Aerodynamic explanation of flight speed limits in hawkmoth-like flapping-wing insects

The hawkmoth is able to sustain a steady hover or level flight at lower speeds. However, previous wind tunnel experiments have suggested that long sequences of steady forward flight are less common at higher flying speeds ($>4.0\mathrm{m}/\mathrm{s}$) despite changes to the flight posture. This flying speed is about one-half of the theoretical prediction based on its body mass. It is unclear why hawkmoths have not been observed achieving steady flight at higher speeds. This paper aims to compare the aerodynamics involved in hawkmoth hovering to those in forward flight. High-speed video recordings and three-dimensional surface reconstruction were used to capture a hawkmoth's wing kinematics when hovering and at forward flight speeds of 2 and 4 m/s. Following reconstruction, the insect model was simulated using an in-house immersed-boundary-method based computational fluid dynamics (CFD) solver. The CFD solver provided a quantitative measure of the force generation, power consumption, and vortex structures generated during sustained flight. These results enabled the analysis of certain trends in how a hawkmoth adjusts flapping kinematics and its associated unsteady aerodynamics changes across different flight speeds. The results show that the moth minimizes drag as flying speed increases, but it immediately loses its lift producing upstroke even at the slow forward flight speed (2 m/s). A significant amount of negative lift is generated during upstrokes at the high forward flying speed (4 m/s). This negative lift in the upstroke potentially reduces maximum sustained flight speeds. This paper provides physical insight into the low-speed flights in flying insects.

Figure 1

Comparison between the hawkmoth meshed model and the real hawkmoth. The hawkmoth mesh shown on the left was used for all simulations in this paper.

Figure 2

Definition of stroke plane angle ($\beta$) and body incline angle ($\chi$). Colored circles in the figure denote the wing leading edge, and the lines present the pitch angle of wing chord at the $0.75R$ along the wingspan. The hawkmoth flapping motion follows a counterclockwise trajectory. The downstroke is pictured in red, and the upstroke is shown in blue. See movie 1 in the Supplemental Material for reconstructed flapping motions [21].

Figure 3

Wing Euler angles definition, including wing stroke ($\psi$), wing deviation ($\varphi$), and wing pitch ($\theta$) angles. The wingtip trajectory during the downstroke is shown in red, and the upstroke is shown in blue. Two snapshots of the left hawkmoth wing are shown—one at the beginning of the flapping cycle and one during the downstroke.

Figure 4

Wing kinematics of a rectangular flapping wing used in solver validation. The benchmark setup is based on Suzuki et al. [8].

Figure 5

Comparison of lift (a) and drag (b) coefficients of flapping rectangular wing between current CFD results and data available in the literature [8, 46, 47].

Figure 6

Validation of the in-house CFD solver used in this paper. Lift and drag forces produced by a hawkmoth's wings and body in hovering flight are plotted against previous experimentally and computationally determined results. Experimental and numerical results obtained by Zheng et al. [15] and Aono et al. [14] are included.

Figure 7

Simulation setup and computational grids applied in the paper. Grids are shown for hovering (left) and forward flight (right).

Figure 8

Lift and drag force production during hovering flight by a single hawkmoth wing. Results are shown for three different grid densities: coarse ($177\times 145\times 177\approx 4.5\times {10}^6$, $\mathrm{\Delta }x\cong 0.0096R$), medium ($225\times 193\times 225\approx 9.8\times {10}^6$, $\mathrm{\Delta }x\cong 0.0044R$), and fine ($257\times 225\times 257\approx 1.5\times {10}^7$, $\mathrm{\Delta }x\cong 0.0019R$).

Figure 9

Images of the hawkmoth while (a) hovering and at forward flight speeds of (b) 2 m/s and (c) 4 m/s. See movie 1 in the Supplemental Material for high-speed video recording [21].

Figure 10

Wing Euler angles for hawkmoth hovering (a) and (d), 2-m/s forward flight (b) and (e), and 4-m/s forward flight (c) and (f). The first row of plots shows the wing pitch ($\theta$) angle at three different locations along the wingspan $R$. The second row shows the wing stroke ($\psi$) and wing deviation ($\varphi$) angles for each flight speed. The downstroke is shaded gray.

Figure 11

Wing chord trajectory for (a) hawkmoth hovering, (b) 2-m/s forward flight, and (c) 4-m/s forward flight. Colored circles in the figure denote the wing leading edge, and the lines present the wing chord at the $0.75R$ along the wingspan. The hawkmoth flapping motion follows a counterclockwise trajectory. The downstroke is pictured in red, and the upstroke is shown in blue. See movie 1 in the Supplemental Material for reconstructed flapping motions [21].

Figure 12

(a)–(c) Lift and (d)–(f) drag force production for hawkmoth hovering, 2-m/s forward flight and 4-m/s forward flight. Force production is shown separately for a single hawkmoth wing and the hawkmoth body. The downstroke is shaded gray.

Figure 13

Instantaneous mass-specific aerodynamic, inertial, and mechanical powers of wing pair for (a) hovering, (b) 2-m/s forward flight, and (c) 4-m/s forward flight. The downstroke is shaded gray.

Figure 14

Wing slices throughout one wingbeat cycle for (a) hawkmoth hovering, (b) 2-m/s forward flight, and (c) 4-m/s forward flight. A total of five slices were taken at $0.11R$, $0.28R$, $0.44R$, $0.60R$, and $0.77R$.

Figure 15

Three-dimensional and two-dimensional views of slice cut vortex structures at middownstroke ($t/T=0.25$) for (a) hawkmoth hovering, (b) 2-m/s forward flight, and (c) 4-m/s forward flight. The 2D view is shown for the chord located at $0.60R$. Net force vectors and arrows representing flapping direction are also included. The black circle is located along the wing's leading edge.

Figure 16

Three-dimensional and two-dimensional views of slice cut vortex structures at midupstroke ($t/T=0.75$) for (a) hawkmoth hovering, (b) 2-m/s forward flight, and (c) 4-m/s forward flight. The 2D view is shown for the chord located at $0.60R$. Net force vectors and arrows representing flapping direction are also included. The black circle is located along the wing's leading edge.

Figure 17

Dual LEV is observed for the hawkmoth forward flight at 2-m/s forward flight. The time instant for this picture is at $t/T=0.79$, around the midupstroke.

Figure 18

$Q$-criterion vortex structures for (a) hawkmoth hovering, (b) 2-m/s forward flight, and (c) 4-m/s forward flight. The isosurface of $Q=10{({\overline{U}}_{\mathrm{tip}}/R)}^2$ was used to plot the vortex structures. See movie 1 in the Supplemental Material for simulated flow animations [21].

Figure 19

Top view of vortex structure generated at the middownstroke ($t/T=0.25$) for (a) hawkmoth hovering, (b) 2-m/s forward flight, and (c) 4-m/s forward flight. The isosurface of $Q=10{({\overline{U}}_{\mathrm{tip}}/R)}^2$ was used to plot the vortex structures.

Figure 20

Stroke-averaged lift force distribution on the wing surfaces during downstroke and upstroke for (a) hovering, (b) 2-m/s forward flight, and (c) 4-m/s forward flight.

Figure 21

Relationship between the advance ratio ($J$) and $F_{\mathrm{vert},\mathrm{d}}$ (% $F_{\mathrm{vert}}$) for different small animal species, including hawkmoth (current study), bumblebee [2, 3], hummingbird [5, 6], and fruit fly [4]. Least-squares trendlines are shown as dashed lines for each species.