The hydrodynamics of eel swimming II. Effect of swimming speed
Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA
e-mail: tytell{at}oeb.harvard.edu
Accepted 10 June 2004
![]() |
Summary |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Key words: eel, Anguilla rostrata, wake structure, particle image velocimetry, fish, fluid dynamics, efficiency, swimming speed, kinematics
![]() |
Introduction |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Nonetheless, the diversity of wakes observed from swimming fishes to some
extent reflects the standard classification of swimming modes
(Breder, 1926). Carangiform and
subcarangiform swimmers produce a single vortex each time the tail changes
direction, resulting in a wavy jet, pointing downstream, between the vortices.
Anguilliform swimmers produce a rather different wake. As originally observed
by Müller et al. (2001
)
and described in detail in Part I of this study
(Tytell and Lauder, 2004
),
eels produce two same-sense vortices each time the tail moves from one side to
the other and do not produce any substantial downstream flow. Connecting these
hydrodynamic differences to kinematic differences remains difficult, in part
because of the diverse morphologies and evolutionary histories of fish with
different swimming modes.
A better way to examine how different body movements affect hydrodynamics
is to examine changes in kinematics and hydrodynamics over a range of speeds
in the same species. Several hydrodynamic studies identify interesting changes
in the wake at different speeds. In particular, Nauen and Lauder
(2002a) described a
substantial reorientation of vortex rings in mackerel wakes as they increased
speed. Clearly, the mackerel must be changing their swimming motions to
produce these hydrodynamic changes. While mackerel swimming kinematics have
been studied at a range of speeds (Videler
and Hess, 1984
; Donley and
Dickson, 2000
), how the kinematics cause this reorientation is not
clear. Also, Drucker and Lauder
(2000
) documented substantial
changes in the wakes of two pectoral fin swimmers - bluegill sunfish and black
surf perch - as they increased swimming speed. Surf perch showed a
reorientation of vortex rings at higher speeds, and bluegill began to generate
an entirely new ring above a certain speed. Pectoral fin kinematics have also
been examined (Webb, 1973
)
separately from the hydrodynamics but, without simultaneous measurements of
fin motions and flow fields, explaining how different kinematics cause the
hydrodynamic changes is difficult.
These previous studies have described how the flow behind various swimming
fishes looks and how it changes with swimming speed, but simultaneous
observation of kinematics and hydrodynamics can begin to explain why the flow
changes the way it does. In the present study, therefore, I examine the
empirical relationship between swimming kinematics and hydrodynamics in
steadily swimming eels, Anguilla rostrata, at a range of speeds from
0.5 to 2 body lengths per second (L s-1).
![]() |
Materials and methods |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
A single laser light sheet, produced using two argon-ion lasers at 4 and 8
W, respectively, was focused 7 mm above the tank bottom. Eels only swam
steadily on the bottom of the flow tank, which required the laser to be this
close to the bottom. A detailed analysis of the flow tank boundary layer was
performed and is reported in Tytell and Lauder
(2004). At this height, the
light sheet illuminated the plane along the dorsoventral midline of the eel
but was above the turbulent boundary layer of the flow tank.
The light sheet and the swimming kinematics were filmed from below using
two high-speed digital cameras, one focused on the eel (RedLake; 250 or 125
Hz, 480x420 pixels) and the other focused on the light sheet behind the
eel (either RedLake or NAC Hi-DCam at 250 Hz, 480x420 pixels or 500 Hz,
1280x1024 pixels, respectively). Additionally, the snout and tail tip
were digitized manually, which allowed a custom Matlab 6.5 (MathWorks, Inc.,
Natick, MA, USA) program to digitize 20 points along the eel midline
automatically. Kinematic parameters, such as tail beat amplitude and
frequency, were calculated from the timing and amplitude of each peak in
lateral excursion along the midline. Following Gillis
(1997), three angles were
calculated for the posterior 5% of the body: its angle relative to the
swimming direction (the tail angle); the angle of its path of motion relative
to the swimming direction (the path angle); and its instantaneous angle of
attack. Strouhal number was also estimated as 2fA/U
(Triantafyllou et al., 1993
),
where f and A are the tail beat frequency and amplitude,
respectively, and U is the swimming speed. Strouhal number has been
shown to be strongly indicative of the force production and efficiency of
flapping foils (Read et al.,
2003
) and may have a similar importance for undulatory
locomotion.
Another Matlab program performed two-pass digital particle image
velocimetry (PIV) as in Hart
(2000) but using a statistical
correlation function (Fincham and
Spedding, 1997
). Vortex centers were digitized manually, and
vortex circulation was calculated by integrating along a contour 8 mm from the
center. Finally, the mean flow was calculated in a 8x8 mm region,
centered 12 mm behind the tail tip.
Force, power and impulse were estimated from both the kinematics and the
flow field. Large-amplitude EBT
(Lighthill, 1971), a reactive
model, was used to estimate thrust and lateral forces and power required to
produce the wake from the kinematic, as follows:
![]() | (1) |
![]() | (2) |
where xb(s,t) and
yb(s,t) are the positions of points along the
midline of an eel facing in the positive x direction in flow with
speed U towards the eel, m is the virtual mass per unit
length, L is the eel's length, t is time and s is
the distance along the midline from head to tail. The body velocities
v and v|| are perpendicular
and parallel to the midline, respectively. In addition, resistive forces were
calculated by summing the quasi-steady drag forces normal and tangential to
the body midline using the true kinematics, in a similar way to Jordan
(1992
). This force is:
![]() | (3) |
where h is the eel's height, is fluid density,
v
and v|| are the fluid
velocities normal and tangential to a segment, taking into account the
segment's own motion, and
is the angle of the segment relative to the
path of motion. The normal and tangential drag coefficients
CD,
and CD,|| were estimated
according to empirical descriptions of turbulent flow normal to a cylinder
(Taylor, 1952
;
Hoerner, 1965
) and parallel to
a flat plate (Hoerner, 1965
),
respectively, under steady conditions:
![]() | (4) |
![]() | (5) |
where Re is Reynolds number. Wake power was not calculated from the resistive model because it does not explicitly account for how power is shed into the wake. Simply integrating power, like force, neglects the fact that fluid must flow over different periods of time into the wake. Without substantially complicating the model, there is no way to calculate wake power.
Lateral impulse from reactive (EBT) and resistive force estimates was
calculated by integrating forces over half a tail beat. These estimates were
compared with the same values measured using PIV. Assuming that vortex pairs
in the wake were separate vortex rings, the ring circulation was also
calculated by integrating along a line equidistant from the vortex pairs. Ring
impulse (Iring) and force (Fring) were
estimated as:
![]() | (6) |
![]() | (7) |
where is the water density,
is the circulation, d is
the distance between the vortex pairs, h is the dorsoventral height
of the eel, and f is the tail beat frequency. Impulse generated at
the tail tip was also estimated from the first moment of vorticity
(Birch and Dickinson, 2003
),
averaged over half a tail beat:
![]() | (8) |
where is the fluid density, r is the position vector from the
tail tip,
is the vorticity vector, and A is the area of the
light sheet. Because only a single horizontal plane was examined, this
expression assumes that vorticity is the same in all horizontal planes over
the height of the eel. Force was estimated by taking the time derivative of
Ivort (Birch and
Dickinson, 2003
). The power required to produce the wake was
determined by integrating the kinetic energy flux through a 80x10 mm
plane, 8 mm behind the eel, and subtracting the kinetic energy flux upstream
of the eel, based on the mean flow velocity. Additionally, a `lateral' power
was estimated by assuming the small and relatively noisy axial component of
velocity was zero and integrating only the lateral velocity contribution to
the kinetic energy flux. Phasing of the wake power was adjusted by
2
xplane/Uf, where xplane
(=8 mm) is the distance between the tail tip and the plane where power was
estimated, to account for the phase lag between when the kinetic energy was
shed at the tail and when it reached xplane.
The cost of producing the wake was estimated by dividing the wake power by the swimming speed. This cost is one component of the total mechanical cost of transport, which also includes the thrust power and the inertial power required to undulate the body.
Forces, powers and impulses were normalized to produce non-dimensional
coefficients by dividing by
SU2,
SU3
and
SLU,
respectively (Schultz and Webb,
2002
; Tytell and Lauder,
2004
), where S is the wetted surface area of the eel,
L is the eel's length and U is the swimming speed.
All statistics were performed in Systat 10.1 (Systat Software, Point
Richmond, CA, USA). All errors listed are standard error. A three-way,
mixed-model analysis of variance (ANOVA;
Milliken and Johnson, 1992)
was performed to compare impulse estimates from PIV and theoretical models.
Forces were not compared directly because of the uncertainty in estimating the
generation time in equations 6, 7. Instead, by comparing impulse, the mean
force output over a tail beat was compared without the problem of when that
force was generated. In the ANOVA, the fixed factors were type of measurement
and swimming speed (slow, moderate and fast), and the random factor was
individual. Measurement type had five values: vortex ring impulse from PIV
(abbreviated as PVR); direct integration of vorticity (PDIV); impulse from the
reactive model (KEBT); impulse from the resistive model (KRES) and the sum of
the reactive and resistive impulses (KBOTH). Four comparisons were planned in
advance: PVR with PDIV, PVR with KEBT, PVR with KRES and PVR with KBOTH.
Because these differences were expected a priori, the same type of
F test used to test for differences among all group members was used
to compare them individually (Milliken and
Johnson, 1992
).
A similar ANOVA was performed to compare mean power estimates but with only three types of measurement: total power from PIV (PTOT); lateral power from PIV (PLAT) and wake power from the reactive (EBT) model (KEBT). Planned comparisons were PTOT with KEBT and PLAT with KEBT.
Other regressions were performed with `individual' as a dummy variable, and
significance tested including it as a random effect
(Milliken and Johnson,
2001).
![]() |
Results |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Kinematics
Because the wake was quite sensitive to changes in swimming movements, the
kinematics were quantified in detail (Fig.
1; Table 1). Tail
beat amplitude and frequency were poorly correlated with swimming speed
(r2=0.372 and 0.572, respectively), particularly at low
speeds, and both varied by as much as 20% in most sets
(S.D.=8%). In addition, amplitude was not significantly
correlated with swimming speed when individual was included as a random effect
(P=0.096; Table 1).
However, at a given swimming speed, amplitude and frequency were approximately
inversely proportional to each other (Fig.
2), so that the mean tail velocity was well correlated with
swimming speed (r2=0.872;
Fig. 1A). This correlation
means that Strouhal number, 2fA/U
(Triantafyllou et al., 1993),
stays approximately constant at 0.324±0.003. No significant change was
observed in Strouhal number with swimming speed
(F1,2=4.39, P=0.171), and the eels seem to
maintain Strouhal number within a swimming speed
(Fig. 2). Individuals do not
have significantly different Strouhal numbers
(F2,268=0.151, P=0.860). Even though amplitude
was not significantly related to swimming speed
(F1,2=8.93, P=0.096), it tended to increase with
swimming speed at all points on the body, increasing fastest at the head
(Fig. 1C). Body wave speed was
tightly correlated with swimming speed (r2=0.957) and
increased slightly faster than the swimming speed
(Fig. 1B; F1,2=26.12, P=0.036). The ratio of swimming speed
to body wave speed, called slip, thus increased from 0.57±0.01 at the
slowest speed to 0.784±0.002 at the highest. Body wave length was, on
average, 0.597±0.005 and did not change significantly with swimming
speed (F1,2=13.48, P=0.067), although it did show
a trend to increase at higher speeds. The largest variation in body wave
length was due to individual variation, resulting in differences of as much as
30% between individuals.
|
|
|
At a given swimming speed, amplitude increased along the body
exponentially. All logarithmic regressions had r2 values
higher than 0.970, while the linear regression r2 values
were always less than 0.2. The lateral (y) position of the midline
could be accurately described as:
![]() | (9) |
where s is the contour length along the midline starting at the
head, A is the tail beat amplitude, is the amplitude growth
rate, L is the body length,
is body wave length, t
is time and V is body wave speed. By this definition, a large
implies that amplitude is low near the head and increases rapidly near the
tail. A smaller
implies more undulation anteriorly. To determine the
parameter at a given swimming speed,
ln[ymax(s)/A] and
ymax/A were regressed on s/L-1
without a constant. Based on the logarithmic regressions,
was equal to
3.90±0.04 at the lowest speed and decreased to 2.25±0.01 at the
highest speed, showing an increase in body amplitude of 420% at the head at
the highest speeds.
The maximum angle of attack of the tail decreased with increasing swimming speed (Fig. 3A). Additionally, at higher swimming speeds, the tail spent a lower fraction of the tail beat with a positive angle of attack (Fig. 3B), decreasing from 0.866±0.003 at the lowest speed to 0.786±0.003 at the highest. The tail generally reached its maximum angle of attack when it had the highest velocity, approximately as it crossed the path of motion.
|
Hydrodynamics
At all steady swimming speeds, the wake retained approximately the same
form. The wake contains lateral jets of fluid, alternating in direction,
separated by one or more vortices or a shear layer
(Fig. 4). 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. This
pattern was consistent at all speeds, even though the strength of the lateral
jet increased at higher speeds (Fig.
5).
|
|
The jet magnitude, direction and diameter were measured at the different swimming speeds (Fig. 5). Jet magnitude increased linearly (r2=0.461) with swimming speed and had a significant slope (Table 1; F1,2=21.29, P=0.044). Neither jet angle nor jet diameter had significant regressions against swimming speed when individual was treated as a random variable (F1,2=0.83 and 4.98, respectively, corresponding to P=0.458 and 0.155). For each individual, jet diameter did tend to increase with swimming speed, which was shown by a significant interaction term between swimming speed and individual (F2,268=24.17, P<0.001). Jet angle, on the other hand, was not significantly different from 90 at any speed (P=0.407), although the jet did have a tendency to point slightly upstream.
Although the jet diameter did not change significantly with swimming speed,
it did have a significant relationship to the body wavelength
(Fig. 6). One might expect that
the jet diameter should be about half of a full wave on the body, because the
bolus of fluid that becomes the jet forms in a half wave
(Tytell and Lauder, 2004).
However, Fig. 6 shows that the
jet diameter is about a quarter wavelength (not significantly different from
0.25; F1,268=1.044, P=0.308) and is significantly
less than half a wavelength (F1,268=133.4,
P<0.001). Individual variation in body wavelength was as much as
30% at a specific swimming speed but, despite this variation, wake jet
diameter remains correlated with body wavelength. For example, the individual
represented by squares and solid lines in
Fig. 6 consistently chose a
longer body wavelength and, as a result, had wider jets than the others, even
at lower swimming speeds.
|
The mean flow from an 8x8 mm region behind the tail tip was regressed on the tail tip velocity (Fig. 7). Tail tip velocity was used as the dependent variable, rather than swimming speed, because it allows variation within a swimming speed to be analyzed but is still highly correlated with swimming speed. Mean axial flow always pointed downstream, away from the eel, and increased linearly with increasing tail velocity (P<0.001, r2=0.299). The mean lateral flow magnitude increased with swimming speed but had a significant nonlinear component. In a quadratic polynomial regression, both the linear and quadratic terms were significant (P<0.001 and P=0.002, respectively), and the constant was not significantly different from zero (P=0.807).
|
The vortices on either side of the lateral jet appear to be part of a small
core vortex ring (Müller et al.,
2001). Thus, by analogy with vortex ring generators (review in
Shariff and Leonard, 1992
),
the total circulation added to the fluid by the tail should be:
![]() | (10) |
where TG is a half tail beat, specifically from maximum
lateral excursion on one side to the other side, and Ut is
the tail tip velocity. Fig. 8
shows the maximum circulation of the primary vortex plotted against
tail. At values of less than
40 cm2
s-1, the two match well but, at higher values,
tail tends to overestimate the measured circulation. A
quadratic polynomial regression between the two had significant linear and
quadratic terms (P<0.001 in both cases). The coefficient of the
linear term was not significantly different from one (P=0.644).
|
The cost of producing the wake increases exponentially with mean tail tip speed (Fig. 9). Again, mean tail speed was used as a proxy for swimming speed to highlight variation within a single swimming speed. Wake energy cost increased as the tail speed increased with an exponent of 1.48±0.03 (r2=0.755, P=0.011). Individuals had significantly different exponents (P<0.001), especially the individual represented by circles, which had an exponent of 2.05±0.08. Because tail velocity is directly proportional to swimming speed, this regression means that wake energy cost also increases with swimming speed to the 1.48 power.
|
Finally, the predictions of Lighthill's reactive EBT
(Lighthill, 1971) and a
resistive model (Taylor, 1952
;
Jordan, 1992
) were compared
with the PIV measurements (Fig.
10; Table 2). All
values were normalized to produce non-dimensional coefficients before
comparison. A three-way mixed-model ANOVA was performed on impulse coefficient
with fixed factors of swimming speed (
0.9, 1.4 and 1.9 L
s-1) and type of measurement (KEBT, KRES and KBOTH vs PVR
and PDIV), and `individual' as a random factor
(Fig. 10A;
Table 2). Because only one
individual swam steadily at the slowest speed (0.55 L
s-1), the above test was required mathematically to exclude this
speed, although it is shown in the figures for visual comparison. Swimming
speed had no significant effect on the measurements (P=0.469) nor did
the differences between types change at different speeds (P=0.189).
Individuals were significantly different (P<0.001). The
measurement types were also significantly different (P<0.001).
A priori planned comparisons were conducted to compare certain
measurement types using F tests
(Quinn and Keough, 2002
). In
particular, vortex ring impulse (PVR) was significantly larger than all other
methods of estimating impulse (P<0.001 in all cases).
|
|
Additionally, the axial force component of Fvort is not significantly different from zero. Based on an ANOVA with speed as the only factor, the axial component does not differ from zero at any speed (F4,270=0.079, P=0.989).
Mean power coefficients were compared in a similar ANOVA as impulse, again
with five types of measurement (KEBT vs PTOT and PLAT;
Fig. 10B;
Table 2). Again, estimates did
not change with swimming speed (P=0.623) nor did the differences
between methods change at different speeds (P=0.331). Individuals
were significantly different (P<0.001). Differences between
measurement types were marginally non-significant (P=0.063). At this
level of significance, planned comparisons can still be conducted
(Quinn and Keough, 2002),
revealing that the mean total power coefficient from PIV (PTOT) is
significantly larger than the reactive power (KEBT; P=0.048), while
mean lateral PIV power coefficient (PLAT) is not significantly different from
the reactive power coefficient (KEBT; P=0.753).
![]() |
Discussion |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Simultaneous kinematic and PIV data were collected at four different
swimming speeds, from 0.5 to 2 L s-1. The kinematics
were consistent with previous data from eels and other anguilliform swimmers
(Gillis, 1997
,
1998
). At all speeds, the wake
resembled that described in Part I of this study: laterally directed jets of
fluid, separated by regions of vorticity, with little downstream flow (Figs
4,
6). The jet increases in
strength at higher swimming speeds and tends to become wider but does not
change angle (Fig. 5). Tail tip
velocity seems to be the kinematic parameter that most affects the flow in the
wake. The circulation of the vortices surrounding the jet increases with
increasing tail tip velocity but seems to level off at the higher speeds. Even
so, the cost of producing this wake increases exponentially at higher tail
velocities, corresponding to higher speeds.
The kinematic data from this study are consistent with Gillis's recent work
on eels (Gillis, 1998). For
example, at 1.0 L s-1, he observed a tail beat frequency
of 2.484±0.007 Hz, a body wave speed of 1.27±0.02 L
s-1 and a tail tip amplitude of
0.08 L, compared with
the values from this study of 2.61±0.08 Hz, 1.34±0.01 L
s-1 and 0.059±0.001 L, respectively. Also, in
Siren intermedia, a salamander that swims in the anguilliform mode,
Gillis (1997
) observed a
similar use of decreasing angles of attack for increasing swimming speed
(Fig. 3A). However, in
Siren, the proportion of the tail beat with positive angles of attack
increases with swimming speed, while for Anguilla the proportion
decreases (Fig. 3B).
Strouhal number, the ratio of mean tail beat speed to swimming speed, has
received increasing attention in recent years as a kinematic parameter that
has a strong effect on hydrodynamics (Triantafyllou et al.,
1993,
2000
;
Taylor et al., 2003
). Flapping
foils reach a peak in efficiency near a Strouhal number of 0.3
(Read et al., 2003
), which may
be related to the instability of the wake for those flapping parameters
(Triantafyllou et al., 1993
).
Eels, like many other fishes, swim with a tail beat amplitude and frequency
near this Strouhal number. In addition, eels maintain a constant Strouhal
number within a single swimming speed (Fig.
2) by varying tail beat frequency inversely with amplitude.
Amplitude and frequency differences primarily represent individual differences
but, because they vary inversely to keep St constant, the variation
may not affect the hydrodynamics substantially. For example, the individual
represented by squares in Fig.
2 consistently chose a higher amplitude and lower frequency than
the others. Strouhal number, on the other hand, was the only kinematic
parameter that did not show a significant difference between individuals
(P=0.860), which probably reflects its hydrodynamic importance.
Because of the physical importance of Strouhal number, it would have been convenient to plot hydrodynamic measurements against it, rather than against swimming speed. Unfortunately, St stays constant. Instead, hydrodynamic variables were usually plotted against tail velocity, as in Figs 6, 7, 8, 9. Variation in tail velocity at a constant flow speed represents changes in Strouhal number, which should have hydrodynamic consequences. Indeed, in each of these plots, the hydrodynamic variable varies with tail velocity both within and between swimming speeds. If the hydrodynamic variables were plotted against swimming speed alone, the variation within a speed would have been lost.
Wake structure
It is intriguing to note that the structure of the eel's wake changes very
little over a nearly fourfold change in speed
(Fig. 4). While the wake jet
increases in strength and tends to increase in size, its angle stays the same,
and no substantial changes in the overall formation pattern were observed.
Even the jet strength has a tendency to stop increasing above 1.5
L s-1, as is seen in the comparable jet magnitudes at 1.35
and 1.94 L s-1 in Fig.
4 and in two individuals in
Fig. 5.
While eels' wakes retain a fairly constant structure over a fourfold speed
range, other fishes change their wakes substantially as they change swimming
speed. For example, mackerel have been observed to reorient their wake jets by
nearly 20° over a twofold speed increase
(Nauen and Lauder, 2002a).
Additionally, labriform swimmers change the angle and strength of the vortex
rings they produce as they swim at higher speeds
(Drucker and Lauder, 2000
).
Bluegill sunfish also change the structure of their wake completely; at low
speeds, they generate a single vortex ring per fin beat, on the downstroke,
but at high speeds, they generate two on the downstroke and the upstroke
(Drucker and Lauder, 2000
).
The reason eel wake structure does not change when that of other fishes
does may be related to differences in how eels and other fishes balance thrust
and drag. As discussed in detail in Tytell and Lauder
(2004), all steadily swimming
fishes must produce no net forward force; i.e. thrust must equal drag. Other
fishes seem to segregate thrust production from drag production, either
spatially, by having the thrust-producing fins functionally separated from the
rest of the body like propellers, or temporally, by producing pulsatile
thrust. This segregation means that evidence of thrust production is visible
in the wake, even though, on average, thrust equals the drag on the body. We
hypothesized in Part I that eels do not have this segregation and therefore
produce no net downstream force within the speed range examined in this study,
indicated by the zero axial component of Fvort
(P=0.989). Thus, the jets must point laterally to maintain zero axial
flow, and the reorientation observed in other fishes is not possible. When
eels accelerate, the net axial force is no longer zero, and the wake jets do
reorient (E.D.T., personal observation).
In this study, however, all eels were swimming steadily, and the morphology
of the wake is fairly constant. It might seem that other hydrodynamic
variables suggest a change in the wake at the highest speeds observed in this
study, or possibly at higher speeds. For example, in
Fig. 7, lateral flow behind the
tail tends to level off at high swimming speed, and in
Fig. 8, tail
overestimates primary vortex circulation at speeds higher than 1.8 L
s-1. I argue, however, that these effects do not represent a
difference in how the wake is generated at high speeds. In the first place,
the cost of producing the wake increases at a constant rate as speed increases
(Fig. 9). The rate is slower
than might be expected from a scaling argument but it does not show any breaks
at different speeds. Additionally, the nonlinear relationship in
Fig. 8 may not represent a true
change in generation mechanism. Fig.
8 was constructed as if the tail was a vortex ring generator.
Piston-based vortex ring generators have an effect referred to as `formation
number': a maximum circulation that can be added to a single vortex ring
(Gharib et al., 1998
). The
formation number is the ratio of the distance the piston travels to its
diameter. When this value is above 4, no more circulation can be added to a
single vortex ring. By analogy, the overestimate of primary vortex circulation
at high speeds may represent a similar effect; that the tail cannot add more
circulation to the primary vortex above 50 cm2 s-1.
Because an eel is not a piston, it is difficult to estimate a value for a
formation number at the tail. Nonetheless, the effect may still exist and may
explain the lack of increase in circulation at high swimming speeds.
Circulation, in turn, is directly tied to the jet velocity between the
vortices. The formation number effect thus may also explain why jet magnitude
and lateral flow level off at high speed, without the need to hypothesize a
change in generation mechanism.
An empirical description of eel swimming
An empirical description of eel locomotion is useful because it relates
simple, easily measured quantities, such as Strouhal number, tail beat
frequency or amplitude, to important hydrodynamic variables. Examining
discrepancies between empirical relationships and those predicted by
theoretical models such as Lighthill's reactive EBT
(Lighthill, 1971) and Taylor's
resistive model (Taylor, 1952
)
may also provide physical insight into swimming mechanics.
Dimensionless constants provide the simplest empirical description of eel
swimming. Over a Reynolds number range from 20 000 to 80 000, impulse and
power coefficients based on PIV both stay approximately constant. Mean vortex
ring impulse coefficient remained at 0.0194±0.0004 across speeds, total
power remained at 0.0377±0.0006 and lateral power was somewhat lower
(0.0157±0.0003). For a 20 cm eel swimming at 1 L
s-1, these coefficients are equivalent to 0.49±0.01 mN
s-1, 191±3 µW and 79±2 µW, respectively. The
lateral vortex ring force coefficient decreased from 0.14±0.02 at 0.549
L s-1 to 0.070±0.003 at 1.88 L
s-1, corresponding to forces of 1.1±0.2 mN and
6.3±0.3 mN.
There was a non-significant trend for both power coefficients to decrease
at higher speeds, as can be seen in Fig.
10B. Additionally, lateral force coefficients also tended to
decrease at higher speeds, because the tail beat frequency increased more
slowly than the length-specific swimming speed. In essence, the same impulse
was produced over a relatively longer period at high speed, resulting in a
lower force coefficient. Data from individuals with a greater size range will
be necessary to establish the constancy of impulse coefficients and the trends
for power and force coefficients more firmly, but, in a general way, these
coefficients can better characterize the hydrodynamic performance of eels
during steady swimming than theoretical models. In a recent paper, Schultz and
Webb (2002) urge the use of
power coefficients, rather than Froude efficiency, as a means of describing
swimming performance.
While Froude propulsive efficiency would be useful to estimate, it requires a measurement of thrust, which cannot be estimated due to the lack of axial flow in the wake. However, changes in the cost of producing the wake (Fig. 9), onecomponent of the total cost of transport, may indicate trends in propulsive efficiency. The cost increases as the tail velocity with the exponent 1.48, which is equivalent to cost increasing with swimming speed with the same exponent. If the power coefficient stayed constant, the cost should increase as swimming speed squared, meaning that the cost of producing the wake increases less quickly than might be expected. In fact, power coefficients do tend to decrease slightly (Fig. 10B), possibly indicating an increase in efficiency at higher speeds.
Kinematics can even provide a more detailed picture of the wake structure.
For example, at all speeds except the highest, an eel's tail functions like a
vortex ring generator (Shariff and
Leonard, 1992), adding circulation to the fluid at a rate
proportional to its velocity squared (Fig.
8). Additionally, the jet diameter is consistently about a quarter
of the total body wavelength, regardless of the substantial individual
variation in body wavelength. Together, these two relationships give a good
idea of the wake structure and can also be combined to produce the wake
impulse.
Impulse and power estimates
Beyond simply describing the empirical relationship of kinematics and
hydrodynamics, a goal of this study was to examine the consistency of
different methods of estimating impulse and power, both directly from the wake
and from the kinematics alone. From the wake, two methods of estimating
impulse were examined. First, the vorticity in the wake was assumed to be part
of a small core vortex ring, and the generation impulse for that ring was
calculated based on the midline circulation of the ring, according to
equations 6, 7. Second, no specific vortical structure was assumed, and the
first moment of vorticity, relative to the tail tip, was integrated over the
plane, according to equation 8. In comparison, from the kinematics alone,
three methods of estimating impulse were explored. Lighthill's reactive EBT
(Lighthill, 1971) and
blade-element resistive models (e.g.
Taylor, 1952
;
Jordan, 1992
) produce force
estimates that can be integrated to produce impulse. Additionally, the sum of
the two kinematic impulses was compared with the PIV estimates. Power, in
turn, was estimated from the PIV data by integrating the kinetic energy flux
convected through a plane behind the eel. A `lateral' power was also
constructed in the same way but ignoring the axial components of flow. These
estimates were compared with the EBT estimate of power shed into the wake. The
resistive model does not account for the way power is shed into the wake and
was therefore excluded from the comparison.
Each of these different methods have potential errors from various sources,
detailed below. Most of the error from PIV comes from the fact that flow in
only a single, horizontal plane was measured. If the geometry of the vortex
ring was different from the oval shape that was assumed, the force could be
over- or underestimated. However, studies that included multiple orthogonal
planes (Drucker and Lauder,
2001; Nauen and Lauder,
2002a
) conclude that wake vortex rings are oval shaped, and the
force estimated from those vortex rings tended to equal the measured drag
force, supporting the validity of this assumption. By contrast, to estimate a
total force from the first moment of vorticity, it was assumed that vorticity
was the same in all planes over the height of the eel and that there was no
vorticity along the other orthogonal axes. Vorticity is actually a vector
quantity (Faber, 1995
); a
horizontal plane allows an estimate of vorticity in the vertical direction.
The same studies with orthogonal planes demonstrated that substantial
vorticity exists in the other directions
(Drucker and Lauder, 2001
;
Nauen and Lauder, 2002a
),
probably resulting in an underestimate of total impulse by directly
integrating vorticity. Birch and Dickinson
(2003
), who successfully used
the first moment of vorticity to estimate lift and drag on an insect wing,
used a system that was configured such that the primary contribution to lift
and drag forces was from spanwise vorticity. For the eel, both the measured
vertical vorticity and the unmeasured axial vorticity combine to produce
lateral forces. The force estimate from the first moment of vorticity does not
include this axial vorticity and thus underestimates total force.
PIV power estimates do not require as many assumptions about the structure
of the flow as do force and impulse estimates, but there may still be errors
because a complete control volume around the eel was not observed. In
principle, power should be estimated by taking the difference between the
kinetic energy passing through two planes, one upstream of the eel's snout and
one downstream of the eel's tail. This method would give an estimate of the
rate at which the eel adds energy to the fluid. Because eels will not swim
with their heads in the light sheet, it was not possible to obtain the flow
upstream of the head. The upstream flow was therefore assumed to be constant
and equal to the mean flow velocity. However, due to turbulent effects from
the boundary layer, the upstream flow may not be constant and, particularly,
may include regions of accelerated or decelerated axial flow due to
quasi-streamwise vortices (Robinson,
1991). Very little lateral flow was observed due to the turbulent
boundary layer or other effects within the flow tank. If quasi-streamwise
vortices do affect the upstream flow, the total PIV power will be affected. In
calculating `lateral' PIV power, all momentum that the eel added to the wake
was assumed to be in the lateral direction. This assumption may be justified
because the eel's axial momentum was not changing. Therefore, it could not
cause the axial fluid momentum to change; it could only cause changes in
lateral fluid momentum. Any fluctuations in axial velocity were therefore
assumed to be the result of turbulence and were ignored.
These PIV estimates were compared with two types of theoretical models.
Both the reactive EBT (Lighthill,
1971) and the resistive model
(Taylor, 1952
;
Blake, 1979
) make assumptions
about the flow. EBT assumes that viscosity is unimportant, which is typical at
high Reynolds number (Faber,
1995
), and that the only substantial force comes from the
acceleration reaction, not from any quasi-steady resistive drag forces
(Lighthill, 1971
;
Daniel, 1984
). The
blade-element resistive model includes those forces but not the acceleration
reaction. It also makes the assumption that individual segments along the
eel's body from its head to its tail do not affect the flow around successive
segments. Although this assumption is clearly false, due to the acceleration
of fluid down the eel's body (Müller
et al., 2001
; Tytell and
Lauder, 2004
), interactions between segments may not cause a
substantial change in the forces (Blake,
1979
). Calculating wake power using the resistive model explicitly
requires violating this assumption, because each fluid element must flow along
the body into the wake. Therefore, power was not estimated using the resistive
model.
Given those potential sources of error, the different methods were compared
using a three-way mixed model ANOVA. The impulse estimated by assuming that
the wake consists of small core vortex rings (PVR), which is hypothesized to
be the most accurate following other wake studies (Drucker and Lauder,
1999,
2001
;
Nauen and Lauder, 2002a
), is
larger than any other method of estimating impulse (P<0.001 in all
cases). Neither the reactive model, the resistive model nor their sum predicts
as much impulse as observed in the wake. Thus, these simple models do not
fully describe the complexity of eel swimming. Nonetheless, both reactive and
resistive impulses are important, making up 33±1% and 16.5±0.5%,
respectively, of the estimated PIV impulse. The remaining
50% may come
from more complex fluid interactions along the body, including
three-dimensional effects and vortex shedding along the dorsal and anal
fin.
By contrast, inviscid theory predicts the `lateral' power with a striking
degree of accuracy. Both the time course and the magnitude of this power are
successfully estimated by EBT alone (Fig.
10; Table 2). This
power was calculated using only the lateral flow component, due to the
impossibility of obtaining a complete control volume around the eel and the
presence of turbulent flow structures that are primarily directed in the axial
direction. EBT estimates wake power as the rate at which fluid kinetic energy
at the tail tip is convected into the wake
(Lighthill, 1971). This
estimate is separate from the estimate of force and thus it is possible for
one to be accurate when the other is not, as observed. Therefore, the power
output can be described accurately by a simple theoretical model, but, despite
this correspondence, neither a reactive model nor a quasi-steady resistive
model fully capture the complexity of force output for a swimming eel.
![]() |
Acknowledgments |
---|
![]() |
References |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Aleyev, Y. G. (1977). Nekton. The Hague: Junk.
Anderson, J. M. (1996). Vortex control for efficient propulsion. Ph.D. Thesis. Dept of Ocean Eng., Massachusetts Institute of Technology.
Birch, J. M. and Dickinson, M. H. (2003). The
influence of wing-wake interactions on the production of aerodynamic forces in
flapping flight. J. Exp. Biol.
206,2257
-2272.
Blake, R. W. (1979). The mechanics of labriform locomotion. I. Labriform locomotion in the angelfish (Pterophyllum eimekei): an analysis of the power stroke. J. Exp. Biol. 82,255 -271.
Breder, C. M. (1926). The locomotion of fishes. Zoologica 4,159 -297.
Daniel, T. L. (1984). Unsteady aspects of aquatic locomotion. Am. Zool. 24,121 -134.
Donley, J. M. and Dickson, K. A. (2000). Swimming kinematics of juvenile kawakawa tuna (Euthynnus affinis) and chub mackerel (Scomber japonicus). J. Exp. Biol. 203,3103 -3116.[Abstract]
Drucker, E. G. and Lauder, G. V. (1999).
Locomotor forces on a swimming fish: three-dimensional vortex wake dynamics
quantified using digital particle image velocimetry. J. Exp.
Biol. 202,2393
-2412.
Drucker, E. G. and Lauder, G. V. (2000). A
hydrodynamic analysis of fish swimming speed: wake structure and locomotor
force in slow and fast labriform swimmers. J. Exp.
Biol. 203,2379
-2393.
Drucker, E. G. and Lauder, G. V. (2001).
Locomotor function of the dorsal fin in teleost fishes: experimental analysis
of wake forces in sunfish. J. Exp. Biol.
204,2943
-2958.
Faber, T. E. (1995). Fluid Dynamics for Physicists. Cambridge: Cambridge University Press.
Fincham, A. M. and Spedding, G. R. (1997). Low cost, high resolution DPIV for measurement of turbulent fluid flow. Exp. Fluids 23,449 -462.[CrossRef]
Gharib, M., Rambod, E. and Shariff, K. (1998). A universal time scale for vortex ring formation. J. Fluid Mech. 360,121 -140.[CrossRef]
Gillis, G. B. (1997). Anguilliform locomotion
in an elongate salamander (Siren intermedia): effects of speed on
axial undulatory movements. J. Exp. Biol.
200,767
-784.
Gillis, G. B. (1998). Environmental effects on
undulatory locomotion in the American eel Anguilla rostrata:
kinematics in water and on land. J. Exp. Biol.
201,949
-961.
Gray, J. (1933). Studies in animal locomotion. I. The movement of fish with special reference to the eel. J. Exp. Biol. 10,88 -104.
Hart, D. P. (2000). PIV error correction. Exp. Fluids 29,13 -22.
Hoerner, S. F. (1965). Fluid-Dynamic Drag. Brick Town, NJ: Hoerner Fluid Dynamics.
Jayne, B. C. and Lauder, G. V. (1995). Speed effects on midline kinematics during steady undulatory swimming of largemouth bass, Micropterus salmoides. J. Exp. Biol. 198,585 -602.[Medline]
Jordan, C. E. (1992). A model of rapid-start swimming at intermediat Reynolds number: Undulatory locomotion in the chaetognath Sagitta elegans. J. Exp. Biol. 163,119 -137.
Lighthill, M. J. (1960). Note on the swimming of slender fish. J. Fluid Mech. 9, 305-317.
Lighthill, M. J. (1971). Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. A 179,125 -138.
Marey, E. J. (1895). Le Mouvement. Paris: Masson.
McCutchen, C. W. (1977). Froude propulsive efficiency of a small fish, measured by wake visualisation. In Scale Effects in Animal Locomotion (ed. T. J. Pedley), pp. 339-363. London: Academic Press.
Milliken, G. A. and Johnson, D. E. (1992). Analysis of Messy Data. 1. Designed Experiments. London: Chapman and Hall.
Milliken, G. A. and Johnson, D. E. (2001). Analysis of Messy Data. 3. Analysis of Covariance. Boca Raton, FL: CRC Press.
Müller, U. K., van den Heuvel, B.-L. E., Stamhuis, E. J.
and Videler, J. J. (1997). Fish foot prints: morphology and
energetics of the wake behind a continuously swimming mullet (Chelon
labrosus risso). J. Exp. Biol.
200,2893
-2906.
Müller, U. K., Smit, J., Stamhuis, E. J. and Videler, J. J. (2001). How the body contributes to the wake in undulatory fish swimming: flow fields of a swimming eel (Anguilla anguilla). J. Exp. Biol. 204,2751 -2762.[Medline]
Nauen, J. C. and Lauder, G. V. (2002a).
Hydrodynamics of caudal fin locomotion by chub mackerel, Scomber
japonicus (Scombridae). J. Exp. Biol.
205,1709
-1724.
Nauen, J. C. and Lauder, G. V. (2002b).
Quantification of the wake of rainbow trout (Oncorhynchus mykiss)
using three-dimensional stereoscopic digital particle image velocimetry.
J. Exp. Biol. 205,3271
-3279.
Pedley, T. J. and Hill, S. J. (1999).
Large-amplitude undulatory fish swimming: fluid mechanics coupled to internal
mechanics. J. Exp. Biol.
202,3431
-3438.
Quinn, G. P. and Keough, M. J. (2002). Experimental Design and Data Analysis for Biologists. Cambridge: Cambridge University Press.
Read, D. A., Hover, F. S. and Triantafyllou, M. S. (2003). Forces on oscillating foils for propulsion and maneuvering. J. Fluids Struct. 17,163 -183.[CrossRef]
Robinson, S. K. (1991). Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23,601 -639.[CrossRef]
Rosen, M. W. (1959). Waterflow About A Swimming Fish. China Lake, CA: US Naval Ordnance Test Station.
Schultz, W. W. and Webb, P. W. (2002). Power requirements of swimming: do new methods resolve old questions? Integ. Comp. Biol. 42,1018 -1025.
Shariff, K. and Leonard, A. (1992). Vortex Rings. Annu. Rev. Fluid Mech. 24,235 -279.[CrossRef]
Taylor, G. (1952). Analysis of the swimming of long and narrow animals. Proc. R. Soc. Lond. A 214,158 -183.
Taylor, G., Nudds, R. and Thomas, A. L. R. (2003). Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature 425, 707.[CrossRef][Medline]
Triantafyllou, G. S., Triantafyllou, M. S. and Grosenbaugh, M. A. (1993). Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluids Struct. 7, 205-224.[CrossRef]
Triantafyllou, M. S., Triantafyllou, G. S. and Yue, D. K. P. (2000). Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32,33 -53.[CrossRef]
Tytell, E. D. and Lauder, G. V. (2004). The
hydrodynamics of eel swimming. I. Wake structure. J. Exp.
Biol. 207,1825
-1841.
Videler, J. J. and Hess, F. (1984). Fast continuous swimming of two pelagic predators, saithe (Pollachius virens) and mackerel (Scomber scombrus): a kinematic analysis. J. Exp. Biol. 109,209 -228.
Webb, P. W. (1973). Kinematics of pectoral fin propulsion in Cymatogaster aggregata. J. Exp. Biol. 59,697 -710.
Webb, P. W. (1975). Hydrodynamics and energetics of fish propulsion. Bull. Fish. Res. Bd. Can. 190,1 -159.
Webb, P. W. (1988). Steady swimming kinematics of tiger musky, an esociform accelerator, and rainbow trout, a generalist cruiser. J. Exp. Biol. 138, 51-69.
Webb, P. W. (1991). Composition and mechanics of routine swimming of rainbow trout, Oncorhynchus mykiss. Can. J. Fish. Aquat. Sci. 48,583 -590.
Webb, P. W. (1992). Is the high cost of body caudal fin undulatory swimming due to increased friction drag or inertial recoil. J. Exp. Biol. 162,157 -166.
Webb, P. W. and Fairchild, A. G. (2001). Performance and maneuverability of three species of teleostean fishes. Can. J. Zool. 79,1866 -1877.[CrossRef]
Weihs, D. (1972). A hydrodynamical analysis of fish turning maneuvers. Proc. R. Soc. Lond. B 182, 59-72.
Wu, T. Y. (1971). Hydromechanics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid. J. Fluid Mech. 46,337 -355.