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First published online June 11, 2007
Journal of Experimental Biology 210, 2181-2191 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.001842
Kinematics, hydrodynamics and energetic advantages of burst-and-coast swimming of koi carps (Cyprinus carpio koi)
1 State Key Laboratory of Precision Measurement Technology and Instruments,
Department of Precision Instruments, Tsinghua University, Beijing 100084,
China
2 The Laboratory for Biomechanics of Animal Locomotion, Graduate University
of Chinese Academy of Sciences, Beijing 100049, China/Department of Mechanics
and Mechanical Engineering, University of Science and Technology of China,
Hefei 230026, China
* Author for correspondence (e-mail: tophow99{at}mails.tsinghua.edu.cn)
Accepted 14 March 2007
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0.242). In the coast
phase, drag coefficient (Cd,coast0.060) is estimated
from swimming speed deceleration. Our estimation suggests that nearly 45% of
energy is saved when burst-and-coast swimming is used by the koi carps
compared with steady swimming at the same mean speed.
Key words: burst-and-coast swimming, kinematics, drag, particle image velocimetry, hydrodynamics, wake, koi carp, Cyprinus carpio koi
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Introduction |
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; Videler,
1993). Previous studies of burst-and-coast swimming behavior have
mainly focused on two aspects. The first is to reveal the flow patterns
generated by a fish that is swimming in burst-and-coast style (reviewed by
Videler, 1993;
Müller et al., 2000). The
second is to discuss the energetic advantages in burst-and-coast swimming
based on theoretical models and/or kinematic data
(Weihs, 1974;
Videler, 1981;
Videler and Weihs, 1982;
Ribak et al., 2005).
Aleyev observed that a dye discharged from the gill slits of a coasting
annular bream (Diplodus annularis) showed no vortices in the wake
(Aleyev, 1977). Similar results
were obtained from photographs of the wake of a zebra danio (Brachydanio
rerio) (McCutchen, 1977).
Müller et al. used two-dimensional digital particle image velocimetry
(DPIV) to obtain a qualitative and quantitative description of the flow
patterns generated by larval and adult zebra danios that were performing
burst-and-coast swimming (Müller et
al., 2000). They reported that the burst phase of burst-and-coast
swimming contained one or more tail flicks. The single or continuous tail
flicks indicated two different burst modes, namely MT mode (multiple tail-beat
mode) and HT mode (half tail-beat mode). But they only showed the flow
patterns generated by single tail flick and disregarded substantial
differences between the flow patterns generated in the two burst modes.
Weihs developed a theoretical model based on a substantial difference in
drag between a rigid body and an actively swimming fish and showed that the
burst-and-coast swimming style could save energy compared with continuous
swimming (Weihs, 1974).
Videler and Weihs used kinematic data to estimate the ratio of the energy
consumed by the burst-and-coast swimming to that consumed by swimming at a
constant speed (Videler and Weihs,
1982). Their estimation also required knowledge of the ratio of
drag during active swimming to drag during coasting. Nevertheless, the ratio
was mostly obtained from theoretical models.
A number of studies (e.g. Weihs,
1980; Fuiman and Webb,
1988; Osse and Drost,
1989) have suggested that burst-and-coast swimming mainly takes
place in the inertial flow regime, which is characterized by a high Reynolds
number (Re) [e.g. Weihs (Weihs,
1980), Re >200; McHenry and Lauder
(McHenry and Lauder, 2005),
Re >1000]. Reynolds number is defined as:
 | (1) |
and µ are, respectively, the density and dynamic viscosity of
water. In the inertial flow regime, inertial force plays a dominant role and
inertial drag is predicted to vary with water density, the square of the
swimming speed and the wetted area, S, of the body
(Batchelor, 1967). Therefore,
the measured drag, D, can be normalized to a dimensionless drag
coefficient, Cd, which is predicted to remain constant
with respect to Re when inertia dominates
(McHenry and Lauder, 2005),
and is represented by:
 | (2) |
;
Blake, 1983;
Videler, 1993) or by towing a
trained live animal (Williams and Kooyman,
1985). An indirect approach is to measure drag from deceleration
of live animals as they coast (Bilo and
Nachtigall, 1980; Stelle et
al., 2000; Johansson,
2003; McHenry and Lauder,
2005). This technique makes use of the fact that during coasting
no thrust is produced; hence the drag should equal the product of body mass
and the rate of deceleration. In addition, some correction is required for the
added mass of water that is accelerated with the body
(Vogel, 1994). However, all
the above estimations of drag coefficient do not concern actively swimming
animals. So they can only be considered as the estimations of `passive drag'.
It is well known that measurements of whole-body drag in actively swimming
animals, so-called `active drag', are extremely difficult to perform. So far,
most of the previous studies on `active drag' have been based on
hydromechanical models (Lighthill,
1971; Webb, 1975;
Ribak et al., 2005). Anderson
et al. obtained friction drag of a swimming scup (Stenotomus
chrysops) by examining the boundary layer
(Anderson et al., 2001). Fish
demonstrated a different approach that allowed the estimation of active drag
(Fish, 1993). The approach was
to estimate the power output of animals during active swimming. In recent
years, DPIV has been demonstrated to be a good approach to estimating the
thrust impulse or power output of swimming animals
(Drucker and Lauder, 1999;
Nauen and Lauder, 2002;
Tytell, 2004). Nevertheless,
DPIV has not yet been applied to estimating active drag. The present study examined body kinematics and flow in the wakes of koi carps (Cyprinus carpio koi) swimming in burst-and-coast style. The differences between the hydrodynamics in the two burst modes were revealed. By measuring the instantaneous speed, we obtained the drag coefficient during coast phase (passive drag). The kinematic data and the DPIV results were linked for the estimation of the drag coefficient during the burst phase (active drag). Consequently, using the drag coefficients, we estimated the energy savings in burst-and-coast swimming compared with steadily swimming at the same mean speed.
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):
 | (3) |
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When the measurement system was in operation, the spontaneous behaviour of the carp was recorded by the two cameras at full frame rate, and the data were transferred to the computer. After the computer calculated the position of the carp in the image caught by camera C, a control signal was sent to the driver of the stage to track the carp, keeping it in the central region of the field of view of camera C. The calculation and control functions were performed by an interface program written in Visual C++ v.6.0. Although the frame rates of the two cameras were not the same, the start and stop operations were synchronized by this program. The program could also record the position of the stage and the time when the computer acquired each image. The tracking duration per trial was 16.7 s, limited by the capacity of the cache memory of the image grabbers (1 GB).
Fish kinematics
We used the auxiliary software of the image grabber to divide and
interpolate the images caught by camera C, which was in interlaced scanning
mode. Consequently, 50 images per second could be obtained. Each image was
binarized using a custom-made program. After clearing the stray points, we
programmed applications to obtain the midline and the geometric center of the
carp. We define the body axis x'
(Fig. 2) as the linear
regression line through the points at the anterior half (from head tip to the
middle) of the midline. It indicates the orientation of the carp's body. The
turn angle (ß) and the lateral displacement (lL) of
the carp were calculated based on the body axis x'. The
instantaneous swimming speed, U, was calculated from the displacement
of the geometric centers between frames. The lateral excursion of the tail tip
(d in Fig. 2) was
calculated as the distance between the x' axis and the tail tip
on the midline.
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Hydrodynamics
We used an `mpiv' toolbox (Mori and
Chang, 2004) to obtain the velocity and vorticity fields of the
flow. The interrogation window we used was 16 pixelsx16 pixels (2.7
mmx2.7 mm), and the overlap between two consecutive windows was 50%. The
velocity field was filtered and smoothed by the functions in the `mpiv'
toolbox. Because the movements of the camera would result in a background
velocity vector field, we marked a position where the flow was not affected by
the carp and used the mean velocity vector there as the background velocity
vector to calibrate the results.
We used the vortex ring model (Milne-Thomson, 1966) and assumed that all
the energy shed by a swimming fish is contained in circular vortex rings.
Illuminating a cross-section through such a ring should yield a flow pattern
consisting of two vortices of opposite rotational senses. Location of the
vortices in the velocity fields was determined by plotting the contours of
vorticity. The morphology of a vortex was described by the vortex center, the
core radius (R0), and the ring radius (R). Vortex
centers were marked manually. Prior to calculating R0 and
R, we defined a coordinate system in which x'' is the
longitudinal axis of the vortex ring through the centers of the pair of
vortices, and axis y'' is perpendicular to x''
(see Fig. 2). Following
Müller et al. (Müller et al.,
1997), we estimated R0 and R by
plotting the profiles of the velocity components u and v
parallel to x'' and y'', respectively
(Fig. 2). Momentum angle
(
) of a vortex pair was the angle between x' and
x''. The angle 
between the jet flow and
x' was obtained as a mean value from the angles of the velocity
vectors in the jet.
Assuming that the vortex rings were small-core circular rings, the impulse,
I, of a single vortex ring can be derived from the ring radius
R according to Milne-Thomson (Milne-Thomson, 1966):
 | (4) |
is the density of freshwater and 
is the mean absolute
value of the circulations () of the pair of vortices. Circulation
is the line integral of the tangential velocity component
(VT) about a curve C enclosing the vortex
(Batchelor, 1967):
 | (5) |
)
according to the mean ring radius (Rm):
 | (6) |
) from
the core radius R0 and the ring radius R
(=R0/R). The parameter with
0<
2 was used by Norbury
(Norbury, 1973) to
characterize a family of vortex rings. For small-core vortices, 0.25
otherwise 0.25<2
(Müller et al.,
2000).
Drag measurements
In order to estimate the drag coefficient on a bursting carp,
Cd,burst, we only consider the accelerating period of the
burst phase and assume that Cd,burst is constant during
the whole burst phase. According to the momentum theorem, the drag impulse
(Id) during this period can be determined by:
 | (7) |
 | (8) |
 | (9) |
 | (10) |
We assumed the wetted area of the body (S) to be equal to the body surface area (Abody). So, after Id is calculated from Eqn 7, Cd,burst can be determined according to Eqn 10.
The drag coefficient on a coasting carp, Cd,coast, can
be calculated according to Bilo and Nachtigall
(Bilo and Nachtigall, 1980) and
McHenry and Lauder (McHenry and Lauder,
2006):
 | (11) |
 | (12) |
Estimation of energy saving
Energy saving in burst-and-coast swimming was usually evaluated by using
the ratio of the energy expended in burst-and-coast swimming
(EB) to the energy expended for swimming steadily at the
same average speed (ES) [ratio
(re)=EB/ES].
EB and ES were estimated by:
 | (13) |
 | (14) |
 | (15) |
Statistical analysis
Sigma Stat (Systat Software Inc., Point Richmond, CA, USA) software was
used for statistical analyses. t-tests were used in the comparisons
of the parameters in the MT mode and the HT mode, except when the Normality
Test of the two samples failed or the variances of the two samples were
significantly different (F-test), in which case, a Mann–Whitney
test was used. Pearson product moment correlation coefficients
(rc) were calculated to test for the relationships between
the coast time t2 and the initial coast speed
(Uf), between the drag coefficient
Cd,coast and the Reynolds number in the coast phase
(Re2), and between the drag coefficient
Cd,burst and the mean acceleration a during the
accelerating period of the burst phase.
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;
McHenry and Lauder, 2005). The
tracking and recording trials were repeated 15 times on carp I, 10 times on
carp II, five times on carp III and 15 times on carp IV. The image sequences
were recorded when the carps were swimming freely, so each of them consisted
of several swimming behaviors. The image sequences showed that the time of
burst-and-coast swimming covered 45–75% (statistic results from 45 image
sequences) of the total swimming time. From the recorded sequences, we
selected 24 bouts of burst-and-coast swimming (eight bouts for carp I, four
for carp II, four for carp III and eight for carp IV) following three
criteria: first, the pectoral fins were kept motionless in the coast phase;
second, the quality of particle images was high enough to yield high density
of original velocity vectors in the wake; third, the shadow created by the
caudal peduncle was visible (to make sure the caudal fin moved through the
light sheet). The following results were all obtained from the 24 bouts.
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Swimming kinematics
The burst-and-coast swimming bouts indicated that the tail moved in two
modes during the burst phase: first, the tail beat at least one cycle (MT
mode; Fig. 3A); second, the
tail performed only one flick, namely a half-cycle beat (HT mode;
Fig. 3B). When the carp burst
in the MT mode, it did not change moving direction visibly (ß in
Table 2) and had no visible
lateral displacement (lL in
Table 2) in the following coast
phase. By contrast, when the carp burst in the HT mode, it turned a certain
angle (ß in Table 2) and
it had a certain lateral displacement (lL in
Table 2) in the following coast
phase. ß and lL in the MT mode and the HT mode were
both significantly different (Table
2). This is in accordance with results reported by Wu et al.
(Wu et al., 2006). The curves
of speed U and lateral excursion d versus time t of
bouts 1 (MT mode) and 13 (HT mode) were plotted in
Fig. 3. The
U–t and d–t curves of the rest
bouts (not plotted here) were similar to the curves shown in
Fig. 3A or
Fig. 3B (the curves of bouts
2–12 were similar to those of bout 1; the curves of bouts 14–24
were similar to those of bout 13). The U–t and
d–t curves indicated that the carp reached the maximum
speed around the moment when the tail tip reached its maximum lateral
excursion in the last flick of the burst phase and started to recoil.
According to the curves, we divided burst-and-coast swimming into burst phase
(t1) and coast phase (t2) and then
divided the burst phase into two periods (accelerating period
t0 and recovery period
t1–t0)
(Fig. 3). No significant
differences were found in the speed variables (Ui,
Um, Uf) between the two modes
(Table 2). The time and
distance variables in burst phase and coast phase are shown in
Fig. 4A,B. In most cases, for
both MT mode and HT mode, coast time t2 is greater than
burst time t1 (Fig.
4A). In the MT mode, t1,
l1 and l2 are more variable than those
in the HT mode (Fig. 4A,B). On
the other hand, variation in t2, v1
(mean speed in burst phase) and v2 (mean speed in coast
phase) appears similar in the MT mode and the HT mode
(Fig. 4B,C). No significant
differences were found in v1 (P=0.726,
t-test) and v2 (P=0.419,
t-test) between the two modes. In addition, there were no significant
relationships between the coast time t2 and the initial
coast speed Uf (rc=0.017,
P=0.938). It seems that the carps control the duration of coast phase
following their own inclinations.
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Hydrodynamics
During the burst phase, the tail generated two types of flow patterns in
the wake: in the MT mode, like the flow pattern generated by a continuously
swimming fish, two vortices were generated per beat cycle and they were at
different sides of the body axis (Fig.
5; but three vortices were visible in the first beat cycle since
the carp started to burst); in the HT mode, a pair of vortices was generated
in a half-cycle beat and the two vortices were at the same side of the body
axis (Fig. 6).
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The flow pattern generated in bout 13 is shown in Fig. 6, which was representative for all the bouts in which the carp burst in the HT mode. In the burst phase, the tail performed only one flick (one half-cycle beat), in which the suction flow at the peduncle induced two vortices (Fig. 6A, vortices 1 and 2) that formed a vortex pair. Vortex 2 shed after vortex 1 (Fig. 6A,B). After shedding, the two vortices that were located at the same side of the body axis moved sideways and backwards at a speed of 10–15 mm s–1 (Fig. 6B–F). In the coasting phase, the flow around the carp (Fig. 6D–F) was similar to that generated in bout 1 (Fig. 5E,F).
Besides the location differences, there were highly significant differences
in momentum angle (P<0.001) and jet angle
(P<0.001) between the vortices generated by the carp bursting in
the two modes (Table 3). When
the carp burst in the MT mode, was 44±19° (mean ±
s.d., N=25) and was 36±17° (mean ± s.d.,
N=25). When the carp burst in the HT mode, was
11±6° (mean ± s.d., N=12) and was
76±12° (mean ± s.d., N=12). These differences
indicated that the backward component of flow generated in the MT mode was
more remarkable than that in the HT mode and the lateral component of flow
generated in the HT mode was more remarkable than that in the MT mode.
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Each pair of vortices represented a cross-section through a vortex ring. To
estimate the impulse of each single vortex ring, we determined its core radius
(R0), ring radius (R) and circulation ()
(Table 3). No significant
differences were found in the wake parameters R0,
R, , I and I' between the two burst
modes (Table 3). The
dimensionless mean core radius (R0/R) was
0.70±0.15 (mean ± s.d., N=37). It suggested that the
carps shed large-core vortices during burst-and-coast swimming. The impulses
estimated according to the small-core model were substantially smaller than
those estimated according to the large-core model
(Table 3).
Drag measurements and energy savings
We obtained the acceleration, a
(Table 2), during the
accelerating period of the burst phase from the instantaneous swimming speed
U(t) and time t by using linear fitting according
to Eqn 8. The correlation coefficients, r, of fitting in the 24 bouts
were all greater than 0.8. The acceleration a in the two burst modes
was not significantly different (Table
2). As discussed above, the dimensionless mean core radius
indicated that the carps shed large-core vortices during burst-and-coast
swimming. So, we assumed that the impulse It was equal to
the impulse I' of the vortex rings shedding in the accelerating
period of the burst phase. According to Eqns 7 and 10, we estimated the drag
coefficient Cd,burst. No significant differences were
found in Cd,burst between the two burst modes
(Table 2). Therefore, we
considered the two burst modes together for Cd,burst:
Cd,burst=0.242±0.083 (mean ± s.d.,
N=24). In addition, the relationship between
Cd,burst and a was tested, but no significant
relationship was found between them (rc=–0.171,
P=0.404).
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No significant differences were found in the ratio of the energy expended during burst-and-coast swimming to the energy expended for swimming steadily at the same average speed when the carp burst in the two different burst modes (P=0.468) (Table 2). Therefore, the two burst modes were considered together for re, and re=0.55±0.18 (mean ± s.d., N=24) (Table 2). It suggested that burst-and-coast swimming saved nearly 45% of energy compared with steady swimming.
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). Our results indicate that the maximum instantaneous speed
during burst-and-coast swimming is reached not at the end of burst phase but
at the moment when the tail reaches the maximum lateral excursion and starts
to recoil. The moment is also when a bound vortex around the inflection point
reaches the tail tip and the vortex is shed. When the maximum speed drops to a
final speed of the burst phase, the carp starts to coast. This is also
mentioned by Müller et al.
(Müller et al., 2000).
The speed curves in our results are similar to that obtained by Ribak et al.
(Ribak et al., 2005) who
studied the burst-and-coast gait of the great cormorant (Phalacrocorax
carbo sinensis).
Wake and impulse estimation
When the koi carps swim in the burst-and-coast style, they burst in two
modes: first, the tail beats at least one cycle (MT mode); second, the tail
beats only a half-cycle (HT mode). When the carp burst in the MT mode, the
flow patterns in the wake are like those generated by a continuously swimming
fish (Müller et al.,
1997; Nauen and Lauder,
2002). The lateral component and backward component of the jet are
both remarkable (=44±19° and =36±17°; mean
± s.d., N=25). Given that the pair of vortices represents the
cross-section of a vortex ring, the carp sheds one vortex ring in each
half-cycle beat when it bursts in the MT mode. The vortex rings are linked and
form a chain. That is why we see three vortices in the first beat cycle
(Fig. 5) but two vortices in
each subsequent beat cycle (Videler et
al., 1999). In addition, the body axis of the carp traverses all
the vortex rings shed so that the vortices in each pair (the cross-section of
each vortex ring) are located at the different sides of the body axis. When
the carp bursts in the HT mode, it also sheds one vortex ring in the
half-cycle beat. Nevertheless, the vortex ring is located on one side of the
body axis of the carp so that the pair of vortices (the cross-section of the
vortex ring) is located at the same side of the body axis. The jet nearly
points to the lateral direction (=11±6° and
=76±12°; mean ± s.d., N=12). Consequently,
the carp turns a visible angle in burst phase and has a certain lateral
movement in coasting phase.
The impulse of vortex rings is estimated according to both the small-core
model and the large-core model. Both models are based on the assumption of
circular vortex rings that might affect the precision of the estimations.
According to Dabiri's conclusion (Dabiri,
2005), the added-mass contribution from fluid surrounding vortices
in the wake should be considered in the estimation of wake impulse. So Eqns 4
and 6 might lead to underestimations of the impulse of vortex rings. In
addition, Fig. 6E,F and
Fig. 7E,F show a jet being shed
from the tail as it finishes motion, but we do not take these into account in
estimating the wake impulse. Because it seemed that this jet was generated
when the body and tail recoiled to straight (period from
t0 to
t1–t0), we only considered the
accelerating period of burst phase (period from 0 to t0)
in estimating Cd,burst. The dimensionless mean core radii
(=0.70±0.15, mean ± s.d., N=37) in our
measurements are much greater than 0.25. It suggests that the vortex rings
generated by koi carps in burst-and-coast swimming have large vortex cores.
Therefore, we considered the impulse I' estimated according to
the large-core models was more reliable than I and used it for the
estimation of Cd,burst and re.
Drag and energy savings
Numerous studies have examined the drag coefficients of the rigid bodies of
fish (reviewed by Webb, 1975;
Blake, 1983) and marine mammals
(Fish, 1998;
Stelle et al., 2000). The drag
coefficient in Webb's dead drag measurements of Salmo gairdneri turns
out to be 0.036. McHenry and Lauder have reported the drag coefficient in
adult Danio rerio: Cd=0.067 from dead drag
measurements and Cd=0.024 from in vivo drag
measurements when the fish is coasting
(McHenry and Lauder, 2005).
The body morphology of these two kinds of fish is close to that of the koi
carps. The drag coefficient Cd,coast=0.060 in our
measurements of coasting carps has a magnitude comparable to the above
results.
Hydrodynamic models suggest that drag increases in a flexing body compared
with a rigid body by a factor of 3–5
(Lighthill, 1971;
Webb et al., 1984). Videler
and Weihs assumed the factor to be 3 and calculated the ratio of the energy
required by burst-and-coast to that required by swimming steadily at the same
average speed (Videler and Weihs,
1982). In our measurements, the Cd,burst of
the carp is 0.242, which is nearly four times Cd,coast.
Our estimate suggests that nearly 45% of energy is saved when burst-and-coast
swimming is used by the koi carps compared with steady swimming at the same
average speed. The great energetic advantage is the probable reason for the
koi carps to use burst-and-coast style very frequently when they swim
freely.
In our estimation, Cd,burst was assumed to be constant and was calculated from Eqns 7 and 10, which do not include the unsteady effects. To obtain more accurate and instantaneous drag coefficients needs further study. The ratio re was estimated based on the assumption that Cd,und was equal to Cd,burst. Due to the added mass effects, there should be some errors in the assumption. Nevertheless, the mean accelerations, a (228±152 mm s–2; mean ± s.d., N=24), in the accelerating period of the burst phase in our experiments are so small that no significant relationship is found between Cd,burst and a (rc=–0.171, P=0.404). In addition, in burst-and-coast swimming of koi carps, the body and tail do not undulate intensively in the burst phase (unlike in fast-start). The assumption that Cd,und is equal to Cd,burst seems to be acceptable in this situation.
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References |
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