During slow flight, bird species vary in their upstroke kinematics using either a ‘flexed wing’ or a distally supinated ‘tip-reversal’ upstroke. Two hypotheses have been presented concerning the function of the tip-reversal upstroke. The first is that this behavior is aerodynamically inactive and serves to minimize drag. The second is that the tip-reversal upstroke is capable of producing significant aerodynamic forces. Here, we explored the aerodynamic capabilities of the tip-reversal upstroke using a well-established propeller method. Rock dove (Columba livia, N=3) wings were spread and dried in postures characteristic of either mid-upstroke or mid-downstroke and spun at in vivo Reynolds numbers to simulate forces experienced during slow flight. We compared 3D wing shape for the propeller and in vivo kinematics, and found reasonable kinematic agreement between methods (mean differences 6.4% of wing length). We found that the wing in the upstroke posture is capable of producing substantial aerodynamic forces. At in vivo angles of attack (66 deg at mid-upstroke, 46 deg at mid-downstroke), the upstroke wings averaged for three birds produced a lift-to-drag ratio of 0.91, and the downstroke wings produced a lift-to-drag ratio of 3.33. Peak lift-to-drag ratio was 2.5 for upstroke and 6.3 for downstroke. Our estimates of total force production during each half-stroke suggest that downstroke produces a force that supports 115% of bodyweight, and during upstroke a forward-directed force (thrust) is produced at 36% of body weight.
In almost all birds flight plays a crucial role in survival, whether in foraging, attracting mates, defending territories, migrating or avoiding predators. In order to accomplish all of these tasks in flight, birds must first take off and accelerate to the appropriate speed. During this acceleration, flying birds appear to have two unique patterns of wing kinematics, associated with wing morphology. The first group fully flexes their wings on the upstroke and has low aspect ratio, rounded wings. The second group uses a ‘tip-reversal’ upstroke, in which the distal wing is heavily supinated, and includes birds with high aspect ratio, pointed wingtips (Brown, 1963; Tobalske, 2000; Tobalske et al., 2003). A notable exception to this pattern is seen in a group of birds designed for explosive take-off, Phasianidae, which have rounded, low aspect ratio wings but a prominent tip-reversal upstroke (Tobalske and Dial, 2000).
The functional significance of the tip-reversal upstroke has been debated for nearly a century. Early flow-visualization evidence suggested that the tip-reversal upstroke is aerodynamically inactive, meaning that it does not produce lift for weight support or thrust (Spedding et al., 1984; Spedding, 1986). However, this is unlikely given dramatic differences in wing kinematics (Brown, 1953; Tobalske and Dial, 1996) between upstroke styles, and observed increases in body acceleration during the upstroke (Brown, 1963; Aldridge, 1987). These early measures of body acceleration do not account for wing inertia in calculating center of mass acceleration, and thus may not accurately reflect performance. Recent evidence for an active upstroke is found in feather loading during the upstroke, indicating a net force directed anteriorly to the bird (Corning and Biewener, 1998), and significant differential pressure on the surfaces of the wing during upstroke through take-off and landing (Usherwood et al., 2005). Similarly, an analysis of instantaneous force changes as measured by accelerometers indicates that the tip-reversal upstroke in a cockatiel (Nymphicus hollandicus) provides 14% the lift of downstroke (Hedrick et al., 2004). If the upstroke provides a portion of the lift or thrust forces necessary for flight, birds that exhibit a tip-reversal upstroke may have a large advantage during slow flight in comparison to birds that flex their wings on the upstroke. Slow flight is energetically costly (Rayner, 1995); induced drag is high, so much of the power required to fly slowly is used to support the animal's weight (Ellington, 1991). Thus, we expect that animals should capitalize on mechanisms that allow them to rapidly accelerate to speeds that require less power output. Therefore, we predict that a tip-reversal upstroke will be aerodynamically active.
Here, we tested for an aerodynamic function of the upstroke in the rock dove (Columba livia, Gmelin 1789, hereafter ‘pigeon’) using a well-established propeller method (Usherwood and Ellington, 2002a; Usherwood and Ellington, 2002b) that has thus far only been used to model wing function during mid-downstroke (Usherwood and Ellington, 2002a; Usherwood and Ellington, 2002b; Altshuler et al., 2004; Usherwood, 2009). We used in vivo kinematics to model the posture of spread wings and to determine the Reynolds number estimated from wingtip angular velocity. We then directly measured lift and drag by mounting the dried-wing propellers on a force plate.
MATERIALS AND METHODS
A captive-bred pigeon was housed in an outdoor aviary at the Field Research Station at Fort Missoula. The bird was trained to fly between horizontal platforms positioned approximately 2 m apart. Prior to kinematic recordings, the bird was marked with black permanent marker at the shoulder, wrist and each primary feather tip, and at three locations on the ventral surface of the body for later digitization. All care and experimental procedures were approved by the University of Montana IACUC.
We recorded the flight of the pigeon using four high-speed cameras: one Photron SA-3 (Photron USA Inc., San Diego, CA, USA), two Photron 1024 PCI and one Fastec Troubleshooter (Fastec Imaging, San Diego, CA, USA), synchronized using a transistor–transistor logic (TTL)-pulse. One camera was placed dorsally and behind the flight path, two were ventrally placed behind and in front of the flight path, and a fourth was placed horizontal to the bird in front of the flight path. Video recordings were made at 1000 Hz with a shutter speed of 1/10,000 s.
To analyze the kinematic data, we recorded a 36-point calibration volume (approximately 30×45×80 cm) for the space between the platforms. A direct linear transformation converted the four camera views into 3D coordinates (MATLAB R2010a, The MathWorks Inc., Natick, MA, USA) (Hedrick, 2008). Markers along the wings were digitized from recorded images to obtain 3D wing kinematics. The third wingbeat cycle following take-off was used.
We performed subsequent analysis using custom scripts in MATLAB. Angular velocity (rad s–1) of the hand wing was calculated by determining a central position in three dimensions between the wrist, the 10th primary feather and the 1st secondary feather, to represent the total hand wing. We then determined the velocity vector of the hand wing, relative to the shoulder of the bird. To determine the angle of attack (α, deg) in global coordinates (accounting for translation), we measured the angle between the velocity vector of the hand wing and the vector normal to the hand-wing plane, and subtracted this from 90 deg.
Wings were removed from three deceased birds (body mass, 446±69 g) and dried in a low temperature oven at 40°C for 1 week. We obtained paired models for each individual, with one wing in mid-upstroke posture and the other in mid-downstroke posture. Each wing was mounted on a brass rod at the proximal edge of the head of the humerus to create a one-blade propeller, with the rod acting as a counterweight (Fig. 1). Wings were photographed and analyzed in ImageJ (v1.43u, National Institutes of Health, Bethesda, MA, USA) and MATLAB to determine wing length and the first, second and third moments of area [following Usherwood and Ellington (Usherwood and Ellington, 2002b)] (Table 1). The second moment of area directly correlates with aerodynamic force, and the third moment with torque (Weis-Fogh, 1973). Thus, these two measurements allow us to account for slight differences in wing shape between birds when calculating mean vertical and horizontal forces (Eqns 1 and 2).
Each wing was spun once at each angle of attack (α), in increments of approximately 10 deg from –10 to 90 deg. Wings were mounted to a brushless DC motor (NEMA 34 motor, Anaheim Automation Inc., Anaheim, CA, USA), and controlled with an Anaheim Automation driver and power supply (DCL 601USB, MBC 12101 and PSA 40V8A), with SMC60WIN software (v. 2.01). Revolutions per minute (r.p.m.) values were based on the mean angular velocity of the wingtip during the middle third of each half-stroke, as determined from in vivo kinematics for upstroke and downstroke. The third wingbeat following take-off of the live pigeon had an angular velocity of 69 rad s–1 for the mid-third of downstroke, and 96 rad s–1 for the mid-third of upstroke, which translated into 376 and 572 r.p.m., respectively.
The propeller shaft was constructed to allow the insertion and manipulation of the brass rod for easy rotation of the wing over a full range of α. We measured α as the acute angle between horizontal and a line described by two points on the wing: midway between the wrist, tip of the 10th primary feather and the tip of the 1st primary feather. We visualized these points using a digital video camera (Photron SA-3 using PFV 3.1.8 software) that was placed horizontal to the wing. We measured α before spinning the wing and during rotation, both to account for deformation due to aerodynamic loading of the wing and to report active α for final analysis. Active α was on average 2±3 deg less than static α.
The wings were mounted so that induced velocities in the wake would generally be directed upwards, away from the force plate. For the downstroke posture, for example, this meant the wing was mounted with its ventral surface facing upwards when α>0 deg. For upstroke posture wings, the secondaries contacted the motor when α<50 deg, so we oriented the wing so that the ventral surface faced downwards. The aerodynamic contributions of the counterweight were accounted for by spinning a rod with double the length of the counterweight, centered on the propeller shaft, at appropriate r.p.m. The resulting measurements were halved and subtracted from observed wing values.
We used a custom-built force plate (15×15 cm, Bertec Corp., Columbus, OH, USA) to measure vertical force along the z-axis and torque about the z-axis due to drag (Usherwood, 2009). The force plate featured a known conversion of 10,000 mN V–1 for force and 800 mN V–1 for torque. Voltage output from the force plate was converted and amplified with a digital gain of 5× using a Bertec model M6810 amplifier. We imported these data to a computer using an ADInstruments PowerLab 8SP A/D converter sampling at 1000 Hz and Chart v5.2 software using a 1 Hz low-pass digital filter (ADInstruments Inc., Colorado Springs, CO, USA). A cardboard cowling (shield) surrounded the base of the motor and the force plate in order to isolate the force plate from air velocities induced by the spinning wings (Fig. 1).
Wing shape during spinning versus flight
To compare deflection of primary feathers during model rotation with in vivo wing shape, we used 3D video recording and analysis on the propeller model (see ‘Kinematics’, above). We videotaped the upstroke and downstroke wing models when spinning with α comparable to in vivo measurements (66 deg for upstroke, 46 deg for downstroke wings). To compare wing shape between the live bird and propeller, digitized points were standardized to a right-hand rule, local coordinate system rooted at the wrist, with the x-axis extending through the 10th primary feather. Conversion from global to local coordinates was accomplished using a series of Euler angle transformations.
We assessed variation for each feather between a model wing (Fig. 2, blue lines) and live animal (red lines) in both the proximal–distal and ventral–dorsal axes. Feather offset was measured as the linear distance in three dimensions from each primary feather tip to its corresponding feather in the live bird, and normalized by dividing by the wing length (Fig. 2). Feathers in upstroke-postured wings differed from in vivo feathers by 1.96±0.97 cm or 6.05±2.75% of wing length. Downstroke posture wings differed by 2.02±0.91 cm or 6.78±3.06% of wing length.
Determining coefficients of lift and drag
Data from the force plate were converted to the coefficient of lift or drag by a series of equations presented by Usherwood and Ellington (Usherwood and Ellington, 2002a). The mean vertical force reported by the force plate (Fv, mN) was converted to the mean vertical force coefficient (Cv): (1) where ρ is the air density (1.07 kg m–3, in Missoula, MT, USA) during the experiments, S2 is the second moment of area (Table 1) and Ω is the angular velocity (rad s–1). Then, we converted the mean torque (Q, mN m) about the z-axis of the plate to a horizontal force coefficient (Ch): (2) where S3 is the third moment of area.
Using both of these values, we determined the dimensionless coefficient of lift (CL) and drag (CD), using a value for the local downwash angle (ε) determined geometrically assuming a triangular distribution of induced velocity along the wing (Usherwood and Ellington, 2002a). This distribution has been confirmed for the propeller model with chukar partridge (Alectoris chukar) wings using particle image velocimetry (PIV) (Heers et al., 2011). Thus, CL was: (3) and CD was: (4)
While in vivo upstroke kinematics qualitatively differ from downstroke kinematics, the tip-reversal upstroke rotates around the wrist, where the pigeon averages an angular velocity of only 6.5 rad s–1. Thus, by positioning the wrist over the axis of rotation in our propeller model, we can assume a constant angular velocity along the hand wing during upstroke, and use the above calculations for both upstroke and downstroke wings.
Estimating in vivo performance
Combining in vivo kinematics with force coefficients extracted from the propeller model allows us to consider the force production of a pigeon during the third wingbeat cycle following take-off using coefficients measured with the propeller. First, assuming a hovering case, we can calculate lift (L) as: (5) and drag (D) as: (6) then, the resultant force (FR) is calculated as: (7)
However, because the bird is moving forward, we must account for both translational velocity (Dickinson and Dickinson, 2004) and Ω along the wing (Ωr). We used a blade element analysis (Osborne, 1951) and expanded Eqn 1 to find the vertical force (Fv): (8) where c is the chord length and VT is the translational velocity of the wing accounting for stroke plane angle. We estimated Ωr directly from kinematic data by dividing the leading edge (defined by a line between shoulder, wrist, and wingtip) into equal segments and taking the Ωr for mean mid-stroke. Wing velocity increased toward the distal wingtip (Fig. 3B, wrist and wingtip). For upstroke, these calculations considered only the hand wing, as the proximal wing is flexed near the body with a low velocity (Fig. 3B, dashed line), and thus was assumed to be aerodynamically inactive. We substituted Ch for Cv in Eqn 8 to estimate Fh. The resultant of these forces then estimates net force production.
We computed polars of mean CL among the three birds sampled as a function of mean CD as well as mean lift-to-drag ratio (CL:CD) as a function of α. To accomplish this, we first interpolated 95 points for CL and CD, to account for subtle differences in α measured between wing samples. We calculated between α=–10 deg and α=85 deg using IGOR Pro (v. 6.01, Wavemetrics, Inc., Beaverton, OR, USA). We then computed a mean and standard deviation among birds for each point in the interpolated series. Herein, we report these means ± s.d.
The third wingbeat cycle following take-off was used in analyses. Angle of attack (Fig. 3A) ranged from 72 deg at the start of downstroke to –68 deg during upstroke. Mean downstroke α=48 deg, and during the mid-third of downstroke (Fig. 3A, blue) mean α=46 deg. Mean upstroke α=–2 deg, but it ranged from –68 to 71 deg. The mean α during the mid-third of upstroke (Fig. 3A, blue) was α=66 deg. Angular velocity (Ω) for the mid-third of downstroke was 69 rad s–1, and for the mid-third of upstroke Ω=96 rad s–1. Tangential velocity of the wingtip varied throughout the stroke cycle, averaging 10.1 m s–1 for mid-downstroke and 8.9 m s–1 for upstroke (Fig. 3B). Maximum wingtip velocity for downstroke was 12.8 m s–1; for upstroke it was 15.3 m s–1. Velocities along the wing measured from proximal to distal grew to a maximum at the wingtip (Fig. 3B, wrist and wingtip). However, during upstroke the proximal wing velocity remained low, likely due to being flexed against the body.
Propeller force coefficients
Coefficients for lift and drag plotted against α for all three birds revealed substantial lift production by the upstroke wings, with a general pattern that followed that of downstroke (Fig. 4). Upstroke-posture wings had a maximum CL of 1.42±0.08 at α=53.8 deg (Fig. 4A), and downstroke wings had a maximum CL of 1.77±0.43 at α=60.8 deg (Fig. 4C). For upstroke wings, the minimum mean CD was 0.18±0.09 at α=7.3 deg (Fig. 4B) while for downstroke wings mean CD was 0.10±0.03 at α=14.3 deg (Fig. 4D).
A diagram of CL as a function of CD revealed that the shape of the CL:CD polars was similar between wing postures (Fig. 5A). The difference in values appears to caused by both a lower CL and a higher CD for upstroke wings, due to the minimum and maximum values seen in Fig. 4. The slopes of the CL:CD polar for upstroke and downstroke wings are almost identical (Fig. 5A), as evidenced by the similar α at the maximum CL:CD (Fig. 5B). The maximum CL:CD for wings in downstroke posture was 6.3, which occurred at α=26.5 deg. Wings in upstroke posture had a lower maximum CL:CD of 2.5, at α=28.0 deg. At measured in vivo α=66 deg during mid-upstroke, the CL:CD is approximately 0.91. At this α, CL=1.33 and CD=1.46. The CL:CD for downstroke posture was 3.33 at the in vivo α=46 deg, with CL=1.54 and CD=0.49.
Bracketing in vivo performance
Various assumptions provided alternative estimates for in vivo force production. We observed minimum values for lift and drag if we assumed a bird was hovering, with no forward translational velocity (Eqns 5 and 6). In this instance, the bird (4.37 N) would produce a resultant force (FR) of 51.3% body weight (2.24 N) with both wings during mid-downstroke (α=46 deg). During mid-upstroke (α=66 deg), the pigeon could produce a FR of 107% body weight (4.71 N).
However, these values likely do not represent in vivo performance: during the third wingbeat following take-off, the bird has already reached a forward velocity of 2.9 m s–1. When including this translation in our estimation by incorporating translational velocity (VT) in a blade-element analysis (Eqn 8), mid-downstroke FR produces 8.7 N, 198% of body weight. With a stroke plane angle of 38 deg during downstroke, the FR is directed in front of the bird but near vertical, at 71 deg from the global horizontal plane, thus providing mostly weight support. During mid-upstroke, FR=5.43 N, 124% body weight. The upstroke stroke plane angle is 21 deg, and FR is oriented in front of the bird, 23 deg above the global horizontal plane, providing mostly thrust. These values probably represent extremes. Consider that mid-downstroke values appear to reflect the kinematics through the majority of downstroke (Fig. 3), while mid-upstroke kinematics reflects only a small portion of the entire half-cycle (Fig. 3). Extrapolating this calculation for the entire downstroke, which is approximately 58% of the wingbeat cycle, suggests that the bird produces a mean force of 115% body weight during downstroke. Throughout the entire upstroke, 42% of the wingbeat cycle, the mean force produced would be 36% bodyweight.
Given the debate over the functional significance of the tip-reversal upstroke (Brown, 1963; Spedding et al., 1984; Spedding, 1986; Aldridge, 1987; Hedrick et al., 2004; Tobalske et al., 2003), our most important result is that the model wings in upstroke posture produced a large amount of aerodynamic force (Figs 4 and 5). In fact, the downstroke and upstroke wings appear to perform similarly, both reaching maximum CL:CD at approximately α=26 deg (Fig. 5). This suggests that during mid-upstroke, significant force can be produced to help support weight or generate thrust, and the magnitude of this force is dependent on α in much the same way as for downstroke.
At in vivo angles of attack, we estimate the upstroke is capable of producing mean force as great as 36% body weight during this half of the stroke cycle. This value is within the range of lift produced by an active upstroke in hovering hummingbirds (33–48%) (Warrick et al., 2005; Warrick et al., 2009). Our results are supported by measures of feather bending during slow flight. Values of strain on primary feathers during upstroke are 43% that of downstroke (Corning and Biewener, 1998), suggesting that upstroke values reported here are not unreasonable for in vivo performance. The upstroke force we estimated also falls near the range of 14% the lift of downstroke measured with accelerometers in cockatiels (Hedrick et al., 2004).
Aerodynamically active upstrokes have been reported in several bat species that exhibit a similar wing-tip reversal pattern. The horseshoe bat (Rhionolophus ferrumenquinum) has an angle of incidence at the wingtip during upstroke of 70 deg (Aldridge, 1986), close to our reported value of 66 deg. Introductory analyses of changes in acceleration found an aerodynamically active upstroke during vertical flight (Aldridge, 1987) and hovering (Aldridge, 1991) of bats. In-depth kinematic analyses accounting for inertial effects will be crucial to validate these findings. More recent analyses of the wake in slow-flying bats using PIV show a vortex ring shed at the end of upstroke (Hedenström et al., 2007; Johannson et al., 2008), indicating both thrust and weight support produced on the upstroke. Further PIV analyses show that the role of upstroke changes to only provide weight support at higher speeds in bats (Wolf et al., 2010). Our results suggest that birds exhibiting tip-reversal upstroke may use aerodynamic mechanisms similar to bats during slow flight.
Our findings indicate that slow-flying pigeons are using their wings at a lower CL:CD than is potentially available to them (Fig. 5A) during both upstroke and downstroke. However, they are operating at α close to, or at, maximum lift performance (Fig. 5B). Peak CL for downstroke was 1.8 at α=61 deg, and for upstroke it was 1.4 at α=54 deg. At such high α on the upstroke, pigeons appear to be operating where lift and drag are almost equivalent (CL:CD=0.91). The resultant force during mid-upstroke is directed forward, 23 deg from global horizontal, due in part to high drag. This suggests that drag may play an important role during the upstroke, possibly to assist in weight support or thrust. This may also be the case for bats in slow flight (Aldridge, 1987; Norberg, 1976), and for chukar partridges during wing-assisted incline running (Tobalske and Dial, 2007).
Our propeller measurements for downstroke were consistent with previously published values for a pigeon wing in downstroke posture. Usherwood reports a maximum CL:CD of 5.4 for a pigeon wing spun on a propeller emulating slow flight (Usherwood, 2009), in comparison to our CL:CD of 6.3 (Fig. 5B). Maximum CL for the same wings was 1.64 (Usherwood, 2009), while our value was 1.77. Our slightly higher values are potentially due to differences in wing morphology (Table 1) and spinning speed. Estimated lift and drag values for downstroke from kinematic analyses of pigeons in slow flight (Berg and Biewener, 2008) yielded values of CL=1.44±0.29 and CD=1.01±0.08, within the bounds of our values. Notably, the s.d. variation for CL and CD between wings increases with higher α in downstroke wings, but equivalent variation did not occur for upstroke wings. During slow flight, unsteady aerodynamic effects likely dominate (Ellington, 1984). At such high α, this may include periodic leading edge vorticity (Warrick et al., 2005) detachment. This flow separation may lead to unpredictable variation in both CL and CD throughout the half-stroke. In contrast, wings in upstroke posture, with smaller observed s.d. at high angles of attack, may reduce these effects via the separation of the primary feathers, allowing individual feathers to function as individual airfoils with lower α (Brown, 1963). Further exploration using flow-visualization would help tease apart differences between in vivo and propeller aerodynamics.
It is important to note that values reported here represent values for mid-stroke. Our estimate of in vivo FR provide a lower bracket for downstroke of 51% bodyweight during hovering, and an upper bracket of 198% when accounting for translation (VT). The resultants from these calculations are directed up 71 deg from global horizontal, indicating the downstroke acts to support weight. The same calculation for upstroke suggests a bracket between 107 and 124% body weight. The upstroke resultant is directed forward, 23 deg from global horizontal due to high levels of drag, suggesting upstroke provides thrust. These values likely provide a reasonable estimate for downstroke. Even then, wing velocity is greatest at mid-stroke, when our estimates are based, and resulting forces should be proportional to the square of wing velocity (Ellington, 1984). The values for upstroke are likely an over-estimate, as we applied static coefficients to a highly unsteady event. In the upstroke, α and wing velocity are highly time variant (Fig. 3), and our estimation occurred at near peak wing velocity. Moreover, wing posture appears to change more dramatically throughout the upstroke when compared with the downstroke, and it may be the mid-stroke posture that is optimal for force production. Our kinematic analysis throughout the stroke cycle (Fig. 3) illustrates that as the hand wing pronates using a rapid ‘flick’ at the end of upstroke, a changing α occurs, as also observed to a lesser extent in the downstroke in the pigeon (Warrick and Dial, 1998) and cockatiel (Hedrick et al., 2002). Such constant variation in wing velocity and α in upstroke is also exhibited by hovering hummingbirds, and it appears that this helps to explain variation in circulation on the wing (Tobalske et al., 2007; Warrick et al., 2009). Thus, our final estimate for net force produced during both halves of the stroke cycle, using Ωr and accounting for the percentage of the total wingbeat cycle, is probably the most accurate for predicting pigeon performance.
Additionally, it is critical to note that the dried wing models do not accurately reflect all aspects of mid-downstroke or mid-upstroke. The preparation process, in which wings are isolated and dried, likely changes the mechanical properties of the wing. Similarly, the posture adopted by the models may not be an accurate representation of actual kinematics, as evidenced in slight differences in feather location between pigeon and propeller measurements (Fig. 2). While our model reflects the forces during an impulse, in reality the wing is morphing throughout most of the stroke cycle. For example, wing flexibility and compliance throughout a stroke significantly increases lift in insects (Mountcastle and Daniel, 2009; Young et al., 2009). During the stroke cycle of the pigeon, significant pronation and supination also occur. This results in wing camber and wing twist, which significantly improve aerodynamic forces in the locust (Young et al., 2009). The propeller model wings are probably unable to fully reflect these characteristics.
Our measure of α on the propeller is the geometric angle of attack, which does not account for induced velocity. It is likely that the angle of incidence (Vogel, 1994) was lower than our reported α. While our calculations of CL and CD account for local induced velocity (ε, Eqns 3 and 4), note that α is reported without accounting for local induced velocity. Thus, for example, a calculation for our mid-upstroke propeller model using a Rankine–Froude momentum jet model (Usherwood and Ellington, 2002a; Vogel, 1994) suggests that local induced velocity at the wingtip would be ∼1.26 m s–1. Therefore, the angle of incidence would be 52 deg, compared with 65 deg for α. To explore the possible impact of induced velocity throughout the stroke cycle, we present a simulated α throughout the stroke cycle (Fig. 3A, dotted line). This rough calculation using the same momentum jet model accounts for induced velocity during the downstroke, assuming that the induced velocity of the downstroke heavily influences the upstroke as it returns through the downstroke wake. In general, these effects appear minimal (∼10 deg), but have most pronounced effects at the downstroke–upstroke transition. Empirical measures of induced velocity would improve understanding of wing function in vivo, but will require methods such as PIV (Spedding and Hedenström, 2009) or hot-wire anemometry (Norberg et al., 1993).
The propeller, while a simplification of the wingbeat, provides new insight into the functional significance of the tip-reversal upstroke in avian flight. It has often been assumed that the tip-reversal posture in birds is simply to reduce drag by allowing air to pass through individual feathers, creating a Venetian blind effect (Brown, 1953; Brown, 1963). Thus far, hummingbirds are the only birds that are known to use an aerodynamically active upstroke during slow flight (Warrick et al., 2005). Our new measurements indicated that tip-reversal upstroke, widespread in birds with pointed wings, might be more similar to the hummingbird wingbeat than previously thought. If an aerodynamically active upstroke does offer advantages, why is it that not all birds use a tip-reversal? A future challenge will be to better understand the trade-offs that have led to many species with rounded wings using a flexed-wing upstroke.
We thank Ashley Heers and Terry Dial for help with the development our propeller model and methods of data analysis. We thank Brandon Jackson, Ashley Heers, Ken Dial and two anonymous reviewers for helpful comments on a draft of this manuscript.
Supported by NSF grants IOS-0923606 and IOS-0919799.
LIST OF SYMBOLS
- chord length
- coefficient of drag
- mean horizontal force coefficient
- coefficient of lift
- mean vertical force coefficient
- horizontal force
- resultant force
- vertical force
- torque about the motor
- radial distance along the wing
- wing length
- wing area
- second moment of area
- third moment of area
- wing translational velocity
- geometric angle of attack
- downwash angle
- air density
- angular velocity
- © 2011.