The hydrodynamics of locomotion at intermediate Reynolds numbers: undulatory swimming in ascidian larvae (Botrylloides sp.)
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
* Author for correspondence at present address: The Museum of Comparative Zoology, Harvard University, 26 Oxford St, Cambridge, MA 02138, USA (e-mail: mchenry{at}fas.harvard.edu)
Accepted 10 October 2002
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Summary |
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Key words: swimming, intermediate Reynolds number, morphology, larvae, ascidian, urochordata, Botrylloides sp
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Introduction |
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Swimmers that are millimeters in length generally operate in a hydrodynamic
regime characterized by Reynolds numbers (Re) between 100
and 103, which is a range referred to as the intermediate
Re in the biological literature (e.g.
Daniel et al., 1992).
Re (Re=
L/µ, where
is mean swimming speed, L is body length,
is density of water,
and µ is dynamic viscosity of water) approximates the ratio of inertial to
viscous forces and suggests how much different fluid forces contribute to
propulsion. At intermediate Re, a swimming body may experience three
types of fluid force: skin friction, form force and the acceleration reaction.
Skin friction and form force are quasi-steady and therefore vary with the
speed of flow. In previous studies on intermediate Re swimming, these
forces have collectively been referred to as the `resistive force' (e.g.
Jordan, 1992
). However, we
will consider these forces separately because the present study is concerned
with how they individually contribute to the generation of thrust and
drag.
Skin friction is generated by the resistance of fluid to shearing. This is
a viscous force, which means that it increases in proportion to the speed of
flow. Skin friction (also called the `resistive force' by
Gray and Hancock, 1955)
dominates the undulatory swimming of spermatozoa
(Re<<100; Gray and
Hancock, 1955
) and nematodes
(Gray and Lissmann, 1964
) and
has been hypothesized to contribute to thrust and drag in the intermediate
Re swimming of larval fish
(Vlyman, 1974
;
Weihs, 1980
) and chaetognaths
(Jordan, 1992
).
The form force is generated by differences in pressure on the surface of
the body and it varies with the square of flow speed
(Granger, 1995). This inviscid
force is equivalent to the resultant of steady-state lift and drag acting on a
body at Re>103. The form force is thought to contribute
to the generation of thrust and drag forces at the intermediate Re
swimming of larval fish (Vlyman,
1974
; Weihs, 1980
)
and may dominate force generation by the fins of adult fish
(Dickinson, 1996
).
The acceleration reaction [also referred to as the `reactive force'
(Lighthill, 1975), the `added
mass' (Nauen and Shadwick,
1999
) and the `added mass inertia'
(Sane and Dickinson, 2001
)] is
generated by accelerating a mass of water around the body and is therefore an
unsteady force (Daniel, 1984
).
This force plays a negligible role in the hydrodynamics of swimming by paired
appendages at Re<101
(Williams, 1994
) but is
considered to be important to undulatory swimming at intermediate Re
(Brackenbury, 2002
;
Jordan, 1992
;
Vlyman, 1974
) and dominant in
some forms of undulatory swimming at Re>103
(Lighthill, 1975
;
Wu, 1971
). Although it is
assumed that the acceleration reaction does not play a role in undulatory
swimming at Re<100
(Gray and Hancock, 1955
), it
is not understood how the magnitude of the acceleration reaction varies across
intermediate Re.
Weihs (1980) proposed a
hydrodynamic model that predicted differences in the hydrodynamics of
undulatory swimming in larval fish at different intermediate Re. He
proposed a viscous regime at Re<101, where viscous skin
friction dominates propulsion, and an inertial regime at
Re>2x102, where inertial form force and the
acceleration reaction are dominant (also see
Weihs, 1974
). For the range of
Re between these domains, thrust and drag were hypothesized to be
generated by a combination of skin friction, form force and the acceleration
reaction. Although frequently cited in research on ontogenetic changes in the
form and function of larval fish (e.g.
Muller and Videler, 1996
;
Webb and Weihs, 1986
), it
remains unclear whether Weihs'
(1980
) theory, which is
founded on measurements of force on rigid physical models, accurately
characterizes the forces that act on an undulating body
(Fuiman and Batty, 1997
).
The present study used a combination of empirical measurements and
mathematical modeling of the larvae of the ascidian Botrylloides sp.
to test whether the hydrodynamics of swimming in these animals is better
characterized by a quasi-steady or an unsteady model. By taking into account
the acceleration reaction, skin friction and form force generated during
swimming, models were used to formulate predictions in terms of the speed of
freely swimming larvae and force generation. By comparing these predictions
with measurements of force and speed, we were able to determine whether larvae
generate thrust and drag by acceleration reaction (the unsteady model) or
strictly by form force and skin friction (the quasi-steady model). Ascidians
are an ideal group for exploring these hydrodynamics because the larvae of
different species span nearly two orders of magnitude in Re [e.g.
5x100 in Ciona intestinalis
(Bone, 1992
);
Re
102 in Distaplia occidentalis
(McHenry, 2001
)].
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Materials and methods |
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Force measurements
Larvae were individually attached to a calibrated glass micropipette tether
in order to measure the forces that they generated during swimming. Each larva
was held at the tip of the tether using light suction
(Fig. 1) from a modified mouth
pipette. This micropipette was anchored at its base with a rubber stopper that
provided a flexible pivot. No bending in the micropipette was visible under a
dissecting microscope when loaded at the tip of the tether. We therefore
assumed that the micropipette was rigid and that deflections at the tip were
due entirely to flexion at the pivot. The small deflections by the tether were
recorded during calibration and larval swimming by a high-speed video camera
(Redlake Imaging PCI Mono/1000S Motionscope, 156 pixelsx320 pixels, 1000
frames s-1) mounted on a compound microscope (Olympus, CHA), which
was placed on its side at a right angle to the micropipette
(Fig. 1). Video recordings of
tether deflections made at the objective of the compound microscope were
translated into radial deflections at the pivot of the micropipette ()
using the following trigonometric relationship:
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|
The tether was modeled as a pendulum, with input force generated by the
tail of a swimming larva (F) at the end and a damped spring at the
pivot (Fig. 2). According to
this model, the moments acting at the pivot were described by the following
equation of motion (based on the equation for a damped pendulum;
Meriam and Kraige, 1997a):
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To calibrate the tether, we measured its stiffness and damping constants in
a dynamic mechanical test. This test consisted of pulling and releasing the
tether and then recording its passive movement over time
(Fig. 2A). The tether
oscillated like an underdamped pendulum
(Meriam and Kraige, 1997a)
with a natural frequency (101 Hz) well outside the range of tail-beat
frequencies expected for ascidian larvae
(McHenry, 2001
). Using the
equation of motion for the tether (equation 2, with F=0), its
oscillations were predictable if the mass and the stiffness and damping
coefficients were known. Conversely, we solved for the stiffness and damping
coefficients from recordings of position and a measurement of the mass of the
tether (see Appendix for details).
We examined how errors in our measurement of stiffness and damping coefficients were predicted to affect calculations of the force generated by larvae (Fig. 2CH). By simulating the input force generated by a larva as a sine wave with an amplitude of 20 µN, we numerically solved equation 2 (using MATLAB, version 6.0, Mathworks) for the position of the tether over time at 1000 Hz (the sampling rate of our recordings). From these simulated recordings of tether position, we then solved equation 2 for F, the force generated by the larva. This circular series of calculations demonstrated that our sampling rate was sufficient to follow rapid changes in input force (Fig. 2C). Furthermore, we found that a minimum of 92% of the instantaneous moments resisting the input force were generated by the stiffness of the tether (i.e. the weight and damping of the tether provided a maximal 8% of the resistance to input force). If the values of stiffness and damping coefficients used in force measurements differed from those used to simulate tether deflections, then measured force did not accurately reflect the timing or magnitude of simulated force (Fig. 2D). This situation is comparable with using inaccurate values of stiffness and damping coefficients for measurements of force in an experiment.
By varying the difference between the stiffness and damping coefficients
used to simulate changes in tether position over time (i.e. the actual
coefficients) and those used for force measurements (i.e. the measured
coefficients), we explored how inaccuracy in measured coefficients was
predicted to alter the timing and magnitude of measured force
(Fig. 2EH). We simulated
changes in force at 18 Hz, to mimic oscillations in force at the tail-beat
frequency (McHenry, 2001), and
at 180 Hz, to simulate rapid changes in force. Within the level of precision
(i.e. ±2 S.D.) of our measurements of stiffness and damping
coefficients, measured force was not predicted to precede or lag behind
simulated force by more than 1 ms, which is just 1.8% of an 18 Hz tail-beat
period (Fig. 2E,F). Error in
the damping coefficient may have caused measurements to overestimate rapidly
changing force by as much as 7.5% (Fig.
2G). Within the precision of measured stiffness coefficients,
measured forces may have differed from actual values by as much as 2.0%
(Fig. 2H). These findings
suggest that our measurements accurately reflect the timing of force generated
by larvae, but the magnitude of force may be inaccurate by as much as
7.5%.
Midline kinematics
The ventral surface of the body was recorded during tethered swimming
(Fig. 1A) with a high-speed
video camera (Redlake Imaging PCI Mono/1000S Motionscope, 320 pixelsx280
pixels, 500 frames s-1) mounted to a dissecting microscope (Wild,
M5A) beneath the glass tank containing the tethered larva. The video signal
from this camera was recorded by the same computer (Dell Precision 410, with
Motionscope 2.14 software, Redlake Imaging) as was used to record micropipette
deflections, which allowed the recordings to be synchronized.
Coordinates describing the shape of the midline of the tail were acquired
from video recordings, and the motion of the tail of larvae of
Botrylloides sp. was characterized using the methodology presented by
McHenry (2001). A macro
program (on an Apple PowerMac G3 with NIH Image, version 1.62) found 20
midline coordinates that were evenly distributed along its length (see
McHenry, 2001
for details). In
order to use the measured kinematics in our hydrodynamic models at any body
length, we normalized all kinematic parameters to the body length of larvae
(L, the distance from the anterior to posterior margins of the body)
and the tail-beat period of their swimming (P; note that asterisks
are used to denote non-dimensionality). According to McHenry
(2001
), the following
equations describe the temporal variation in the change in the position of the
inflection point along the length of the tail (z*), the
curvature of the tail between inflection points (
*), and the
trunk angle (
, the angle between the longitudinal axis of the trunk and
the third midline coordinate, located at 0.15 tail lengths posterior to the
intersection point of the trunk and tail):
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Morphology and mechanics of the body
We measured the shape of the body to provide parameter values for our
calculations of fluid forces and to estimate the body mass, center of mass and
its moment of inertia. The peripheral shape of the body was measured (with NIH
Image version 1.62 on an Apple PowerMac G3) using digital still images of
larvae from dorsal and lateral views that were captured on computer (7100/80
PowerPC Macintosh with Rasterops 24XLTV frame grabber) using a video camera
(Sony, DXC-151A) mounted on a dissecting microscope (Nikon, SMZ-10A). These
images had a spatial resolution of 640 pixelsx480 pixels, with each
pixel representing approximately a 6 µm square with an 8-bit grayscale
intensity value. Coordinates along the peripheral shape of the body were
isolated by thresholding the image (i.e. converting from grayscale to binary;
Russ, 1999). We found
coordinates at 50 points evenly spaced along the length of the trunk and 50
points evenly spaced along the length of the tail (using MATLAB). From images
of the lateral view, we used the same method to measure the dorso-ventral
margins of the trunk, cellular tail and tail element. By the same method, we
measured the width of the trunk from the dorsal view.
By assuming that the trunk was elliptical in cross-section and that the
cellular region of the tail was circular in cross-section, we calculated the
body mass, center of mass and moment of inertia using a program written in
MATLAB from reconstructions of the body's volume. These calculations divided
the volume of the body into small volumetric elements (each having a volume of
wi, where i is the element number) with
the position of each element's center located at xi and
yi coordinates with respect to the body's coordinate
system. This system has its origin at the intersection between the trunk and
tail, its x-axis running through the anterior-most point on the
trunk, and its orthogonal y-axis oriented to the left of the body, on
the frontal plane (as in McHenry,
2001
). The tail fin was assumed to be rectangular in
cross-section, with a thickness of 0.002 body lengths (measured from camera
lucida drawings of tail cross-sections;
Grave, 1934
;
Grave and Woodbridge, 1924
).
The mass of the body was calculated as the product of the tissue density
(
body) and the sum of volumetric elements that comprise the
body:
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In order to test the effect of tissue density, we ran simulations (see
`Modeling free swimming' below) with the mean kinematics and morphometrics at
high tissue density (body=1.250 g ml-1, the density
of an echinopluteus larva of an echinoid with calcareous spicules;
(Pennington and Emlet, 1986
)
and low tissue density (
body=1.024 g ml-1, the
density of seawater at 20°C; Vogel,
1981
). All other simulations were run with a tissue density
typical of marine invertebrate larvae not possessing a rigid skeleton
(
body=1.100 g ml-1;
Pennington and Emlet,
1986
).
Kinematics of freely swimming larvae
Freely swimming larvae were filmed simultaneously with two digital
high-speed video cameras (recording at 500 frames s-1) using the
methodology described by McHenry and Strother
(in press). These cameras
(Redlake PCI Mono/100S Motionscope, 320 pixelsx280 pixels per camera,
each equipped with a 50 mm macro lens, Sigma) were directed orthogonally and
both were focused on a small volume (1 cm3) of water in the center
of an aquarium (with inner dimensions of 3 cm width x 3 cm depth x
6 cm height). Larvae were illuminated from the side with two fiberoptic lamps
(Cole Parmer 9741-50).
We recorded the swimming speed of larvae by tracking, in three dimensions,
the movement of the intersection between the trunk and tail during swimming
sequences. From the mean values of swimming speed (), we
calculated a Re of the body for freely swimming larvae using the
following equation:
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Hydrodynamic forces and moments generated by the tail
We modeled the hydrodynamics of the tail using a blade-element approach
that divided the length of the tail into 50 tail elements and calculated the
force generated by each of these elements. Each element was dorso-ventrally
oriented, meaning that the length of each element ran from the dorsal to the
ventral margins of the fin. For each instant of time in a swimming sequence,
the force acting on each element (Ej, where
j is the tail element number) was calculated by assuming that it
generated the same force as a comparably sized flat plate moving with the same
kinematics. Our models assume that each tail element generates force that is
independent of neighboring elements. This neglects any influence that flow
generated along the length of the body may have on force generation. The total
force generated by such a plate is the sum of as many as three forces: the
acceleration reaction (Eja), skin friction
(Ejs) and the form force
(Ejf; Fig.
3). The contribution of each of these forces to the total force
and moment instantaneously generated by the tail was calculated by taking the
sum of forces and moments generated by all elements (see Appendix). Dividing
the tail into 75 and 100 tail elements did not generate predictions of forces
or moments that were noticeably different from predictions generated with 50
tail elements, but models with 25 tail elements did generate predictions
different from models with 50 elements. Therefore, we ran all simulations with
50 tail elements.
|
We modeled the swimming of larvae with both quasi-steady and unsteady models. In the quasi-steady model, the force generated by the tail (F) was calculated as the sum of skin friction (Fs) and the form force (Ff; F=Ff+Fs), and the total moment (M) was calculated as the sum of moments generated by skin friction (Ms) and the form force (Mf; M=Mf+Ms). According to this model, the force acting on a tail element is equal to the sum of the form force and skin friction acting on the element (Ej=Ejf+Ejs; Fig. 3B). In the unsteady model, the force generated by the tail was calculated as the sum of all three forces (F=Ff+Fs+Fa, where Fa is the acceleration reaction generated by the tail), and the total moment was calculated as the sum of moments generated by all three forces (M=Mf+Ms+Ma, where Ma is the moment generated by the acceleration reaction). According to the unsteady model, the force acting on a tail element is equal to the sum of the form force, skin friction and acceleration reaction (Ej=Ejf+Ejs+Eja; Fig. 3C).
The acceleration reaction
The acceleration reaction generated by a tail element was calculated as the
product of the added mass coefficient (cja),
the density of water () and the component of the rate of change in the
velocity of the element that acts in the direction normal to the element's
surface and lies on the frontal plane of the body
(Vjnorm;
Lighthill, 1975
):
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Skin friction
At Re<102, skin friction may generate force that is
both normal and tangent to a surface. Therefore, the equation for skin
friction on a tail element combines analytical approximations for skin
friction acting tangent (Schlichting,
1979) and normal (Hoerner,
1965
) to the surface of a flat plate:
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Form force
The form force acts normal to a surface and varies with the square of flow
speed, as expressed by the following equation
(Batchelor, 1967):
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The contribution of the form force to the total force acting on a flat
plate is predicted to change with Re
(Fig. 3D). Using the form of
the curve-fit equation for changes in the force coefficient on a sphere at
different Re given by White
(1991), the following equation
gives the force coefficient generated by both form force and skin friction
(cjs+f norm) over intermediate Re
(100<Re<103):
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Hydrodynamic forces and moments generated by the trunk
The force acting on the trunk (T) was assumed to be the same as that
acting on a sphere with the same kinematics and a diameter equal to the length
of the trunk. At intermediate Re, this force is equal to the sum of
skin friction (Ts) and the form force
(Tf). The form force varies with the square of the velocity
of the trunk (P; Batchelor,
1967):
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Modeling free swimming
Using the equations that describe the hydrodynamics of swimming, we modeled
the dynamics of free swimming to calculate predicted movement by the center of
mass of a swimming ascidian larva. The acceleration of the body (A) was
calculated as the sum of hydrodynamic forces acting on the body, divided by
body mass:
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We calculated the percentage of thrust and drag generated by the form force
and skin friction in order to evaluate the relative importance of these forces
to propulsion. This percentage was calculated individually for the trunk and
tail and for both thrust and drag. For example, the following equation was
used to calculate the percentage of thrust generated by the form force on the
tail (Hf tail):
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In order to examine how the relative magnitude of form force and skin friction changes with the Re of the body, we ran a series of simulations using model larvae of different body lengths. Each simulation used the mean morphometrics and kinematic parameter values. The non-dimensional morphometrics and kinematics were scaled to the mean measured tail-beat period and the body length used in the simulation. This means that animations of the body movements in the model appeared identical for all simulations (i.e. models were kinematically and geometrically similar), despite being different sizes.
Statistical comparisons between measurements and predictions
We tested our mathematical models by comparing the measured forces and
swimming speeds of larvae with model predictions. We measured the mean thrust
(force directed towards the anterior) and lateral force generated by a
tethered larva and used our model to predict those forces using the same
kinematics as measured for the tethered larva and the mean body dimensions.
Such measurements and model predictions were made for a number of larvae, and
a paired Student's t-test (Sokal
and Rohlf, 1995) was used to compare measured and predicted
forces. Such comparisons were made with predictions from both the quasi-steady
model and the unsteady model.
Predictions of mean swimming speeds from both models were compared with
measurements of speed. Model predictions of swimming speed were generated
using the mean body dimensions and the tail kinematics of individual larvae
measured during tethered swimming. This assumes that the midline kinematics of
freely swimming larvae were not dramatically different from that of tethered
larvae. Mean swimming speeds were measured on a different sample of freely
swimming larvae, and an unpaired t-test
(Sokal and Rohlf, 1995) was
used to compare predictions of swimming speed with measurements. We verified
that samples did not violate the assumption of a normal distribution by
testing samples with a KolomogorovSmirnov test (samples with
P>0.05 were considered to be normally distributed).
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Results |
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The force predictions by the quasi-steady model more closely matched the timing of measurements than those of the unsteady model (Fig. 7). The force predicted by the quasi-steady model (F=Ff+Fs) oscillated in phase with measured lateral forces. However, the unsteady model (F=Ff+Fs+Fa) predicted peaks of force generation by the acceleration reaction acting in the direction opposite to the measured force (Figs 5, 7). At instants of high tail speed, the form force was large and was followed by the acceleration reaction acting in the opposite direction as the tail decelerated and reversed direction. Although both models accurately predicted mean forces (Table 2), the timing of force production suggests that the acceleration reaction does not generate propulsive force in the swimming of ascidian larvae.
|
Freely swimming larvae
Simulations of free swimming allowed the body of larvae to rotate and
translate in response to the hydrodynamic forces generated by the body. As
such movement could contribute to the flow encountered by a swimming larva,
the forces generated by freely swimming larvae were not assumed to be the same
as those generated by tethered larvae. Therefore, simulations of free swimming
were a closer approximation of the dynamics of freely swimming larvae and
provided a test for whether the results of tethering experiments apply to
freely swimming larvae.
The results of these simulations support the result from tethering
experiments that the acceleration reaction does not play a role in the
hydrodynamics of swimming. The quasi-steady model
(F=Ff+Fs) predicted a mean swimming speed
that was statistically indistinguishable from measured mean swimming speed. By
contrast, the unsteady model
(F=Ff+Fs+Fa) predicted a
mean swimming speed that was significantly different from measurements
(Table 2). We found small
(<4%) differences in predicted mean speed between models using a high
(body=1.250 g ml-1) and low
(
body=1.024 g ml-1) tissue density, suggesting that
any inaccuracy in the tissue density used for simulations
(
body=1.100 g ml-1) had a negligible effect on
predictions.
Reynolds number values varied among different regions of the body (Table 3). The mean Reynolds number for the whole body (Re=7.7x101) was larger than the Reynolds number for the trunk (Rea=2.8x101) because the whole body is greater in length than the length of just the trunk. The mean height-specific Reynolds number (Rejl) and the mean position-specific Reynolds number (Rejs) were larger towards the posterior (Table 3).
|
Hydrodynamics at 100<Re<102
Predictions by the quasi-steady model showed how thrust and drag may be
generated differently by form force and skin friction at different
Re. At Re100, both thrust and drag were
predicted to be dominated (>95%) by skin friction acting on the trunk and
tail (Fig. 8A,B). At
Re
101, most drag (63%) was generated by skin friction
acting on the trunk, and most thrust (69%) was generated by skin friction
acting on the tail (Fig. 8C,D).
At Re
102, drag was generated by a combination of skin
friction and form force, but thrust was generated almost entirely by form
force acting on the tail (Fig.
8E,F). By running simulations throughout the intermediate
Re range (100<Re<102), we found
that form force gradually dominates thrust generation (up to 98%) with
increasing Re. Although the proportion of drag generated by form
force increases with Re, skin friction generates a greater proportion
of drag (>62%) than does form force, even at
Re
102.
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Discussion |
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This discrepancy between our results and Jordan
(1992) on the relative
importance of the acceleration reaction may be reconciled if the acceleration
reaction coefficient varies with Re. The acceleration reaction is the
product of the acceleration reaction coefficient (which depends on the height
of the tail element), the density of water and the acceleration of a tail
element (equation 12). Both Jordan
(1992
) and the present study
used the standard inviscid approximation (equation 13) for the acceleration
reaction coefficient (used in elongated body theory;
Lighthill, 1975
). However,
chaetognaths attain Re
103 and more rapid tail
accelerations than ascidian larvae. If the actual acceleration reaction
coefficient is lower than the inviscid approximation at the Re of
ascidian larvae (Re
102), then predictions of the
acceleration reaction would be smaller in magnitude. The chaetognath may still
generate sizeable acceleration reaction in this regime by beating its tail
with relatively high accelerations.
Although swimming at Re>102 has not been reported
among ascidian larvae, numerous vertebrate and invertebrate species do swim in
this regime. We predict that as Re approaches 103, the
acceleration reaction contributes more to the generation of thrust in
undulatory swimming. Although it remains unclear how the magnitude of the
acceleration reaction changes with Re, the unsteady models proposed
here (F=Ff+Fs+Fa)
and elsewhere (Jordan, 1992;
Vlyman, 1974
) should
approximate the hydrodynamics of undulatory swimming at
Re
103.
Skin friction and form force
In support of prior work (e.g. Fuiman
and Batty, 1997; Jordan,
1992
; Vlyman,
1974
; Webb and Weihs,
1986
; Weihs,
1980
), our quasi-steady model
(F=Ff+Fs) predicted that the
relative magnitude of inertial and viscous forces is different at different
Re. At Re
10°, skin friction (acting on both the
trunk and tail; Fig. 8)
dominated the generation of thrust and drag
(Fig. 9). This result is
consistent with the viscous regime proposed by Weihs
(1980
) for swimming at
Re<101. Also in accordance with Weihs
(1980
) are the findings that
form force contributes more to thrust and drag at high Re than at low
Re (Fig. 9) and that
thrust (Fig. 8) is dominated by
form force at Re
102. However, it is surprising that
drag was generated more by skin friction than form force at
Re
102 (Figs
8,
9). Contrary to Weihs'
(1980
) proposal for an
inertial regime at Re>2x102, this result suggests
that the fluid forces that contribute to thrust are not necessarily the same
forces that generate drag. This is unlike swimming in spermatozoa (at
Re<<100), where both thrust and drag are dominated by
skin friction acting on both the trunk and flagellum
(Gray and Hancock, 1955
), or
some adult fish (at Re>>102), where thrust and drag are
both dominated by the acceleration reaction
(Lighthill, 1975
;
Wu, 1971
).
|
Our results suggest that ontogenetic or behavioral changes in Re
cause gradual changes in the relative contribution of skin friction and form
force to thrust and drag. As pointed out by Weihs
(1980), differences in
intermediate Re within an order of magnitude generally do not suggest
large hydrodynamic differences. Although it has been heuristically useful to
consider the differences between viscous and inertial regimes (e.g.
Webb and Weihs, 1986
), it is
valuable to recognize that these domains are at opposite ends of a continuum
spanning three orders of magnitude in Re. This distinction makes it
unlikely that larval fish grow through a hydrodynamic `threshold' where
inertial forces come to dominate the hydrodynamics of swimming in an abrupt
transition with changing Re (e.g.
Muller and Videler, 1996
).
In summary, our results suggest that the acceleration reaction does not
play a large role in the hydrodynamics of steady undulatory swimming at
intermediate Re (100<Re<102).
Our quasi-steady model predicted that thrust and drag are generated primarily
by skin friction at low Re (Re100) and that
form force generates a greater proportion of thrust and drag at high
Re than at low Re. Although thrust is generated primarily by
form force at Re
102, drag is generated more by skin
friction than form force in this regime. Unlike swimming at
Re>102 and Re<100, the fluid
forces that generate thrust cannot be assumed to be the same as those that
generate drag at intermediate Reynolds numbers.
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Appendix |
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![]() | (29) |
![]() | (30) |
![]() | (31) |
![]() | (32) |
Calculating tail force
The total force generated by the tail of a larva was calculated as the sum
of forces acting on all elements of the tail. For example, the total
acceleration reaction generated by the tail was found as the sum of
acceleration reaction forces acting on tail elements:
![]() | (33) |
![]() | (34) |
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Acknowledgments |
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References |
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