Article Figures & Tables
Figures
- Fig. 1.
Wing design and the experimental method. (A) The wing planform used for all experiments. The wing was scaled directly from a Drosophila melanogaster wing and equipped with multiple slots at the base to allow the axis of rotation to be changed. The leading edge corresponds to a non-dimensional rotational axis (x̂0) value of 0, whereas the trailing edge corresponds to a value of 1. (B,C) Two-dimensional cartoons showing the kinematics of rotation and translation. The wing translates from left to right at a velocity Ut and rotates about a fixed axis of rotation with varying angular velocity, ω. The leading edge of the wing is indicated by a filled circle. (D,E) Kinematic variables as a function of time. Translational velocity is shown in blue, rotational velocity in red. Data are shown for two representative rotational velocities of 1.5 rad s-1 (D) and 0.667 rad s-1 (E). In both cases, the translational velocity is 0.272 m s-1. (F,G) Net aerodynamic forces as a function of time. The continuous red line indicates the measured forces and the dotted red line indicates the quasi-steady translational estimates. The difference between these traces (double-headed arrow) is used to calculate rotational force coefficients over the shaded region. The early peaks in both force traces are due to inertial transients caused by rapid acceleration of the wing at the start of each trial. Similar inertial effects also occur as a result of rapid rotational acceleration, as is evident in the force traces. Although detectable, these effects are small in comparison with the circulatory forces.
- Fig. 2.
Variation in rotational coefficient (Crot,exp) with angular velocity and axis of rotation. (A) Rotational coefficients versus angular velocity (ω; rad s-1) for each axis of rotation. The axes of rotation used for each series of measurements are given in blue. Non-dimensional values of angular velocity (
) are presented in red under the
corresponding dimensional value. The vertical dashed lines signify the section
through this plot at values of
of 0.166 and 0.374, for which the representative regressions are provided in
B. , non-dimensional axis
of rotation. (B) Rotational coefficient versus axis of rotation. Two
representative plots of rotational force coefficients at two values of
, 0.166 (filled circles) and
0.374 (open circles), are plotted, with the corresponding regression lines for
each. The red line indicates the prediction given by equation 9. (C) A series
of regression lines for representative values of angular velocity (numbers in
blue) plotted together with the quasi-steady prediction (red line) from
equation 9. - Fig. 3.
Validation of the revised quasi-steady model for representative kinematic patterns. (A) Advanced rotation. (B) Symmetrical rotation. (C) Delayed rotation. (Ai,Bi,Ci) Two-dimensional cartoons of the three-dimensional kinematics and the translational (broken line) and rotational (solid line) velocities as a function of non-dimensional time within the stroke cycle. The red arrows signify the net force measured on the surface of the flapping wing. (Aii,Bii,Cii) Drag; (Aiii,Biii,Ciii) Lift. The red lines indicate the measured forces, the green lines indicate the quasi-steady translational component based on empirical measurements, the purple lines indicate the quasi-steady rotational component and the blue lines indicate calculated estimates of added mass. In Aiv,v, Biv,v and Civ,v, the summed quasi-steady predictions for drag (Aiv,Biv,Civ) and lift (Av,Bv,Cv) are shown in black and compared with the measured values (red lines) for the corresponding kinematics.
- Fig. 4.
Isolation of wake capture. (A) Advanced rotation. (B) Symmetrical rotation. (C) Delayed rotation. In all panels, the red line represents the measured forces and the black line represents the sum of the added mass inertia, translational force and rotational force. The force due to wake capture is represented by a blue line and is calculated as the difference between the measured drag (top panels) and lift (bottom panels) and the corresponding quasi-steady estimates.
Tables
- Table 1.
Parameters from Crot,exp versus x̂0 regressions for various values of ω
Angular velocity, (ω) (rad s-1) Non-dimensional angular velocity, ( ) Slope x̂0-intercept Crot,exp-intercept r2 0.677 0.166 -1.21*** 0.57*** 0.69 0.80 0.762 0.187 -1.35** 0.56** 0.75 0.71 0.847 0.208 -1.38** 0.65** 0.90 0.67 0.931 0.229 -1.84** 0.62** 1.14 0.87 1.016 0.249 -2.24** 0.62** 1.39 0.94 1.101 0.270 -2.49* 0.64* 1.59 0.93 1.185 0.291 -2.75 0.64* 1.75 0.95 1.270 0.312 -2.85 0.68 1.93 0.95 1.354 0.333 -3.13 0.68 2.13 0.97 1.439 0.353 -3.26 0.70 2.27 0.97 1.523 0.374 -3.51 0.69 2.43 0.97 -
Values for the slope and x-intercept that are significantly different from the theoretical predictions based on equation 9 (slope=-π, x̂0-intercept=0.75) are marked with asterisks (*P<0.05, **P<0.01, ***P<0.001).
x̂0, axis of rotation; Crot,exp, experimental rotational force coefficient.
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