The reverse flight of a monarch butterfly (Danaus plexippus) is characterized by a weight-supporting upstroke and postural changes

Abstract

Butterflies are agile fliers which use inactive and active upstrokes (US). The active US plays a secondary role to the downstroke (DS), generating both thrust and negative vertical force. However, whether their active halfstroke function is fixed or facultative has not been clarified. We showed that during multiple backward flights of an individual, postural adjustments via body angles greater than 90°, with pitch-down and pitch-up motions in the DS and US, respectively, reoriented the stroke plane and caused the reversal of the aerodynamic functions of the halfstrokes compared with forward flight. The US and DS primarily provided weight support and horizontal force, respectively, and a leading edge vortex (LEV) was formed in both halfstrokes. The US's LEV was a Class II LEV extending from wingtip to wingtip, previously reported albeit during the DS in forward flight. The US's net force contribution increased from 32% in forward to 60% in backward flight. Likewise, US weight support increased from 8 to 85%. Despite different trajectories, body postures and force orientations among flight sequences in the global frame, the halfstroke-average forces pointed in a uniform direction relative to the body in both forward and backward flight.

1. Introduction

Under the influence of their ecosystem, volant insects use different techniques for aerial prowess via wing and body kinematic adjustments and deformations, varied wing shapes and configurations, and the use of aerodynamic mechanisms which are distinct in the translational and rotational phases [1–5]. Butterflies are among the most agile insects, having developed this ability to evade predators [6]. Butterflies have a low wing loading (bodyweight to wing area ratio) and, with each wingbeat, cover substantial distances while changing their flight trajectory considerably. Their varied body motion is characterized by changes in abdominal deformation which has a rigid phase relationship with wing motion [7], and body orientation [8–10], the latter which affects the wing aerodynamics substantially on a halfstroke basis.

The flapping frequency (n_w) is low, and the timescales of flapping and body motion are similar (signified by high advance ratios (J) in forward flight) [11]. Moreover, body rotations (particularly pitch) are exaggerated. Thus, both the body and wing motion of butterflies play pivotal roles in their flight performance in comparison to other insects, such as dragonflies, and so on. To inspire insect-like robots [12,13] and to clarify the underlying physics of flight, several studies have evaluated the behaviours that delineate the flight envelope of butterflies such as hovering, take-off, flap-gliding, forward, climbing and turning flights [6,9,10,14].

Ellington [15] described the novel use of a vertical stroke plane during the take-off and hovering flight of a cabbage butterfly (Pieris braisscae) in a seminal work. The wings were strongly supinated in the upstroke (US), while large angles of attack (AoA) characterized the downstroke (DS). Accordingly, the vertical force was produced by pressure drag in the DS, signified by the shedding of a large vortex ring from the wingtips, while the US produced little force. Associated with the wing kinematic adjustments were alterations in the body kinematics, whereby the thorax was horizontal and vertical in the DS and US, respectively. Sunada et al. [9] also arrived at similar conclusions to Ellington [15], by analysing the aerodynamics, in addition to the kinematics, of a related species (Pieris melete). They [9] showed that the variation in the body posture influences the stroke plane orientation and is the key mechanism in take-off flight. The aerodynamic torques from the wings, which raise the thorax, and the moments generated by abdominal deflection to suppress thoracic motion, modulated stroke plane inclination.

Using smoke visualization under free-flight conditions, Srygley & Thomas [16] showed that butterflies (Vanessa atalanta) use unsteady aerodynamic mechanisms. However, butterflies could switch the wing aerodynamics either on a stroke-by-stroke basis or a flight mode basis. The generation of large forces was signified qualitatively by the presence of a leading edge vortex (LEV), connected from tip to tip over the thorax (Class II LEV) during the DS. In some strokes, additional circulation was generated during the dorsal contact between the wings (clap and peel), as well as during wake capture.

Using computational fluid dynamics (CFD) simulations, Zheng et al. [17] investigated the forward flight of Vanessa cardui, with constant inflow (i.e. the body was artificially tethered in the CFD simulation). They reported that wing deformation, especially wing twist, improves the lift-to-power ratio in flight [17]. For a different species (Kallima inachus), Fei & Yang [18] studied the effect of transient body motion and noted that failing to consider the variation of flight speed within the stroke cycle may lead to an overestimate of lift or an underestimate of the flight speed and thrust. In another study, Fei & Yang [8] investigated the influence of body rotations and reported that in addition to stroke plane modulation, body rotations controlled the shed vortex ring orientation during each halfstroke. Moreover, the initial body angle determined the obliqueness of flight from the horizontal.

A common denominator in the studies above was the dominance of the DS in force production. Since butterflies move their wings in a steeply inclined stroke plane in hovering and forward flight [11,17,19], the halfstroke kinematics and forces are asymmetric [3,15]. Butterflies generate 59–74% of the net aerodynamic forces during the DS, and the US is less dominant [17,18,20]. The DS is always aerodynamically active, whereas the US can switch between an active and inactive/near-inactive US [16]. The inactive US occurs when the airstreams around the insect are undisturbed by the wing, and no substantial momentum is imparted to the flow. Feathering the US also reduces the force generated in that halfstroke, and attached flows dominate [21,22]. Conversely, if active, the US imparts momentum to the flow. The DS generates the force for weight support, while the US provides the thrust force for flight. However, body postural changes can affect changes in wing function and aerodynamics as the authors have noted in a previous work [23]. Nowhere in the literature has evidence been presented that the US (ventrodorsal stroke [3]) generates substantial forces for weight support of butterflies.

Here, we show that when J is negative (backward flight), an active (weight-supporting) US is present, thus, providing a new insight into the wing function and versatility of butterflies. Backward flight of several body lengths (L) has been observed among hawkmoths [24,25], although it has not been documented among butterflies previously. Backward translation may sometimes occur in take-off [26]; however, the displacement is small (about 0.6 l) compared with 3–7 l in this work. Thus, it is not regarded as backward flight. Moreover, the mechanics and aerodynamics of backward flight are unknown for any Lepidoptera species. In this study, backward flight was appropriated for take-off from vertical and horizontal surfaces, ‘normal’ and accelerating flight, obstacle avoidance, turning and escape response by the monarch butterfly (D. plexippus) in the laboratory. Nectar-feeding volant organisms may use backward flight to render immediate turning after feeding unnecessary [3,27]. We are interested in understanding the techniques of force generation in the backward flight of a butterfly, namely, the aerodynamic functions of the halfstrokes, the role of the wing pairs and the importance of body motion in the overall flight performance. Also, we are interested in the kinematic changes or trends that may differentiate backward flight from other flight modes. To this end, we extracted quantitative data (both transient body motion [18] and wing deformation [17]) from the three-dimensional (3D) surface reconstruction of high-speed videos. Afterward, a high-fidelity CFD solver was used to compute flight forces and flow features.

2. Material and methods

2.1. Insects, experimental set-up and three-dimensional surface reconstruction

The monarch butterfly (D. plexippus) was selected because of its availability. We captured monarchs outside and transported them to the laboratory for video recording. The insects were placed in a filming area comprising of a take-off platform and two sidewalls, where they initiated flight voluntarily (figure 1a). We did not train the butterflies to fly backward. We used three high-speed cameras (Photron Fastcam SA3 60 K; Photron USA, Inc., San Diego, CA, USA) arranged orthogonally, recording at 1000 frames s⁻¹, with a shutter speed of 1/20 000 s and 1024 × 1024 pixel resolution, to record their flight. Since the monarch's n_w was approximately 9 Hz, the cameras captured about 110 images per stroke, of which 25 instants were reconstructed. Previous work [10] had only captured 11 images per stroke. For different species (V. cardui) with n_w of 20 Hz, 100 images were recorded and nine instants reconstructed per stroke [17]. Backward flight typically ended when the insect transitioned to another flight mode, or came to rest in the shooting area. We captured seven backward flight sequences for an individual whose morphological parameters are documented in table 1 (see electronic supplementary material for videos). Because of insufficient data on forward flight of D. plexippus in the literature, we also recorded two forward flight sequences of the same individual to aid our discussion.

Figure 1.

Butterfly in backward flight (a). Experimental set-up. (b) Montage of a typical stroke. (c) (i–iii) Template-based reconstruction. (iv) Morphological parameters labelled on an anecdotal butterfly.

Table 1.

Morphological parameters. Wing parameters are reported for one wing. Length and mass measurements uncertainties are ±1 mm and ±1 mg, respectively. m, mass; L, body length; R, wing length; S, wing area; FW, forewing; HW, hindwing.

m (mg)	L (mm)	RFW (mm)	SFW (mm2)	RHW (mm)	SHW (mm2)
500	35	51	800	35	753

After motion capture, we performed 3D reconstruction in Autodesk Maya (Autodesk, San Raphael, CA, USA) using a template-based technique [28] which had been successfully applied to four-winged [29] and functionally two-winged insects [30]. The template comprised of a solid body and two membranous wings. The body template, comprising of triangular meshes (figure 1c), was matched to the high-speed images via control of a skeleton embedded within the template. The wing template comprised of 30 nodes which were matched to the images by following the patterns on the butterfly's wings, ensuring that wing deformation was also captured. We modelled the forewing (FW) and hindwing (HW) as one piece (figure 1c(iv)) since their overlap and sliding during flight was minimal, similar to past studies [17,18,20]. Afterward, the wing surface was exponentially sub-divided (while keeping the coordinates of the initial 30 nodes fixed) using Catmull–Clark subdivision and triangulated for CFD simulations. To account for the FW's contribution to force generation, we deleted the unstructured meshes of the HW region.

2.2. Kinematic definitions

In a body-fixed reference frame such as shown in figure 2a, the stroke plane is the least-squares plane based on wing root and tip coordinates (figure 2b). The stroke plane angle relative to body $({\beta }_b)$ was measured relative to the longitudinal axis $({\hat{e}}_l)$, which is the line connecting the head and hinge point of the abdomen. A coordinate system was placed at each wing root (x_w, y_w, z_w), and the stroke plane wing kinematics were measured (figure 2b). The Euler angles, flap (ϕ), deviation (θ) and pitch (ψ) denote the rigid wing orientation in the mean stoke plane. ϕ is the back and forth motion of the wing. $\theta$ is the angle between the wing and its projection on the stroke plane. $\psi$ is the angle between the wing chord and the mean stroke plane, and ${\psi }_{DS}$ is less than 90°. The wing chord is a line extending from the leading edge to the trailing edge of the wing. The effective AoA (α_eff) is the angle between the wing chord and vector sum (U_eff) of both body (U_b) and wing velocities (U_w), where U_b and U_w are the time derivatives of the body and wing displacements, respectively (figure 2c).

Figure 2.

Kinematics quantification. (a) The body normal (n̂) is the crossproduct of ê_l and ê_la (a vector pointing laterally and perpendicular to ê_l). (ê_l, n̂, ê_la) form a body-fixed coordinate. (b) Wing Euler angle definitions. (ê_l, n̂, ê_la) is rotated around ê_la until it coincides with normal to the stroke plane (n̂_sp) and translated to each wing root to form (x_w, y_w, z_w). (c) Additional kinematics definitions. Star symbol denotes the front end of the stroke plane. Symbols definitions are found in the text.

J, indicating the ratio of the body to wing velocity, was negated to indicate backward flight. Vis-à-vis body kinematics, the abdominal angle (θ_t) is the angle between ${\hat{e}}_l$ and the vector pointing from the hinge of the abdomen to its tip $({\hat{e}}_a)$. The dominant frequency of both the body and wing angles for each flight sequence was computed using the fast Fourier transform (FFT) in Tecplot 360 (Tecplot, Bellevue, WA).

2.3. Computational fluid dynamics simulation set-up

After 3D reconstruction, we used an in-house immersed boundary method CFD solver for simulations. A detailed exposition on the algorithm has been published [31] with validations [32]. The incompressible Navier–Stokes equation is:

$$\mathrm{\nabla }\cdot \mathbf{u}\hspace{0.17em}=0;\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}=-\frac{1}{\rho }\mathrm{\nabla }p+\upsilon {\mathrm{\nabla }}^2\mathbf{u},$$ 2.1

where $\mathbf{u}$ is the velocity vector, $\rho$ is the density, $\upsilon$ is the kinematic viscosity and p is the pressure, which was solved using a finite difference method with second-order accuracy in space and a fractional step method for time-stepping. The momentum equation was solved using a second-order Adams–Bashforth scheme for the convective terms and an implicit Crank–Nicolson scheme for the diffusive terms. The simulation domain size was 30R × 30R × 30R, and the size of the dense regions was selected based on the range of the motion of each butterfly trajectory and designed to resolve the vortex structures. The insect (unstructured triangular meshes (figure 3a)) moved freely in the domain (figure 3b) based on the motions obtained from the high-speed images. The domain boundary conditions (BCs) of both the pressure and velocity were homogeneous Neumann BC, i.e. zero gradient. The Reynolds number was defined as $Re=\overline{U}R/\upsilon$, where $\overline{U}$ is the body speed, and R is the FW length, following Kang et al.’s [10] work. Re ranged between 1700 and 3300. Each simulation required four processor cores on a single computational node on a high-performance computing cluster. Each stroke comprised of 1440 non-dimensional time steps ($\mathrm{\Delta }t$ = 6.94 × 10⁻⁴). For the fine grids, a stroke (for a model with one body and two wings) took about 4 days to run. A grid refinement study is shown for a stroke in one of the flight sequences in this study (figure 3c and table 2). The fine grids were deemed sufficient for our following analyses.

Figure 3.

Computational set-up: (a) 3456 and 4160 triangular elements define each wing (red) and body (blue). (b) Illustration of background mesh-grids used for CFD simulation coarsened five times in each direction. (x, y, z) represents the global coordinate system. +x, backward; +y, upward and +z, left (towards the reader) directions, respectively, (c) CFD grid refinement. F_v and F_h are the forces in the +y and +x directions, respectively.

Table 2.

Mesh set-ups and corresponding forces. $\mathrm{\Delta }x,\hspace{0.17em}\mathrm{\Delta }y,\hspace{0.17em}\mathrm{\Delta }z$-dense region's spacing.

	meshes	$\mathrm{\Delta }x,\hspace{0.17em}\mathrm{\Delta }y,\hspace{0.17em}\mathrm{\Delta }z$ (10−2 mm)	${\overline{\mathit{F}}}_{\mathbf{V}}$ (mN)	${\mathit{F}}_{{\mathbf{V}}_{\mathbf{m}\mathbf{a}\mathbf{x}}}$ (mN)	${\overline{\mathit{F}}}_{\mathbf{H}}$ (mN)	${\mathit{F}}_{{\mathbf{H}}_{\mathbf{m}\mathbf{a}\mathbf{x}}}$ (mN)
coarse	217 × 265 × 169	91 44 73	4.18	9.59	−0.81	14.2
fine	281 × 345 × 217	68 33 56	4.08	9.17	−0.86	14.1
finer	385 × 353 × 241	49 32 49	3.95	8.95	−0.86	13.5

3. Results

3.1. Kinematics

3.1.1. Body kinematics

The body kinematics are reported in figure 4 and table 3. As expected, the flight trajectories greatly varied from horizontal to oblique backward flight (figure 4a). By way of example, the time history of body angle (χ) and velocity during some backward flight sequences are displayed in figure 4b,c, respectively.

Figure 4.

Body kinematics. (a) Displacement of all backward flight sequences in the global frame (see figure 3b) and x₀ and y₀ are the initial positions of the insect. (b) Body and tail angles and (c) flight velocities for (i) B1, (ii) B2 and (iii) B4. Grey shading denotes the DS.

Table 3.

Body kinematics. ${\phi }_{\mathrm{b},\mathrm{w}}$ and ${\phi }_{\mathrm{t},\mathrm{w}}$ are the phase shifts between $\chi$ and $\varphi$, and ${\theta }_{\mathrm{t}}$ and $\varphi$, respectively, and calculated only when the frequencies of the two parameters were similar. +${\phi }_{\mathrm{b},\mathrm{w}}$ means that $\chi$ leads $\varphi$. +${\phi }_{\mathrm{t},\mathrm{w}}$ means that $\varphi$ leads ${\theta }_{\mathrm{t}}$. ID is the identifier of each flight sequence; B, backward; F, forward. $\overline{\mathrm{\Delta }\chi }$ is the change in $\chi$ measured at the start and end of each halfstroke, then, averaged for all halfstrokes [11]. Positive and negative values indicate pitch-up and pitch-down, respectively. Other symbol definitions are found in the text. We scared the butterfly to initiate flight.

ID	duration	$\overline{\chi }$	${\overline{\chi }}_{\mathrm{DS}}$	${\overline{\mathrm{\Delta }\chi }}_{\mathrm{DS}}$	${\overline{\chi }}_{\mathrm{US}}$	${\overline{\mathrm{\Delta }\chi }}_{\mathrm{US}}$	${\overline{\theta }}_{\mathrm{t}}$	$n_{\mathrm{b}}$	$n_{\mathrm{t}}$	${\phi }_{\mathrm{b},\mathrm{w}}$	${\phi }_{\mathrm{t},\mathrm{w}}$
	(ms)	(°)	(°)	(°)	(°)	(°)	(°)	(Hz)	(Hz)	(°)	(°)
B1	440	85	82	−9	87	16	51	10.2	9.09	44.6	181.4
B2	301	106	100	−1	111	17	51	7.01	9.97	—	199.0
B3a	147	119	116	1	133	33	28	6.81	13.6	—	—
B4	270	99	91	−17	103	12	42	7.41	7.41	39.6	142.5
B5	190	107	103	6	112	10	43	5.24	15.7	—	—
B6	220	108	100	−13	114	16	60	9.09	9.09	59.0	157.4
B7	290	108	103	−3	114	12	56	3.44	10.3	—	191.6
F1	101	28	24	−5	32	13	15	9.91	9.91	36.1	160.0
F2	101	50	40	11	61	16	14	9.91	9.91	—	195.2

^aWe scared the butterfly to initiate flight.

A steep body posture was maintained throughout backward flight, and body motion was characterized by substantial abdominal and body rotations within each wing beat (figure 4b). The body pitched-down and pitched-up during the DS and US, respectively. In contrast to the body rotation, the abdomen pitched-up and pitched-down in the DS and US, respectively. Both the body (n_b) and tail (n_t) frequencies were less than 15 Hz and sometimes had the same value. However, n_t and n_w were more closely linked. The grey shadings denoting the DS in figure 4b usually coincided with the tail but not necessarily with the body. The body pitching motion led the wing motion by about 40°–60° (figure 4b(i,iii) and table 3), whereas the wings and tail moved out of phase by approximately 180° (table 3). $U_{\mathrm{b}}$ also varied on a halfstroke basis with a mean value less than 1 m s⁻¹ (figure 4c). During the DS, the backward velocity increased while the vertical velocity decreased. The reverse occurred during the US.

3.1.2. Wing kinematics

The kinematics of the left and right wings were averaged, and the time histories were presented in figure 5 and summarized in table 4. ${\beta }_{\mathrm{b}}$ was approximately 90°. The stroke plane angle relative to the horizontal (β_h) was inclined (less than 40°), with the front end being above the horizontal (see star symbol in figure 2c). ${\beta }_{\mathrm{h}}$ was more inclined during the US than in the DS (greater than 10°) as a result of $\overline{\chi }$ being higher by about 10° in the US (table 3). The US-to-DS duration ratio (US:DS, [11]) varied among flights, but the US duration was usually longer than the DSs. Likewise, the US velocities were larger.

Figure 5.

Backward flight wing kinematics. Pooled mean ± s.d. (shaded) of (a) the backward flight wing angles. ψ is shown at 0.25R (proximally, ψ_p) and 0.75R (distally, ψ_d). (b) The effective angle of attack for backward flight. α_eff is shown at 0.25R (α_eff,p) and 0.75R (α_eff,d). A discontinuity exists due to the half stroke averaging process (see Ros et al. [33] for a similar artefact).

Table 4.

Wing kinematics parameters.

											${\psi }_{\mathrm{p}}$ (°)		${\psi }_{\mathrm{d}}$ (°)		${\alpha }_{\mathrm{eff},\mathrm{p}}$ (°)		${\alpha }_{\mathrm{eff},\mathrm{d}}$ (°)
ID	J	${\overline{U}}_b$ (m s−1)	US:DS	${\left({\displaystyle \frac{{\overline{U}}_{\mathrm{US}}}{{\overline{U}}_{\mathrm{DS}}}}\right)}_{eff}^2$	${\overline{\beta }}_{b,DS}$ (°)	${\overline{\beta }}_{b,US}$ (°)	${\overline{\beta }}_{h,DS}$ (°)	${\overline{\beta }}_{h,US}$ (°)	$n_w$ (Hz)	Φ (°)	DS	US	DS	US	DS	US	DS	US
B1	0.29	0.44	1.07	1.33	82	82	0	5	9.08	148	79	108	72	124	44	41	53	37
B2	0.28	0.54	1.06	1.62	85	85	16	26	9.97	135	78	108	72	130	44	41	58	33
B3a	0.28	0.83	0.82	—	—	—	—	—	13.6	142	—	—	—	—	—	—	—	—
B4	0.27	0.42	1.36	0.96	87	83	4	20	7.41	142	79	102	76	120	85	65	72	51
B5	0.30	0.57	0.85	1.71	78	78	25	35	10.6	115	81	109	73	128	44	54	56	42
B6	0.29	0.54	1.45	1.04	86	83	14	32	9.06	144	76	119	77	99	69	53	67	44
B7	0.40	0.82	1.25	1.39	85	82	18	32	10.3	132	77	107	73	125	55	49	58	39
F1	0.90	1.75	0.61	1.14	93	93	69	61	9.92	103	81	95	76	105	20	22	28	30
F2	0.42	0.91	0.74	1.35	78	91	38	30	9.92	124	78	103	67	122	50	37	52	37

^aEscape manoeuvre J =U_b/2Φn_wR.

The wings flapped with large amplitudes (approx. 130°) and with high pitch angles $\psi$ in both halfstrokes (figure 5a). ${\psi }_{\mathrm{US}}$ differed between the proximal location (0.25R) and the distal location (0.75R), indicating that the wings were twisted. ${\alpha }_{\mathrm{eff}}$ was measured at the same locations as $\psi$ (figure 5b). In general, the proximal effective AoA $({\alpha }_{\mathrm{eff},\mathrm{p}})$ was larger than the distal effective AoA $({\alpha }_{\mathrm{eff},\mathrm{d}})$ in the US similar to $\psi$, while the reverse trend occurred in the DS. Both $\psi$ and ${\alpha }_{\mathrm{eff}}$ were larger in the DS.

3.1.3. Aerodynamic force and power

The aerodynamic forces and flow features were obtained from the CFD simulation (§2.3). The aerodynamic force was obtained from integrating the pressure and shear stress on the wing. The aerodynamic power is defined as $p_{\mathrm{aero}}=-\int \int (\stackrel{\rightharpoonup }{\sigma }\cdot \stackrel{\rightharpoonup }{n})\stackrel{\rightharpoonup }{u}\text{d}s$, where $\stackrel{\rightharpoonup }{\sigma }$ is the stress tensor, $\stackrel{\rightharpoonup }{u}$ is the velocity of the fluid adjacent to the wing, and $\stackrel{\rightharpoonup }{n}$ and ds are the unit normal and area of each element, respectively. The muscle-mass-specific aerodynamic power is $p_{\mathrm{aero}}^{\ast }=\hspace{0.17em}{p}_{\mathrm{aero}}/M_m$, and the muscle mass (M_m) is 30% of the body mass [34]. For validation, we compared the wing generated ${\overline{F}}_V$ from CFD simulation with the force ${\overline{F}}_{V,i}$ estimated from the vertical acceleration of the insect body from a selected flight (b1, (figure 4a)) as follows:

$${\overline{F}}_{V,i}=m({\overline{a}}_V+g),$$ 3.1

where m is the body mass, g is the acceleration due to gravity (9.81 m s⁻²) and ${\overline{a}}_V$ is the average upward acceleration (0.8 m s⁻²) obtained from the time derivative of the body velocity (figure 4c(i)). ${\overline{F}}_{V,i}$ was 5.3 mN (1.08 × bodyweight), which was similar to the CFD simulation result (5.3 mN, 1.08 × bodyweight, table 5).

Table 5.

Force magnitude, orientation and muscle-mass-specific power consumption. B3 was excluded.

ID	${\overline{\mathit{F}}}_{\mathbf{D}\mathbf{S}}$	${\overline{\mathit{F}}}_{\mathbf{U}\mathbf{S}}$	${\overline{\mathit{F}}}_{\mathbf{V}\mathbf{,}\mathbf{D}\mathbf{S}}$	${\overline{\mathit{F}}}_{\mathbf{V}\mathbf{,}\mathbf{U}\mathbf{S}}$	${\overline{\mathit{\gamma }}}_{\mathbf{D}\mathbf{S}}$	${\overline{\mathit{\gamma }}}_{\mathbf{U}\mathbf{S}}$	${\overline{\mathit{\xi }}}_{\mathbf{D}\mathbf{S}}$	${\overline{\mathit{\xi }}}_{\mathbf{U}\mathbf{S}}$	${\mathit{p}}_{\mathbf{a}\mathbf{e}\mathbf{r}\mathbf{o}}^{\ast }$
							(°)	(°)	(W kg−1)
B1	1.70	1.69	0.88	1.28	146	50	64	−37	85.3
B2	1.90	2.24	0.35	2.06	171	74	70	−38	100.4
B4	1.46	1.35	0.58	1.05	156	53	65	−50	62.9
B5	1.34	2.59	−0.03	2.32	177	72	64	−41	107.0
B6	2.11	2.33	0.35	2.04	168	65	68	−49	120.8
B7	1.75	2.22	0.22	2.03	175	72	79	−42	98.7
F1	2.42	1.05	2.27	−0.31	79	215	77	−66	65.0
F2	2.35	1.17	2.14	0.45	72	151	69	−32	86.9

Across the backward flight sequences, the DS generated horizontal forces primarily but also vertical force in some of the strokes and negative vertical force at times. The US produced vertical forces for weight support primarily but also generated forward horizontal forces (figure 6a). The forward force reduced the backward impulse of the DS but not enough to prevent backward flight (figure 6b). $p_{\mathrm{aero}}^{\ast }$ ranged between 63 and 121 W kg⁻¹, which was within the scope of values measured for forward flight [17] (figure 6c).

Figure 6.

Force production and aerodynamic power. Pooled mean ± s.d. (shaded). Grey shading denotes the DS.

3.1.4. Force orientation

In the global frame, the halfstroke-average aerodynamic force vectors during backward fight are presented in the x–y plane (figure 7a). The DS-average forces $({\overline{F}}_{\mathrm{DS}})$ pointed backward (+x direction) and upward (+y direction), while the US-average forces $({\overline{F}}_{\mathrm{US}})$ pointed upward and forward (−x direction). The angle between the force and −x direction is denoted as $\gamma$. ${\overline{\gamma }}_{\mathrm{DS}}$ and ${\overline{\gamma }}_{\mathrm{US}}$ were 166° and 64°, respectively.

Figure 7.

Force orientation. (a) Halfstroke-average force in the global frame (measured relative to −x): (i) downstroke and (ii) upstroke. (b) Halfstroke-average force relative to ê_l, (i) downstroke and (ii) upstroke. Dashed lines denote average values. The coloured arrows are the backward flight results from table 5. Black vectors with white arrowhead, body longitudinal axis; red vectors, upstroke forces; green vectors, downstroke forces.

Figure 7b shows the force orientation relative to ${\hat{e}}_l$. Despite the different trajectories in the global frame (figure 4a) and large variation in the force orientation in the global frame (figure 7a), the force vectors were clumped up in a consistent direction during each halfstroke. The angle between the force and ${\hat{e}}_l$ is denoted as $\xi$. The halfstroke forces were only produced in the anterior side of the body (half-disc from 90° to −90°, clockwise; figure 7b). Furthermore, the forces in dorsoventral stroke (DS) were produced in the dorsal side (half-disc from 0° to 180°, anticlockwise), with the major component pointing towards $\hat{n}$ $({\overline{\xi }}_{\mathrm{DS}}={68}^{\circ })$. The forces produced in the ventrodorsal stroke (upstroke) were produced in the ventral side (half-disc from 0° to 180°, clockwise), pointing in between the longitudinal axis and the dorsoventral axis $({\overline{\xi }}_{\mathrm{US}}=-{43}^{\circ })$ The range of variation in the mean force vector relative to the body among flight sequences was 15° (DS), and 13° (US) was within the scope of values (±20°) recorded for biological fliers as well as helicopters, that appropriate force vectoring [33].

3.1.5. Three-dimensional flow features

We elucidated the flow features around the butterfly in backward flight using the isosurface of the Q-criterion. The evolution of the flow features over a representative stroke is shown for sequence B1 (figure 8). The flow videos are shown in the electronic supplementary material.

Figure 8.

Flow structures in a selected stroke. The isosurface of Q (Q = 1600) is coloured by non- dimensional pressure. (a) Top row (i–iv) is the DS flow at t/T = 0.13, 0.25, 0.38 and 0.48, respectively. (b) Bottom row (i–iv) is the US flow at t/T = 0.63, 0.75, 0.88 and 0.98. The flow is coloured by the coefficient of pressure (C_p = (p − p_∞)/0.5ρŪ_eff²). TEV, trailing edge vortex; TV, tip vortex; RV, root vortex; FW, forewing; HW, hindwing.

An LEV was formed following the separation of the wing pairs at the start of the DS (figure 8a(i)). The LEV, which is characterized by low-pressure regions around the leading edge of the wing, grew and remained attached on the wing surface for most of the stroke, shedding only at wing reversal. The LEV fed into a strong tip vortex. The LEV was present on the FW but absent on the HW. Attached vortex structures could be seen on the periphery of the HW, but their contribution towards force production may not be substantial.

The simulation (figure 8) was performed similarly to Zheng et al. [17] in that only the wings were simulated. However, when the body was placed in the simulation of one of the flight sequences (B4, figure 9a(ii)), the LEV extended from tip to tip across the thorax in the US and resembled a Class II LEV [35], albeit in the US. The vortex topology changes because the body acts as a focal point where the vortices from both wings can connect as one. The root vortices disappeared, and the flow that otherwise would have been lost was harnessed by the body, forming the LEV on the thorax (figure 9b). In the absence of the body, this does not occur (figure 9c,d). An isolated body generates little force, approximately 0.3% of the wing net forces per stroke. However, in the presence of the wing, the body generated about 2.5% of the wing net forces per stroke and more vertical force was produced in the US. Although the force increase was substantial for the body, the wing forces were not significantly influenced by the body's presence (2 and 4% increase in vertical and net forces, respectively). Both figures 8a and 9a show a strong interaction between the wing pairs, which could potentially enhance force production. Nonetheless, true ‘clap and peel’ was only observed in one stroke in sequence B4.

Figure 9.

Mid-stroke flow structures of sequence B4. (a) Three-dimensional flow at (i) mid-DS and (ii) mid-US. (b) Two-dimensional slices at mid-US to elucidate the LEV that runs across the body (Class II LEV). (c) Three-dimensional flow at (i) mid-DS and (ii) mid-US for isolated wings. (d) Two-dimensional slices at mid-US to show that the Class II LEV does not form in the absence of the body. Isolated body and isolated wings are superimposed here. Contours of non-dimensional spanwise vorticity and velocity vectors are shown.

3.1.6. Fore and hindwings contribution to force generation

Butterfly wing motion is driven by the action of the FWs although both wing pairs are well developed. The HW of the monarch contributes 52% of the total wing area [6,12]. During flight, the wing pairs are uncoupled but overlap effectively acting as a single airfoil. Here, we investigate the contribution of each wing pair to the force production in backward flight by simulating an FW-only (FO) and the FW + HW (ALL) case. The HW-only (HO) case is not simulated to prevent flow separation at the HW's leading edge, which is improbable under most free-flight conditions (figures 8 and 9). The time histories are shown in figure 10. The FW contributed about 65% of the net forces for flight and generated 70% of weight support. The HW generated both vertical (figure 10a) and backward force production but also increased the negative (forward) horizontal force produced in the US (figure 10b).

Figure 10.

Contribution of the fore and hindwings to force production. The mean forces of all complete half strokes are plotted. Solid lines, ALL case; dashed lines, FO case.

4. Discussion and conclusion

As a means of examining the wing halfstroke function and aerodynamics of butterflies, we investigated a previously uncharacterized flight mode. Backward flight was used for retreating from walls, take-off from the ground, for ‘normal’ and accelerating flight, obstacle avoidance, turning and as an escape response. Among nectar-feeding Lepidopterans, short backward flight had been previously observed in hawk moths such as Manduca sexta [25,36] and Macroglossum stellatarum [24]. Uncontrollable backward flight was elicited when the flagella of the antenna were severed, while controlled backward flight occurred in response to visual cues [24,36]. Here, the backward flight of the monarch was self-motivated without visual cues or alteration to the sensory organs. We discuss our findings in light of the current literature.

In the flight sequences captured, backward flight was accomplished by maintaining a steep body posture which oscillated around a mean value that was greater than 90°. However, not all the flights involved large-amplitude pitching of the body (figure 4b). ${\overline{\chi }}_{\mathrm{DS}}$ was about 10° less than ${\overline{\chi }}_{\mathrm{US}}$. Likewise, for hovering [15], take-off [9], climbing [10] and forward flight [18], $\overline{\chi }$ was less during the DS. The steep body angle technique for backward flight is shared among vastly different organisms such as hummingbirds (50–75°) [37], dragonflies (85–95°, [23]; 100°, [38]), water lily beetles (50–70°) [39], cockchafer beetles [40] and cicadas (86–130°; A. T. Bode-Oke 2019, unpublished data), although without the high-amplitude body rotations, and indicates that body postural adjustments are necessary for reverse flight for different flying species.

A steep body angle, however, does not indicate that every time a butterfly assumes that posture, it flies backward. Monarchs appropriate a steep posture (mean: 65°; minimum: 31°, maximum: 85°) in steep vertical climbs (A. T. Bode-Oke 2019, unpublished data). Leaf butterflies (K. inachus) also possess large $\chi$ (60°–90°) in forward flight. Kallima inachus relies on drastic body rotation in each halfstroke, so that the forward (thrust) force of the US counteracts the backward (drag) force of the DS, to preclude backward flight.

Body rotations have been reported to perform the function of wing rotation due to reduced wing degrees of freedom of a butterfly [8,41]. Although substantial body oscillation occurred in backward flight with $|\overline{\mathrm{\Delta }\chi }|$ < 25° (figure 4b(i) and table 3), larger body rotation in each halfstroke was not necessary (figure 4b(i)). Thus, effecting wing rotation using the body motion only, as previously described [8,41], may not completely characterize the backward flight of the monarch in our study since it can supinate the FWs to an extent (figure 5). Instead, the US is characterized by wing supination and a pitch-up motion of the body to aid the wings. It is also interesting to note that body rotation leads the wing motion (figure 4b), which may also signify the importance of body motion on the wing motion.

Comparing backward to forward flight kinematics listed in tables 4 and 6, both Φ and n were similar. Likewise, ${\overline{\beta }}_{\mathrm{b}}$ was comparable, indicating that there was no substantial change in its inclination though the flight modes are distinct in terms of wing aerodynamic function and body posture. The effect of fixed ${\overline{\beta }}_b$ influenced the force orientation and is discussed later. Backward flight was slower than forward flight (approx. 1–2 m s⁻¹, table 6) and also occurs at low speeds for other fliers; 1 m s⁻¹ for dragonflies [23], cicadas and Delfly II MAV [42], 1.5 m s⁻¹ for bumblebees [43] and 2 m s⁻¹ for hummingbirds [27]. The upper limit of backward flight speeds in the wild is still unknown for any animal.

Table 8.

Halfstroke contribution to the net force and force type produced predominantly in each halfstroke, for insects which use an inclined stroke plane.

insect	flight mode	DS force (%)	DS force type	US force (%)	US force type	reference
butterfly	forward	74	vertical	26	horizontal	Zheng et al. [17]
		75	vertical	25	horizontal	Yokoyama et al. [20]
		59	vertical	41	horizontal	Fei & Yang [18]
	backward	40	horizontal	60	vertical	current study
cicada	forward	90	vertical	10	horizontal	Liu et al. [48]
		80	vertical	20	horizontal	Wan et al. [30]
dragonfly	forward	84	vertical	16	horizontal	Bode-Oke et al. [21]
		75	vertical	25	horizontal	Sato & Azuma [49]
	backward	33	horizontal	67	vertical	Bode-Oke et al. [23]
	forward	80	vertical	20	horizontal	Azuma & Watanabe [50]
		67	vertical	33	horizontal	Hefler et al. [51]
fruit fly	forward	61	vertical	39	horizontal	Meng & Sun [52]
hawkmoth	forward	80	vertical	20	horizontal	Willmott et al. [53]
locust	forward	84	vertical	14	horizontal	Young et al. [54]

Table 6.

Forward versus backward flight of the butterfly. −J is backward flight, +J is forward flight. $\overline{F}$ is normalized by body weight.

species	$\mathit{J}$	${\overline{\mathit{U}}}_{\mathbf{b}}$	$\overline{\mathit{\chi }}$	${\mathit{\beta }}_{\mathbf{b}}$	${\mathit{\beta }}_{\mathbf{h}}$	US:DS				${\overline{\mathit{F}}}_{\mathbf{D}\mathbf{S}}$	${\overline{\mathit{F}}}_{\mathbf{U}\mathbf{S}}$	${\overline{\mathit{F}}}_{\mathbf{V},\mathbf{D}\mathbf{S}}$	${\overline{\mathit{F}}}_{\mathbf{V},\mathbf{U}\mathbf{S}}$			${\mathit{p}}_{\mathbf{a}\mathbf{e}\mathbf{r}\mathbf{o}}^{\ast }$	reference
		(m/s)	(°)	(°)	(°)		$\mathbf{\Phi }$ (°)	${\mathit{\psi }}_{\mathit{d},\mathbf{D}\mathbf{S}}$ (°)	${\mathit{\psi }}_{\mathit{d},\mathbf{U}\mathbf{S}}$ (°)					${\mathit{\xi }}_{\mathbf{D}\mathbf{S}}$ (°)	${\xi }_{\mathrm{US}}$ (°)	(W kg−1)
Vanessa cardui	0.4	1.14	6	85	79	0.77	120	—	—	2.46	1.16	2.37	−0.87	85	−56	120.7	Zheng et al. [17]
Kallima inachus	0.5	0.7	77	—	—	1.0	115	90	90	3.02	2.10	1.93	−0.03	69	−85	—	Fei & Yang [18]
a	0.4–2.0	0.55– 3.16	3–32	83	67	0.5–1.35	61–144	—	—	—	—	0.97–1.41	—	—	—	—	Dudley [11,44]
Parantica sita	1.2	1.6	28	—	—	1.0	92	—	—	2.27	0.74	2.19	0.23	55	−33	—	Yokoyama et al. [20]
Danaus plexipus	0.4 –0.9	0.91– 1.75	27–51	85–93	34–65	0.61–0.74	103–124	67–76	105–122	2.35–2.42	1.05–1.17	2.14–2.27	−0.03 to 0.45	69–77	−32 to −66	65.0–86.9	current study
Danaus plexipus	−0.3 to −0.4	−0.44 to −0.83	85–119	78–85	3–30	0.82–1.45	115–148	72–77	99–130	1.34–2.11	1.34–2.59	−0.03–0.88	1.05– 2.32	64–79	−37 to −50	62.9–120.8	current study

^aSee table 1 in [11] for the species information. Dudley's [11] data are the only result based on quasi-steady aerodynamic analysis.

Comparing backward flight of the individual butterfly in our study with those of other insects indicates similarities in kinematics. The US duration is longer than the DS, likewise the effective velocities (table 7 and [23]). Whereas α_eff is slightly lower in the US, the higher US velocity compensates for the lower US AoA. ${\overline{\beta }}_{\mathrm{h}}$ is smaller than angles recorded for dragonflies and cicadas. Because butterfly ${\overline{\beta }}_{\mathrm{b}}$ is higher than those of other fliers, a higher body angle has to be appropriated to achieve similar force partition in both halfstrokes as those insects, for example, dragonflies. Even with the larger $\chi$ substantial forward force was produced in the US due to the inability to supinate the whole (fore + hind) wing (figures 6b and 10b). Supination may aid the FWs to have more directional control of the forces than the HWs. If $\chi$ is too large, the flight may become uncontrollable.

Table 7.

Backward flight of different organisms in free flight. ${\alpha }_{\mathrm{eff}}$ is measured at 0.75R. *, forewings; †, hindwings.

animal	–J	${\overline{U}}_{\mathrm{b}}$ (m s−1)	$\overline{\chi }$ (°)	${\beta }_{\mathrm{b}}$ (°)	${\beta }_{\mathrm{h}}$ (°)	US:DS	${\alpha }_{{\mathbf{e}\mathbf{f}\mathbf{f}}_{,}\mathbf{D}\mathbf{S}}$ (°)	${\alpha }_{\mathrm{eff},\mathrm{US}}$ (°)	${\overline{F}}_{\mathrm{DS}}$	${\overline{F}}_{\mathrm{US}}$	reference
cockchafer beetle	—	−1.2	101	—	—	—	—	—	—	—	Schneider [40]
hummingbird	0.3	−1.5	51–75	57–71	−15 to 6	0.93–1.14	—	—	—	—	Sapir & Dudley [27]
dragonfly*	0.3	−1.0	90	35	47	1.15	25	21	1.4	2.2	Bode-Oke et al. [23]
dragonfly†	0.3	−1.0	90	35	47	1.20	27	27	2.1	3.2	Bode-Oke et al. [23]
water lily beetle	—	—	50–70	40–50	0–30	—	—	—	—	—	Mukundarajan et al. [39]
delfly II	—	−1.0	70–100	—	—	—	—	—	—	—	Caetano et al. [42]
butterfly	0.3– 0.4	−0.4 to −0.8	85–119	78–86	0–35	0.82–1.45	53–72	33–51	1.3–2.1	1.3–2.6	current study

Butterflies can generate as much as six times the body weight during the DS in forward and climbing flight [14,17,18]. During the US, however, the butterfly generates little to no vertical force in forward flight [8]. If any vertical force, the US generates negative vertical force (table 6). Negative vertical force results from the inability to supinate the whole wing completely in a steeply inclined stroke plane. The US net forces are also smaller compared with the DS; a consequence of the steep stoke plane which induces a force asymmetry, and also wing feathering. When large forces are produced in the US in forward flight, the forces are generated to counteract the drag generated during the DS. This is particularly common in species such as K. inchus that maintain large body angles in the DS (approx. 70°) [18].

Our results indicate that peak vertical force in backward flight was about three times the body weight (figure 6). The US produced vertical forces primarily, while the DS generated horizontal (backward) force primarily. Negative vertical force was rare, and when it occurred, it occurred during the DS (figure 7a(i) and table 5). On average, the US force was higher than the DS's. The wing AoA was substantial in both halfstrokes (table 4) and sufficient to form an LEV. Although the US AoA was lower than the DS's, the US wing velocity was higher.

In insect flight, the production of large lift is attributed to an LEV. Wing–wing interactions are also important in butterfly flight, although ‘clap and peel’ does not occur in every stroke. Ancel et al. [12] showed that during gliding flight at different AoA for different butterfly wing shapes, the LEV was restricted to the FW surface. Srygley & Taylor [16] reported the absence of an LEV in V. atalanta in steady forward flight in a wind tunnel. During forward flight with acceleration, however, an LEV was formed on the wing surface, typically extending from tip to tip. Zheng et al. [17] and Yokoyama et al. [20] also observed LEVs from their CFD simulations. J indicates the ratio of steady to unsteady effects in flapping flight. For butterflies, the body motion is unsteady (J changes continually) [18]. Since $|\overline{J}|$ ranged between 0.3 and 0.4 here, unsteady effects were important. We observed LEVs in both halfstrokes. The LEV extended from tip to tip, when the body was considered in the simulations, matching previous observations of a Class II LEV in forward flight [16].

Unlike previous works, a strong LEV was formed during the US. The LEV in both halfstrokes was absent on the HW, and the FW dominated force production, generating about 65% of the net force and 70% of the bodyweight. In functionally two-winged insects with smaller HWs, it had been shown that the LEV resides on the FW, and the HW does not affect its formation [45]. We expect that the same applies to butterflies. Although the HW has been reported to be unnecessary for flight, our results indicate that without wing kinematic adjustments of the FW such as an increase in frequency as observed when the HWs were ablated [6], weight support during backward flight may not be possible (figure 10).

Since the US carried the bodyweight primarily, we compared the mechanism of US vertical force generation in forward and backward flight. The assumption of an active US is dependent on what part of the wing, either the ventral or dorsal, the oncoming flow strikes [46,47]. Srygley & Thomas [16] verified these assumptions [46] using smoke visualization. They identified both a US that provided weight support (positive loading) and one that counteracted weight support (negative loading). Although only a wingtip vortex was visualized in the negative loaded US in Srygley & Thomas's study [16], CFD simulations indicate that an LEV, with a reversed sign of circulation, may reside on the wing on the ventral surface [17]. This US LEV was smaller than the DS's and contributed to negative vertical force and thrust forces [17]. In the case of the positively loaded US, an LEV of the same sign of circulation as DS rested on the dorsal surface of the wings. The body of the butterfly was horizontal, and the wings were also negatively cambered but not strongly supinated. The raised leading edge, however, possessed a high enough AoA, so that the incoming flow separated over the wing forming the vortex. The stagnation point was located on the wing's ventral surface and indicated that the US was lifting (see both fig. 4d and the supplementary video in [16]). The occurrence of a positively loaded or weight-supporting US, however, is rare. It occurred only in one US at the inception of flight in Srygley & Thomas's study [16] and has not been reported in any other forward free-flight studies [17,18,20]. The positively loaded US may be an artefact of the incoming flow or unique kinematics of V. atalanta in the experiment.

The mechanics of the US in our study is different because the LEV was stably attached on the ventral surface, indicating that the sense of circulation of the DS and US was reversed. The US loading was changed due to the reorientation of the body posture. We also did not observe any inactive US in any of the flight sequences. Since the forces generated during the US support the insect's weight, feathering the US is not a viable option in backward flight.

As aforementioned, butterflies elicit backward flight by notable postural changes reminiscent of the backward flight observed in other species [23,27]. A butterfly is likely not to elicit backward flight with a horizontal posture because the range of motion of the wing stroke plane is constrained within a narrow range relative to the body, in the anterior side, due to reduced wing degree of freedom (tables 3 and 6). Although the wing forces varied considerably in the global frame (figure 7a), the angle between the body and the aerodynamic force vector was relatively constant (figure 7b). To generalize our findings, we compared our results with the forward flight data of the same individual monarch and literature values for other butterfly species presented in table 6 (figure 11). Similar to backward flight, the forward flight forces were constrained in the anterior side. The DS forces were constrained in the dorsal side with the forces tilted towards the dorsoventral axis. Likewise, the US forces were produced in the ventral side in between the longitudinal axis and the dorsoventral axis. The uniformity of the results in forward and backward flight indicate that this monarch butterfly controls its flight forces using postural changes, at least within the range of speeds −1 to 2 m s⁻¹ or −0.4 ≤ J ≤ 0.9, here. This finding does not indicate that every aspect of forward and backward kinematics is the same. For example, the body angles are much higher in backward flight (table 6). Furthermore, the body motion is different, with an increase in horizontal speed occurring during DS and reduction in the US (figure 4c). The reverse happens in forward flight [8,18]. The upstroke duration is also longer in backward flight. The wing aerodynamic functions in the global frame are reversed between forward and backward flight. Because the US carries the weight in backward flight, there is also increase in its force magnitude. Nevertheless, the wing kinematics in the stroke plane are similar (at least for the monarch butterfly; table 6), and the force orientation is in a uniform direction relative to the body (local frame) in both flight scenarios.

Figure 11.

Butterfly flight forces point in similar directions relative to the body in rectilinear flight. (a) Dorsoventral stroke (downstroke). (b) Ventrodorsal stroke (upstroke). The arrows are the backward flight results from table 5. Shaded sectors of the circle are the forward flight results of the monarch (tables 5 and 6). The dashed lines are the forward flight results of other butterfly species from the work of Zheng et al. [17], Fei & Yang [18] and Yokoyama et al. [20] (table 6).

Data accessibility

This article has no additional data.

Authors' contributions

A.T.B.-O. carried out the reconstruction, simulations and analysis, and drafted the initial manuscript. H.D. oversaw the experimental and computational work. Both authors contributed to the final paper.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by the National Science Foundation (grant no. CBET-1313217) to H.D. and the University of Virginia 2018/2019 Mechanical and Aerospace Engineering Graduate Fellowship award to A.T.B.-O.