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. 2015 Apr 6;12(105):20150051.
doi: 10.1098/rsif.2015.0051.

Power reduction and the radial limit of stall delay in revolving wings of different aspect ratio

Jan W Kruyt  1 , GertJan F van Heijst  2 , Douglas L Altshuler  3 , David Lentink  4
Affiliations

Power reduction and the radial limit of stall delay in revolving wings of different aspect ratio

Jan W Kruyt et al. J R Soc Interface. .

Abstract

Airplanes and helicopters use high aspect ratio wings to reduce the power required to fly, but must operate at low angle of attack to prevent flow separation and stall. Animals capable of slow sustained flight, such as hummingbirds, have low aspect ratio wings and flap their wings at high angle of attack without stalling. Instead, they generate an attached vortex along the leading edge of the wing that elevates lift. Previous studies have demonstrated that this vortex and high lift can be reproduced by revolving the animal wing at the same angle of attack. How do flapping and revolving animal wings delay stall and reduce power? It has been hypothesized that stall delay derives from having a short radial distance between the shoulder joint and wing tip, measured in chord lengths. This non-dimensional measure of wing length represents the relative magnitude of inertial forces versus rotational accelerations operating in the boundary layer of revolving and flapping wings. Here we show for a suite of aspect ratios, which represent both animal and aircraft wings, that the attachment of the leading edge vortex on a revolving wing is determined by wing aspect ratio, defined with respect to the centre of revolution. At high angle of attack, the vortex remains attached when the local radius is shorter than four chord lengths and separates outboard on higher aspect ratio wings. This radial stall limit explains why revolving high aspect ratio wings (of helicopters) require less power compared with low aspect ratio wings (of hummingbirds) at low angle of attack and vice versa at high angle of attack.

Keywords: aspect ratio; hover; hummingbird; leading edge vortex; revolving wing; stall delay.

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Figures

Figure 1.
Figure 1.
Revolving hummingbird wings generate a stable LEV, and can be modelled with a rectangular model wing. (a) The average vorticity concentration above a Calypte anna wing at Re = 13 000 reveals an attached LEV at 30° and 45° angle of attack (α). (b) The distribution of vortex lift coefficient, Cl, calculated based on local circulation and chord length, shows vortex lift drops when radius, r, divided by chord length, c, is larger than approximately 3 (line, average; points, instantaneous values). (c) A single rectangular model hummingbird wing with aspect ratio 3.5 and 6% camber (see Material and methods) generates somewhat more lift (CL) and drag (CD) than a single C. anna wing (Re = 14 000; n = 5). The effect of a double versus single winged rotor is a small reduction in lift and an increase in drag. We contrasted double winged rotor measurements with aspect ratio 3.5 (solid red line; Re = 11 000) with 3.0 (dotted light red line; Re = 13 000) and 4.0 (dashed light red line; Re = 13 000) to show the effect on lift and drag. (d) The lift and drag values of the rectangular model hummingbird wing fall in the range of lift and drag values generated by the wings of 12 hummingbird species. Hummingbird lift-drag data have been adapted from Kruyt et al. [3].
Figure 2.
Figure 2.
Aspect ratio 10 model wings perform better below 20° angle of attack, whereas aspect ratio 4 wings do better above 20°. (a–d) Lift, drag, glide ratio, and power factor versus angle of attack as a function of aspect ratio, averaged over Re = 9000–25 000. Aspect ratio 4 wings combine near maximal lift, and intermediate drag, which maximizes power factor beyond 20°. The relative difference between minimum and maximum lift (e) drag (f) glide ratio (g) and power factor (h) among wings is substantial (green line, std calibration accuracy for reference). The optimal aspect ratio to obtain maximum (red) versus minimum (blue) lift (i), drag (j), glide ratio (k) and power factor (l), depends on angle of attack. The colour intensity corresponds with the p-value of the Wilcoxon rank-sum test for aspect ratio at constant angle of attack (black area; see the electronic supplementary material, figure S4).
Figure 3.
Figure 3.
Model wings with aspect ratio 4, close to values for hummingbird wings, perform well at incidence beyond 20°. (a) High aspect ratio wings best approximate the resultant force angle predicted by the quasi-steady model for flat plates, because of their relatively high drag. (b) Reynolds averaged power factor among aspect ratio wings is compared with the wing that attains maximum power factor at every incidence (100%). This normalized power factor shows that aspect ratio 10 wings perform best below 20° angle of attack, but do poorly at higher incidence; aspect ratio 4 wings provide the best trade-off above 20°.
Figure 4.
Figure 4.
The LEV remains attached and lift is elevated at radii up to four chord lengths at 45° (highlighted with a green bar). Vorticity is integrated (excluding the lower surface) at Re = 13 000 to approximate lift of model wings. (a,b) The average local vortex lift coefficient, Cl, steeply decreases for r/c > 4, whereas standard deviation increases. At the first station the hub is visible, which increases std artificially (semitransparent). (c) The vortex lift distribution is Re insensitive (thin lines, Re = 5000, 13 000, 25 000; thick line, average) and similar to C. anna (figure 1b). (d) When the angle of attack of an aspect ratio 10 wing increases from 30° to 45° vortex lift increases inboard of r/c ∼ 4 and lift starts to fluctuate outboard (line, average; points, instantaneous values). (e) Average vorticity concentration reveals an attached LEV inboard of r/c ∼ 4. Outboard vortices detach from the leading (yellow) and trailing edge (blue).
Figure 5.
Figure 5.
At high angle of attack the near wake is thick at radii beyond four chord lengths, because the flow separates from the upper surface. Distributions of horizontal speed (Vhor) at 1/8th chord behind the revolving wing (local wing speed, V0(r)) at Re = 13 000 illustrate this. (a) At 15° the wake is formed by an attached boundary layer, which thins with radially increasing local Re for aspect ratio 10 wings (above)—but not for aspect ratio 4 wings at Re 5000 (below left) and Re 13 000 (below right). The lack of radial effect due to Re on aspect ratio 4 wings help show that r/c effects dominate over local Re on the aspect ratio 10 wing at 30°. (b) The wake velocity profile at 30° thickens and fluctuates outboard (line, average; area, std; radial position, see (a)). (c) Outboard of r/c ∼ 4 the revolving wings drag a large volume of stagnant air along at 45° (clipped at 1/8th chord beyond trailing edge). (d) The mean R/c of hummingbirds (n = 65), other birds (n = 117), bats (n = 39) and insects (n = 98) is between 3 and 4 [11], facilitating stall delay.
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