First published online October 7, 2004
Journal of Experimental Biology 207, 3945-3958 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.01258
Hydrodynamics of surface swimming in leopard frogs (Rana pipiens)
L. Christoffer Johansson1,* and
George V. Lauder2
1 Dept of Theoretical Ecology, Lund University, Ecology Building, SE-223 62
Lund, Sweden
2 Dept of Organismic and Evolutionary Biology, Harvard University, 26 Oxford
Street, Cambridge, MA 02138, USA
View larger version (114K):
[in a new window]
|
Fig. 2. Images from five stages of synchronous kicks to show basic leg and foot
motion, with the ventral and caudal view from the same sequence. The lateral
view is from an additional sequence, synchronized relative to the position of
the legs and feet as seen from the ventral view. Note that the ankles are
crossed and overlap as seen in the posterior view during the middle of the
stroke, a phenomenon we saw frequently in these frogs. The feet are more or
less perpendicular to the swimming direction throughout the majority of the
kick. Only at the end of the kick are the feet swept medially, aligning the
feet with the swimming direction. Numbers indicate the timing (s) of the image
relative to the start image.
|
|
View larger version (118K):
[in a new window]
|
Fig. 3. Images from asynchronous kicks to show basic leg and foot motion, with the
ventral and lateral view from the same sequence. The caudal view is from an
additional sequence, synchronized relative to the position of the legs and
feet as seen from the ventral view. Numbers indicate the timing (s) of the
image relative to the start image.
|
|
View larger version (28K):
[in a new window]
|
Fig. 4. Three-dimensional kinematics of the foot (relative to a stationary flow),
of a representative synchronous kick as seen from the ventral (A), lateral (B)
and caudal (C) views. Data points are separated by 0.004 s. Note the relative
excursions of the different toes.
|
|
View larger version (21K):
[in a new window]
|
Fig. 5. Swimming velocity and the forward velocity of the third toe of (A) an
asynchronous and (B) a powerful synchronous kick. Note the difference in
magnitude of body velocity. In the synchronous kick, the foot is moving
backwards relative to the water throughout nearly the entire acceleration
phase of the kick, while in the asynchronous kick the body starts to
decelerate when the foot is still moving backwards relative to the water. The
inserted images illustrate the posture of the frog at the time indicated by
the position of the waist of the frog.
|
|
View larger version (69K):
[in a new window]
|
Fig. 6. Velocity vectors from an asynchronous kick based on a ventral view
(horizontal light sheet; xz plane) showing the buildup and shedding
of the vortex ring generated by the foot. The reference vector equals 1 m
s–1. The four images are separated by 0.04 s. Velocity
vectors overlying the foot represent foot movement and illustrate the velocity
of the foot relative to the surrounding fluid. Free-stream flow velocity has
been subtracted from each vector.
|
|
View larger version (82K):
[in a new window]
|
Fig. 7. Velocity vectors from an asynchronous kick based on a lateral view
(vertical light sheet; xy plane). The reference vector equals 1 m
s–1. The four images are separated by 0.04 s, showing the
buildup and shedding of the vortex ring generated by the foot. The lower
vortex center is shed briefly before the dorsal vortex center, and a discrete
vortex ring with a posteroventral central jet flow of high velocity is shed
into the water. Free-stream flow velocity has been subtracted from each
vector.
|
|
View larger version (53K):
[in a new window]
|
Fig. 8. Velocity vectors from a synchronous kick based on a ventral view
(horizontal light sheet; xz plane) plotted on a vorticity background.
The reference vector equals 1 m s–1. Red indicates
counter-clockwise and blue indicates clockwise vorticity
(s–1). The three images are separated by 0.04 s, showing the
build up of the vortex ring shed by the feet. No central jet is visible
between the feet, but instead each foot produces its own vortex ring with a
corresponding posteriorly directed jet. The forward directed vectors (to the
left of the image) and the associated vorticity are caused by the feet still
being in the image. Free-stream flow velocity has been subtracted from each
vector.
|
|
View larger version (54K):
[in a new window]
|
Fig. 9. Velocity vectors from a synchronous kick based on a lateral view (vertical
light sheet; xy plane) plotted on a vorticity background. The
reference vector equals 1 m s–1. Red indicates
counter-clockwise and blue indicates clockwise vorticity
(s–1). The three images are separated by 0.04 s, showing the
buildup of the vortex ring shed by the foot. In the two upper images, the foot
is still in the image, affecting the estimation of the vector fields. In the
top image, the vorticity centers are upstream of the foot, while in the middle
image the foot is the source of the vectors pointing upstream just upstream of
the vortex ring. Free-stream flow velocity has been subtracted from each
vector.
|
|
View larger version (23K):
[in a new window]
|
Fig. 10. Transect of the vortex wake of a synchronous kick (A) and the corresponding
velocity profile perpendicular to the transect (B). Positive velocity is
forward, showing an augmented forward velocity between the vortex rings. There
is no backward-directed central jet produced at the end of the kick as the
feet come together. If anything, there may be an interaction between the
vortex rings produced during the kick, increasing the forward velocity between
the vortex rings. The forward velocity between the rings may also be partly
due to the drag wake of the body.
|
|
View larger version (12K):
[in a new window]
|
Fig. 11. (A) Forward-directed thrust per foot relative to forward swimming velocity,
with higher thrust and velocity for synchronous kicks relative to asynchronous
kicks. (B) Mass-specific momentum per foot relative to forward velocity and
(C) duration of the power stroke, with the horizontal line in C representing
the approximate average mass-specific momentum per foot for Rana
esculenta calculated from fig.
3 in Nauwelaerts and Aerts
(2003 ).
|
|
View larger version (18K):
[in a new window]
|
Fig. 12. The drag/wave drag
[Cd/(d/l)2] of a streamlined
object moving at a depth (h) relative to its length (l) as a
function of the Froude number (Fr). The arrows (black, synchronous
kicks; gray, asynchronous kicks) indicate the average Froude number at which
the frog swims during asynchronous and synchronous kicking, calculated from
the velocity (U) and length of the outstretched frog at the end of
the kick. Graph modified from Hoerner
(1965). g,
gravitational acceleration.
|
|
© The Company of Biologists Ltd 2004
), vortex ring angle (ß) and swimming direction
(
) in the lateral (top) and ventral (bottom) views, used to measure the
vortex wake and body kinematics of swimming leopard frogs. Subscripts v and h
represent light sheet orientation during the DPIV experiments, i.e. vertical
and horizontal, respectively.