Unsteady aerodynamic characteristics of a translating rigid wing at low Reynolds number

Due to its inherent simplicity, the rigid thin flat-plate has been the subject of many research efforts in the fluid dynamics community. Building upon the rich literature of accelerating finite wings in the low Reynolds number flight regime,^1–5 we consider here the streamwise component of rectilinear acceleration, over a range of rates and incidence angles. Despite the span of extant literature, there are still many complex flow phenomena occurring in this flight regime that are not presently well understood, such as transient separation, transition, and reattachment.

Of recent popularity has been the application of flapping-wing flight, whether in natural flyers^6,7 or manmade vehicles of comparable scale. Observing natural fliers (especially those capable of hover) has provided insight into efficient low Reynolds number flight mechanics and kinematics. Evidence of wing rigidity for the smallest flyers^8–10 has met with numerous elegant demonstrations of wing flexibility at higher Reynolds numbers.^11–13 While capturing structural flexibility is essential for understanding the actual sectional incidence-angle history and the resulting aerodynamic performance, the rigid-wing abstraction remains inviting for investigations of the role of principal vortical structures, the imposed kinematics, and the resulting aerodynamic forces. Of principal interest for flight-control applications of flapping wings¹⁴ is understanding the extent to which the lift history is quasi-steady with motion history. Studies mapping the in-flight wing kinematics of natural fliers^11,15 have found that the kinematics typically consist of translational, rotational, and pitching motions. For in-depth studies of the fundamental fluid mechanics of such flight, it is common to isolate the individual wing motions to analyze them separately. Such is done in the present study in the form of a linearly translating rigid wing accelerating from rest.

Wagner’s classical 2D attached-flow solution¹⁶ for impulsive-start (equivalently impulsive change in free-stream speed, after subtracting buoyancy or inertial effects¹⁷), or impulsive change in angle-of attack, must of course be modified for experimentally realizable acceleration rates. This was done for 15° incidence-angle by Pitt-Ford and Babinsky,¹⁸ who argued that while bound circulation on the plate was essentially zero, vorticity ascribed to the leading-edge vortex (LEV) is responsible for producing circulation not unlike that of the Wagner function. Further, Pitt-Ford and Babinsky, following Brennan¹⁹ and Lamb,²⁰ obtained a solution for the apparent-mass contribution during the acceleration-phase of the plate’s motion. In Wagner’s treatment, this would have been a delta-function. Here we extend Pitt-Ford’s treatment to higher incidence-angles and slower accelerations, and propose a kinematic scaling of lift history versus acceleration-rate.

The so-called “formation number” has been introduced to assess saturation of shed-vortex strength for plates moving rectilinearly normal to themselves,^21–23 at incidence angle²⁴ and in rotational motion. For the latter, experimental^25,26 and computational²⁷ work has shown a leading edge vortex that is steady with respect to the plate for arbitrarily long time after the plate completes its acceleration from rest and continues revolving at steady rate. However, in the rectilinearly translating plate, the leading edge vortex detaches from the plate and is followed by successive such vortices after a steady relative free-stream has been achieved.²⁸ The qualitative size and kinematics of the leading edge vortex is relatively well documented in the aforementioned literature. However, the exact contribution to lift from the leading edge vortex is at present unclear. This study aims to take steps in rectifying this uncertainty.

In the present work, we compare particle image velocimetry (PIV) and qualitative flow visualization with direct measurement of lift, drag, and pitching moment for a range of smoothed rectilinear acceleration profiles, where the acceleration (apart from endpoint smoothing) is constant with wall-clock time for early time and then goes to zero. The goal is to segregate circulatory from noncirculatory contributions by tracking leading-edge and trailing-edge vortex trajectories, and to attempt to collapse lift coefficient variations for a range of acceleration rates. The philosophical goal is a closed-form or reduced-order model purely derived from kinematics, with no internal-state variables²⁹ or evolution-equations for shed vortices.³⁰ 

Experiments were performed at the U.S. Air Force Research Laboratory (AFRL) at Wright Patterson Air Force Base in the Horizontal Free-Surface Water Tunnel, shown in Figure 1. The water tunnel has a 4:1 contraction ratio, a test section 46 cm wide by 61 cm high, and a speed range of 3–45 cm/s. A three-degree-of-freedom motion rig is fitted to the tunnel and consists of a triplet of H2W linear motors, driven by AMC DigiFlex servo-drives controlled by a Galil DMC 4040 4-channel card. Each channel is carefully configured with user-selected proportional/integral/derivative (PID) constants. Each motor is programmed independently so that the desired angle of attack and horizontal position time-history of the model are converted to position commands for each linear motor. This allows for single degree-of-freedom motions such as pure linear translation. Shown in Figure 1, the wing used in this study was a thin, rectangular flat plate with aspect ratio (AR) 4 and a 75 mm chord. To address the possibility of tunnel blockage, a study was performed on several different wing sizes, and it was confirmed that the present wing geometry produces negligible blockage effects.

Figure 2 shows the velocity profiles describing the wing’s kinematics. Velocity profiles were generated using the Eldredge function,³⁰ which provides a nearly trapezoidal profile with respect to time and includes a smoothing parameter to round the corners of the profile, which minimizes rig vibrations. Purely surging motions did not require significant corner smoothing and the velocity profiles were made as nearly trapezoidal as the motors would allow. During all of the experiments presented here (with the exception of static measurements), the tunnel was not running and was used as a towing tank. The wing was driven in pure rectilinear translation at a fixed angle of attack, α. All velocity profiles are linear with respect to time, accelerating to a target velocity and carrying out the rest of the motion at this terminal speed. To prevent excessive vibrations of the model, placing undesired stress on both the driving motors and the model, a smoothing transient during the beginning and end of the acceleration phase was applied. The distance over which the wing accelerates, s_a, is normalized by its chord length, c, and determines the length of the acceleration region. Figure 2 illustrates the parameter space of the investigation into the effects of wing acceleration, ranging from a slowly accelerating case over 6 chord-lengths of wing travel to a fast case accelerating over 0.125 chords. Additionally, an angle of attack study was performed on the s_a/c = 1 case for α = 5°, 7°, 8°, 10°, 15°, 20°, 30°, 35°, and 45°. Because this study is solely focused on the effect of the startup transient and subsequent constant-speed translation, the deceleration region is not of present importance. Thus, all figures in Secs. II B and II C will be truncated at s/c = 14, the point at which wing deceleration begins.

Several measurement techniques were used in the present work: direct force measurements, dye flow visualization, and PIV. Force measurements were taken with an ATI Nano25 6-component load cell connected to the plate’s center plane via a 2.25 in tall bracket. The manufacturer’s quoted uncertainty bounds for the Nano25 load cell are ≤0.28 N, which is commensurate with the highest 95% confidence interval in dimensional normal force presented here. Force measurements were recorded at 1 kHz and low-pass filtered in hardware at f = 18 Hz. A moving average of 101 points was then taken to remove residual rig vibrations from the processed data for visual clarity and qualitative observation of trends. All quantitative analysis, however, was performed using un-averaged data. Finally, a fourth-order Chebychev II low-pass filter with a −20 dB attenuation of the stopband was applied. The low-pass filter cutoff frequency is lower than the frequency of the non-circulatory load spikes, but instead of completely eliminating them with the stopband, the signal is attenuated so they are still present in the data but of lower magnitude. To offset the time-shift applied to the data in the passband, a forwardbackward filtering technique was applied using the MATLAB command filtfilt, eliminating any phase shifts in the filtering process. All force measurements were performed at a chord-based Reynolds number of 20 000.

Flow visualization of the leading edge vortex was performed using fluorescent dye and planar laser fluorescence. A high concentration of Rhodamine 6G in water was injected at the leading and trailing edges at the 3/4 semispan location by a positive-displacement pump with a prescribed volumetric infusion rate. The dye was injected via a set of 0.5 mm internal diameter rigid lines glued to the bottom surface of the plate, which can be seen as the metallic tubing in Figure 1, and was illuminated by a Nd:YLF 527 nm pulsed laser sheet of 2 mm thickness at 50 Hz. Images were recorded with a PCO DiMax high-speed camera through a Nikon PC-E 45 mm Micro-lens. A Tiffen orange #21 filter removed the reflected 527 nm laser light. Since the dye fluorescence wavelength was 566 nm, the resulting images present only the flow structures that contained the dye. The camera and laser operation were externally synchronized to the wing motion via a Quantum Composer timing box. These images also provide semi-quantitative data, such as the relative size of vortices and the vortex shedding frequency. All flow visualization experiments were performed at a chord-based Reynolds number of 2500.

PIV measurements were performed using the same laser and camera setup to illuminate 5 μm Vestosint seeding particles. Images were acquired at 667 Hz, i.e., dt = 1.5 ms, and all measurements were performed at Re = 20 000. PIV processing was performed on Lavisons DaVis 8.1 software. A multipass, variable window-size processing method was implemented. The first pass used a 48 × 48 pixel window with 1:1 square weighting and 50% overlap. The next two passes used 32 × 32 pixel windows with 1:1 circular weighting and 75% overlap. Vector post-processing was kept to a minimum by only utilizing a remove-and-replace median filter of 2 standard deviations.

Force and moment measurements were obtained directly from the force balance. Nondimensionalizing the forces and moments using the terminal velocity, U, results in

To further understand the physical sources of lift during the acceleration phase, force measurements were broken down into two components: circulatory and non-circulatory. When a body accelerates through a fluid, it experiences an inertial force often referred to as an “added-mass” effect. This non-circulatory force can be thought of as the reaction force by the added mass of the fluid being accelerated by the wing. Pitt Ford¹⁸ provides a complex potential analysis, via the unsteady Blasius equation and including the inherent assumptions of attached, inviscid, and incompressible flow, for an accelerating rigid flat plate in pure translation and derives the lift coefficient contribution from added-mass as

where α is the plate’s angle of incidence and U is the plate’s instantaneous velocity. For the case of constant acceleration, the case in this study, this can be simplified to

where a is the number of chord-lengths traveled during acceleration. From Eq. (4), it is evident that the effect of added mass increases with angle of attack. When the wing ceases acceleration, however, the added-mass term goes to zero. The remaining “circulatory” lift may consist of contributions from both bound circulation and leading edge vortices. However, Pitt Ford¹⁸ concluded that at early time in the trajectory of a flat plate at large angle of attack surging from rest, the bound circulation is negligibly small and, therefore, the lift must be caused only by external vortices and non-circulatory effects. Thus, the focus on physical sources of lift will remain on the added mass term during acceleration and the formation of leading and trailing edge vortices throughout translation.

The canonical comparison for impulsively started wings is to Wagner’s problem, which accounts for delay in circulation development on an impulsively started flat plate.¹⁶ Jones’ approximation³¹ algebraically expresses Wagner’s analytical function for lift produced by finite aspect ratio, impulsively started wings as

where

and s is the distance traveled in chord-lengths.

A common method for identifying a vortex center is to use the Γ₁ equation proposed by Graftieaux et al.³² This function, applied to the velocity fields from PIV measurements, characterizes the extent to which the flow streamlines are circular around a point, P, and is given as

where S is the area of integration and θ is the angle between point P and the velocity vector at dS. The value of Γ₁ at each point in the vector field is the sine of the angle between the vector from point P to a nearby velocity vector and the velocity vector itself. Thus, a Γ₁ value of 1 indicates purely tangential flow and one that is highly rotational about point P. This equation can be applied to the entire velocity field to yield a scalar field ranging from −1 to +1, where the sign indicates direction of rotation. Typically, Γ₁ is thresholded to a value of $Γ 1 ≥0.6$ to retain only the areas of strong rotation and that has been done in this study. The location of the vortex center can then be calculated as the centroid of the thresholded Γ₁ regions.

As explained by Manar et al.,³³ a benefit of implementing this method is that it incorporates spatial averaging that attenuates measurement noise, resulting in smooth, contiguous regions that ease the vortex identification process.

The circulation on the upper surface of the wing was calculated using the “overwing box” method described by Manar et al. This method employs a rectangular box extending from the trailing edge to slightly forward of the leading edge and a height of 1 chord in the wing-normal direction. At each time step, all vorticity within this box is integrated to obtain a time history of circulation. The present implementation of this method makes no attempt at distinguishing flow structures, and thus provides a “total circulation” contained above the wing. This method was chosen based on its simplicity, robustness, and ease of implementation for any wing geometry and kinematics.

The primary focus of this study was to identify the effect of wing kinematics on the forces and moments produced by translating low aspect ratio wings. Observation and quantification of flow structures help to explain the trends in the time histories and provide insight into individual contributors to lift. Sections III A–III C will explore the effects of angle of attack and acceleration distance, s_a/c, on coefficients of lift and pitching moment for a rigid flat plate wing. Results are also compared to steady-state values as measured on a static wing in a freestream provided by the water tunnel to determine when it becomes reasonable to apply a quasi-steady approximation.

The effect of angle of attack on force coefficients was investigated to ascertain the existence of high lift on translating wings, pinpoint the range of incidence angles over which vortex lift becomes important, and evaluate the time scales over which it persists. Time histories of lift, drag, and pitching moment were measured and normalized with respect to the wing’s terminal velocity.

The flat plate wing was set at a fixed angle of attack and accelerated to its terminal velocity over 1 chord-length of travel. Angles of attack ranged from 5° to 45°. The resulting lift force histories are given in Figure 3. At low angles of attack, α = 5°-10°, the lift coefficient curves agree well with Wagner’s prediction (shown in the figure as Jones’ approximation³¹). At these low angles of attack, the flow is largely attached (see Figure 4), as is assumed in Wagner’s analysis. In an attached flow, no leading edge vortex forms and thus there is no vortex lift. At higher angles of attack (α ≥ 15°), however, the flow separates at the leading edge, a leading edge vortex forms on the suction side of the wing (Figures 4 and 5), and the lift coefficient experiences a significant increase (Figure 3).

Several noteworthy phenomena occur over the course of the wing translation. As the wing accelerates from rest, lift increases steeply until reaching a maximum at exactly 1 chord-length of travel (i.e., when the wing stops accelerating). While the wing is accelerating, both inertial and circulatory effects contribute to lift production, i.e., the total lift measured is the sum of that due to added mass as well as the growth of circulation around the wing or that of the leading edge vortex. Past this point, the leading edge vortex that started to form at the start of the wing motion continues to grow, but the added mass force goes to zero. The continued growth leads to a local maximum around s/c = 2-3 for this 1c acceleration case, with the local maximum not appearing for α ≤ 15° and tending to occur earlier at higher angles of attack.

As the wing continues to translate, lift decreases as the LEV becomes less coherent and moves down the wing towards the trailing edge, and the trailing edge shear layer curls around and into the wake of the wing (see Figure 5, s/c = 5.5). At this stage, flow over the upper surface of the wing is arguably more a region of recirculating flow than a coherent vortex. Note that although the described flowfield is clearly shown in the PIV image in Figure 5, the corresponding dye visualization does not at first glance provide a similar result. Remember, dye is only injected at the 3/4 span, aligned with a thin laser sheet as the interrogation plane. As long as the flow is sufficiently two-dimensional (as is the case early in the wing motion), laser-fluorescent dye visualization provides an accurate representation of the flow structures, making it an excellent heuristic tool for PIV. However, upon the onset of three-dimensionality (e.g., spanwise flow), dye is occasionally pulled out of the interrogation plane and is not present in the resulting image. Thus, the presence (or absence) of dye in the flowfield images indicates possible locations of spanwise flow.

At high angles of attack, the flow is largely separated over the entire upper surface of the wing (see Figure 4), leading to unsteady flow structures shedding and reforming throughout the wing motion. Following the local lift minima around s/c = 5.5 for the high angle of attack cases, clockwise circulation about the leading edge regains dominance over the wing surface (see Figure 5, s/c = 7.75), resulting in a second lift peak. These periodic flowfield oscillations appear to continue while the force histories gradually decline towards their respective steady state values.

Force measurements for long convective times were acquired using a stationary wing at fixed angle of attack with the tunnel operating at a speed corresponding to Re = 20 000 (see static values in Figure 3). Measurements were obtained for −45° ≤ α ≤ 45° in increments of 2°. Force measurements at each angle of attack were averaged over 26 chords traveled before slowly incrementing by 2° to the next angle. The measured static lift for α = ± 10° was in good agreement with Prandtl’s lifting line theory. Since lifting line theory only accounts for bound circulation, it makes sense that there is good agreement for cases in which the flow is attached and there is little to no leading edge vortex formation. This is also the region in which Wagner’s effect matched well with the unsteady lift after long convective time in Figure 3.

The purpose of extending the travel distance out to many chord-lengths was to assess the short- and long-term effects of transient disturbances, e.g., wing acceleration and gusts, on force coefficients. Comparing the lift produced at the end of the 14 chord-length surge motion to that of its corresponding steady state value provides such an assessment of convergence. Although the unsteady lift forces in Figure 3 appear to be converging in the direction of the static measurements, Figure 3 shows that at s/c = 14 the unsteady forces are still 20%-50% larger than static values for α > 20°. For cases with largely separated flow (α > 15°), the effect of a transient disturbance has a lasting effect on force coefficients.

To gain insight into the effect of transient disturbances on the flowfield and force histories, experiments were run at fixed angle of attack with widely varying acceleration profiles. Prior to this study, it was hypothesized that the transient high lift experienced at the start of wing motion would decrease and converge to a steady-state value after a few chord-lengths of travel (∼5-8) past the end of acceleration. A wide range of acceleration distances were tested and each case runs to 14 chord-lengths of travel, the maximum achievable distance in the AFRL water tunnel.

Figure 6(a) shows the time-history of the lift coefficient for a wing at α = 45° and s_a/c = {0.125, 0.25, 0.5, 1, 2, 3, 4, 5, 6}. Focusing on the acceleration phase, the measured C_L for each case appears to increase proportionally to (or maintain the same shape as) its prescribed motion profile, refer to Figure 2, illustrating a dependence on acceleration profile for lift histories. To further investigate the acceleration phase and attempt to collapse the lift curves during their acceleration transients, Figure 6(b) suggests a rescaling of the x-axis of Figure 6(a) by each respective acceleration distance, s_a/c. Applying the new scaling, each acceleration phase occurs within 0 ≤ s/s_a ≤ 1. Additionally, the added mass term from Eq. (4) is subtracted from the measured lift to isolate the growth of only the circulatory force component. Due to rig vibrations in cases where s_a/c < 1, accelerations faster than s_a/c = 1 have been omitted from the figure. Figure 6(b) shows that the circulatory lift curves collapse nicely when s/s_a ≤ 0.4, indicating a region where circulation growth is independent of acceleration profile. After this collapsed region, the curves diverge slightly and peak at s/s_a = 1. This result suggests that a main factor corresponding to the difference in lift production between the cases (at least during the acceleration phase) is an increased circulatory force component. This concept will be further addressed in Sec. III C.

Once the wing ceases acceleration, the added mass contribution goes away, leaving only circulatory forces responsible for lift production. (The minor oscillations in the beginning of the faster motions are due to physical vibrations of the rig.) Following the peak at the end of acceleration, all curves show a decrease in lift, reaching a minimum around s/c = 5.5–6, except for the s_a/c = 5 and 6 cases, which reach minimums around s/c = 9.5–10. For s_a/c ≤ 4, the lift once again increases to a second local maximum around s/c = 7.5–8. After the second lift peak, all acceleration cases gradually decrease in lift until the end of the motion. The motion ceases at s/c = 14 due to the physical limitations of the tunnel. Based on prior work,²⁸ it is expected that the strength and coherence of the leading edge vortex contributes significantly to the overall lift production.

Looking further into the dynamics and force contribution of the leading edge vortex, the moment coefficient was also examined. Figure 7(a) shows the pitching moment about the mid-chord and quarter-chord for the same range of acceleration distances. Dye flow and PIV images in Figure 7(b) correspond to the local maxima and minima of the moment coefficient about the mid-chord. Since all acceleration profiles with considerable lift oscillations appear to have the same trend, flowfield images are given only for s_a/c = 1. For visual clarity, the flow images in Figure 7(b) have been rotated by 45°, orienting the normal force on the wing vertically upward. Positive vorticity, shown in blue, can be attributed to the leading edge vortex and clockwise rotation from the leading edge shear layer. Negative vorticity, shown in red, typically manifests from the trailing edge and contributes counter-clockwise vorticity to the flow field. This figure shows how the formation, shedding, and reformation of the leading edge vortex shifts the location of the dominant low pressure region chord-wise along the wing. Consequently, this moves the location of the center of pressure, affecting the moment coefficient.

The mid-chord pitching moment reaches a maximum around 1c, when the wing ceases acceleration and the leading edge vortex is strong, coherent, and attached to the leading edge of the wing. The clockwise circulation from this LEV is located forward of the mid-chord (observed in both dye flow and vorticity in Figure 7(b)), thus producing a positive pitching moment. This maximum, however, is short-lived. The LEV subsequently convects off the wing surface and downstream, during which counter-clockwise circulation from the trailing edge appears and interacts with the leading edge vortex. This effect is clearly illustrated in the vorticity field at s/c = 5 in Figure 7(b). This phenomena is not, however, observable in the corresponding dye flow visualization, and thus presents a limitation in relying solely on dye injection to visualize flow fields. The competition between the leading and trailing edge vortices leads to a lower resultant force and one that is located closer to the mid-chord, explaining the minimum in moment coefficient. The LEV formation and convection repeats once more as the clockwise vorticity from the leading edge regains dominance over the wing surface (see Figure 7(b), s/c = 7.2) and moves the resulting low pressure region (and resultant force) farther forward of the mid-chord, leading to a second local maximum.

For the study presented here, angle of attack remains fixed, but the shedding of LEVs over the suction surface can be viewed as potentially changing the effective angle of attack of the wing. It is of interest to note that as the center of pressure moves chord-wise along the wing, the pitching moment coefficient does not change sign, contrary to classical dynamic stall,³⁴ during which there is a sharp change in sign of the pitching moment.^35–37 In the case presented here, there are dips in lift and moment coefficients corresponding to shedding LEVs, but not to the extent one sees in the case of dynamic stall.

Looking more closely at two cases, a “fast case” (s_a/c = 1) and a “slow case” (s_a/c = 6), we first consider the force contributions from circulatory and non-circulatory effects during the acceleration of the wing. To isolate the effect of the circulatory forces on lift in the acceleration region, the non-circulatory added-mass force from Eq. (3) was removed from the measured force data, as shown in Figure 8. Even after removing the theoretical added-mass force from the two lift curves, there is still a ΔC_L between the cases, (e.g., at s/c = 1, C_Lsa/c=1 = 1.6, and C_Lsa/c=6 = 0.3) implying that the difference in lift during the transient phase of each acceleration profile is not completely due to inertial forces, but that circulatory forces provide a significant portion of the measured lift, even during wing acceleration. In both of these cases, wing acceleration is high enough that the added mass force contribution is large (- - line in Figure 8), but circulatory forces (-.- line in Figure 8) quickly build to be much larger than the added mass force. Immediately following the end of acceleration, it appears that the circulatory force aligns directly with measured force, further confirming that lift from this fully separated flow is dominated exclusively by circulatory forces. Visualizing the vortex trajectory and quantifying its strength will provide insight into how the evolution of the LEV affects the shape and magnitude of the measured force and moment histories.

The vortex tracking algorithm described earlier, which identifies local extrema in the Γ₁ function to locate vortex centers, is implemented to identify the leading edge vortex trajectory throughout the wing motion. As shown previously, at some point in the motion, the vortex breaks down and loses its coherent structure. After this vortex dissipation, the tracking algorithm can no longer accurately locate the correct vortex center. Thus, the tracking method presented here is only carried out until the first LEV can no longer be detected.

Figure 9(a) plots vortex trajectories in the lab-fixed and wing-fixed frames for the fast and slow cases. It was previously hypothesized that, because the two cases have vastly different force histories, their vortex trajectories would also show notable differences. However, Figure 9(a) shows that the fast and slow surging cases actually produce nearly identical vortex trajectories with respect to convective time. The leading edge vortex forms immediately upon startup and translates linearly over the first chord of travel. At s/c ≈ 1.25, both LEVs take an immediate departure from the wing surface, illustrated by Figure 9(c), remaining at a relatively constant X′ location (Figure 9(b)).

Vortex trajectory provides the instantaneous position of the vortex, but it does not contain information about its size or strength. Figure 10 provides visualization of the vortex in the form of vorticity plots, showing a clear difference between the two cases. The fast surge case forms a tight, coherent vortex upon startup, and the vortex remains this way through s/c = 1.5. The slow case, however, forms a much weaker leading edge vortex that quickly dissipates as it convects from the wing surface. This point is reiterated quantitatively by Figure 11, which shows normalized total circulation above the wing for both the fast and slow wing kinematics. As expected from the larger vortex with higher vorticity (as seen in Figures 10(a) and 10(b)), the fast case produces consistently higher circulation throughout the motion. As stated previously, force production during these particular wing kinematics are largely driven by circulatory forces, which manifest themselves largely in the strength of the leading edge vortex. This analysis shows that despite having nearly identical LEV trajectories, the fast and slow surging cases produce vortices of different strengths, explaining the large difference in force histories.

This study examined the effects of acceleration distance and incidence angle on the lift coefficient, pitching moment, and flow structures on a rigid flat plate. Force measurements were acquired via a submersible load cell and provided the force histories over 14 chord-lengths of travel. Force histories highlighted several distinct locations of interest where lift and moment coefficients reached local maxima and minima. Through dye flow visualization and particle image velocimetry, these points were found to be associated with the formation and subsequent shedding of leading edge vortices, which contribute significantly to lift generation. A unique vortex tracking method was employed to track the trajectory of the leading edge vortex formed upon startup. It was observed that the “fast” and “slow” cases produced nearly identical vortex trajectories. Circulation measurements confirmed that the faster acceleration case generates more vorticity in the form of a tighter, more coherent vortex and produces significantly more circulation than the slower acceleration case, which provides an explanation for the difference in force production. Future studies will include a similarly detailed analysis on the significance of the trailing edge vortex and other physical mechanisms, which may have considerable effects on lift production.

An underlying hypothesis in this study was that 8 chord-lengths would be a sufficient travel distance over which the wing would eventually produce a steady-state lift, i.e., the lift value produced in a fully developed flow. However, a comparison to static force measurements has shown that for cases in which there is large flow separation and formation of leading edge vortices (i.e., α > 15°), 14 chord-lengths traveled was not sufficient to reach fully developed flow, implying that the startup transient somehow influences the flowfield long after the actual disturbance.

This work was supported by the Air Force Office of Scientific Research (AFOSR) under AFOSR Award No. FA9550-12-1-0251 and the U.S. Army Research Laboratory under the Micro Autonomous Systems and Technology (CTA-MAST) program. The authors would also like to thank Andrew Lind and Albert Medina for their support and contributions to this work.