke plane constant (orange) while it travels forward (blue). It can be demonstrated that a constant actuator disc area corresponds with a nearly constant stroke amplitude along the path sg at the radius of gyration Rg. The relative curvature of the path of the wing sections with respect to its chord length is again measured by R/c, the single-wing aspect ratio. Graphically we can directly infer that the paths traveled at more outward radii are much less curved and therefore the corresponding 3D rotational accelerations will diminish.
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