The hydrodynamics of eel swimming : I. Wake structure
Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA
* Author for correspondence (e-mail: tytell{at}oeb.harvard.edu)
Accepted 8 March 2004
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Summary |
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Key words: eel, Anguilla rostrata, wake structure, particle image velocimetry, fish, swimming, fluid dynamics, efficiency
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Introduction |
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Most modern studies on the hydrodynamics of fish swimming have been done on
carangiform swimmers. These fishes tend to have fusiform or laterally
compressed bodies, often with a pronounced caudal peduncle. The greatest
lateral excursions occur near the peduncle and the caudal fin
(Webb, 1975), although there
may be some yawing motions at the head
(Donley and Dickson, 2000
). In
addition, researchers have distinguished several gradations of carangiform
swimming, from subcarangiform, in which a greater proportion of the body
undulates, to thunniform, in which the tail moves largely independently of the
body (Webb, 1975
). While
swimming, carangiform fishes produce a series of vertical linked vortex rings,
angled to the swimming direction
(Müller et al., 1997
;
Triantafyllou et al., 2000
;
Drucker and Lauder, 2001
;
Nauen and Lauder, 2002a
).
The hydrodynamics of anguilliform swimming have been studied much less.
Like the eel, after which this mode is named, anguilliform swimmers tend to be
elongate with little or no narrowing at the caudal peduncle. This lack of
separation between the body and tail is particularly extreme in eels, in which
the dorsal, caudal and anal fins effectively form a continuous median fin
(Helfman et al., 1997). In
other anguilliform swimmers, such as sharks and needlefish, the fins are more
separated and there may be a slight narrowing at the caudal peduncle
(Liao, 2002
). They undulate
from one-third to almost all of their bodies, depending on speed, often with
one or more complete waves present at a time
(Gillis, 1998
). These extra
undulations, relative to carangiform swimmers, must affect the flow around
their bodies and in the wake, but the effect is not well understood.
Lighthill's elongated body theory (referred to here as EBT) offers some
insight into the possible effect of different kinematics
(Lighthill, 1971). He argues
that the carangiform mode is more efficient, because his theory predicts that
thrust is produced only at the trailing edge of the tail. Therefore, any extra
body undulation is wasted energy, and efficient swimmers should undulate as
little of their body as possible. Indeed, many pelagic predators considered
highly efficient (Lighthill,
1970
; Barrett et al.,
1999
) are thunniform swimmers and hold their bodies relatively
straight. EBT, however, is a simple model, and neglects many effects,
including viscous forces, which could enable thrust production along the
length of an anguilliform fish's body
(Taneda and Tomonari, 1974
;
Shen et al., 2003
).
Only two recent studies (Carling et al.,
1998; Müller et al.,
2001
) address the hydrodynamics of eel-like swimming, and they
offer divergent conclusions. Müller et al.
(2001
) used particle image
velocimetry (PIV; Willert and Gharib,
1991
) to observe the flow fields around freely swimming juvenile
eels. Based on their observations, they hypothesized that eels' wakes consist
of unlinked vortex rings moving laterally
(Fig. 1A). They proposed that
eels shed two separate same-sign vortices because of a lag between the
stop/start vortex (solid arrows in Fig.
1A), shed when the tail changes direction, and centers of rotation
that progress down the body, which they termed `proto-vortices' (broken arrows
in Fig. 1A). They did not
observe a downstream jet behind the tail, which is typical of carangiform
wakes (e.g. Nauen and Lauder,
2002a
). Due to the difficulty of working with freely swimming
eels, Müller and colleagues did not evaluate the effects of different
swimming speeds on the wake structure. Also, the mechanical significance of
the difference between carangiform wakes and the wake they observed for eels
remains unclear.
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By contrast, Carling et al.
(1998) used a two-dimensional
computational fluid dynamic model to estimate the flow fields behind a
self-propelled `eel-like' anguilliform swimmer. Their calculations indicated a
single, large vortex ring wrapping around the eel, with the eel in the center,
producing upstream flow behind the eel
(Fig. 1B). These results
suggest that eels produce thrust almost exclusively along the body, but not at
the tail tip, which seems to result mostly in drag. Carling's model, while
tested thoroughly in several standard test cases
(Carling, 2002
), has not been
verified on living eels.
Several important questions remain. Which of these two views of
anguilliform wake flow patterns are correct? What are the quantitative
differences between anguilliform and carangiform wakes? How do these
differences affect the swimming performance? How efficient hydrodynamically is
anguilliform swimming relative to carangiform swimming? In particular,
Lighthill's (1970) argument
for the inefficiency of anguilliform swimming leads to an inconsistency: eels
migrate thousands of kilometers without feeding
(van Ginneken and van den Thillart,
2000
), and many anguilliform sharks swim constantly
(Donley and Shadwick, 2003
).
It is unlikely that such proficient swimmers are highly inefficient. In fact,
a recent study of swimming energetics found that the physiological cost of
migration for eels was low (van Ginneken
and van den Thillart, 2000
).
In the present study, therefore, we examine in detail the wake of the
American eel, Anguilla rostrata, swimming steadily at a single speed.
The flow around anguilliform swimmers is compared with both previous models
and with previous data from carangiform swimmers. We propose a new explanation
of the hydrodynamic differences between anguilliform and carangiform swimming,
emphasizing the importance of carangiform swimmers' narrow caudal peduncle and
propeller-like caudal fin over the importance of differences in kinematics. In
addition, we provide the first quantitative comparison of the predictions of
EBT (Lighthill, 1971) to
empirical forces estimated using PIV and demonstrate a partial correlation.
Finally, we examine the efficiency and power output for steadily swimming
eels.
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Materials and methods |
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All data were taken from eels swimming at 1.4 L
s-1, ranging from 1.30 to 1.44 L s-1. In total,
the swimming kinematics for 415 tail beats were analyzed. The hydrodynamics of
118 of these were examined. Considerable effort was expended to analyze only
truly steady swimming sequences; all sequences analyzed had a maximum
variation in velocity under ±5%; in most cases, the velocity varied by
less than ±3%; and the S.D. in velocity over all sets was
2%. In addition, most sequences involved 10 or more sequential steady tail
beats. During the experiments, an eel was gently maneuvered into position
using a wooden probe. Care was taken to remove the probe completely from the
region around the eel before data were taken.
The eels were filmed from below through a mirror inclined at 45° with
two high-speed cameras, one to record the swimming kinematics and one to film
the light sheet for PIV (Fig.
2). An approximately 30 cm-wide horizontal light sheet was
projected 7 mm above the bottom of the tank, along the dorso-ventral midline
of the eels, using two argon-ion lasers operating at 4 W and 8 W,
respectively. The light from the two lasers was combined optically to form a
single large light sheet. The eels' swimming kinematics were recorded using a
RedLake digital camera at 250 or 125 frames s-1. For PIV, a
close-up view of the light sheet was filmed using either a RedLake digital
camera at 250 frames s-1 at 480 pixex420 pixel resolution or
a NAC Hi-DCam at 500 frames s-1 at 1280 pixex1024 pixel
resolution. A six-point calibration between the two cameras allowed positions
to be converted between the two images with an error of
0.5 mm using a
linear rotation and scaling transformation (Matlab 6.1 imtransform routine;
Mathworks, Inc., Natick, MA, USA)
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Kinematics
Eel outlines and midlines were digitized automatically using a custom
Matlab 6.1 (Mathworks, Inc.) program. The positions of the head and tail were
identified manually. The eel midline was then located by performing a 1-D
cross-correlation analysis along transects between the head and the tail, to
find the bright region with a width corresponding to the known width of the
eel. This technique produced fewer errors resulting from the presence of PIV
particles in portions of the images than thresholding-based techniques used in
previous kinematic studies (e.g. Tytell
and Lauder, 2002). A similar method located the edges of the eel's
image. Twenty points were identified along the midline and were simultaneously
smoothed temporally and spatially using a 2-D tensor product spline (Matlab's
spaps routine), a two-dimensional analog of an optimal method (`MSE' method in
Walker, 1998
). The tensor
product spline, however, does not allow a direct specification of the mean
error on the data as in the 1-D version. Thus, the smoothing values were
initially set at 0.5 pixel, the limit of measurement accuracy from the video,
and adjusted manually until a good fit was reached. This resulted in a mean
distance between the smoothed and measured values of less than 0.3 pixel
(approximately 0.2 mm).
Kinematic variables, including amplitude at each body point, tail beat frequency, body wave length and body wave velocity, were calculated by finding the peaks in the lateral excursion of each point over time. A Matlab program automatically located the peaks based on the midlines, as estimated above. The amplitude, frequency and wave length were determined by the timing, position and height of the peaks. Average side-to-side tail velocity was estimated as 4A/f, where A is amplitude and f is frequency. The body wave velocity was determined by the slope of the line fitted to the wave peak position (in distance down the body) and the time of that wave peak.
The forces and power required for swimming were calculated using
large-amplitude EBT (Lighthill,
1971). The time-varying thrust (Fthrust) and
lateral force (Flateral) are:
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![]() | (2) |
![]() | (3) |
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Additionally, the position of the proto-vortices along the eel's body was
estimated according to Müller et al.
(2001) by searching for the
points along the body where lift (Flift) equals zero,
defined using small-amplitude EBT
(Lighthill, 1960
) as:
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Hydrodynamics
High-resolution PIV was performed using a custom Matlab 6.1 program in two
passes using a standard statistical cross-correlation
(Fincham and Spedding, 1997)
and a Hart (2000
) error
correction technique with an integer pixel estimate of the velocity between
passes (as in Westerweel et al.,
1997
; Hart, 1999
).
PIV interrogation regions were about 5 mmx5 mm and 2.5 mmx2.5 mm
in coarse and fine pass, respectively, with search regions of 9 mmx9 mm
and 3.5 mmx3.5 mm. For the lower resolution video, this produced a
matrix 68x78 vectors, and for the higher resolution, 100x125
vectors. Data were smoothed and interpolated onto a regular grid using an
adaptive Gaussian window algorithm with the optimal window size (23 mm
for these data; Agüí and
Jiménez, 1987
; Spedding
and Rignot, 1993
), being careful to note the inherently uneven
spacing of PIV data (Spedding and Rignot,
1993
). The Gaussian window method was used because it provides
good results (Fincham and Spedding,
1997
) while being simple and fast when applied to such large
matrices of vectors.
Boundary layers and background flow
Because eels swam on the bottom of the flow tank, we made a series of
measurements to quantify the flow regime in this part of the flow tank and to
be certain that we were observing free-stream flow. At all swimming speeds,
the PIV light sheet was above the flow tank boundary layer, which was
turbulent. The boundary layer was quantified using a vertical light sheet,
showing that the boundary layer thickness () was equal to
7 mm at
the slowest flow speed used (Fig.
4). The boundary layer changes from laminar to turbulent just
below that speed, indicating that the boundary layers in all data sets were
turbulent. At speeds above this transition, the boundary layer is always
thinner, decreasing proportionally to the free-stream velocity to
the1/5 power (Schlichting,
1979
). Thus, at the highest speed used,
40 cm s-1,
we estimated the boundary layer to be
5.5 mm thick.
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Because the boundary layer was turbulent, the background flow was complex.
Turbulent boundary layers are characterized by a range of relatively
long-lived, coherent structures that rise up out of the boundary layer region
(Robinson, 1991). In
particular, structures called `quasi-streamwise vortices' were common. In our
data, quasi-streamwise vortices, which are vortex lines oriented approximately
parallel to the flow (Robinson,
1991
), were visible as streamwise regions of slower or angled flow
(Fig. 4C). Conveniently, they
were consistent over a duration of many minutes.
The consistency of the turbulent structures enabled us to subtract their effect from the flow. For each swimming speed, we took 50 flow fields without the eel present. These fields were then averaged to estimate a mean background flow, which was subtracted from the wake data to remove the turbulent effects. The background velocity changed spatially by as much as 13% cm-1 but changed over time by only about 0.1% s-1 (Fig. 4C).
Wake analysis
Wakes were only analyzed when the kinematics remained steady for at least
three tail beats. Most wakes analyzed included between five and 15 consecutive
steady tail beats. Phase-averaged wake vector fields were produced by
averaging frames corresponding to the same tail-beat phase, dividing the tail
beat into 20 steps. These phase-averaged fields are instructive for
visualization, but no quantitative values were measured from them.
Vortex centers were digitized manually. Location of the vortices in the
vector fields was aided by plotting the discriminant for complex eigenvalues
(DCEV; Vollmers, 1993;
Stamhuis and Videler, 1995
):
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Assuming that the two vortices on either side of the lateral jet are the
cores of a small-core vortex ring, the impulse (I) of the ring was
estimated as (Faber, 1995):
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The power (P) that the eel added to the fluid was determined by
integrating across a plane approximately 8 mm behind the tail tip:
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![]() | (8) |
Force, impulse and power were both normalized to produce force and power
coefficients. Using coefficients is important because it makes these values
comparable between eels of different sizes and between the present and other
studies (Schultz and Webb,
2002). The normalization factors for force and power were the
standard
SU2 and
SU3, respectively, where S is the
wetted surface area of the eel (Faber,
1995
). No standard normalization exists for impulse, however.
Since impulse is in units of force x time, we chose to normalize by the
standard characteristic force
SU2 and a
characteristic time L/U, resulting in an impulse
normalization factor of
SLU.
Statistics
The kinematics in the data set used for PIV measurements were compared with
those in the complete data set to make sure that the swimming behavior in the
selected data was typical. A mixed-model multivariate analysis of variance
(MANOVA; Zar, 1999) was
performed on the kinematic variables, including the individual as a random
effect and which set the data came from (i.e. the PIV or complete data sets)
as a fixed effect. The kinematic differences between individuals in the PIV
data set were also assessed using a MANOVA including only the effect of
individual variation.
The changes in wake morphology over time were examined by regressing individual wake morphology parameters on tail-beat phase, including the individual as a random effect. Significant slopes were determined by testing the significance of variation in time over the variation due to the interaction by individuals with time, as in a mixed-model analysis of variance (ANOVA).
A repeated-measures ANOVA (Zar,
1999) was performed to compare the initial circulation of the
primary vortex to the sum of the circulations of the primary and secondary
vortices, after they divided. Circulation at two different times was the
repeated measure, which allowed the early primary vortex circulation to be
compared with the sum of the primary and secondary vortex circulations later
in time. The individual was included as a random factor
(Zar, 1999
).
Finally, mixed-model ANOVAs were used to compare force, power and impulse
estimates based on EBT (Lighthill,
1971) with those values measured using PIV. The individual was
again included as a random factor.
All analyses were performed using Systat 10.2 (Systat Software, Inc., Point Richmond, CA, USA). All error values that are reported are standard error and include the number of data points, where appropriate.
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Results |
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To verify that the sequences chosen for hydrodynamic analysis were typical of overall swimming performance, we examined a larger data set containing 415 tail beats taken under the same conditions but in which the PIV data were not quantitatively analyzed. A MANOVA on four parameters (tail-beat amplitude and frequency, amplitude growth parameter, body wave length and slip) that completely define the kinematics did not show a significant difference between the larger data set and that used for hydrodynamic analysis (Wilk's lambda=0.978; F4,409=1.858; P=0.101).
Swimming kinematics varied significantly among individuals. In most
variables, individuals differed from one another by less than 10%. However,
one individual consistently chose to swim with a higher amplitude (about 13%
higher) and lower frequency (about 25% lower) than the others. Another
individual used a longer body wave (about 20% longer) than the others. By
contrast, wave velocity differed very little among individuals; all were
within 5% of each other. While these differences were highly significant
(MANOVA: Wilk's lambda=0.0139; F10,200=149.5;
P<0.001), most studies of this nature have significant variation
among individuals (e.g. Shaffer and
Lauder, 1985).
Given the average swimming kinematics, the predicted position of the proto-vortex was calculated analytically using equation 4. The proto-vortex is shed off the tail 16 ms after the tail reaches its maximum lateral excursion, or 5.1% of a tail-beat cycle later.
Hydrodynamics
In all 11 individuals, the wake consisted of lateral jets of fluid,
alternating in direction, separated by one or more vortices or a shear layer
(Fig. 5). Each time the tail
changes direction, it sheds a stop/start vortex. As the tail moves to the
other side, a low pressure region develops in the posterior quarter of the
body, sucking a bolus of fluid laterally. The bolus is shed off the tail,
stretching the stop/start vortex into an unstable shear layer, which
eventually rolls up into two or more separate, same-sign vortices. The wake
generally contains more total power than is predicted by large-amplitude EBT
(Lighthill, 1971). These
features are analyzed in detail below, focusing on the three individuals
chosen for detailed quantitative study.
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Wake morphology
The stop/start vortex, shed when the tail changes direction, is designated
the `primary' vortex. The vortex formed later, when the shear layer rolls up,
is called the `secondary' vortex. The primary vortex from one half tail beat
and the secondary vortex from the next form the edges of each lateral jet.
These two vortices appear to be the cores of a small-core vortex ring.
However, without velocity data from the planes perpendicular to the one used
in the present study, it is not certain that the vortices truly form a ring.
To emphasize this difference, we will not call this region a vortex ring;
instead, we term it the lateral jet.
To address how the wake changes over time, wake morphology parameters were
regressed individually on tail-beat phase and individual, treating the
individual as a random factor. In general, the wake widens over time and
becomes weaker. The distance between the primary and secondary vortex
increases at 0.12 L T-1, where T is a
tail-beat period (F1,2=28.7; P=0.033), during the
approximately 1.5 T in which the wake was visible. The diameter of
the lateral jet, however, stays approximately constant at 0.21 L
throughout time (F1,2=0.370; P=0.605). The two
vortices on either side of the lateral jet (the `vortex ring') stay parallel
to the swimming direction (F1,2=0.037; P=0.864),
but the lateral jet itself is inclined slightly upstream, with an angle of
87° (significantly different from 90°; P<0.001). There is
a trend for the jet to rotate downstream over time, but it is not significant
(F1,2=1.860; P=0.306). The peak velocities in the
jet decrease significantly over time (F1,2=24.0;
P=0.039), diminishing by
15% over a half tail beat, from 0.45 to
0.38 L s-1. By contrast, the circulation measured through
the center of the lateral jet does not change over time
(F1,2=1.536; P=0.349), remaining at
2490±10 cm2 s-1.
To illustrate the rolling up of the unstable shear layer, we took
cross-sections through the primary and secondary vortices over time
(Fig. 6A). The idealized
profile through a single Rankine vortex
(Faber, 1995) is shown above
the first profile and a profile through two same-sign vortices is shown below
the last profile. Additionally, Fig.
6B shows cross-sections across the lateral jet over time, with an
ideal profile through a small-core vortex ring.
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The circulations of the primary and secondary vortices both decrease over
time. In principle, total circulation should remain constant, implying that
the sum of the two circulations should not change over time. A
repeated-measures ANOVA (Zar,
1999) in which the repeated measure was tail-beat phase divided
into early and late regions shows that the initial circulation of the primary
vortex alone, 3300 cm2 s-1, is not significantly
different from the sum of the primary and secondary circulations in the end,
1910 cm2 s-1 and 1520 cm2 s-1,
respectively (F1,89=1.471; P=0.228).
To examine how the wake is generated, flow close to the bodies of the eels was examined. Fig. 7 shows a typical flow pattern near the body of an eel over the course of a tail beat. In the first three frames shown, a strong suction region develops near the tail, pulling a bolus of fluid laterally. This bolus will become the lateral jet in the wake. Proto-vortices are visible (shown with red and blue arrows), but their vorticity is very low (generally less than ±5 s-1).
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Finally, for comparison with the computational model of Carling et al.
(1998), we computed an average
flow behind the eel, averaged over many tail beats. The computational model
predicts a net velocity deficit behind the eel that could be obscured by the
temporal variations in the observed flow.
Fig. 8 shows the flow behind an
eel averaged over 14 tail beats with axial flow magnitude shown in color. On
average, momentum in the wake was elevated above free-stream momentum by
between 2.84 and 6.65 kg mm s-2 at planes 25 mm and 95 mm,
respectively, behind the tail.
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Force, impulse and power
Impulses were calculated from PIV using
equation 6 byassuming that the
observed vortex cores were part of a small-core vortex ring. In
equation 6, rather than using the
circulation of one of the cores, which vary over time and are sensitive to
digitization error, we chose to use the circulation measured through the
center of the lateral jet, which is constant over time and fairly robust to
digitization error. Thus, the impulse coefficient for the lateral jets was
0.0217±0.0004, corresponding to an impulse in a 20 cm eel of 0.76 mN s.
From this value, given that the lateral jet was generated over half a period,
the lateral force coefficient was 0.097±0.001 (4.64 mN in a 20 cm eel).
Fig. 9A shows a typical trace
of lateral force from EBT with the average force estimates from PIV
superimposed; Table 2 displays
the same comparison numerically.
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Power was also measured in the wake at a plane approximately 8 mm downstream of the tail tip. Both total power, including both velocity components, and `lateral' power, including only the lateral (v) velocity component, were calculated. Fig. 9B shows a typical trace of power over time. The total power coefficient was, on average, 0.023±0.002 (303 µW in a 20 cm eel). Lateral power was usually less than half of the total power and was equal to 0.0151±0.0003 (198 µW). Table 2 summarizes the comparison of force, impulse and power measurements from PIV with those calculated via EBT.
In general, EBT underestimates force and power as measured by PIV, although for certain values the two match well (Fig. 9). Both the impulse and the total wake power estimated by PIV and EBT are highly significantly different (P<0.001 in both cases; Table 2). However, the mean force from the PIV measurements matches the peak lateral force estimated by EBT (P=0.182). Additionally, the power estimated using only the lateral component of velocity is not significantly different from the total EBT estimate, in both maximum (PÅ1.000) and mean values (P=0.693). The shape of these two power curves is also visually quite similar (Fig. 9B).
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Discussion |
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The lateral jets are produced along the body, just anterior to the tail tip. In particular, when the tail has reached its maximum lateral excursion, and thus has zero velocity, the point 0.15 L anterior to the tail has reached a high lateral velocity (60.2±0.6% of the swimming velocity). This substantial velocity difference along the eel's body seems to result in a strong suction region that pulls fluid laterally. Once the tail changes direction, it sheds a stop/start vortex (the primary vortex) and begins to shed a bolus of fluid to form a lateral jet. Each full tail beat produces two jets, one to each side, and two vortices separating them.
Because the velocities in successive lateral jets are large and in opposite
directions, a substantial shear layer is present between the jets, with
shearing rates of as much as 90 s-1. This shear layer is unstable
and breaks down into two or more vortices (the secondary vortices), probably
through a KelvinHelmholtz instability
(Faber, 1995). This
instability develops gradually (Fig.
6A), resulting in a fully formed secondary vortex about one full
cycle later. Classic hydrodynamic theory predicts that a
KelvinHelmholtz instability should result in vortices with a spacing
approximately equal to 4
, where
is the shear layer
thickness (Faber, 1995
).
Before the shear layer breaks down,
is approximately 3 mm, giving a
predicted vortex spacing of 37 mm, which is close to the 2030 mm
spacing observed when the secondary vortex is fully formed. Additionally, the
theory suggests that many vortices with this spacing could be formed. Indeed,
another secondary vortex is occasionally formed at about twice the distance
from the primary vortex.
When the jets are fully developed, they point almost directly laterally,
meaning that very little flow is directed axially. Previous studies of caudal
fin swimming (e.g. Müller et al.,
1997; Lauder and Drucker,
2002
; Nauen and Lauder,
2002a
) have interpreted axial downstream flow as evidence for the
production of thrust and have found that estimates of the thrust from PIV
approximately match the estimated drag on the fish
(Lauder and Drucker, 2002
).
This balance also held true for fish swimming using their pectoral fins
(Drucker and Lauder, 1999
). If
downstream velocity is evidence for thrust, where is the thrust signature in
the eel wake?
Because the eels in the present study were swimming steadily, without any substantial accelerations, the net force on the animal must be zero and, thus, the net force measurable in the wake should also be zero. Equivalently, because the momentum of the eel is not changing, there must be no net change in fluid momentum. Thus, while somewhat counter-intuitive, it is physically reasonable that no downstream momentum jet would be evident in the wake. It is important to think of the eel as producing thrust and drag simultaneously. If one could separate thrust from drag, one would see fluid being accelerated down the eel's body, as it produces thrust. At the same time, however, the drag along the eel's body is removing momentum from the fluid. In combination, the two effects cancel each other out, producing no net change in downstream fluid momentum as long as the eel is swimming steadily. All the lateral momentum observed in the wake also cancels out and is simply evidence of wasted energy.
If thrust and drag balance exactly, why did we observe a small increase in
downstream momentum immediately behind the tail
(Fig. 8)? Probably, this
increase is offset by an increase in the opposite direction at the eel's
snout. In still water, an eel swimming forward would push some fluid out of
its way with its snout, increasing the upstream fluid momentum
(Long et al., 2002). For
forces to balance, this upstream increase must be matched by a small
downstream increase at the tail, as we observed. The eel's snout adds upstream
momentum at a rate proportional to
Uahead, where
ahead is an area at the snout, representing a force in the
order of 5 mN. The extra downstream momentum in the wake represents forces
between 3.5 and 7.5 mN, which are roughly in agreement. We thus argue that the
additional downstream momentum observed in the eel wake
(Fig. 8) is necessary to fully
conserve momentum and is not evidence for thrust. A complete control volume
around the eel would resolve this question fully, but eels would not swim
steadily with their heads in the light sheet, preventing us from performing
that additional experiment.
It is important to note that the lack of net change in momentum is not
equivalent to `leaving no footprints', as hypothesized by Ahlborn et al.
(1991). The `footprints' of an
eel are the lateral jets. In principle, at 100% efficiency, as Ahlborn et al.
(1991
) suggested, all power
would go into producing forward motion, and none would go into producing a
wake. The fact that an eel does leave a wake, or footprints, is evidence that
they are not completely efficient.
This momentum balance described above must be true for all steady swimming,
including previous studies that have observed a strong downstream jet during
carangiform and pectoral fin swimming
(Müller et al., 1997;
Drucker and Lauder, 2000
;
Lauder and Drucker, 2002
;
Nauen and Lauder, 2002a
). It
is our hypothesis that these previously studied fishes display some spatial or
temporal separation between thrust and drag production that allows momentum to
balance on average over a tail beat, while still producing a downstream jet
indicating thrust. The apparent discrepancy between this study and these
previous ones is easiest to explain for pectoral fin swimmers. Drucker and
Lauder (2000
) observed a
downstream jet from pectoral fin swimming in bluegill sunfish (Lepomis
macrochirus) and surf perch (Embiotoca jacksoni) that
represented enough force to balance the experimentally measured drag. Unlike
eels, bluegill and surf perch rely solely on their pectoral fins for thrust in
the speed range examined. Pectoral fins effectively produce only thrust and
little drag, relative to the body, which is held nearly motionless at these
swimming speeds and produces only drag. The spatial separation between the
thrust-producing pectoral fins and the drag-producing body allows accurate
measurement of thrust from the pectoral fins alone, as Drucker and Lauder
(2000
) found. Nonetheless, if
one were to examine a control volume around the entire fish, the net fluid
momentum change would be zero. The situation is somewhat like that of an
outboard propeller on a boat: the body, like a boat's hull, generates most of
the drag and negligible thrust, and the pectoral fins, like propellers,
generate all of the thrust with negligible drag.
For carangiform caudal fin swimmers, the situation is more complicated, but
previous results should still be valid. For many fishes, the outboard motor
analogy may still be appropriate. Because carangiform swimmers move their
anterior body relatively little compared to the caudal fin
(Webb, 1975;
Jayne and Lauder, 1995
;
Donley and Dickson, 2000
), very
little thrust can be generated anterior to the caudal peduncle. Flow also does
not separate along the body (Anderson et
al., 2000
) but rather converges on the caudal peduncle
(Nauen and Lauder, 2000
). As
fluid moves along the body, drag removes momentum, but this low-momentum flow
is concentrated at the caudal peduncle. The dorsal and ventral portions of the
caudal fin are therefore exposed primarily to undisturbed free-stream flow.
Except at the very center of the fin, the caudal fin thus may also function
like an outboard motor, producing almost entirely thrust with very little
drag. Probably the analogy is most valid for fishes such as mackerels and
tunas that have a very narrow caudal peduncle and a large caudal fin. Indeed,
in their study of chub mackerel (Scomber japonicus), Nauen and Lauder
(2002a
) found that thrust
measured from the downstream jet roughly balanced experimentally measured drag
(although drag measurements were difficult to make accurately).
For carangiform swimmers with less pronounced caudal peduncles, the
outboard motor analogy may break down somewhat, but differences in swimming
kinematics between them and anguilliform swimmers may explain why thrust wakes
were still observed (e.g. in Müller
et al., 1997; Drucker and
Lauder, 2001
). We speculate that anguilliform swimmers may produce
thrust more continuously over time than carangiform swimmers. For a steadily
swimming fish, thrust need only balance drag on average over a full tail beat.
If thrust is produced in a very pulsatile way, it may briefly exceed drag to
such an extent that it would be evident in the wake. According to a reactive
inviscid theory such as Lighthill's EBT
(Lighthill, 1971
), thrust is
only produced at the tail tip (or other sharp trailing edges). Evaluation of
the EBT equation for thrust generated at the tail tip
(equation 1) results in a
pulsatile force. However, these equations do not include possible thrust from
the body anterior to the tail due to viscous effects. Recent direct numerical
simulations showed that an infinitely long waving plate can produce thrust
(Shen et al., 2003
), in
support of previous experimental observations
(Taneda and Tomonari, 1974
;
Techet, 2001
). Like a waving
plate, the short wavelength undulations along an eel's body can produce thrust
smoothly to even out the pulsatile force from the tail tip. In particular,
since a full wavelength is present on the eel's body, a portion of the body is
moving and likely producing force out of phase with the tail tip. The majority
of thrust may still be produced in the posterior regions of the eel's body,
where we saw accelerated flow (Fig.
7) but, even so, some regions of the posterior body are moving out
of phase with the tail tip, helping to smooth out pulsatile thrust. For
carangiform swimmers, unlike eels, the long wavelength body undulations do not
contain out-of-phase motions at sufficient amplitude and may tend to reinforce
the pulsatile thrust from the tail (Webb,
1975
). Therefore, at certain points in a carangiform swimmer's
tail beat, thrust may exceed drag to produce a thrust wake, even though the
two forces balance on average. For eels, thrust and drag may balance more
evenly over time. Note that Fig.
9 does not contradict this statement.
Fig. 9 shows that lateral force
and power are pulsatile, but axial force was not measurable and `axial power',
constructed in a similar way to `lateral power', remains fairly constant and
small over the tail beat.
The importance of shape
We speculate that the novel wake structure of swimming eels is highly
dependent on their shape and that differences in shape, along with differences
in kinematics, may be one of the primary distinctions between anguilliform and
carangiform swimming. In particular, eels do not have a narrow caudal
peduncle, whereas most carangiform swimmers do. The large lateral jets develop
in the suction region centered around 85% of body length. Both
anguilliform and carangiform swimmers have a substantial undulation amplitude
this close to the tail, even though the kinematics on the anterior body differ
substantially. For example, both chub mackerel and kawakawa tuna
(Euthynnus affinis) have amplitudes of
4% of body length at 0.85
L (Donley and Dickson,
2000
), and largemouth bass (Micropterus salmoides) have
amplitudes of
4.5% at 0.85 L
(Jayne and Lauder, 1995
),
comparable with the 4.4% we measured in eels. However, most carangiform
swimmers are different from eels because they have a narrow caudal peduncle
around 0.85 L. If their body shape were more similar to that of eels,
it is likely that a substantial suction could develop there in the same way as
in eels. The narrowness of the peduncle, however, probably prevents such
suction from developing. Even if a mackerel, for example, swam using the same
kinematics as an eel, its wake would probably differ from an eel's due to the
differences in body shape. In fact, recent results from an engineering study
of rectangular flapping membranes indicate that simple shape differences, such
as the ratio of flapping amplitude to body height, can determine whether the
wake is a linked vortex ring wake, as observed in carangiform swimmers, or an
unlinked ring wake, as in eels (J. Buchholz, personal communication).
Clearly, this effect in fishes is more complicated than a simple ratio and
probably depends on how narrow the peduncle is, relative to the size of the
body and tail. It would therefore be strongly affected by the wide range of
body shapes in fishes. Wakes, therefore, probably show a gradation from those
of mackerel (Nauen and Lauder,
2002a), for example, which have very narrow peduncles but large
caudal fins, to those of eels, which have no narrowing at the peduncle at
all.
Efficiency of anguilliform swimming
One of the goals of the present study was to evaluate the efficiency of
anguilliform swimming relative to carangiform swimming. However, for steady
swimming, efficiency is very hard to evaluate. Froude efficiency () is
usually written, neglecting inertial forces, as:
![]() | (9) |
Specifically, EBT can be used to calculate this thrust value using
equation 1, which can be combined
with the wake power estimate from PIV to produce an efficiency. The estimated
mean thrust is 0.83 mN, and the measured wake power is between 198 and 303
µW, resulting in efficiency estimates between 0.43 and 0.54. Additionally,
EBT can also estimate the efficiency directly. This value,
EBT, is usually written as
1
(VU)/V, where V is
the body wave velocity (Lighthill,
1970
). According to this method, EBT estimates
EBT
as 0.865±0.001. However, since EBT usually underestimated the total
power in the wake (Table 2),
the first range, 0.430.54, is probably the more realistic estimate.
Anguilliform swimming has been hypothesized to be inefficient
(Lighthill, 1970;
Webb, 1975
). Our measurements,
however, indicate a swimming efficiency of around 0.5, or potentially as high
as 0.87, depending on how it is calculated. Because of the difficulties of
estimating efficiency from a steadily swimming fish, it is difficult to
compare this value with previously reported values, which range from 0.74 to
0.97 (Drucker and Lauder,
2001
; Müller et al.,
2001
; Nauen and Lauder,
2002a
,b
).
Comparison with previous studies of anguilliform swimming
Müller et al. (2001)
first observed the wakes of swimming eels and noted their unusual structure.
They showed that two vortices were produced per half tail beat and that the
jet between successive vortices was primarily lateral. Their observations are,
in general, quite similar to ours. With our higher resolution PIV, we are able
to propose a different mechanism for generating the wake. Additionally, our
data allowed a much more detailed examination of the balance of thrust and
drag and the Froude efficiency of steady swimming, which have been
controversial (Schultz and Webb,
2002
).
Nonetheless, there are some important differences between our findings and
those of Müller et al.
(2001). They hypothesized that
the double vortex structure resulted from a phase lag between the vorticity
shed from the tail and circulation produced along the body, which they termed
proto-vortices. Although proto-vortices were evident along the body
(Fig. 7), their vorticity was
much lower than the vorticity of the secondary vortex. The vorticity in the
proto-vortices along the body is generally less than 5 s-1, while
the secondary vortex peak vorticity was often more that 15 s-1.
Müller et al. (2001
) also
observed that fluid velocity increases along the body linearly from head to
tail. By contrast, we observed relatively little increase in fluid velocity
until the last 30% of body length, where the fluid bolus is generated
(Fig. 7). Finally, we
calculated the phase difference between the shedding of stop/start vortices
and the shedding of proto-vortices off the tail. The difference was only
5% of a tail beat cycle, so any proto-vorticity is likely to simply add
to the stop/start vortex, which is forming at almost the same time, rather
than create a separate vortex.
It is somewhat surprising that we saw so much less fluid velocity along the
body than Müller et al.
(2001) did. While the eels
analyzed in detail in the present study, at 20 cm long, were more than twice
as long as those in Muller's study, we examined the wake of a 12 cm eel
qualitatively and found the same pattern as in the larger eels. The eels in
Müller's study seemed to show greater undulation amplitude along the
body, particularly near the head, than the eels in our study. This amplitude
difference may explain the stronger fluid flow near the body but it also
suggests that Müller's eels may have been accelerating slightly, because
increased anterior undulation is often found in accelerating eels (E. D.
Tytell, manuscript in preparation). Additionally, they document a slight
downstream component to the jets
(Müller et al., 2001
),
another indication of acceleration (E. D. Tytell, manuscript in
preparation).
The other model examined in the present study, Carling and colleagues'
computational fluid dynamic model for an 8 cm-long anguilliform swimmer
(Carling et al., 1998), is not
supported by our data or those of Müller et al.
(2001
). Carling's model
predicts a substantially reduced velocity immediately behind the tail, as if
the eel were sucking fluid along with it as it swam
(Fig. 1B). Even averaged over
many tail beats, we did not observe any reduced velocity in the wake; in fact,
immediately behind the tail, the flow is accelerated downstream
(Fig. 8). Somewhat
surprisingly, we observed that momentum in the far wake, 95 mm from the eel's
tail, was greater than that in the near wake, 25 mm from the tail. We
speculate that this effect is due to three-dimensional reorientation or
contraction of the wake, similar to that in the far-field wake of a hovering
insect (Ellington, 1984
).
Nonetheless, it seems clear that axial wake momentum is downstream, the
opposite of what the model predicted
(Carling et al., 1998
).
Additionally, their model does not predict the complex vortical structures and
lateral jets that we consistently observed in all individuals covering a
length range from 12 to 23 cm. While we did not observe the wake of an 8
cmindividual, the size they modeled, Müller et al.
(2001
) examined one that size
and observed a wake similar to those we observed in larger individuals and
quite different from Carling and colleagues' predictions
(Carling et al., 1998
).
To continue the exploration of the hydrodynamic differences between
different modes of swimming, future studies should be careful to include
detailed kinematics. Small differences in kinematics may cause substantial
changes in flow, as we noted in the differences between Müller et al.
(2001) and our study. This
effect may prove useful, however: small kinematic differences as fishes change
swimming speed may induce large hydrodynamic changes, as seen in pectoral fin
swimming (Drucker and Lauder,
2000
). Examining both effects simultaneously will help to
elucidate the mechanical effect of changing kinematics with swimming speed and
between different swimming modes.
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Acknowledgments |
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References |
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