Wing deformation improves aerodynamic performance of forward flight of bluebottle flies flying in a flight mill

Abstract

Insect wings are flexible structures that exhibit deformations of complex spatiotemporal patterns. Existing studies on wing deformation underscore the indispensable role of wing deformation in enhancing aerodynamic performance. Here, we investigated forward flight in bluebottle flies, flying semi-freely in a magnetic flight mill; we quantified wing surface deformation using high-speed videography and marker-less surface reconstruction and studied the effects on aerodynamic forces, power and efficiency using computational fluid dynamics. The results showed that flies’ wings exhibited substantial camber near the wing root and twisted along the wingspan, as they were coupled effects of deflection primarily about the claval flexion line. Such deflection was more substantial for supination during the upstroke when most thrust was produced. Compared with deformed wings, the undeformed wings generated 59–98% of thrust and 54–87% of thrust efficiency (i.e. ratio of thrust and power). Wing twist moved the aerodynamic centre of pressure proximally and posteriorly, likely improving aerodynamic efficiency.

1. Introduction

Insect wings are considered as smart, flexible structures that can deform substantially during flight, exhibiting time-varying, three-dimensional spatial patterns that are essential for optimizing the aerodynamic functions [1]. These deformations are often quantified in terms of camber and twist similar to those used in aircraft wing designs [2]. However, instead of being designed as permanent morphological features, they result from complex interactions among wing structural properties (e.g. inertia, venation and corrugation), aerodynamic pressure distribution on the wing surface [3] and active musculoskeletal control at the wing hinge. It has been shown that wing deformation has profound implications on the flight performance of both insects [4–6] and flapping-wing micro aerial vehicles (MAVs) [7].

In recent decades, the advancement of high-speed videography has enabled direct filming and quantitative measurement of wing surface deformations. To measure the deformations, it is common to use high-speed cameras to record the displacement of either morphological features or artificially introduced markers on the wings, from which the wing surface deformation can be reconstructed [8,9]. Despite the success of these methods for measuring near hovering or tethered flight, it remains a challenging task to fully reconstruct the intricate details of wing deformations in free-flying insects. This is mainly because these deformations have fast-varying, three-dimensional spatiotemporal patterns; they also occur on miniature, partially transparent wings that flap at high frequency and move rapidly with the insect in space.

Efforts to understand the aerodynamics of insect wing deformation are also hindered by the difficulties in reproducing the deformation kinematics in experiments using dynamically scaled [10] or at-scale [11] robotic wings. While these methods are commonly used to study the aerodynamics of flapping wings based on rigid wings, it is more challenging in scaling or matching both fluid and structural characteristics of the insect wings (e.g. Reynolds number, wing-fluid density ratio and flexural stiffness). In addition, it remains infeasible to develop an at-scale artificial wing that has structural complexity and distributed actuation at the wing hinge to emulate the deformation observed in insect flight. Furthermore, the highly unsteady and rotational nature of the flapping wing aerodynamics [12,13] renders it even more challenging to identify the contributions of wing deformation. Therefore, the aerodynamic mechanisms of insect flight (e.g. leading-edge vortices) and the relevant mathematical models (e.g. quasi-steady aerodynamic models [14]) are historically investigated assuming rigid wings [15,16]; and it remains unclear how wing deformations affect these mechanisms or revise the modelling.

To quantitatively assess the effects of wing deformation, the most successful approach to date, arguably, is using computational fluid dynamics (CFD) simulation [17–20]. CFD methods rely on reconstructed wing deformation kinematics to specify boundary conditions, solve the Navier–Stokes equations accordingly and provide rich data on the wing surface pressure, aerodynamic forces, flow field and vorticity patterns [18,21,22]. Compared with experimental studies of insects or flapping-wing MAVs, CFD methods allow researchers to more conveniently model and simulate the spatiotemporal deformation patterns of the wings and identify their aerodynamic effects by comparing with artificially removed deformation patterns (e.g. rigid versus deforming wings [23], deforming wings with camber removed [5], deforming wings with reinforced veins [24]). However, owing to the aforementioned difficulties in reconstructing the wing’s deformation patterns, most CFD studies focus on simplified model wings with limited reconstruction fidelity.

Previous efforts on understanding of wing flexibility and deformations focus primarily on hovering or low-speed flight where lift production and efficiency are considered [4,17]. It has been shown that flexible wings can enhance lift capacity [25] and reduce power requirements [24,26]. Flexible wings also increase lift-to-drag and lift-to-power ratios [27] and conceivably increase passive flight stability [28]. In forward flight, insect wings must keep lift equal to its own body weight while overcoming the drag and with thrust to sustain forward motion. To date, it is unclear whether the effects of wing deformation identified in hovering or low-speed flight remain similar in fast-forward flight.

Limited studies on the effects of wing deformation in forward flight have been conducted in insects of relatively large body sizes with two pairs of wings, which exhibited substantial wing deformation originating from the combined forewing and hindwing (e.g. locust [5], butterfly [20] and beetles [29]). Young et al. fully reconstructed the kinematics of deformed wings of a tethered locust in a wind tunnel and identified the effects of wing camber and twist using CFD simulations by comparing the deformed wings with wings with camber and/or twist artificially removed [5]. They found that deformed wings enhanced aerodynamic efficiency by vectoring aerodynamic force and by reducing flow separation. Also using CFD simulation, Zheng et al. showed that wing twist, instead of camber, was essential in achieving high aerodynamic efficiency of a free-flying butterfly in forward flight [20]. Le et al. used CFD to study the aerodynamics of a beetle in free-forward flight with wing flexibility and the effects of the elytron [29]. They show that flexible wings reduce flow separation on the leading edge and thus increase aerodynamic power efficiency. Furthermore, the elytron produces augmented lift by interacting with flexible wings, which is essential for weight support. Nevertheless, it remains unclear whether the aerodynamic advantages identified in large insects with two pairs of wings remain significant in insects of smaller body size with only one pair of wings, such as flies.

In this work, we hypothesize that wing deformation remains a source of considerable aerodynamic benefits for flying insects of small body size in forward flight. To test this hypothesis, we combined flight mill experiments in forward flight of bluebottle flies with a novel surface reconstruction method and CFD simulation to systematically study the aerodynamic effects of wing deformation. Specifically, we first measured and reconstructed the time-varying, three-dimensional wing spatial deformation patterns in bluebottle flies flying semi-freely in a magnetic-levitated flight mill (MAGLEV; see §2.1). To reconstruct the wing surface deformation, we introduced a surface reconstruction method (see §2.3). Next, we use a CFD solver to simulate the three-dimensional flows around the flies and calculate the surface pressure distribution and the aerodynamic force and power of the wings. To assess the contributions of wing deformation, we also performed CFD simulation on a flat-plate wing model (with wing deformations artificially removed). In addition, the CFD results were also compared with those predicted by an advance-ratio-revised quasi-steady model [30], thereby to shed light on the effects of unsteady mechanisms.

2. Methods

2.1. Experimental apparatus

The forward flight of bluebottle flies (4–7 days old, Calliphora vomitoria, unspecified sex) (n = 4, labelled as BBF #k, k = 1, 2, 3, 4) flying in the MAGLEV flight mill was recorded using three high-speed cameras (Fastcam Mini UX100, Photron, Japan) (figure 1a). Here, bluebottle flies were selected as they exhibit a wide range of flight speeds and impressive aerodynamic performance [31,32]. Their small body size is also more suitable for the compact flight mill used in this study and warrants the use of the marker-less wing kinematics reconstruction method. In the meanwhile, they are easily accessible commercially at a relatively low cost (Mantisplace, Olmsted Falls, OH, USA) and cultured in the laboratory. Details of the flight mill can be found in our previous work [33], and a brief overview is provided here. The rotating shaft of the flight mill was sandwiched between two permanent magnets, and it was levitated magnetically by a pair of electromagnets. Bluebottle flies were glued with UV adhesive (4305, Loctite Corp., CT, USA) to one end of the shaft with approximately zero body pitch angle (definition of the body pitch angle ($\chi$) is presented in figure 1b; $\chi$ = 5.0° ± 1.2°) and flew at a nearly constant speed in an annular corridor enclosed by a pair of inner (diameter 203.2 mm) and outer (diameter 304.8 mm) cylindrical walls; both walls were covered with grating patterns (50.4 mm square wave). Three high-speed cameras were positioned to capture the flies’ motion at a resolution of 1280 × 1024 pixels, a frame rate of 4000 s⁻¹ and a shutter rate of 8000 s^–1. All three cameras were calibrated using direct linear transformation (DLT) [34] for body and wing kinematics extraction and wing surface deformation reconstructions (see §2.3).

Figure 1.

Experimental apparatus, coordinate frames and kinematic variables. (a) Apparatus of MAGLEV flight mill. (b) Definition of the inclined angle of the stroke plane (β) and the body pitch angle (χ). The stroke plane frame (X_s, Y_s and Z_s) was defined based on β. (c) Anatomical landmarks (blue circles) were used for body and wing kinematic extraction. (d) Definition of body frame (X_b, Y_b and Z_b), wing frame (X_w, Y_w and Z_w), stroke plane (shaded area), wing stroke (ϕ), wing deviation (θ) and wing pitching (ψ). (e) Definition of wing deformation. Wing spanwise vector ($\stackrel{⃑}{R}$) was defined by wing root (${\stackrel{⃑}{V}}_{root}$) to wing tip (${\stackrel{⃑}{V}}_{tip}$). Wing chordwise vector (${\stackrel{⃑}{C}}_{i\widehat{R}}$) at different spanwise locations was defined by point on trailing edge (${\stackrel{⃑}{V}}_{trailing}$) to point on leading edge (${\stackrel{⃑}{V}}_{leading}$), where $i\widehat{R}$ was the normalized spanwise location ranging from $0\times \widehat{R}$ (${\stackrel{⃑}{V}}_{root}$) to $1\times \widehat{R}$ (${\stackrel{⃑}{V}}_{tip}$). ${\stackrel{⃑}{C}}_{i\widehat{R}}$ was also perpendicular to $\stackrel{⃑}{R}$ . Wing camber at different spanwise location was defined by ratio of the height of the chord ($L_h$) to the length of the chord ($L_C$). Wing twist ($\theta$) was defined by the angle between wing normal plane at $0.1\widehat{R}$ (${\pi }_N=\stackrel{⃑}{R}\times {\stackrel{⃑}{C}}_{0.1\widehat{R}}$) and the plane at $i\widehat{R}$ .

Note that the flies’ body pitching angle in this study (approximately 5°) is lower than what is typically observed in free flight (greater than 15° [35–37]). Previous works from the authors and others have suggested that flies’ forward flight conforms to the helicopter model, i.e. they have limited angle change of cycle-averaged wing aerodynamic forces relative to the body, and therefore have to rely on body pitch adjustment to vector the wing aerodynamic forces to regulate flight speed [33,38,39]. As a result, there exists an inverse linear relationship between the body pitch angle and forward speed [33], i.e. the smaller the pitch angle, the faster the flight speed. Therefore, by manually adjusting the body pitch angle as in our experiment, we can enable the flies to fly at different speeds. Here, we chose the smallest pitch angle that was tested previously to maximize forward velocity [33].

2.2. Definitions of wing kinematic and surface deformations

Flies’ wing kinematics are defined in figure 1b–d. We defined the body roll axis (X_b) as a vector from the thorax–abdomen junction to head–thorax junction, body pitch axis (Y_b) as a vector from the right-wing base to the left-wing base and the body yaw axis (Z_b) perpendicular to X_b and Y_b (figure 1c). Body frame was determined by X_b, Y_b and Z_b. We defined the wing spanwise axis (Y_w) as a vector from the wing base to the wing tip. Normal axis (X_w) was defined by wing plane that was determined by Y_w and location of vein CuA₁, and wing chordwise axis (Z_w) was defined by cross-product of X_w and Y_w (figure 1c). Wing frame was therefore determined by X_w, Y_w and Z_w. Projection of each wing tip trajectory onto the X_b–Z_b plane was fitted as a line using the least squares method. The average inclined angle of the two fitted lines (left and right wings) was used as the inclined angle (β) of the stroke plane (figure 1d), and a stroke plane frame (X_s, Y_s and Z_s) was defined by rotating the body frame about Y_b until X_s was parallel to the stroke plane (figure 1b). Then wing kinematics were defined relative to the fly’s stroke plane frame. The Euler angles (stroke position ϕ, stroke deviation θ and wing pitching ψ) were calculated from the wing rotation matrices [40] from stroke plane frame to wing frames (figure 1d). Wing frames were defined at each local spanwise location as shown in figure 1e. First, we defined $\stackrel{⃑}{R}$ as the spanwise vector directing from the wing root to the wing tip ($\stackrel{⃑}{R}={\stackrel{⃑}{V}}_{tip}-{\stackrel{⃑}{V}}_{root}$). Then, a wing normal plane ($P_{N,i\widehat{R}}$) was defined by cross-product of $\stackrel{⃑}{R}$ and ${\stackrel{⃑}{C}}_{i\widehat{R}}$ , where ${\stackrel{⃑}{C}}_{i\widehat{R}}$ was defined as chord vector spanning from trailing edge to leading edge ($i$ was normalized spanwise location ranging from 0 (wing root) to 1 (wing tip)). ${\stackrel{⃑}{C}}_{i\widehat{R}}$ also lies on the wing cross-sectional plane perpendicular to $\stackrel{⃑}{R}$, located at $i\bullet \widehat{R}$ (where $\widehat{R}=\stackrel{⃑}{R}/‖\stackrel{⃑}{R}‖$). ${\stackrel{⃑}{C}}_{i\widehat{R}}$ vector at different spanwise location (e.g. 0.2$\widehat{R}$ , 0.5$\widehat{R}$ and 0.8$\widehat{R}$) provides local wing pitching angles and was used to quantify wing twist.

Twist angle ${\theta }_{i\widehat{R}}$ was defined as the angle between vectors ${\stackrel{⃑}{C}}_{i\widehat{R}}$ and ${\stackrel{⃑}{C}}_{0.1\widehat{R}}$, with ${\stackrel{⃑}{C}}_{0.1\widehat{R}}$ serving as the reference (lightest blue plane, figure 1e). Note that the definitions of wing twist angle and pitch angle are identical except the difference in their references (twist angle ${\theta }_{i\widehat{R}}$ was defined relative to plane at $0.1\widehat{R}$ and wing pitching was defined relative to wing stroke plane frame). Wing camber was defined as $C_i=L_c/L_h$ (figure 1e), where $L_c=\left|{\stackrel{⃑}{C}}_{i\stackrel{⃑}{R}}\right|$ was the chord length between the leading edge and the trailing edge, and $L_h$ was the chord height (i.e. the maximum distance between ${\stackrel{⃑}{C}}_{i\stackrel{⃑}{R}}$ and wing profile on the wing normal plane $P_{N,i\widehat{R}}$). Note that the concave shape on the ventral side of the wing represented a positive camber and the concave shape on the dorsal side represented a negative camber.

2.3. Kinematics extraction and deformation reconstruction

The wing kinematics and surface deformation were reconstructed using commercial software MAYA^® (Autodesk Inc., San Rafael, CA, USA). We tracked the body movement and reconstructed wing deformation and kinematics for one wingbeat cycle. The forward flying speed was derived by tracking the body anatomical landmarks on the three-dimensional model, where we assumed no body deformation during flight. We also ignored lateral and vertical translations and body orientations as they are negligible compared with forward translation. We used the ‘rigging’ module in MAYA, which employs a (virtual) joint-dependent local deformation (JLD) mapping algorithm to reconstruct the wing surface deformation based on the nature of the virtual joint movement [41] (note that the virtual joints described here are the control points for deformation reconstruction, as opposed to insects’ anatomical joints). The ‘rigging’ module allowed us to emulate the entire wing deformation by adjusting the positions and orientations of the virtual joints (see figure 2 and examples in electronic supplementary material, video S1).

Figure 2.

Wing kinematics/deformation reconstruction. The wing was retrieved and scanned to be recreated on a three-dimensional realistic housefly model. The wing models were assumed to be zero-thickness flat plates. The three-dimensional fly model was then imported into a virtual environment in MAYA^® with camera in place as the replica of the real experiment setup. The projection centres and image planes were recreated using the DLT coefficients obtained from the image sequence analysis program (DLTdv6 and easyWand [34]). The three-dimensional fly body and wing surface were rigged to model the position, orientation and deformation through manipulating the virtual joints.

Specifically, we first created a three-dimensional fly model, and then adjusted its body size and wing silhouettes to match those of bluebottle flies in the high-speed video frames. The membranous wings and virtual joints were designed on the wing surface in spanwise and chordwise directions (see figure 2, rigging/virtual joints) according to the flies’ wing venation. We imported the high-speed video frames and recreated a virtual camera scene with DLT coefficients obtained from camera calibration using the image sequence analysis program DLTdv6 [34], thereby simulating the three cameras’ focuses and projection planes with perspective projection (video frames from each camera were warped on the projection planes). The three-dimensional rigged model was then brought back into the virtual camera scene, then we manually adjusted the positions and orientations of the virtual joints designed on the body and the wings to match the silhouettes of the fly in all three high-speed video frames. This process was considered finished when the projections of the three-dimensional body and wing model onto the projection frame had covered the largest possible area of background silhouettes (i.e. no further improvements can be made according to visual inspection). To validate our method, we used DLTdv8 [34] to capture the locations of three wing points of both the original images (regarded as ground truth) and the reconstructed wing model of BBF #1. The absolute error is shown in electronic supplementary material, figure S1. The maximum error is 0.15 mm (less than 1.5% of the body length), and the average error over a wing stroke cycle is 0.05 mm (0.5% of the body length). To further analyse the wing deformation kinematics, we exported the reconstructed wings from MAYA^® to MATLAB (MathWorks, Natick, MA, USA). Then, the wing kinematics and deformation variation were analysed in MATLAB. The temporal sequences of the kinematics data were fitted to a five-term Fourier model (‘fourier5’) using the ‘fit’ function in MATLAB to remove the high-frequency noises.

To contrast and quantify the contribution of wing deformation on aerodynamic performance, an undeformed flat-plate wing model was developed. The wing deformation was removed in this model by removing the orientations of the virtual joints (figure 2) on the reconstructed deformed wing, leaving the wing a flat plate. The flat plate was determined by three points: the wing root, wing tip and the tip of vein CuA₁ (figure 1c). This method aimed to capture the wing’s largest area and minimize kinematic variations at the root and tip (the location of CuA₁ is found in fig. 9.2 of [42]).

2.4. Computational fluid dynamics simulation and quasi-steady evaluation

Based on the reconstructed body and wing kinematics, we solved three-dimensional unsteady incompressible flow with a finite difference Cartesian grid-based immersed boundary method [43]. The reconstructed wing kinematics for a single wingbeat cycle for each fly were repeated four times for the simulation. The incompressible flow is governed by Navier–Stokes equations:

$${\displaystyle \mathrm{\nabla }\cdot u=0;\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\cdot \mathrm{\nabla }u=-\frac{1}{\rho }\mathrm{\nabla }\mathrm{p}+v{\mathrm{\nabla }}^{2}u,}$$ (2.1)

where $u$ denotes the velocity vector in the Cartesian coordinate system, $t$ stands for time, $\rho$ is air density, $\mathrm{p}$ is pressure and $v$ is kinematic viscosity. The Navier–Stokes equations are discretized in the Eulerian perspective on a Cartesian grid and integrated over time using the fractional step method. The boundary conditions are enforced through a ghost-cell procedure. Details about this method and validation of the CFD solver can be found in [43,44].

The computational domain size was 13c × 13c × 11c (c is average wing chord length) with a non-uniform Cartesian grid at 192 × 160 × 256 resolutions. A cubic region spanning 1.4c × 1.2c × 1.1c in size around the fly was designed with uniform grids at 0.009c grid spacing to resolve near-field vortex structures. The computational grid configuration is shown in electronic supplementary material, figure S2. A constant inflow velocity boundary condition and a zero-gradient outflow boundary condition were provided on the left- and right-hand boundaries, respectively. The zero-stress boundary condition was used at all lateral boundaries. The homogeneous Neumann condition was used for pressure at all boundaries. The boundary conditions for the fly were prescribed based on the surface reconstruction results. The instantaneous aerodynamic forces were calculated through surface integration of normal and shear stress acting on the wing and body, respectively. The lift and thrust were computed by projecting the total force onto the vertical and forward direction. The resultant forces were smoothed using ‘filtfilt’ function in MATLAB to remove the high-frequency noise.

The Reynolds number in the simulation was defined by $Re=\stackrel{-}{c}U/v$, where $\stackrel{-}{c}$ is mean chord length and $U$ is forward flight speed. Kinematic viscosity was 1.56 × 10⁻⁵ m² s⁻¹ for air at room temperature of 27°C. The advance ratio was defined as $J=U/\left(2\mathrm{\varphi }fR\right)$, where $\mathrm{\varphi }$ is stroke amplitude, $f$ is flapping frequency and $R$ is wing length. An example of simulation results is provided in electronic supplementary material, video S2.

In addition to the CFD simulation, we also calculated the aerodynamic forces, moments and power using blade element analysis and a quasi-steady model. The aerodynamic forces are the sum of force components owing to three mechanisms: (i) wing translational forces owing to delayed stall, (ii) rotational lift, and (iii) added mass, the details of which are found in Cheng et al. [45]. The lift and drag coefficients used in the quasi-steady model were based on those reported by Dickson and Dickinson [30]. The wing kinematics at 0.5$\widehat{R}$ were used for the quasi-steady model.

3. Results

3.1. Flight speed and wing kinematics

All four flies performed forward flight for over 10 laps in the flight mill. The average speed ranged from 0.59 (BBF #4) to 1.24 m s⁻¹ (BBF #1) (mean 0.98 ± 0.27 m s⁻¹, table 1). The time series of wing stroke, deviation and pitching of a single wingbeat are shown in figure 3. The flapping frequencies ranged from 140.35 (BBF #4) to 175.82 Hz (BBF #2) (mean 156.90 ± 13.68 Hz), and the wing stroke amplitude was the smallest with BBF #4 (96.07°) and the largest with BBF #1 (143.99°) (table 1). The wing deviation was relatively minor for all flies; the smallest deviation was for BBF #2 (16.24°) and the largest deviation was for BBF #4 (29.14°) (table 1).

Table 1.

Mass, morphological data and kinematic data of bluebottle flies from the experiment.

morphological/kinematic data	BBF #1	BBF #2	BBF #3	BBF #4	mean ± s.d.
weight (mg)	48.5	37.5	43.5	28.5	39.50 ± 8.60
body length (cm)	1.04	1.12	1.05	0.99	1.05 ± 0.05
wing length (cm)	0.80	0.76	0.75	0.72	0.76 ± 0.03
forward speed (m s−1)	1.24	1.21	0.88	0.59	0.98 ± 0.27
wingbeat frequency (Hz)	148.16	175.82	163.27	140.35	156.90 ± 13.68
body pitch angle (°)	4.00	5.82	3.85	6.16	4.96 ± 1.04
advance ratio	0.23	0.22	0.19	0.17	0.20 ± 0.02
wing stroke amplitude (°)	143.99	135.28	102.43	96.07	119.44 ± 23.73
wing deviation (°)	21.32	16.24	28.23	29.14	23.73 ± 6.09

Figure 3.

Wing kinematics in a wing stroke cycle. (a–d) The wing stroke angles for each individual. (e–h) The deviation angle. The pitching angle at different spanwise locations for (i–l) the left wing and (m–p) the right wing. The colour darkness indicates the spanwise locations. $\widehat{t}$ is the normalized time for a stroke cycle. The shaded areas indicate the downstroke, while the unshaded areas are the upstroke.

The wing pitching angle differed substantially from wing root to wing tip during upstroke (figure 3i–p), corresponding to significant wing twist (figure 4a–h). Note that pitching angle = 0° is when the wing normal plane is perpendicular to the stroke plane; therefore, having the wing approximately perpendicular to the incoming flow or 90° angle of attack (AoA). All wing chords showed comparable positive pitching (or pronation) during majority of the downstroke (${\displaystyle 0<\hat{t}\lesssim 0.5}$) but increased negative pitching (or supination) from root to tip during upstroke (${\displaystyle 0.5\lesssim \hat{t}<1}$). Overall, the pitching motion is more pronounced near the distal part of the wing (darker coloured curves, figure 4i–p), indicating that AoA is lower distally, especially during upstroke.

Figure 4.

Twist and camber plots in a wing stroke cycle. The twist angle (relative to twist angle at a spanwise location 0.1$\widehat{R}$) of the (a–d) left and (e–h) right wing. The camber percentage of the (i–l) left wing and (m–p) right wing of BBF #1–4. The colour darkness indicates the spanwise locations. $\widehat{t}$ is the normalized time for a stroke cycle. The shaded areas indicate the downstroke, while the unshaded areas are the upstroke.

3.2. Wing surface deformations

3.2.1. Twist

The wing twist was defined as the time-varying angle between a local wing chord and the reference chord at 0.1$\widehat{R}$ (figure 4a–h). Substantial wing twist (positive) resembling the washout design in aircraft wings was observed during the entire upstroke for all flies. The peak upstroke twist ranged from 24.43° (BBF #4, figure 4d) to 39.65° (BBF #2, figure 4f). There was only moderate wing twist (negative) during the early phase of the downstroke, and the switch from negative to positive twist preceded ventral stroke reversal, the timing of which varied among flies. One exception is BBF #4, whose right-wing twist did not switch signs (figure 4h).

3.2.2. Camber

Camber followed a similar temporal profile to the twist, being significant during upstroke (negative) and minor during downstroke (positive) (figure 4i–p). For both left and right wings, negative camber during upstroke is strongest proximally near the wing root, which corresponded to a pocket shape (or concave) facing towards the airflow. Peak negative camber was established in the beginning of the upstroke near the wing root and was maintained over the majority upstroke (with a moderate decrease) before it switched to positive camber at the beginning of downstroke. BBF #2 has the largest negative camber among all individuals, being 35.23% (figure 4n) and BBF #3 has the smallest camber among all flies, which is 17.47% (figure 4o).

There was a strong correlation between twist and camber, which suggests that they were coupled effects of a wing surface deflection that occurred primarily about the claval flexion line (i.e. one of the primary flexion lines on insect wings that allows the wings to deform during flight [1]), as illustrated in figure 5a. Both the twist and camber were most pronounced during the upstroke, with an example presented in figure 5b. A comparison of the deformed wing and rigid wing is also provided in electronic supplementary material, figure S3.

Figure 5.

(a) Summary of the wing deflection of bluebottle flies during forward flight. The major deflection occurred about the claval flexion line, which created camber at the wing root and twist at the wing tip. The rigid wing plane was used to determine the wing kinematics of both deformed and undeformed wings. (b) Reconstructed model of BBF #1 during the upstroke. Green line marks the edge of the wing to show wing twist. Pink line shows camber of the right wing. The claval flexion lines are marked with blue lines.

3.3. Aerodynamic performance

3.3.1. Thrust

Aerodynamic thrust was mainly generated during upstroke (figure 6a–d). It peaked during the translational phase when camber and twist were most significant (figure 4). CFD showed that flexible wings generated the largest average thrust per wing stroke, while rigid wings generated 59–98% and the quasi-steady model estimated 26–90% of thrust (table 2). In BBF #2, the quasi-steady model even predicted drag during downstroke (figure 6b, green line).

Figure 6.

Instantaneous thrust (a–d), lift (e–h), aerodynamic power (i–l) and wing motion with force vector (m–p) of the four BBFs. The forces are estimated by CFD with flexible wing (red lines and arrows) and rigid wing (blue lines and arrows), and the quasi-steady model (green lines and arrows). The dotted lines in (a–l) represent the average force and power of a stroke cycle. The shaded areas indicate the downstroke, while the unshaded areas are the upstroke.

Table 2.

Aerodynamic data of bluebottle flies from the CFD simulation of flexible and rigid wings and the quasi-steady model.

aerodynamic data	BBF #1			BBF #2			BBF #3			BBF #4
model	flexible	rigid	QS	flexible	rigid	QS	flexible	rigid	QS	flexible	rigid	QS
thrust (mN)	0.180	0.144	0.163	0.359	0.335	0.095	0.083	0.048	0.024	0.044	0.043	0.013
lift (mN)	0.363	0.326	0.267	0.409	0.423	0.400	0.285	0.303	0.155	0.141	0.150	0.105
power (mW)	1.415	1.302	1.609	2.354	2.809	3.199	1.307	1.430	1.002	0.742	0.713	0.454
thrust efficiency (mN mW−1)	0.127	0.111	0.101	0.153	0.119	0.030	0.063	0.034	0.024	0.059	0.060	0.029
lift efficiency (mN mW−1)	0.256	0.250	0.166	0.174	0.150	0.125	0.166	0.218	0.212	0.190	0.210	0.232
thrust ratioa	1	0.800	0.902	1	0.931	0.264	1	0.586	0.286	1	0.982	0.297
lift ratioa	1	0.898	0.734	1	1.034	0.977	1	1.06	0.543	1	1.064	0.748
thrust efficiency ratioa	1	0.874	0.795	1	0.778	0.196	1	0.540	0.381	1	1.017	0.492
lift efficiency ratioa	1	0.977	0.648	1	0.862	0.718	1	1.313	1.277	1	1.105	1.221

^a Normalized by CFD simulation results of flexible wings.

3.3.2. Lift

Aerodynamic lift was mainly generated during the downstroke, and peaked near the middle of the translational phase (figure 6e–h). CFD estimated similar average lift per wing stroke between flexible wings and rigid wings (figure 6e–h and table 2). The estimated lift was less than the body weights of the flies, suggesting that the flies were taking advantage of the flight mill tether for their weight support. The quasi-steady model estimated 54–98% of lift force compared with the CFD data of flexible wings (table 2).

3.3.3. Power

The total aerodynamic power expenditure peaked during the two translational phases in downstroke and upstroke while it was nearly negligible during pitching phases (supination and pronation). The powers estimated by CFD and quasi-steady model were comparable (table 2). We also evaluated efficiencies for thrust and lift generation, defined by averaged thrust or lift divided by power. The thrust efficiency of rigid wings was only 54–87% compared with that of flexible wings except BBF #4 where thrust efficiency was similar for both cases (table 2). The quasi-steady model generally underestimated thrust efficiency by 20–80%. The lift efficiency estimated by CFD and the quasi-steady model were comparable (table 2).

The wing pressure and force distributions at five instants of a wingbeat cycle, together with the wing camber and twist at four wing spanwise locations are illustrated in figure 7. We selected the cycle instants that correspond to different behaviours of wing motion. The 4% instant is the timing when the wing started downstroke motion; the 22% instant is the middle of downstroke; the 45% instant is the timing of stroke reversal; the 63% instant is nearly the middle of upstroke; the 90% instant is the timing when the wing is finishing upstroke. Comparing deformed and rigid wings, the centre of pressure of the deformed wing moved more proximally at 22% and 63% wingbeat cycle (figure 7c), where lift and thrust were near the peak, respectively (figure 7b). This was likely the result of wing twist (leading to a higher angle of attack proximally), which was significant at 22% and 63% wingbeat cycle (figure 7b).

Figure 7.

Time course of wing camber, wing twist and pressure distribution in a stroke cycle, with BBF #2 used as an example. (a) Illustration of a fly flying to the right at different percentages of a stroke cycle. (b) Force vectors owing to surface pressure distribution (dorsal force vectors represented in red arrows and ventral force vectors represented in blue arrows) at different spanwise locations in flexible (left column) and rigid (right column) settings. The larger arrow represents the total force, and its length indicates the relative magnitude of the total force. The two columns of the force vectors correspond to the cycle percentage on top in (a). (c) Pressure distribution of the wing at different time instances in flexible (left) and rigid (right) settings. The pressure distributions of the wing correspond to the force vectors and cycle percentage on top in (a,b). The red and grey dots are the instantaneous centres of pressure (CoP). Note that CoPs of rigid wings are also plotted on flexible wings for comparison.

4. Discussion

In this work, we quantified wing surface deformations of bluebottle flies using a marker-less surface reconstruction method and studied their effects on aerodynamic forces, power and efficiency. Compared with previous results on aerodynamic effects of wing deformation in flying insects of larger body size and with two pairs of wings (e.g. locusts [5] and butterflies [20]) or in hovering flight [25,26], our results support the hypothesis that wing deformation remains a source of considerable aerodynamic advantages in fast-forward flight in bluebottle flies of smaller body size with one pair of wings.

4.1. Wing twist and camber were coupled effects primarily about claval flexion line

Our results show that there was a strong correlation between the temporal profiles of camber and twist, suggesting that they were coupled effects of a wing surface deflection that occurred primarily about the claval flexion line (figure 5). Wing deformation about the claval flexion line can be explained parsimoniously by relative movement between the anal area of the wing (i.e. proximal and posterior) and the rest area. At the wing proximal region, relative movement between the anterior and posterior area led to camber, while relative movement between the proximal and distal area led to twist. This flexion was strongest during ventral stroke reversal (supination) and persisted throughout the middle of the upstroke (when thrust was produced). Both wing twist and camber switched directions between downstroke and upstroke, as the deflection between anal wing area and the remigium about the claval flexion line was bidirectional. However, this deflection was more biased towards supination, leading to more significant twist and camber during upstroke, which suggested a more critical role of wing deformation for thrust generation than lift production. Note that flies were not required to support their body weight when tethered in flight mill, and thrust production was likely more critical for forward flight in flight mill.

4.2. Wing twist moved aerodynamic centre of pressure proximally and posteriorly

The bidirectional wing twist led to a consistent ‘washout’ wing configuration for both up and down strokes, as wing root was kept at a higher angle of attack than wing tip. Washout wings are common designs among fixed wing [46] and rotatory wing aircrafts [47]. It helps a fixed wing aircraft to experience stall first at wing root and therefore to prevent rapid rolling motion, making the aeroplane more stable. On the other hand, a washout helicopter blade can redistribute the lift over the blade and reduce the induced power, rendering higher aerodynamic efficiency [47]. For bluebottle flies, although wing strokes have a significantly higher angle of attack than fixed or rotary wings (figure 3i–p), our results show that wing twist helped move the centre of pressure more proximally (towards wing root) and posteriorly (towards the trailing edge) (figure 7, 22% and 63% wingbeat cycle). A more proximal centre of pressure corresponds to a lower moment of the arm about the wing hinge, and therefore leading to lower torque and power required for producing thrust, which explains the improved thrust efficiencies with the presence of wing camber and twist. This is consistent with the previous findings that the twist seems to have little effect on aerodynamic force [27], but enhances power economy [5,17].

4.3. Stronger wing camber near the wing root suggests the potential role of active musculoskeletal force in wing deformation

From the observed wing deformation in bluebottle flies, the region near the wing root exhibited a more significant wing camber. Given the relatively low aerodynamic pressure (figure 7c) experienced by the proximal area of the wing, the pronounced deformation near the wing root is in favour of active musculoskeletal force in producing the camber, alongside the passive structural responses, instead of the role of aerodynamic force. This observation aligns with deformation patterns observed in other species such as butterflies [20], bumblebees [48], hoverflys [2], desert locusts [2] and hawkmoths [17].

The wing camber also altered the force vector (both magnitude and direction), especially during the beginning of the upstroke phase, where most of the thrust was generated (figure 6m–n). The force vectors of the deformed wing pointed more toward the thrust direction (figure 7, 63% wingbeat cycle). Moreover, during ventral stroke reversals, when drag instead of thrust was produced, the cambered wing showed a pocket shape facing back (or convex) towards the airflow (figure 7, 45% wingbeat cycle), which reduced the effective wing area, and thus the drag (figure 6a–d). In addition, previous work using CFD has shown that wing camber can improve lift by approximately 10–20% [4,27], which differs from our findings (no improvement). This is possibly owing to the small camber (<10%) during the downstroke, where most lift is generated (figure 6e–h).

4.4. Quasi-steady model is likely inadequate for predicting force and power for wings with substantial deformations

In this study, prediction by the quasi-steady model captured the general trend of the lower order temporal profile of force and power; however, it failed to accurately predict the higher order features and peaks, and also severely underestimated cycle-averaged magnitudes (figure 6a–h and table 2). While the quasi-steady model included the effects of delayed stall, rotational circulation and added mass inertia [14,15], it may still miss unsteady mechanisms, such as wing–wake interaction [49], wing–body interactions [50] or aeroelastic damping [51]. These mechanisms are highly dependent on size, morphology, wing and body kinematics and wing structural properties, which renders the modelling almost infeasible owing to the large number of variables included. Furthermore, the predictions of the quasi-steady model are very sensitive to the effective AoA [52], while it remains difficult to estimate AoA for wings with substantial camber and twist. Despite the sensitivity, results from the quasi-steady model with different AoA (±5° to the AoA from the wing kinematics) still failed to match the data from CFD simulation (electronic supplementary material, tables S1 and S2).

4.5. Limitations and future work

The magnetic flight mill in this study provides an alternative approach to wind tunnels to study insect forward flight in controlled laboratory settings. However, there are two major differences between the two methodologies. First, insects are able to fly at a self-determined speed in the flight mill, while they are forced to fly at a speed near the prescribed wind speed in the wind tunnel. Second, while in the wind tunnel insects are untethered, they remained tethered in the flight mill despite flying semi-freely in the forward direction; as a result, they are constrained in their body pitch angle (5° in the current work) and may not be able to fly at a preferred body angle. Therefore, the flies could attempt to pitch up during flight. Future work might focus on providing flies a free body pitch by attaching a micro-ball bearing between the flies and the angle pin (figure 1a) [33].

In this study, we defined the flat plate with three points: the wing root, wing tip and the tip of vein CuA₁ to capture the largest possible area of the wing. However, there could be other alternatives (using other wing features) to define the flat plate, which could produce slightly different orientations of the flat plate that alter the wing angle of attack and aerodynamic performance accordingly. Future work can focus on testing the sensitivity of aerodynamic performance on small variations of flat-plate orientation and systematically identifying the best flat-plate representation of a deformed wing. In the meanwhile, the manual adjustment of the virtual joints in the experimental procedure is time consuming and requires a considerable amount of training, and errors between the actual wing kinematics and reconstructed wing kinematics cannot be completely removed (electronic supplementary material, figure S1). Leveraging the state-of-the-art deep learning techniques could potentially expedite the process and further reduce the error of reconstruction [53].

Finally, note that the fine details of wing deformation, such as venation, flexion lines, relief (e.g. corrugation [54]), were not well resolved by our reconstruction. Instead, our wing deformation reconstruction captured the spatiotemporal wing surface patterns that can be mainly described by camber and twist, in terms of both their temporal patterns (as functions of time) and spatial patterns (as functions of spanwise location) (figure 4). Therefore, the spatiotemporal surface patterns captured in this work suggest the possibility of using a simpler parametrized model of wing deformation by wing camber and twist. To develop such parametrized mathematical models of wing deformation, one would need to propose function approximations for both the spatial and temporal patterns of camber and twist that capture various modes of wing deformation (e.g. those observed in figure 4). In future work, we expect the need to develop and compare candidate models that use different levels of approximation; and to reveal the proper level of approximation that captures the aerodynamic effects, CFD simulation for each of the candidate models will need to be conducted and the results of which compared.

Ethics

This work did not require ethical approval from a human subject or animal welfare committee.

Data accessibility

All data generated are available in the paper or its electronic supplementary material [55].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors’ contributions

S.-J.H.: conceptualization, formal analysis, investigation, methodology, visualization, writing—original draft, writing—review and editing; H.D.: formal analysis, investigation, visualization, writing—original draft, writing—review and editing; J.W.: investigation, methodology, software, writing—review and editing; H.D.: investigation, methodology, software, writing—review and editing; B.C.: conceptualization, funding acquisition, methodology, project administration, resources, supervision, writing—original draft, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

Funding for this research includes National Science Foundation (CMMI 1554429 to B.C.).