First published online February 20, 2004
Journal of Experimental Biology 207, 1063-1072 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.00848
Force production and flow structure of the leading edge vortex on flapping wings at high and low Reynolds numbers
James M. Birch*,
William B. Dickson
and
Michael H. Dickinson,
Department of Integrative Biology, University of California,
Berkeley, CA 94720, USA
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Fig. 1. Comparison of translational force coefficients at Re=120 and
Re=1400. (A) We rapidly accelerated the wing from rest to a constant
tip velocity of 0.26 m s–1. The angle of attack (AOA) was
increased between trials in 10° increments. Labels to the right of the
forces in the first panel indicate angles of attack. At both Re, an
initial transient peak was followed by stable force generation. (B)
Coefficients of lift (CL) and drag
(CD) averaged between the broken lines in A. The polars
form two concentric arcs with values measured at high Re around the
outermost arc. (C) Net force coefficients increase with angle of attack, with
greater increases at high Re. (D) The angle of the net force vector
quickly reaches 90°, indicating pressure forces dominate at both low and
high Re.
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Fig. 2. Vorticity measurements at both Re. (A) Side views of wing at
0.65R (R is the length of one wing) at mid-downstroke. Wing
is moving to the left at an angle of attack of 45°. Note the stronger and
larger leading edge vortex at the higher Re. (B) Circulation around
the wing as a function of wing length. The vertical line at 0.65R
represents the position of the pseudocolor plots in A. The area of greatest
vorticity shifts slightly towards the wingtip at high Re, occurring
at 0.65R versus 0.49R at low Re.
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Fig. 4. The magnitude and distribution of axial flow is dependent on Re.
The top schematic shows the position of the three side view panels
(0.45R, 0.55R and 0.65R) at each Re. Flows
are captured from a wing at mid-downstroke starting from rest. Columns 1 and 3
show the sectional velocity field as arrows (uy and
ux) superimposed over a pseudo-color plot of axial
velocity (uz). Next to each column are plotted the
ux and uz values along the gray broken
transect from A to B shown in i. The left two columns show flow at low
Re. Note how the maximum axial flow (uz) at low
Re occurs farther behind the vortex center than at high Re.
Also, at high Re, flow near the leading edge is much more
complicated, with a stronger axial flow component.
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Fig. 5. Vorticity and velocity when viewed from behind. Color represents vorticity,
arrows represent velocity (arrow scale upper right). Panels i–iv show
successive slices starting just behind the leading edge (i) and moving toward
the trailing edge in 1 cm increments (see inset at top). The solid horizontal
line indicates the laser sheet intersection with the wing; vectors above this
line are above and behind the wing, vectors below represent fluid movement as
seen through the wing (i.e. below and in front of the wing). Columns
(A–D and E–H) represent two experimental protocols with identical
wing size, flapping frequency and kinematic pattern; only oil viscosity and
Re are different.
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Fig. 6. Velocity vectors from the rear at Re 1400. These slices were
captured in an identical fashion to those in
Fig. 5 except that they were
spaced every 0.5 cm with the camera closer to the wing. The wing tip is to the
left, the leading edge (top) is into the page, the trailing edge (bottom) is
out of the page, and the wing is sweeping away from the viewer and is caught
during mid-downstroke. Note the localized high velocity movement of fluid in A
and B, possibly representing the front edge of the spiral vortex. By E, the
laser sheet is capturing the rear of the spiraling leading edge vortex.
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Fig. 7. Photographs near mid-downstroke using bubble rake. Left column (A,B) at
Re=120. Right column (C,D) at Re=1400. Full wing views (A,C)
taken at approximately mid-downstroke when wing is parallel to camera.
Close-ups of the leading edge (B,D) show the growth of flow within the core of
the leading edge vortex. Note the lack of a tight helix at low Re
(B).
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© The Company of Biologists Ltd 2004
z) and chordwise (
z) in the Kutta–Zhukovski equation,
sectional lift values at both Re are maximum near 0.65R,
with high Re generating more lift along the outer third of the
wing.