Stereoscopic particle image velocimetry measurements of the three-dimensional flow field of a descending autorotating mahogany seed (Swietenia macrophylla)

Seed dispersal distances are a fundamental aspect of plant evolution and conservation (Howe and Smallwood, 1982; Willson, 1993). In order to increase the distance of dispersal, samaras, or winged seeds, descend very slowly by spinning. Mahogany seeds initiate autorotation about their center of mass almost immediately after being released from the host tree by creating a prominent leading-edge vortex (LEV) that is similar to the flow structures that are responsible for the high lift generated by the wings of hovering insects such as fruit flies (Birch and Dickinson, 2003; Fry et al., 2005), hawkmoths (Van den Berg and Ellington, 1997a; Van den Berg and Ellington, 1997b), butterflies (Brodsky, 1991; Willmott et al., 1997), birds (Usherwood, 2009; Usherwood et al., 2005) and bats (Hedenström et al., 2009; Muijres et al., 2008). Ever since Maxworthy (Maxworthy, 1979; Maxworthy, 1981) discovered the LEV on the wings of a scaled-up model of the hawkmoth Manduca sexta, the attached LEV has been identified as the responsible for the production of high lift forces and has been subjected to considerable investigation (Birch et al., 2004; Bomphrey et al., 2006; Liu et al., 1998; Wu and Sun, 2004).

Two-dimensional (2D) studies have shown that the growth of the LEV begins at the start of translation and continues until the vortex becomes unstable, detaches from the leading edge (LE) and is shed into the wake (Dickinson and Götz, 1993). Several studies have been performed with the aim of characterizing the unsteady aerodynamics of insect flight (Brodsky, 1991; Ellington et al., 1996; Grodnitsky and Morozov, 1993; Somps and Luttges, 1985; Thomas et al., 2004). Dickinson et al. (Dickinson et al., 1999) performed direct measurements of the forces produced by flapping wings and suggested that the enhanced aerodynamic performance of insects results from three distinct, yet interactive mechanisms: delayed stall, rotational circulation and wake capture. Usherwood and Ellington (Usherwood and Ellington, 2002) determined the steady aerodynamic performance of wings in revolution at low and high Reynolds numbers (Re) from 1100 to 26,000 and reported that their aerodynamics are quite insensitive to variations in both wing morphology and kinematics, and that aspect ratio has remarkably little influence on aerodynamic force coefficients at low to moderate angles of attack. Birch and Dickinson (Birch and Dickinson, 2001) systematically mapped the flow generated by a dynamically scaled model insect at Reynolds numbers matching the flows relevant for most insects, and found that the structure of this spanwise flow takes different forms depending on the value of the Reynolds number. Their results demonstrate that flapping wings do not generate a spiral vortex akin to that produced by delta-wing aircraft. Also, they found that limiting spanwise flow with fences and edge baffles did not cause detachment of the LEV. Their findings support the hypothesis that induced downward flow from tip vortices and wake vorticity reduce the effective angle of attack and attenuate the growth of the LEV. Ellington (Ellington, 1984) proposed a scheme to include wing rotation with translation in quasi-steady models. Because the lift in insect flight was significantly higher than expected on the basis of quasi-steady aerodynamics, he suggested that important unsteady flow phenomena play a major role in the force generation process. Dickinson (Dickinson, 1994) investigated rotational parameters by measuring forces on the model wings of Drosophila. His results show that axis of rotation, rotation speed and angle of attack during translation are of great importance in the force development during each stroke. Sane and Dickinson (Sane and Dickinson, 2002) characterized the effect of wing rotation on the production of aerodynamic forces by a flapping airfoil and proposed a revised and improved model that incorporates rotational effects into a translational quasi-steady model of flapping flight. Lentink and Dickinson (Lentink and Dickinson, 2009) performed flow measurements and flow visualizations on revolving fly wings by employing air bubble visualization. Their results indicate that the LEV is stabilized by the ‘quasi-steady’ centripetal and Coriolis accelerations that are present at low Rossby numbers and result from the propeller-like sweep of the wing. Their findings support the hypothesis that the spanwise flow balances the formation of vorticity at the LE by draining it into the tip, and the authors conclude that force augmentation through stably attached LEVs represent a convergent solution for the generation of high lift forces. Lan and Sun (Lan and Sun, 2001) numerically investigated the aerodynamic properties of a wing of relatively small aspect ratio during typical unsteady rotational motions at low Reynolds number. Their results show that the airfoil in small Reynolds number flow can produce a large aerodynamic force, as effective as in large Reynolds number flow. Bomphrey et al. (Bomphrey et al., 2009) performed experiments using smoke visualization to describe the flow topology of free-flying bumblebees and suggested that wing rotation might play an important role in the formation of the LEV. Lentink and Gerritsma (Lentink and Gerritsma, 2003) studied numerically the aerodynamic performance of plunging airfoils at Re ~100, to assess the role of airfoil shape, concluding that the thin cambered airfoil outperformed a thick airfoil with respect to thrust coefficient and propulsive efficiency. Garmann et al. (Garmann et al., 2012) performed numerical simulations to examine the vortex formation, breakdown and effect of induced angle of attack about a finite-aspect-ratio revolving wing in quiescent flow for a range of Reynolds numbers (200–60,000). The authors determined that for the Reynolds numbers examined, the overall vortex structure including the attachment of a LEV was mostly insensitive to transitional effects despite the apparent vortex breakdown and shear layer instabilities observed at higher Reynolds numbers. Also, although the LEV was found to weaken with increased angle of attack, it still remained attached to the wing surface.

Particle image velocimetry (PIV)-based methods have been extensively used to study the airflows induced by the flapping wings of animals in free and tethered flight and by fixed mechanical model flappers. Lu and Shen (Lu and Shen, 2008) studied experimentally the 3D flow structures and evolution of the LEV on a flapping wing in the hovering condition. Their results showed that the LEV system is a collection of four vortical elements: one primary vortex and three minor vortices. Lu et al. (Lu et al., 2006) performed dye flow visualization and PIV measurements on flapping wings to reveal the detailed features of the LEV region and confirmed the existence of dual LEVs. Ansari et al. (Ansari et al., 2009) studied experimentally the flow around a constant-speed rotating wing and described the flow features of the LEV for two Reynolds numbers (500 and 15,000). The aerodynamic characteristics of autorotating seeds have been analyzed theoretically and experimentally (Azuma and Yasuda, 1989; Yasuda and Azuma, 1997). Recently, Lentink et al. (Lentink et al., 2009) identified the LEV as the high-lift source in rotary seeds and described its detailed flow structure in terms of vorticity, spanwise flow and vorticity transport in and behind the vortex for spanwise regions at 25, 50 and 75% span. The LEV involves LE flow separation that reattaches to the seed and forms a region of high vorticity concentration. Once the flow field separates at the LE and vortex generation occurs, a large suction force accounts for the generation of large lift forces. More recently, Ozen and Rockwell (Ozen and Rockwell, 2012) studied experimentally the effects of rotation of a rectangular plate and characterized the LEV in relation to the overall flow structure to assess the effects of centripetal and Coriolis forces in the absence of rotational acceleration effects for Reynolds numbers from 3600 to 14,500. Their results show that a stable LEV exists for values of effective angle of attack ranging from 30 to 75 deg, and that at a given value of angle of attack, the scale and form of the LEV is relatively insensitive to Reynolds number.

The aim of the present investigation is to perform a detailed description of the flow pattern of autorotating mahogany seeds using stereoscopic digital PIV (DPIV) and evaluate their aerodynamic performance. Because the unsteady flow field of wind-dispersed seeds exhibit complex flow structures that feature the formation of LEVs, results are particularly presented to provide a detailed description of the mechanism responsible for its prolonged attachment via the instantaneous 3D velocity field and 2D velocity and vorticity contour maps.

Mahogany seeds (Swietenia macrophylla King) were obtained from the tropical woods of Veracruz, Mexico. The samples were manually collected immediately after they had fallen naturally. Prior to the experiments, the samples were preserved in a humidifier to prevent the evaporation of moisture. The experiments were carried out in a low-speed, vertical wind tunnel (0.6×0.6×0.5 m Plexiglas working section) crafted to study the flow and kinematics of descending mahogany seeds. A schematic diagram of the experimental setup of the vertical wind tunnel and DPIV system is shown in Fig. 1. Flow uniformity was achieved by placing flow straighteners at the tunnel's inlet and outlet sections. The measured turbulence intensity of the free-stream velocity was less than 1%. Free-stream velocity was adjusted with the terminal velocity of the seeds so that they spin at a stationary height. The Reynolds number based on the seeds' terminal velocity (, where U_∞ is upstream uniform free-stream velocity or descent speed, is the mean chord length of the seed and ν is the kinematic viscosity) ranged from 1474 to 1866, which corresponds to a range of airspeed of U_∞=1.2–1.6 m s⁻¹. Because samara seeds descend vertically with a small side-slip, in order to perform measurements at the same spanwise locations, the seeds were initially fixed by means of a 0.3-mm-thick fishing line (Araty Superflex fishing line, Sao Paulo, Brazil) that went through the wing's center of rotation. To verify that the limited degrees of freedom and the friction exerted by the fishing line do not have a significant influence on the flight kinematics and the angular velocity of the freely falling seeds, rate of descent, rotational speed and coning angle were compared against those obtained with the same seeds spinning around a fishing line. As no appreciable changes in the mean values were obtained for any of the samples, it was assumed that there was no artifact introduced by fixing the seeds to the fishing line. To minimize the scatter of laser light, the seeds and the fishing line were painted mat-black and monochromatic filters appropriate for the laser wavelength were installed on the lenses to minimize stray light from other sources. The small amount of ink increased the weight of the seeds by an average of 1%. To reduce errors, the laser and the cameras were kept stationary and the wind tunnel was mounted on a positioning translation system. By moving it in a direction perpendicular to the laser sheet, measurements at the various spanwise locations of the seed were accomplished without displacing the DPIV apparatus.

Velocity field measurements were performed using a Dantec Dynamics stereoscopic DPIV system (Skovlunde, Copenhagen, Denmark). The plane was illuminated with a 200 mJ double pulse Nd:YAG laser (New Wave Gemini, Fremont, CA, USA) at a wavelength of 532 nm. All measurements were performed using a pulse separation interval of 50 μs. The laser was equipped with a cylindrical lens system that produced a diverging light sheet parallel to the flow direction with a thickness of ~6 mm. Dantec Dynamics software (DynamicStudio version 3.0.69) was used to control the cameras and laser for DPIV data acquisition. Laser/camera synchronization was achieved using an automated trigger equipped with a position sensitive infrared detector and a digital delay/pulse generator (Sharp GP2Y0A21, Las Vegas, NV, USA) so that all image pairs were recorded when the seed was aligned perpendicular to the laser sheet. Fig. 2 illustrates the configuration of both cameras, the illuminated plane by the laser sheet and the location of the seed when the pulse generator triggers synchronized image acquisition and laser pulses. Also, the orientation of the Cartesian rectangular coordinates is shown, with x, y and z in the streamwise, transverse and spanwise directions, respectively. Two 12 bit CCD double-frame digital cameras (HiSense MKII, Skovlunde, Copenhagen, Denmark) with a resolution of 1344×1024 pixels and equipped with 50 mm lenses on Scheimpflug mounts were used to record image pairs. Because the size of the seeds varies from one sample to another, the spanwise separations were obtained by displacing the laser sheet to distances that correspond to 0.25R, 0.3R, 0.5R, 0.75R and 0.875R, where R is the wingspan (see Fig. 3). Measurements were also taken at the wingbase and the wingtip. Seeding was supplied from a smoke generator (Antari Z-1500II Fog Machine, Taiwan, ROC) placed at the tunnel intake, and seeding quantity was regulated by monitoring the output from the DPIV system (particle size 1 μm). Because the seeds initiated a rotational motion immediately after being exposed to a vertical flow, it was possible to fix the seeds in a specified position of the test section by adjusting the flow speed. By repeating the DPIV measurements for 20 samples using 10 sets of 50 image pairs per spanwise section for each seed, instantaneous 3D velocity vectors and 2D velocity and vorticity contours were obtained.

For each image pair captured, 32×32 pixel interrogation areas with 50% vertical overlap and 25% horizontal overlap were used. An adaptive-correlation PIV algorithm with five high-accuracy subpixel refinement steps yielded a 55×63 array of vectors. Data validation was carried out using peak validation with a minimum peak height relative to peak 2 of 1.15. Local neighborhood validation was performed using a moving average validation algorithm with an acceptance factor of 0.12. A 3×3 filter was used to smooth the vector fields in order to clearly define the instantaneous 3D flow structures. Deleted values were filled using an interpolation method based on the surrounding vectors from a 3×3 nearest neighbor matrix, resulting in an average percentage of spurious vectors of roughly 2%. The computed 3D vector and vorticity fields for each spanwise location were displayed using Tecplot 360 (Bellevue, WA, USA).

The geometry and kinematics of rotary seeds have been described previously (Azuma and Yasuda, 1989; Greene and Johnson, 1990; Minami and Azuma, 2003; Yasuda and Azuma, 1997). To characterize representative kinematics, images of the rotary seeds spinning at a stationary height during free fall were recorded using a high-speed digital camera (Redlake, HG-100K/HG-LE, Pasadena, CA, USA) at a frame rate of 250 frames s⁻¹, as this was sufficient to resolve the rotational motion of the seeds. Images were captured on a homogeneously illuminated white background to obtain good contrast. The coning angle (γ) was computed from static images in which the seeds lie orthogonal to the camera direction. Because the seed's dimensions along the spanwise direction are known, the coning angle was inferred using the projected length of the seed's total length along the horizontal plane (Varshney et al., 2012). An accurate spinning rate of autorotation was inferred by measuring the elapsed time between consecutive images where the seed lies orthogonal to the camera direction. Table 1 shows the measured geometry and kinematics of the seeds. Results are displayed as means ± s.d., and the sample size was N=20.

When displaying 3D flows in the plane, the flow structure varies depending on the location of the observer relative to the wing. For all of the images shown, the origin is located at the lower right corner, as in Fig. 2. For all spanwise locations, the origin of the z-coordinate is located at the central part of the plane illuminated by the laser sheet, where z is positive in the direction towards the wingbase and negative in the direction towards the wingtip. Velocity and vorticity contours are displayed in the x–y plane located at z=0, and they are a 2D projection of the real 3D vectors using velocity components in the plane of interrogation. For all images presented, the brown contour denotes the size and location of the wing section within the flow. Red/yellow coloration represents positive vorticity or clockwise fluid rotation, while blue/purple coloration indicates negative or counterclockwise rotation. Here, green regions reflect a lack of rotational motion.

Fig. 4 shows the instantaneous vector map and velocity and vorticity fields at the wingbase. Note that for this spanwise region the laser light sheet does not illuminate the region below the LE when the pulse generator triggers synchronized image acquisition and laser pulses. Therefore, a mask has been applied to the region with bad stereoscopic PIV correlation. The free-stream flow direction is vertical upwards and parallel to the x–y plane (upper left image), and the unperturbed free stream can be seen at the lower left and right corners of the latter. Although the spanwise velocity at this position is over 1 m s⁻¹, which is plenty for flow separation under the right conditions, no flow separation occurs on top of the airfoil and therefore no LEV exists for this spanwise region. However, it can be seen that the sheer layers above the LE and the trailing edge (TE) of the airfoil serve as a vorticity source, because two regions of relatively small vorticity with opposite-sign rotation are generated (lower right image). However, although their sizes are relatively similar, the left-hand clockwise vorticity on top of the LE is stronger than the right-hand counterclockwise vorticity on top of the TE. The two regions of vorticity located at the upper right corner of this image illustrate how vorticity from a previous revolution is shed into the wake.

Fig. 5 shows how for a spanwise location of 0.25R, the flow separates at the LE and a small LEV appears above the airfoil. Here, a mask has been applied at the lower left region of the velocity and vorticity contours, as noise from the PIV signal that is coming from the scattered laser light reached a peak at this spanwise location. Nonetheless, the detailed flow structure is well resolved above the airfoil. For clarity, instead of displaying the instantaneous 3D velocity vectors, the corresponding sectional streamlines are depicted (upper left image) spiraling into a critical point, which is a discernible distinction for the presence of the LEV, with its core highlighted by the red dotted line. The LEV core refers to the region of the curve demarcated by limit cycles defining the LEV trajectory. A slight increase in the region of clockwise vorticity on top of the LE and a dramatic drop in the region of counter-clockwise vorticity next to the TE demonstrate that flow separation at the LE creates and feeds vorticity to the LEV. A strong spanwise flow towards the seed tip has been measured. The resulting flow strain ∂W/∂Z close to 100 s⁻¹ stretches the LEV, thus increasing the LEV intensity and decreasing its size, promoting the LEV to be maintained at the seed's upper surface. This velocity gradient in the spanwise direction is mainly generated by the centrifugal force of the spinning seed.

Fig. 6 displays the instantaneous flow pattern and the instantaneous velocity and vorticity contour plots at a spanwise location of 0.5R. For this spanwise region, a well-defined LEV is located on top of the airfoil. Here, the W velocity component above the LE is more than twice the value of the free-stream velocity. Also, a region of concentrated vorticity is consistent with a growing LEV, and an excellent match between the location of the maximum out-of-plane vorticity in the local flow field and the location of the maximum spanwise velocity was found. Inspection of the instantaneous velocity vectors and contour plots readily reveals that the influence of the spanwise flow through a spiral vortex extends to a region downstream with respect to the free stream, as can be seen from the U velocity contour plot (top middle image).

Fig. 7 shows the instantaneous 3D velocity vector field, the instantaneous sectional streamlines, and the instantaneous velocity and vorticity fields at 0.75R. For this spanwise region, the size of the LEV core, highlighted by the dotted red line, reaches a maximum and extends to approximately the size of the wing chord. Again, the largest value of the spanwise velocity (normal to the laser sheet) is observed below the core axis. Hence, the spanwise pressure distribution is affected accordingly, causing the lift force to reach a maximum at the outer third of the wing. This supports Norberg's analysis of Acer platanoides (Norway maple) samaras, where the maximum lift force measured is located in the region that corresponds to the outer third of the wingspan (Norberg, 1973).

The instantaneous flow pattern at 0.875R is shown in Fig. 8. Again, the instantaneous 3D velocity vector field, the instantaneous sectional streamlines, and the U, U_ϕ and W velocity and vorticity fields are displayed. For this spanwise region, the sectional streamlines (upper middle image) spiraling into a critical point demonstrate that the LEV is still bound to the wing surface. However, despite its reduction in size and less coherent structure, the flow keeps separating at the LE creating vorticity. Although its size is smaller than for 0.75R and the magnitude of the spanwise flow starts to drop, its value is still up to twice the value of the free-stream velocity. Also, the reduction in the size of the LEV core highlighted by the red dotted line (upper middle image) affects the vorticity field accordingly, and its value increases when compared against its value for 0.75R.

Fig. 9 shows the instantaneous 3D velocity vectors, and the U, U_ϕ and W velocity and vorticity contours at the wingtip. Here, the wingtip is located at the region with the largest value of the tangential velocity component, which in this case corresponds to 2.7 m s⁻¹. The instantaneous velocity vectors illustrate how for this spanwise location, no LEV is present at the wingtip, suggesting that the LEV re-orientation/inflection into the tip vortex takes place in a spanwise region between 0.9R and the wingtip. This is supported by the displayed velocity and vorticity contours at the wingtip, as there is no sign of a long trail of vorticity in the seed's wake and the instantaneous velocity vectors show that the free stream is only perturbed by the wingtip at this location.

In this subsection, the flow patterns for each spanwise location shown previously are displayed using a non-inertial frame. In order to show the flow structure in an inertial frame, the non-inertial frame is transformed to an inertial frame by adding the local rotational velocity of the wing at each spanwise location. In this way, the vortical flow structure is obtained for stationary wing experiments. It should be noted that both the W velocity and vorticity fields remain unchanged when using an inertial frame.

Fig. 10 shows the instantaneous flow field and contour maps at the wingbase. Here, a slight deflection of the instantaneous velocity pattern is observed due to an increase in the local rotational velocity.

Fig. 11 shows the instantaneous flow field at 0.25R. Comparing these results with those in Fig. 5, an increase in rotation relative to translation affects the pattern accordingly, generating circulatory forces above the wing, which in turn enhance spanwise flow.

The flow pattern at 0.5R in an inertial frame is shown in Fig. 12. Clearly, a remarkable change in the flow pattern is observed, whereas the value of the local rotational velocity matches the value of the free-stream velocity with a consequent increase in the total velocity defined in the following subsection. Hence, the incoming flow of the free stream no longer has an upward direction. Instead, the instantaneous 3D velocity vectors in the upper left image of Fig. 12 have an orientation of ~45 deg with respect to the horizontal. Also, clockwise fluid rotation reaches a maximum at the LEV core, which coincides with the location of the maximum spanwise velocity.

The instantaneous 3D velocity vector field, velocity and vorticity fields at 0.75R in an inertial frame are shown in Fig. 13. For this spanwise region, a very well-defined vortex core that reaches its maximum stability is defined by the spiraling flow pattern. It is to be noted that the size and strength of the vortex core at this spanwise location are consistent with the maximum measured value of the tipward spanwise flow above the airfoil, which is three times the value of the free-stream velocity. Also, it is observed that an increase in the size of the LEV core and a decrease in the value of positive vorticity take place. By inspecting the measured flow pattern at each spanwise location, it is clear that a gradual increase in the size, strength and structure of the LEV along the length of the wing confirm that the LEV structure is that of an outboard spiraling straight cone pattern.

The instantaneous 3D velocity vector field, velocity and vorticity contours at 0.875R in an inertial frame are shown in Fig. 14. Here, the rotation of speed at the LEV core slows down, its strength and stability become weaker, and a decrease of the spanwise velocity is observed. In Fig. 15, the measured flow at the wingtip in an inertial frame shows how the local rotational velocity is maximum, reaching a value of almost three times the free-stream velocity.

In this work, the detailed flow structure of the LEV of descending autorotating mahogany seeds (S. macrophylla) has been revealed through stereoscopic DPIV with high spatial resolution. Flow measurements have been performed to quantitatively measure the airflow and assess the underlying flow structures and spiraling nature of the LEV. Our experimental findings demonstrate that the LEV is responsible for the generation of strong spanwise flow above the airfoil, which in turn produces a steep increase in the pressure differential across the wing, enhancing lift generation. Results show that the separated boundary layer at the LE rolls up to form the LEV. The latter has a conical straight shape and increases in size along the spanwise direction, starting at 0.25R and reaching its maximum stability and spanwise velocity at 0.75R. At a wingspan region between 0.9R and the wingtip, the strength and stability of the vortex core decreases and the LEV is reoriented into the tip vortex, thus convecting away the vorticity. We observed that for all wingspan regions demarcated by a stably attached LEV, all of the velocity and vorticity plots show a peak of local spanwise velocity and vorticity at the LEV core. We have shown that the LEV has two different regions, each with distinguished features. One is the vortex core (with a large spiral structure) and the other is an outer vortex region with small spanwise flow. Based directly on the measurements of the 3D flow field, the aerodynamic performance of mahogany seeds in autorotational flight during optimal flight conditions has been correlated with the aerodynamic force they produce, which allows an understanding of the role of the LEV in the lift-generation process associated with strong spanwise flow towards the wingtip. The measured mean thrust coefficients were calculated as the average of the corresponding instantaneous forces over the entire wing, and our data reveal that the measured aerodynamic forces yield results that are very similar to those of other autorotating seeds.

In the present study, the airflow of autorotating mahogany seeds (S. macrophylla) was quantitatively measured using stereoscopic DPIV. In addition, evaluation of their aerodynamic performance has been presented. The results of the present study show that high flow strain and strong spanwise flow within the LEV core, with velocities as high as 148% of the wingtip speed, are responsible for augmenting lift forces. Based on the measured flow patterns, the following conclusions can be drawn:

1. No LEV is present at the wingbase.

2. At 0.25R, the flow separates at the LE, a region of positive (clockwise) vorticity is generated, the LEV appears above the airfoil, and the separated flow reattaches close to the TE.

3. The LEV increases in size and strength while spiraling outward towards the wingtip with a straight cone pattern, which is responsible for enhancing lift generation. The strong flow strain in the spanwise direction stretches the LEV, thus increasing its intensity and promoting the attachment of the latter against the airfoil.

4. The LEV core and spanwise velocity reach a plateau at 0.75R and decrease in size and strength at 0.875R.

5. In a spanwise region between 0.9R and the wingtip, the LEV is reoriented into the tip vortex.

The authors gratefully thank Systelectro S.A. de C.V. for providing technical assistance with the DPIV system. The authors would like to express their sincere thanks to Ruben Mil and Uriel Suárez for their assistance in the experimental tests.

 
• A_SD

  disc area swept by rotating wing

 
• c

  normalized local chord length

 
• 

  mean wing chord length

 
• c_L

  normalized local lift coefficient

 
• C_L

  local lift coefficient

 
• C_T

  mean thrust of vertical force coefficient based on the wing planform area

 
• D

  drag

 
• DPIV

  digital particle image velocimetry

 
• g

  gravitational acceleration

 
• l

  wingspan

 
• L

  lift

 
• LE

  leading edge

 
• LEV

  leading-edge vortex

 
• M

  wing mass

 
• PIV

  particle image velocimetry

 
• r

  local radial distance from the wingbase

 
• R

  distance from the wingbase to the wingtip

 
• R_D

  distance from the seed's center of rotation to the wingtip

 
• Re

  Reynolds number based on the mean wing chord length

 
• R_W

  distance from the seed's center of rotation to the wingbase

 
• S

  wing planform area

 
• T

  thrust

 
• 

  mean thrust force

 
• TE

  trailing edge

 
• U

  velocity in the x-direction

 
• U_∞

  upstream uniform free-stream velocity (descent speed)

 
• U_T

  relative speed of wing element or total velocity

 
• U_ϕ

  velocity in the tangential direction

 
• W

  velocity in the z-direction

 
• W_L

  wing loading based on the planform area

 
• W_LD

  wing loading based on the disc area

 
• x, y, z

  Cartesian coordinate system

 
• β

  angle between the horizontal and the relative velocity direction of the flow

 
• ε

  R_W/R_D

 
• γ

  coning angle

 
• ν

  kinematic viscosity

 
• ξ

  r/R_D

 
• ρ

  fluid density

 
• σ

  solidity

 
• ω

  out-of-plane vorticity (normal to the x–y plane)

 
• Ω

  rotational speed or spinning rate