The hydrodynamics of locomotion at intermediate Reynolds numbers: undulatory swimming in ascidian larvae (Botrylloides sp.)
Matthew J. McHenry1,*,
Emanuel Azizi2 and
James A. Strother1
1 Department of Integrative Biology, University of California, Berkeley, CA
94720, USA
2 Organismic and Evolutionary Biology Program, 221 Morrill Science Center,
University of Massachusetts, Amherst, MA 01003, USA
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Fig. 1. The experimental set-up for tethering experiments. (A) We recorded the tail
motion of a larva and the deflections of the tether to which a larva was
attached. The larva is illustrated with the orientation that allowed for the
recording of lateral forces: the longitudinal axis of the body is
perpendicular to the direction of deflections. To measure thrust, the
longitudinal axis was aligned parallel to the deflections of the tether. (B)
The ventral perspective of a larva was recorded with video camera #1 mounted
to a dissecting microscope mounted beneath the glass tank. (C) Deflections of
the glass tether ( ) were recorded by video camera #2 mounted to a
compound microscope.
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Fig. 2. The precision and accuracy of force measurements. (A) An example of the
measurements of the passive movement of the tether after being pulled and
released. These data were used to measure the coefficients of stiffness and
damping (see Materials and methods). (B) A free-body diagram illustrates the
forces acting on the tether during an experiment. The input force generated by
a swimming larva is resisted by a component of the weight of the tether and
the stiffness (illustrated by the spring) and damping of the pivot
(illustrated by the dashpot). (C,D) Measurements of input force (filled
circles) at 1000 Hz from the deflections of a tether (not shown) calculated
from the simulated changes in force (red lines). (C) The input force measured
from deflection measurements using accurate values for the stiffness and
damping coefficients. (D) The input force measured using a damping coefficient
that is less than the actual value (by 2x10-6 Nms
rad-1). (E,F) The time lag between simulated and measured input
force for varying degrees of error in the damping (E) and stiffness (F)
coefficients. (G,H) The ratio of maximum measured to maximum simulated input
force for varying degrees of error in the damping (G) and stiffness (H)
coefficients.
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Fig. 3. Schematic drawing of the quasi-steady and unsteady hydrodynamic models. (A)
The force generated by a single tail element (Ej)
is drawn on the silhouette of the body of a larva from a dorsal perspective.
The force generated by this element has components acting towards thrust and
laterally. The force generated by the whole tail was calculated
instantaneously as the sum of force generated by all tail elements. The
position vector of the element (Rj) with respect to
the center of mass describes the lever arm used by the tail element to
generate a moment about the center of mass. (B,C) Each of the models is
illustrated by the vectors that comprise the force generated by the tail
element. (B) The force acting on tail elements (Ej)
in the quasi-steady model was calculated as the sum of the form force
(Ejf) and skin friction
(Ejs). (C) The force acting on tail elements in the
unsteady model was the sum of the quasi-steady forces and the acceleration
reaction (Eja). (D) The coefficient of force acting
normal to the surface of a flat plate (Cjnorm)
oriented normal to flow. The form force (in green; see equation 21) is found
as the difference between the total force (in black; see equation 18) and the
force generated by skin friction (in violet; see equation 19). The total force
is generated primarily by form force at height-specific Reynolds numbers
(Rejl) of  103, skin friction is
dominant at Rejl<100, but the
normal force is a combination of the two at intermediate Re
values.
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Fig. 5. Measured and predicted forces for a tethered larva. (A) The legend for the
direction of force data (B—H). Violet traces represent lateral forces
that are directed to the right of the body when negative and to the left when
positive. Green traces show force along the antero-posterior axis of the trunk
that is directed toward the anterior when positive and toward the posterior
when negative. (B—H). As in Fig.
4, the vertical gray bands show when the trunk angle ( ) is
directed toward the left side of the body, and white bands occur when the
trunk angle is directed to the right. Note that (B) the form force and (C) the
acceleration reaction are on the same scale as the measured force, but both
(D) the tail inertia force and (E) skin friction are plotted on smaller
scales. (F) The tail force predicted for quasi-steady
(F=Ff+Fs, heavy line) and unsteady models
(F=Ff+Fs+Fa, thin line)
illustrate the differences between these models. (G) Force generated by the
larva against the tether, in the lateral direction for unfiltered (points) and
filtered (line) data (see text for details). (H) Variation in trunk angle with
time (the line is filtered data and the points are unfiltered).
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Fig. 6. The phase relationship between lateral force generation and tail
kinematics. Positive values are directed to the left of the body and negative
values are directed to the right. Each blue curve shows the mean values over
four tail beats for a single larva, with time normalized to the tail-beat
period. The black solid lines represent the mean, and the black dotted lines
represent ±1 S.D. for all larvae (N=11). As in
Fig. 4, the gray band shows
when the trunk angle is directed towards the left side of the body, and white
bands show when the trunk angle is directed to the right. Measurements of
lateral force (A) are plotted above trunk angle (B).
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Fig. 7. Comparison of predicted and measured lateral forces. Graphs to the left (A
and C) show mean measured lateral forces (dark gray line) ±1 S.D.
(light gray fill) of measured forces (the same data as in
Fig. 5A; N=11) and the
mean (solid black line) ±1 S.D. (dotted black line) of predicted
lateral forces for the same 11 larvae. Graphs on the right (B and D) present
the same data, but the measured forces are plotted against predicted forces
for each instant of time in the tail-beat cycle. Points vary in color from
blue to red as the tail-beat cycle progresses. The green regression line was
calculated by a least-squares solution to a linear curve fit of the data
(slope=0.32, y-intercept=0, r2=0.50 in B;
slope=0.13, y-intercept=0, r2=0.05 in D). The
gray line has a slope of 1, which represents a perfect match between measured
and predicted data. (A,B) The forces predicted for the lateral force by the
quasi-steady model compared with measurements. (C,D) The forces predicted for
the lateral force by the unsteady model compared with measurements.
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Fig. 8. Thrust and drag predicted by the quasi-steady model to act on the body of a
freely swimming larva at different Reynolds numbers (Re). The graphs
on the left (A,C,E) show a representative time series of the skin friction
(violet lines) and form force (green lines) acting on the trunk and tail for
approximately 4.5 tail beats with the same non-dimensional tail kinematics and
different body lengths. Thrust acts in the positive direction and drag acts in
the negative direction. The total force is the sum of skin friction and the
form force. The graphs on the right (B,D,F) illustrate the percentage of the
total thrust and drag that is generated by skin friction and form force that
acts on the trunk and tail. Error bars denote ±1 S.D., which is
variation generated by running simulations with different kinematic patterns
(N=5). Re was varied by changing the body length of model
larvae.
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Fig. 9. The percentage of thrust and drag generated by skin friction and form force
predicted by the quasi-steady model. Reynolds number of the whole body
(Re) was varied by running a series of simulations over a range of
body lengths. Lines show the percentage of (A) thrust and (B) drag generated
by skin friction (violet) and form force (green).
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© The Company of Biologists Ltd 2003
*). The trunk angle was positive when
the tail was bent to the right and negative when bent to the left of the body.
(B) Changes in the midline of the tail of a larva over a single tail beat.
Notice that as time progresses (to the right), inflection points (filled
circles) move down the midline in the posterior direction (away from the base
of the tail). (C-E) Points represent measurements of each kinematic parameter
and the curves are found by a least-squares fit to functions described in the
Materials and methods. (C) The curvature of bends between inflection points
(with a period 
) in both concave-left and concave-right bends (having
an amplitude 
). (D) The propagation of inflection points begins at the
tail base with a phase lag of 
with respect to a zero value of the trunk
angle. (E) The trunk angle oscillates with time (with a period P).
The vertical gray bands show when the trunk angle is directed towards the left
side of the body, and white bands occur when the trunk angle is directed to
the right.