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The influence of wing–wake interactions on the production of aerodynamic forces in flapping flight

James M. Birch^* and Michael H. Dickinson

Department of Integrative Biology, University of California, Berkeley, CA 94720, USA

View larger version (27K): [in a new window]   Fig. 1. Experimental apparatus and procedure. (A) Each arm of the robotic insect consists of a flapping wing controlled by three stepper motors. (B) Wing position arranged as two-dimensional projections. The line represents the wing chord, with the circle indicating the leading edge. Numbers indicate the value of . (C) Mini-YAG lasers create a 2.5 mm-thick sheet of light at 0.65R. By seeding the oil with small air bubbles and calculating their movement between two laser flashes separated by a 2 ms delay, we captured fluid velocity (u) in the x and y direction and subsequently calculated z vorticity (). (D) To collect time series data, we timed the firing of the laser and the starting position of the wing so that the wing was in front of the camera at the desired phase of the stroke (i). After collecting an image at that position, the laser delay was increased by 100 ms, and the starting position of the wing was set backwards so that the wing passed in front of the camera and the flow was illuminated at a later point in the stroke cycle (ii). This laser timing/wing positioning adjustment was repeated to reconstruct a complete series of flow visualizations.

View larger version (56K): [in a new window]   Fig. 2. Convention for displaying fluid velocity. (A) From the wing's frame of reference, the wing (gray bar) is moving to the left, with instantaneous fluid motion indicated by the arrows. The length of the calibration arrow represents 0.40 m s-1 in all panels. Field of view is approximately 0.28 mx0.28 m. (B) The identical velocity plot with vorticity added in pseudocolor. Blue represents clockwise vorticity, red represents counter-clockwise vorticity. (C) Fluid motion with wing speed subtracted and only every other vector plotted. This convention will be used throughout the paper.

View larger version (44K): [in a new window]   Fig. 3. Time history of aerodynamic forces during one complete stroke cycle. (A) Net force. The red line is the net force during the first stroke; the blue line is the force during the fourth stroke. The black line is the difference in net force between the fourth and first strokes. At the start of the downstroke, the fourth stroke generates considerably more force than the first stroke but generates less force as translation continues. Note that these differences nearly vanish during the upstroke of each stroke, since by the start of the first upstroke, the dynamics of the wake have reached a limit cycle. The gray box represents the downstroke and the time during which the digital particle image velocimetry (DPIV) images in Figs 4, 5, 9 and 10 were captured. (B) Lift and drag for stroke one. (C) Lift and drag for stroke four.

View larger version (120K): [in a new window]   Fig. 4. Flow visualization and force generation during the first downstroke. The first 16 frames are 100 ms apart; numbers in the upper left-hand corners represent the value of . In this representation, fluid moves to the right over the wing (white bar) and, for clarity, velocity vectors are shown undersampled by a factor of 2. For scale, the length of the black calibration arrow in the first panel equals 0.40 m s-1. Pseudocolor represents vorticity; length of the red arrows represents the instantaneous total force. Note the slow build-up of the leading edge vortex (blue region; =0.03–0.10) and the shedding of the starting vortex from the trailing edge (red region; =0.09–0.19). Near =0.40, the wing begins to slow for stroke reversal, rotates and sheds trailing edge vorticity (red region).

View larger version (119K): [in a new window]   Fig. 5. Flow visualization and force generation during the downstroke of stroke four. Panels represent identical time points as in Fig. 4. Note the difference between this stroke and stroke one at frames =0.00–0.14 due to the wing traveling through the shed vorticity of the prior upstroke. Note also the concomitant increase in force (compare red arrow length at =0.05–0.10 between strokes one and four). Flow vector lengths are to the same scale as in Fig. 4.

View larger version (18K): [in a new window]   Fig. 6. Side views show tip vorticity just after stroke reversal. (A) As illustrated by this cartoon looking down on the wing from above, the tip vortex is curved and roughly follows the sweep of the wing tip. (B) Images captured at this time show two regions of clockwise vorticity (blue) beneath the wing: the shed rotational starting vortex and an oblique slice of the tip vortex.

View larger version (19K): [in a new window]   Fig. 7. Diagrammatic summary of wake dynamics. Each panel shows the growth and shedding of three vortices. Warm tones (reds) represent counter-clockwise vorticity; cool tones (blues) represent clockwise vorticity. Lighter shades represent vorticity that was generated in the previous stroke. Numbers in the bottom right-hand corners indicate the proportion of the stroke cycle completed. Arrow length is proportional to instantaneous fluid velocity. USL, underwing shear layer; LEV, leading edge vortex; TSV, translational starting vortex; RSV, rotational starting vortex.

View larger version (16K): [in a new window]   Fig. 8. Predictions of equation 1 and measured forces give similar time histories at the start of stroke one. The x-axis covers the first 16% of stroke one. Blue traces (left-hand y-axes) show measured lift and drag forces. Red circles (right-hand y-axes) plot values of sectional lift and drag calculated from the vortex moment calculation in equation 1. Predictions beyond =0.15 were unreliable because starting vorticity moves out of the visualized frame.

View larger version (110K): [in a new window]   Fig. 9. Subtractive reconstruction of wake velocity fields. Pseudocolor represents the magnitude of the velocity difference between strokes four and one. Arrows indicate direction and magnitude of flow. Note the two regions of high fluid velocity from =0.00 to 0.05, which represent the influence of the shed vorticity from the prior stroke. Also note the downward component of velocity in the flow vectors from =0.09 to 0.33, indicating that through most of the stroke, stroke four encounters fluid at a lower aerodynamic angle of attack than during stroke one. After the stroke has completed 38% of a cycle (=0.38–0.48), there is no difference in fluid velocity between the first and fourth strokes. Flow vector lengths are to the same scale as in Fig. 4.

View larger version (122K): [in a new window]   Fig. 10. Subtractive reconstruction of wake vorticity fields. Pseudocolor represents the magnitude of the difference in span-wise vorticity between strokes four and one. Arrows, representing velocity differences, are identical to those in Fig. 9. Note the greater amount of both clockwise and counter-clockwise vorticity early in the stroke (indicating greater vorticity during stroke four), and the diminution of any vorticity difference near the end of the stroke (=0.38–0.48.)

View larger version (20K): [in a new window]   Fig. 11. The wake of prior strokes results in both an increase in fluid velocity and a reduction in the aerodynamic angle of attack (aero). (A) Representative panel with selected area from which we estimated the mean angle of attack and mean resultant velocity for both the first and fourth strokes. This region retained approximately the same area through wing rotation and flipped to the right side of the panel when the wing reversed direction. (B) Mean resultant velocity in the same region shows higher values during stroke four than stroke one. The green line shows fluid velocity predicted from wing kinematics. (C) Because of the downwash, aero is much less during the fourth stroke (blue) than the first (red). This difference nearly disappears during the upstroke when downwash is present in both strokes. The green line represents the angle of attack programmed into the robot (geom); the red and blue lines represent the measured aero from the first and fourth strokes, respectively. The inset shows how each angle of attack is measured. (D) Comparison among measured net force (solid blue line), predictions of a quasi-steady translational model (green line) and predictions of a quasi-steady translational model corrected using the time courses of velocity and aero shown as blue traces in B and C (broken blue line). Traces show the mean values of a fourth down- and upstroke for both measured and predicted values. The corrected quasi-steady model correctly predicts a drop in force due to downwash (=0.08–0.18) but not an early increase due to wake capture (=0–0.08).

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