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First published online November 30, 2007
Journal of Experimental Biology 210, 4379-4389 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.009787
Routine turning maneuvers of koi carp Cyprinus carpio koi: effects of turning rate on kinematics and hydrodynamics
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
3 Department of Modern Mechanics, University of Science and Technology of
China, Hefei 230026, China
* Author for correspondence (e-mail: tophow99{at}mails.tsinghua.edu.cn)
Accepted 16 October 2007
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Key words: routine turn, single-beat turn, kinematics, particle image velocimetry, hydrodynamics, wake, koi carp, Cyprinus carpio koi
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Introduction |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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; Donley and Dickson,
2000) and hydrodynamics
(Müller et al., 1997;
Müller et al., 2001;
Nauen and Lauder, 2002;
Tytell and Lauder, 2004) of
steady swimming or on kinematics while examining rapid maneuvering [i.e. fast
starts (Domenici and Blake,
1997; Spierts and Leeuwen,
1999; Domenici et al.,
2004)]. However, watching fish swimming in nature reveals that
most fishes rarely swim steadily or perform fast maneuvers; instead, they
frequently perform some moderate maneuvers
(Webb, 1991;
Wu et al., 2007). Routine
turning maneuvers (spontaneous turns), in particular, are key elements of such
moderate maneuvers. Due to complications and difficulties in quantification,
routine turning maneuvers have not received detailed attention
(Webb and Fairchild,
2001).
Routine turning maneuvers can be divided into median and/or paired fins
(MPF) turns and body and/or caudal fin (BCF) turns, according to the
propulsion patterns of fish (Webb,
1998; Gerstner,
1999). Experimental studies of the two types of turns have
traditionally focused on the kinematics and maneuverability
(Gerstner, 1999;
Budick and O'Malley, 2000;
Walker, 2000;
Webb and Fairchild, 2001). In
recent years, digital particle image velocimetry (DPIV) has been employed to
investigate the flows generated by the fish performing BCF turns
(Wolfgang et al., 1999;
Sakakibara et al., 2004) or
MPF turns (Drucker and Lauder,
2001). Drucker and Lauder studied the wake dynamics of MPF turns
in bluegill sunfish in detail and clarified the hydrodynamic mechanism of MPF
turns (Drucker and Lauder,
2001). However, the quantitative analysis in experimental
hydrodynamics of BCF turns has not yet been well documented. Furthermore, the
empirical relationship between kinematics and hydrodynamics remains yet an
unstudied aspect of BCF turns.
Examining the changes in kinematics
(Jayne and Lauder, 1995;
Donley and Dickson, 2000) and
hydrodynamics (Drucker and Lauder,
2000; Nauen and Lauder,
2002; Tytell,
2004) over a range of speeds is an effective way to study steady
swimming of fish. Changing the swimming speed of the fish in these experiments
was accomplished by controlling the flow speed in the flume. Unfortunately, it
is extremely difficult to force fish to turn at a precisely controlled and
repeatable speed and/or turning rate. Webb studied the influences of swimming
speed and acceleration on turning radius by placing fish in predation
situations (Webb, 1983).
However, the effect of speed and/or turning rate has not been examined when
studying the kinematics and hydrodynamics of spontaneous turns of fish.
A video tracking system for simultaneous measurements of kinematics and
flow of a freely swimming fish (Wu et al.,
2006; Wu et al.,
2007) offers a suitable approach to studying spontaneous maneuvers
of fish. In the present study, we employ this method to quantify the
kinematics and flow patterns of koi carp Cyprinus carpio koi
performing routine BCF turns. As reported elsewhere
(Gerstner, 1999), routine BCF
turns include cruising turns and single-beat turns. In our experiments,
single-beat turns occurred much more frequently than cruising turns, and the
sample size of cruising turns was small. Consequently, only the single-beat
turns were considered in the present study. The selected single-beat turns
cover a large range of turning rates and linear regressions were used to
analyze the effects of turning rate on kinematics and hydrodynamics.
Furthermore, two types of flow patterns generated in routine single-beat turns
were characterized, and the related kinematics was analyzed.
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;
Wu et al., 2007).
Kinematics
Each image from camera C was binarized by using a custom-made computer
program. After clearing the stray points, we programmed applications to obtain
the midline and the geometric center of the carp. The geometric center was
approximately treated as the center of mass (COM). We define the body axis
x' (Fig. 1) as
the linear regression line through the points at the anterior half (from nose
to the middle) of the midline. It indicates the heading of the carp. The
change in orientation of axis x' was defined as the turning
angle (β in Fig. 1) of the
carp. The lateral excursion of the tail tip (d in
Fig. 1) was calculated as
distance between x' axis and the tail tip on the midline. Its
maximum value, dmax, represents the bending amplitude of
the carp during a turn. Turning radius (Rt) was defined as
the radius of the trajectory of COM during a turn. The moving speed of the
carp at the onset of a turn was denoted by U0, and was
calculated based on the displacement of the COM.
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Like C-start escape responses (Weihs,
1972), routine single-beat turns can also be divided into two main
stages, based on kinematics (stage 1 and stage 2). Stage 1 involves a bend to
one side, causing a C-shaped curve in the body, while stage 2 corresponds to
the return flip of the body and tail. Duration of stage 1
(t1) was defined as the time between the onset of bend and
the onset of return flip of the tail. Duration of stage 2
(t2) was defined as the time between the end of stage 1
and the instant when the return flip finished. Besides the kinematic variables
given above, turning angles (β1, β2), mean
turning rates (
1, 2) and net changes of
speed (
U1, U2) were used
to characterize the kinematics of a routine single-beat turn. The subscripts
of the symbols indicate which stage the variables are used for.
β1 always has a positive sign. The sign of β2
is positive if the rotation in stage 2 is in the same direction as that in
stage 1; otherwise, it is negative. The sign convention for
1 and 2 is similarly defined.
U1 and U2 were
calculated by considering only the magnitude of the moving speed of the carp.
When the tail beats in water, it is held at an angle (angle of attack) to its
line of motion (Bainbridge,
1963). In the present study, the angle of attack of the tail was
only considered for the region at the tail tip. This angle at the beginning,
the middle and the end of stage 2 were denoted by 
2b,
2m and 2e, respectively.
Hydrodynamics
The velocity and vorticity fields of the flow were obtained using an `mpiv'
toolbox (Mori and Chang,
2004). The interrogation window we used was 24 pixelsx24
pixels (4.1 mmx4.1 mm), and the overlap between two consecutive windows
was 50%. The velocity field was filtered and smoothed by the functions in the
`mpiv' toolbox.
To estimate the momentum shed by a carp during turning, we used the vortex
ring model (Milne-Thomson, 1966) and assumed that all the energy shed by the
carp is contained in oval-shaped vortex rings. Illuminating a cross section
through such a ring should yield a flow pattern consisting of two vortices of
opposite rotational senses (Fig.
1). Locations of the vortices were determined by plotting the
contours of vorticity. The momentum of a vortex ring can be expressed
(Tytell and Lauder, 2004) as:
 | (1) |
is the density of fresh water, 
is the mean absolute value
of the circulations () of the pair of vortices, h is the
height of the ring, equivalent to the tail's height, and D is the
diameter in the plane of the light sheet. Circulation is the line
integral of the tangential velocity component about a curve enclosing the
vortex. The momentum I in stages 1 and 2 are denoted by
I1 and I2, respectively. The angle
between the jet flow and x'
(Fig. 1), namely the jet angle,
indicates the direction of the momentum of the vortex ring. The jet angles in
stages 1 and 2 is denoted as 1 and 2,
respectively. Jet angle 1 was determined at the end of stage
1, by which time paired vortices were fully developed. Jet angle
2 was determined at the end of stage 2.
To estimate the impulsive moment obtained from the beat of the body and
tail in stage 1 (Li1), we defined two parameters, distance
lc and angle 
(Fig. 1).
lc is the distance between the COM of the carp and the
trailing edge of the tail, while is the angle between the jet and the
line through the COM of the carp and the trailing edge of the tail. Both
lc and were considered as the mean values in stage
1. Li1 can be estimated as:
 | (2) |
 | (3) |
), the
moment of inertia of the carp (J) was estimated based on the
assumption that the density is homogeneous throughout the carp's body. In
estimating the added moment of inertia of entrained liquid
(Ja), we assumed that the body of the carp was a flattened
ellipsoid. The semi-major axis (a) of the ellipsoid was considered to
be equal to half the distance from the nose to the tail tip of the carp, the
horizontal semi-minor axis (b) was assumed to be equal to half the
maximum width of the carp, and the vertical semi-minor axis (c) was
assumed to be equal to half the maximum height of the carp. Based on the
assumptions above, Ja can be calculated
(Saunders, 1957):
 | (4) |
is
the density of water.
Statistical analysis
Sigma Stat (Systat Software Inc., Point Richmond, CA, USA) software was
used for statistical analyses. The individuals we selected were similar in
body size, so we expected no differences in the kinematic and hydrodynamic
variables among the individuals; this was confirmed by one-way analysis of
variance (ANOVA). Regressions were performed on the kinematic and hydrodynamic
variables to examine the effects of turning rate. We also used one-way ANOVA
to compare the kinematic variables of the turns with two different types of
wakes. All data are presented as means ± s.e.m. (standard error).
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). According to
the recorded swimming sequences, 41 single-beat turns were selected by two
criteria: (1) the quality of particle images were high enough to yield a high
density of original velocity vectors in the wake; (2) the shadow created by
the body and caudal peduncle was visible (to make sure the body and caudal fin
moved through the light sheet). Mean turning rates in stage 1
(1) of the 41 turns covered a range from 88 to 1050 deg.
s–1, and were divided into four turning rate categories: slow
(0–200 deg. s–1), moderate (200–400 deg.
s–1), fast (400–600 deg. s–1) and very
fast (>600 deg. s–1). The sample sizes of the four
categories indicate that slow and moderate routine turns occur more frequently
than fast and very fast routines turns when the koi carp are swimming
spontaneously (Fig. 2).
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Kinematics
As shown in Fig. 3A, the
turning angle in stage 1 (β1) had a significant linear
relationship with the mean turning rate in stage 1 (1)
(r2=0.887, P<0.001;
Table 1), while the turning
angle in stage 2 (β2) did not (r2=0.063,
P=0.114; Table 1).
Obviously, β1 was always much greater than β2
in magnitude (Table 1),
indicating that the changes in heading were mainly accomplished in stage 1
when the carp performed a routine single-beat turn. Thus the mean turning rate
in stage 1 (1) reflected the intensity of the whole turn,
which is why we examine the effects of 1 instead of the
turning rate of the whole turn. The duration of stage 1
(t1) seemed to remain stable in the single-beat turns with
different 1 (P=0.125). Consequently,
β1 was dominated by 1. The duration of stage
2 (t2) and the mean turning rate in stage 2
(2) also did not seem to be affected by 1
significantly (P=0.09 and P=0.135, respectively). In
addition, the relationship between the bending amplitude
(dmax) and 1 was examined
(Fig. 3B,
Table 1). The faster the carp
turned, the more tightly its body bent.
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The moving speeds of the carp at the onsets of the turns
(U0) covered a range from –0.76 to 1.58 L
s–1 (Fig. 3C),
and they showed very poor relationship with 1
(r2=0.00007, P=0.958). However, both the net
changes of speed in stage 1 and stage 2 (U1 and
U2) showed significant linear relationship with
1 (Fig. 3E,F,
Table 1). The turning radii of
the turns (Rt) covered a range from 0.08 to 0.55
L (Fig. 3D). The
relationship between Rt and U0 was
examined by using linear regression and no significant relationship was found
(P=0.157). In addition, the relationship between
Rt and 1 was marginally non-significant
(P=0.066). However, when 1 was greater than 600
deg. s–1, namely in the very fast category, the turning radii
distributed in a relatively narrow range and were relatively smaller than
those of the turns with lower turning rates
(Fig. 3D). Hence, we considered
the slow, moderate and fast categories as one group and the very fast category
as another group. Significant differences were found in the turning radius
between the two groups (P=0.023).
Hydrodynamics
Two types of wakes were found when the carp performed routine single-beat
turns: wake type I (Fig. 4) and
wake type II (Fig. 5).
Fig. 4 gives a two-dimensional
view of the type I wake. The start and stop vortices in this plane indicated
the formation and evolvement of the vortex rings. When the carp started to
bend, suction and pressure flows formed a bound vortex (vortex 1 in
Fig. 4B) around the inflection
points of the body. At the same time, the suction flow at the caudal peduncle
induced another vortex (vortex 2 in Fig.
4B). When the tail reached the maximum excursion and recoiled
subsequently, the two vortices shed and formed a vortex pair in the wake
(Fig. 4C,D). The jet between
the two vortices, named side jet, was directed nearly laterally, so the
momentum must help the carp to turn. During the recoil of the body and tail,
another suction flow was formed at the caudal peduncle and two more vortices
(vortices 3 and 4) were induced (Fig.
4D). At the end of stage 2, vortices 3 and 4 shed and formed
another vortex pair (Fig.
4E,F). The jet between the two vortices, so-called thrust jet, had
a considerable rearward component. In comparison with the wake of type I, only
one vortex pair (vortices 1 and 2 in Fig.
5) was visible in the wake of type II. The vortex pair was
generated in stage 1 (Fig.
5B–D) while no substantial vortices were formed in stage 2
(Fig. 5D–F).
Consequently, in the wake of type II, only the side jet was substantial and
the thrust jet was missing. Only some weak flow moved following the tail in
stage 2 (Fig. 5D–F).
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2b) did not show significant differences between the turns
with the two types of wakes (P=0.497;
Table 2), but the angle of
attack of the tail at the middle and the end of stage 2 (2m
and 2e) did (P=0.002 and P<0.001,
respectively; Table 2). The
differences in the angle of attack of the tail might also result in
differences in the generated flows. The initial speed, U0,
showed no significant difference in the turns with two types of wakes
(P=0.62). However, due to the thrust jet, the net changes of speeds
in stage 2 (U2) in the turns with wake type I were
significantly greater than those in the turns with wake type II
(P<0.001). In stage 2, the speed of the carp increased
0.64±0.11 L s–1 (N=20) with the
thrust jet but decreased 0.2±0.04 L s–1
(N=21) without the thrust jet.
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The effects of turning rate on hydrodynamics were examined in the same way
as those in kinematics. The momentum of the vortex ring shed in stage 1
(I1) and the corresponding jet angle (1)
both increased linearly with the mean turning rate (1) in
stage 1 (Table 1,
Fig. 6A), but the angle
(1–β1), namely the angle between the
side jet and the carp's initial orientation of a turn (see
–β in Fig.
1), remained stable when the mean turning rate increased
(P=0.68; Table 1).
This indicates that, before a carp performs a routine single-beat turn, the
orientation of the side jet generated in the forthcoming turn can be
estimated. As described above in Kinematics, the bending amplitude
increased with the turning rate, so that the distance from the COM to the
trailing edge of the tail (lc) and
J1+Ja1 (normalized such that the
maximum value of J1+Ja1 during a turn
was unity) decreased linearly with the increasing 1
(r2=0.856, P<0.001;
Table 1). However, the angle
between the jet and the line joining the COM of the carp and the
trailing edge of the tail remained stable when the turning rate increased
(P=0.136; Table 1). In
routine single-beat turns with various turning rates, was close to
90° (=95.3±1.3°, N=41). This is favorable for
maximizing the impulsive moment and improving the turning efficiency according
to Eqn 2. The impulsive moment
obtained from the beat of the body and tail in stage 1
(Li1) and the mean angular momentum of the carp in stage 1
(Lj1) both increased linearly with increasing
1 (Table 1,
Fig. 6B), but the slope of
Li1 was greater than that of Lj1.
Li1 was always greater than Lj1, and
(Li1–Lj1) mainly reflected the
loss of angular momentum because of the drag and the acceleration reaction.
The ratio of (Li1–Lj1) to
Li1 was not affected by 1
(P=0.318; Table 1),
and its value (0.52±0.03; N=41) indicated that the angular
momentum loss caused by drag and the acceleration reaction was a remarkable
part of the total angular momentum exerted.
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Considering only turns with wake type I, the momentum of the vortex ring
shed in stage 2 (I2) increased linearly with the turning
intensity while the corresponding jet angle (2) decreased
linearly (Table 1,
Fig. 7A,B). The jet angle
(2) covered a range from 4° to 56°
(Fig. 7B). In most cases, both
the lateral and posterior components of the thrust jet were remarkable. With
increasing turning intensity, the proportion of the posterior component
increased while the proportion of the lateral component decreased.
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). The
authors suggested that the differences in the relationship between the
duration of stage 1 and the turning rate are possibly subject to neural
controls (Domenici et al.,
2004). Notice that routine turns in our study are spontaneous, but
escape maneuvers of spiny dogfish are stimulated. The causes of the
differences between these two types of maneuvers in the relationship between
the duration of stage 1 and the turning rate need additional electromyographic
studies for clarification.
Previous studies have revealed that the turning radius is not related to
the swimming speed in several species of fishes (reviewed by
Domenici and Blake, 1997). The
same conclusion can be drawn for the koi carp. Moreover, the turning radius is
also not affected by the turning rate (except very fast turning rate), in
accordance with other studies (Gerstner,
1999). If the slow, moderate and fast categories are considered as
one group, however, and the very fast category is considered as another group,
the turning radius in the latter group is significantly smaller than in the
former. This is probably because the turning rate in the very fast category is
close to the magnitude of that in fast-starts, and the turning radii in
fast-starts are usually smaller than those in routine turns
(Webb and Fairchild,
2001).
Our results show that the turning angle in stage 1 accounts for the greatest portion of the total turning angle of a routine single-beat turn. Moreover, the duration of stage 1 remains stable in various routine single-beat turns. Consequently, the turning angle of a single-beat turn is dominated by the turning rate in stage 1. The moving speeds of the carp at the onset of turns cover a wide range. The appearance of a single-beat turn is not affected by the swimming speed. Moreover, the turning rate of a turn and the swimming speed before the turn are independent; however, the net changes of speed in stage 1 and stage 2 are both shown to have linear relationships with the turning rate in stage 1.
Effects of turning rate on hydrodynamics
The angle between the side jet and the carp's initial orientation of a turn
(1–β1) is always close to 90°
(94.2±3.1°, N=41), and is substantially independent of the
turning rate. The same conclusion can be drawn from escape maneuvers in
damsel-fly larvae (Enallagma cyathigerum L.)
(Brackenbury, 2003), and it
means that the orientation of the side jet in the forthcoming turn can be
predicted. Another interesting result concerning the side jet is that the
angle between the jet and the line through the COM of the carp and the
trailing edge of the tail is also close to 90° (95.3±1.3°,
N=41). It is helpful for the carp to maximize the torque so as to
improve the turning efficiency.
In stage 1, the impulsive moment obtained from the beat of the body and
tail (Li1) and the mean angular momentum of the carp
(Lj1) both increased linearly with the increasing turning
rate, and the slope of Li1 is greater than that of
Lj1 (Fig.
6B). But the ratio of
(Li1–Lj1) to
Li1 (0.52±0.03, N=41) remains stable in
the turns with a large range of turning rates. The difference between
Li1 and Lj1 is rather large, and
mainly caused by drag and the acceleration reaction, but may also be affected
by direct measurement error and errors in estimation. For instance, (1) the
smoothing operations in our flow measurements may reduce the measured
circulation (Fincham and Spedding,
1997) and the spatial resolution of our DPIV system is not very
high, so that it may contribute to measurement error in estimating
circulation. (2) The shape of the vortex ring is not measured precisely, so
Eqn 1 is based on the small core
vortex ring model, which may not be strictly appropriate
(Dabiri, 2005), and therefore
may lead to an over- or underestimation of the impulse of flow. (3) As
described in the Hydrodynamics section of Materials and methods, the
moment of inertia of the carp was estimated approximately based on an
assumption, which may have led to substantial errors. (4) Additionally, the
assumption in estimating the added moment of inertia of entrained liquid will
also lead to some errors. Even so, Li1 and
Lj1 show an agreement in magnitude in the turns with
various turning rates. But to make a more accurate comparison of the two
parameters needs more precise measurements and more accurate estimations.
As discussed elsewhere (Wolfgang et
al., 1999; Wu et al.,
2006), the vortex ring shed in stage 2 helps the fish to
accelerate. With the increase in turning intensity, the momentum exerted on
the carp in stage 2 increases and the jet angle in stage 2 decreases (tends to
zero). Both changes boost the acceleration in stage 2.
Two types of wakes and related kinematics
In steady swimming, several species of fishes change their wakes
substantially as they change swimming speed
(Nauen and Lauder, 2002;
Drucker and Lauder, 2000). At
low speeds, bluegill sunfish 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). In the present study, two types of flow patterns are found
in the wake of routine single-beat turns of koi carp Cyprinus carpio
koi. The difference between the two types of wakes is whether or not a
vortex ring and the thrust jet are generated in stage 2. Consequently, the
related kinematics shows a large difference between the turns with the types
of wakes (i.e. the net change of speed in stage 2). As mentioned above, the
spatial resolution of our DPIV system is not very high and may prevent
detection of additional vortices in the wake of type I. Referring to the error
report of `mpiv' toolbox' (Mori and Chang,
2004), however, the error of velocity in our flow measurements is
about 0.1 mm s–1. Therefore, if a vortex ring does shed in
stage 2 but is not detected, it must be much smaller in size, of much weaker
vorticity and without thrust jet in comparison with the vortex ring shed in
stage 1. In this case, the conclusion drawn above about the two types of wakes
remains tenable.
The flow patterns generated by the fast and very fast turns have thrust
jets, whereas the flow patterns without thrust jet appear frequently in the
slow and moderate turns. The wake type of a routine single-beat turn probably
depends on the recoil speed of the tail in stage 2. It seems that when the
recoil speed of the tail is greater than a critical value (in the present
study, it is 2 L s–1), wake type I appears;
otherwise, wake type II appears (Fig.
8A). Unfortunately, there is an exception to this rule. (As shown
in Fig. 8A, in one turn, the
recoil speed of the tail is 1.3 L s–1, but it
generates the wake of type 1.) Moreover, although the momentum
I2 seems to increases with recoil speed of the tail
(Fig. 8B), the correlation
between them is not significant (r2=0.171,
P=0.07). Therefore, although the carp can control the recoil speed of
the tail to influence the flow patterns substantially, the recoil speed should
not be the only factor that affects the flow patterns, and the bending
amplitude is probably another one, because it determines the recoil amplitude
of the body and tail in stage 2. They show significant differences between the
turns with the two wake types (P<0.001;
Table 2). Furthermore, the
momentum I2 increases with the bending amplitude
significantly (r2=0.38, P=0.004;
Fig. 8C). However, the bending
is accomplished in stage 1, so that by stage 2 the carp cannot control the
bending amplitude to control the flow patterns. The angle of attack of the
tail is another potential factor that the carp can control to influence the
flow patterns. Just as Bainbridge suggested, the angle of attack of the tail
is an essential factor determining the magnitude of thrust
(Bainbridge, 1963). In the
present study, although 2b shows no significant difference
between the turns with the two types of wakes, the significant differences in
2m and 2e between the two types of turns
(Table 2) indicate that in
stage 2, the mean angles of attack of the tail in the turns with wake type I
are substantially greater than those in the turns of wake type II. It seems
that the greater angle of attack of the tail, the more likely the carp will
generate thrust jet in stage 2 of a turn. On the other hand, it is necessary
to realize that the angles of attack of the tail are different for different
regions. Moreover, when the tail flicks, it performs a complex
three-dimensional motion (Lauder,
2000). Therefore, more detailed three-dimensional kinematics of
the tail is necessary for examining the effects of tail motion on the
generated flow patterns.
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Center of mass
It is difficult to obtain the real COM of a fish when it performs a turning
maneuver. Several investigators (Webb,
1976; Domenici and Blake,
1991; Domenici et al.,
2004) assumed the COM of a stretched-straight body is the same as
that of a bent body, namely, the COM is always at a fixed position on the
midline (typically, 0.33–0.39 L from the nose). Obviously,
however, the COM of a fish will move away from the midline when the fish bends
its body. Additionally, the more tightly the fish bends, the more substantial
the error is in the estimation of the COM. In the present study, the density
of fish is assumed to be homogeneous and the geometric center is used instead
of the COM. The geometric centers for the stretched-straight carp in our
experiments are on the midline, at 0.35–0.37 L from the nose.
They are close to the results reported for the COM values of fishes that have
similar body shape to koi carp (Webb,
1975; Webb, 1976).
It seems that using the geometric center instead of the COM in the present
study is acceptable. Furthermore, geometric center should be more precise than
a fixed point on the midline of a fish in estimating the COM for studying of
turning maneuvers of fish.
Concluding remarks
In routine single-beat turns of koi carp, the turning angle in stage 1
accounts for the greatest portion of the total turning angle. The durations of
stage 1 remain relatively stable over a nearly tenfold change in turning rate.
Consequently, the turning angle is dominated by the turning rate in stage 1.
The turning radius is not related to the swimming speed. Moreover, except in
very fast turns, the turning radius is also not affected by the turning
rate.
The angle between the side jet and the carp's initial orientation of a turn is always close to 90°, substantially independent of the increasing turning rate. Thus the orientation of the side jet in the forthcoming turn can be predicted. The angle between the jet and the line joining the COM of the carp and the trailing edge of the tail (mean value in stage 1) is also always close to 90°. It is helpful for the carp to maximize the torque so as to improve the turning efficiency. In stage 1, the impulsive moment obtained from the beat of the body and tail and the mean angular momentum of the carp show an agreement in magnitude.
Two types of flow patterns are found in the wake of routine single-beat turns. The difference between the two types of wakes is whether or not a vortex ring and the thrust jet are generated in stage 2. The recoil speed of the tail, the bending amplitude of the turn, and the angle of attack of the tail are three probable factors influencing the flow patterns in stage 2.
List of symbols and abbreviations
1
2
U1
U2
2b
2m
2e
1
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