Koi carps frequently swim in burst-and-coast style, which consists of a burst phase and a coast phase. We quantify the swimming kinematics and the flow patterns generated by the carps in burst-and-coast swimming. In the burst phase, the carps burst in two modes: in the first, the tail beats for at least one cycle (multiple tail-beat mode); in the second, the tail beats for only a half-cycle (half tail-beat mode). The carp generates a vortex ring in each half-cycle beat. The vortex rings generated during bursting in multiple tail-beat mode form a linked chain, but only one vortex ring is generated in half tail-beat mode. The wake morphologies, such as momentum angle and jet angle, also show much difference between the two modes. In the burst phase,the kinematic data and the impulse obtained from the wake are linked to obtain the drag coefficient (Cd,burst≈0.242). In the coast phase, drag coefficient (Cd,coast≈0.060) 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.

Burst-and-coast swimming behavior consists of cyclic bursts of swimming movements followed by a coast phase in which the body is kept motionless and straight (Weihs and Webb,1983; 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:
\[\ Re={\rho}UL{/}{\mu},\]
(1)
where U is swimming speed, L is the length of the swimmer,and ρ 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:
\[\ C_{\mathrm{d}}=(2D){/}({\rho}SU^{2}).\]
(2)
The Cd of a rigid body has been measured for different animals from direct force measurements of dead animals (reviewed by Webb, 1975; 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.

Animals and body morphometrics

The experiments were performed on koi carps (Cyprinus carpio koiL.), who swim in burst-and-coast style frequently. The carps were housed in the laboratory on a 14 h:10 h light:dark cycle at a temperature of 26±2°C. The body length, L, of each carp was measured from the images taken by camera C (Fig. 1). Based on body length, the body surface area(Abody) of each carp was estimated according to McHenry and Lauder (McHenry and Lauder,2006):
\[\ A_{\mathrm{body}}=3.06{\times}10^{-1}L^{2.16}.\]
(3)
The volume (V) and mass (m) of each carp was measured directly.

Experimental setup and procedure

The experimental setup is shown in Fig. 1A. The tank (100 cm long × 40 cm wide × 30 cm high)has a working volume (50 cm long × 25 cm wide × 10 cm high) that is confined by wire grids covered with fine mesh. Underneath the tank there is a translation stage driven by two servo motors (SM1 and SM2) in two orthogonal directions. The translation stage has a travel range of 50 cm ×30 cm, a position accuracy of 5 μm, a maximum speed of 0.5 m s–1and a maximum acceleration of 10 m s–2. The servo motors are controlled by a control box (CB) that is connected to a computer (PC). A continuous wave laser beam (100 mW) of wavelength 0.532 μm is expanded by two cylindrical lenses (CL1 and CL2) to generate a light sheet. A mirror (M)fixed on the bracket (B) at the end of the Y rail is used to reflect the light sheet into the tank. The flow is visualized by seeding the water with nylon particles (40–70 μm, 1.05 g cm–3) to reflect the laser light. The stage moves the mirror M in the x direction,tracking the carp, so that the flow in the wake is illuminated by the light sheet. Two CCD cameras (C and HSC), used for obtaining the ventral views of carp, are mounted on the stage to track the carp. Camera C (MTV-1881EX; 25 frames s–1, 768 pixels ×576 pixels; Mintron Inc.,Taipei, China), with a lens of 16 mm focal length and a red-glass optical filter, is used to record the swimming kinematics. Camera HSC (AM1101; 100 frames s–1, 640 pixels ×480 pixels; Joinhope Ltd.,Beijing, China), with a lens of 25 mm focal length and an optical band-pass filter with a wavelength of 534±20 nm, is used to record the particle images in the wake. Because the working area was simultaneously illuminated by the uniform light source and the laser light sheet, the two optical filters were placed in front of the two cameras to improve their imaging qualities. Actually, camera C is a little inclined to make sure that the centres of the fields of view of the two cameras are matched. Example images taken by cameras C and HSC are shown in Fig. 1B,C, respectively. All images recorded by the two cameras are sent to the computer (PC) by two image grabbers.

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).

Fig. 1.

(A) Schematic drawing of the video tracking and recording setup. A white paper sheet (WP) covering the tank that is illuminated by a halogen lamp (HL)is used to provide an even background for the high-contrast images of the carp's silhouette. The emission spectrum of the halogen lamp peaks in the red to yellow region. The flow around the carp is exposed by an expanded light sheet from a continuous wave (CW) laser. A high-speed CCD camera (HSC) fitted with an optical band-pass filter was used to catch the particle images (image in C), and another CCD camera fitted with a red glass plate was used to catch the swimming kinematics (image in B). We set up a global coordinate system with its origin coinciding with the end of the Y rail and the beginning of the X rail (the side of a rail where a motor is fastened is defined as the beginning).

Fig. 1.

(A) Schematic drawing of the video tracking and recording setup. A white paper sheet (WP) covering the tank that is illuminated by a halogen lamp (HL)is used to provide an even background for the high-contrast images of the carp's silhouette. The emission spectrum of the halogen lamp peaks in the red to yellow region. The flow around the carp is exposed by an expanded light sheet from a continuous wave (CW) laser. A high-speed CCD camera (HSC) fitted with an optical band-pass filter was used to catch the particle images (image in C), and another CCD camera fitted with a red glass plate was used to catch the swimming kinematics (image in B). We set up a global coordinate system with its origin coinciding with the end of the Y rail and the beginning of the X rail (the side of a rail where a motor is fastened is defined as the beginning).

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 (β) 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.

The burst phase is considered to be the period between the onset of tail beat and the instant when the body reaches a stretched straight posture(Fig. 3). The coast phase is defined as the period when the body stays straight until the carp starts another burst or ceases moving forward(Fig. 3). The burst phase(covering time t1 and distance l1) can be divided into two periods. During the first period, namely the accelerating period (covering time t0 and distance l0), the carp accelerates from an initial speed(Ui) to a maximum speed (Um). During the second period, namely the recovery period, it decelerates from the maximum speed to a final speed (Uf), which is also the initial coast speed. The coast time is denoted by t2 and the corresponding distance is l2.

Fig. 2.

(A) Coordinate system and notation used to describe the kinematics and wake of the fish. The bold arrow indicates the jet flow. See List of symbols for definitions. R0 and R were estimated by plotting the profiles of the velocity components u (B) and v (C)parallel to x″ and y″, respectively.

Fig. 2.

(A) Coordinate system and notation used to describe the kinematics and wake of the fish. The bold arrow indicates the jet flow. See List of symbols for definitions. R0 and R were estimated by plotting the profiles of the velocity components u (B) and v (C)parallel to x″ and y″, respectively.

Fig. 3.

The speed, U, and lateral excursion, d, versus time, t. (A) U and d curves in bout 1. (B) U and d curves in bout 13. See List of symbols for definitions of Uf, Um and Ui.

Fig. 3.

The speed, U, and lateral excursion, d, versus time, t. (A) U and d curves in bout 1. (B) U and d curves in bout 13. See List of symbols for definitions of Uf, Um and Ui.

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 pixels×16 pixels (2.7 mm×2.7 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 vparallel to x″ and y″, respectively(Fig. 2). Momentum angle(φ) of a vortex pair was the angle between x′ and x″. The angle α 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):
\[\ I={\rho}{\bar{{\Gamma}}}{\pi}R^{2},\]
(4)
where ρ is the density of freshwater and Γ is the mean absolute value of the circulations (Γ) of the pair of vortices. CirculationΓ is the line integral of the tangential velocity component(VT) about a curve C enclosing the vortex(Batchelor, 1967):
\[\ {\Gamma}={{\oint}_{\mathrm{C}}}\mathbf{V}_{\mathbf{T}}\mathrm{d}\mathbf{l},\]
(5)
where dl is the differential element along the curve C. Considering that the vortex ring shed by the carp may not meet the assumption of small-core, we determined the impulse I′ of the large-core vortex ring following Müller et al.(Müller et al., 2000)according to the mean ring radius (Rm):
\[\ I^{{^\prime}}={\rho}{\bar{{\Gamma}}}{\pi}R_{\mathrm{m}}^{2}.\]
(6)
Rm was estimated from the area occupied by elevated vorticity. We also derived the dimensionless mean core radius (ϵ) from the core radius R0 and the ring radius R(ϵ=R0/R). The parameter ϵ with 0<ϵ≤√2 was used by Norbury(Norbury, 1973) to characterize a family of vortex rings. For small-core vortices, ϵ≤0.25 otherwise 0.25<ϵ≤√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:
\[\ I_{\mathrm{d}}=I_{\mathrm{t}}-(m+k{\rho}V)a{\ }t_{0},\]
(7)
where It is the thrust impulse obtained from the flow, m and V are the mass and volume of the carp, respectively, k is the added mass coefficient, and a is the mean acceleration obtained during the accelerating period of the burst phase. Assuming the acceleration was constant, we determined a by using a linear fit to the data pairs [time t, instantaneous swimming speed U(t)] obtained throughout the accelerating period of the burst phase:
\[\ U(t)=at+U_{\mathrm{il}},\]
(8)
where Ui1 is the initial speed of burst (fitted value). The drag impulse Id was also the integration of drag D with respect to time t:
\[\ I_{\mathrm{d}}={{\int}_{0}^{t_{0}}}D{\ }\mathrm{d}t.\]
(9)
Considering Eqn 2, we can express Id by:
\[\ \begin{array}{rl}I_{\mathrm{d}}&={{\int}_{0}^{t_{0}}}\frac{1}{2}C_{\mathrm{d},\mathrm{burst}}{\rho}SU^{2}\mathrm{d}t\\&=\frac{1}{2}C_{\mathrm{d},\mathrm{burst}}{\rho}S{{\int}_{0}^{t_{0}}}(U_{\mathrm{il}}+at)^{2}\mathrm{d}t\\&=\frac{1}{2}C_{\mathrm{d},\mathrm{burst}}{\rho}S(U_{\mathrm{il}}^{2}t_{0}+aU_{\mathrm{il}}t_{0}^{2}+\frac{1}{3}a^{2}t_{0}^{3}).\end{array}\]
(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):
\[\ C_{\mathrm{d},\mathrm{coast}}=2j(m+k{\rho}V){/}({\rho}S),\]
(11)
where j is the slope of the regression of U–1versus t and is obtained according to:
\[\ \frac{1}{U(t)}=jt+\frac{1}{U_{\mathrm{i}2}},\]
(12)
where Ui2 is the initial coast speed (fitted value).

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:
\[\ E_{\mathrm{B}}=(I_{\mathrm{t}}{/}t_{0})l_{0}\]
(13)
and
\[\ E_{\mathrm{S}}=\frac{1}{2}C_{\mathrm{d},\mathrm{und}}{\rho}SU_{\mathrm{S}}^{2}(l_{1}+l_{2}),\]
(14)
where Cd,und is the drag coefficient of the steadily swimming carp. US is the speed of the carp and is equal to the average speed[(l1+l2)/(t1+t2)]of burst-and-coast swimming. Consequently, re can be derived according to:
\[\ r_{\mathrm{e}}=\frac{E_{\mathrm{B}}}{E_{\mathrm{S}}}=\frac{2I_{\mathrm{t}}l_{0}(t_{1}+t_{2})^{2}}{C_{\mathrm{d},\mathrm{und}}{\rho}S(l_{1}+l_{2})^{3}t_{0}}.\]
(15)
Considering that the body of the carp undulated both in the burst phase of burst-and-coast swimming and in steady swimming, we assumed that Cd,und was equal to Cd,burst.

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.

Four koi carps (carp I, II, III and IV) were collected for the experiments. The morphometrics of the four carps are shown in Table 1. The fineness ratios(length/maximum width of a body) of the carps are nearly 6. So, we used the added mass coefficient k=0.045(Azuma, 1992; 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.

Table 1.

Morphometrics of the carps

L (mm)S (mm2)V (103 mm3)m (g)
Carp I 55.6 1799 2.2 2.4 
Carp II 55.2 1771 2.0 2.1 
Carp III 51.8 1544 1.8 1.9 
Carp IV 59.2 2060 3.0 3.0 
Mean ± s.d. 55.5±3.0 1794±211 2.25±0.53 2.35±0.48 
L (mm)S (mm2)V (103 mm3)m (g)
Carp I 55.6 1799 2.2 2.4 
Carp II 55.2 1771 2.0 2.1 
Carp III 51.8 1544 1.8 1.9 
Carp IV 59.2 2060 3.0 3.0 
Mean ± s.d. 55.5±3.0 1794±211 2.25±0.53 2.35±0.48 

See List of symbols for definitions.

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 (β 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 (β in Table 2) and it had a certain lateral displacement (lL in Table 2) in the following coast phase. β 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 Ut and dt 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 Ut and dt 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 t1t0)(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.

Table 2.

Kinematic variables and drag measurements

MT modeHT modeTestTotal
β (deg.) 3.0±1.8 15.3±7.8 P<0.001* — 
lL (mm) 0.4±0.2 2.1±1.5 P<0.001* — 
Ui (mm s-136±23 42±14 P=0.475 39±19 
Um (mm s-178±32 69±19 P=0.435 73±26 
Uf (mm s-167±32 54±17 P=0.239 61±26 
a (mm s-2236±164 220±147 P=0.908* 228±152 
Re1(1032.5±1.6 2.7±0.9 P=0.718 2.6±1.3 
Re2(1032.7±1.3 2.5±1.1 P=0.664 2.6±1.2 
Cd,burst 0.242±0.084 0.242±0.085 P=0.991 0.242±0.083 
Cd,coast 0.059±0.014 0.060±0.012 P=0.915 0.060±0.013 
re 0.57±0.21 0.52±0.15 P=0.468 0.55±0.18 
N 12 12 — 24 
MT modeHT modeTestTotal
β (deg.) 3.0±1.8 15.3±7.8 P<0.001* — 
lL (mm) 0.4±0.2 2.1±1.5 P<0.001* — 
Ui (mm s-136±23 42±14 P=0.475 39±19 
Um (mm s-178±32 69±19 P=0.435 73±26 
Uf (mm s-167±32 54±17 P=0.239 61±26 
a (mm s-2236±164 220±147 P=0.908* 228±152 
Re1(1032.5±1.6 2.7±0.9 P=0.718 2.6±1.3 
Re2(1032.7±1.3 2.5±1.1 P=0.664 2.6±1.2 
Cd,burst 0.242±0.084 0.242±0.085 P=0.991 0.242±0.083 
Cd,coast 0.059±0.014 0.060±0.012 P=0.915 0.060±0.013 
re 0.57±0.21 0.52±0.15 P=0.468 0.55±0.18 
N 12 12 — 24 

Values are presented as means ± s.d. *Indicates Mann—Whitney test; all other tests are t-tests. See List of symbols for definitions.

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).

Fig. 5 gives a two-dimensional view of the flow generated by the carp in bout 1. The flow pattern shown was representative for all the bouts in which the carp burst in the MT mode. In the burst phase, the tail performed two flicks (two half-cycle beats), a flick to its right side (t=0–0.10 s) and then a flick to its left side (t=0.10–0.24 s). As the body and tail moved in an undulatory form, the alternating suction and pressure flows formed a bound vortex around the inflection points of the body/tail(Fig. 5A, vortex 2; Fig. 5B, vortex 3). The bound vortex shed when the inflection point reached the tail tip(Fig. 5B, vortex 2; Fig. 5C, vortex 3). In the first half-cycle beat, the suction flow at the peduncle, where the flow followed the lateral movement of the posterior body, induced another vortex(Fig. 5A, vortex 1) that formed a vortex pair together with vortex 2 (Fig. 5B). During the second half-cycle beat, vortex 2 was reinforced,and formed another vortex pair together with vortex 3(Fig. 5C). The vortices of each vortex pair were located at different sides of the body axis and moved backwards at a speed of 8–12 mm s–1(Fig. 5B–F). Vortex 1 was weak and was soon destroyed in the second half-cycle beat. In the coast phase,no significant flow was generated in the wake and only some weak flow adjacent to the carp body was visible (Fig. 5E,F).

Fig. 4.

Histograms of time, distance and speed variables in the burst phase and the coast phase. All the variables, including (A) t1 and t2, (B) l1 and l2and (C) v1 and v2, are presented independently for the two burst-and-coast swimming modes (MT and HT). The values are plotted as means ± s.d. (N=12).

Fig. 4.

Histograms of time, distance and speed variables in the burst phase and the coast phase. All the variables, including (A) t1 and t2, (B) l1 and l2and (C) v1 and v2, are presented independently for the two burst-and-coast swimming modes (MT and HT). The values are plotted as means ± s.d. (N=12).

Fig. 5.

Flow patterns generated in bout 1. A–F show the evolvement of vorticity and flow velocity vector fields at different times.

Fig. 5.

Flow patterns generated in bout 1. A–F show the evolvement of vorticity and flow velocity vector fields at different times.

Fig. 6.

Same as in Fig. 5, but for flow patterns generated in bout 13.

Fig. 6.

Same as in Fig. 5, but for flow patterns generated in bout 13.

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 φ (P<0.001) and jet angle α(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, φ was 44±19° (mean ±s.d., N=25) and α was 36±17° (mean ± s.d., N=25). When the carp burst in the HT mode, φ was 11±6° (mean ± s.d., N=12) and α 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.

Table 3.

Wake and vortex variables

MT modeHT modeTestTotal
\({\bar{R}}_{0}(\mathrm{mm})\)
 
3.0±0.5 3.2±0.4 P=0.346 3.1±0.4 
R (mm) 4.6±1.0 4.7±1.2 P=0.855 4.6±1.1 
Rm (mm) 6.9±0.9 7.0±0.9 P=0.678 6.9±0.9 
φ (deg.) 44±19 11±6 P<0.001* — 
α (deg.) 36±17 76±12 P<0.001 — 
\({\bar{{\Gamma}}}(\mathrm{mm}^{2}{\ }\mathrm{s}^{-1})\)
 
851±440 854±345 P=0.984 852±407 
I (10-5 Ns) 6.4±4.6 7.2±6.9 P=0.948* 6.7±5.4 
I′ (10-5 Ns) 14.3±10.1 14.4±9.7 P=0.858* 14.3±9.8 
N 25 12 — 37 
MT modeHT modeTestTotal
\({\bar{R}}_{0}(\mathrm{mm})\)
 
3.0±0.5 3.2±0.4 P=0.346 3.1±0.4 
R (mm) 4.6±1.0 4.7±1.2 P=0.855 4.6±1.1 
Rm (mm) 6.9±0.9 7.0±0.9 P=0.678 6.9±0.9 
φ (deg.) 44±19 11±6 P<0.001* — 
α (deg.) 36±17 76±12 P<0.001 — 
\({\bar{{\Gamma}}}(\mathrm{mm}^{2}{\ }\mathrm{s}^{-1})\)
 
851±440 854±345 P=0.984 852±407 
I (10-5 Ns) 6.4±4.6 7.2±6.9 P=0.948* 6.7±5.4 
I′ (10-5 Ns) 14.3±10.1 14.4±9.7 P=0.858* 14.3±9.8 
N 25 12 — 37 

Values are presented as means ± s.d. *Indicates Mann—Whitney test; all other tests are t-tests. 25 vortex pairs from 12 burst-and-coast bouts. At least two vortex pairs are generated when the carps burst in the MT mode. See List of symbols for definitions.

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 (Γ)(Table 3). No significant differences were found in the wake parameters R0, R, Γ, I and I′ between the two burst modes (Table 3). The dimensionless mean core radius ϵ (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 ϵindicated 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).

Fig. 7.

Estimations of drag coefficients during the coast phase of the carps. (A)and (B) are for reciprocals of measured instantaneous speeds (U)during the coast phase in bouts 1 and 13, respectively.

Fig. 7.

Estimations of drag coefficients during the coast phase of the carps. (A)and (B) are for reciprocals of measured instantaneous speeds (U)during the coast phase in bouts 1 and 13, respectively.

After applying curve-fitting Eqn 12(Fig. 7), we calculated Cd,coast during coasting from Eqn 11 and found that Cd,coast tended to decrease while Re2increased (rc=–0.640, P<0.001). Note that Cd,coast in the bouts of the two different burst modes was not significantly different (P=0.915)(Table 2) and that Cd,coast can be expressed by 0.060±0.013 (mean± s.d., N=24) with consideration of the total 24 bouts.

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.

Kinematics

Our experimentally measured speed versus time curves of the koi carps during burst-and-coast swimming show some small discrepancies with the results of Videler (Videler and Weihs,1982). 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 (φ=44±19° and α=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 (φ=11±6° andα=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 t1t0), we only considered the accelerating period of burst phase (period from 0 to t0)in estimating Cd,burst. The dimensionless mean core radii(ϵ=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.

     
  • a

    mean acceleration during the accelerating period of the burst phase

  •  
  • Abody

    body surface area of the carp

  •  
  • C

    closed curve around the vortex

  •  
  • Cd

    inertial drag coefficient (general)

  •  
  • Cd,burst

    drag coefficient in the burst phase, calculated using the large-core vortex model

  •  
  • Cd,coast

    drag coefficient in the coast phase

  •  
  • Cd,und

    drag coefficient in steady swimming

  •  
  • d

    lateral excursion of the tail tip

  •  
  • D

    drag

  •  
  • dl

    differential element along the curve C

  •  
  • EB

    energy expended in burst-and-coast swimming

  •  
  • ES

    energy expended in steady swimming at the same average speed as in burst-and-coast swimming

  •  
  • I

    impulse of the vortex ring, calculated using the small-core vortex ring model

  •  
  • I′

    impulse of the vortex ring, calculated using the large-core vortex ring model

  •  
  • Id

    drag impulse

  •  
  • It

    thrust impulse obtained from the flow

  •  
  • j

    slope of the regression of U–1versus time

  •  
  • k

    added mass coefficient

  •  
  • L

    body length of the carp

  •  
  • l0

    distance traversed during the accelerating period of the burst phase

  •  
  • l1

    distance traversed during the burst phase

  •  
  • l2

    distance traversed during the coast phase

  •  
  • lL

    lateral displacement of the carp in the coast phase

  •  
  • m

    body mass of the carp

  •  
  • r

    correlation coefficients of fitting

  •  
  • R

    vortex ring radius

  •  
  • Rm

    mean ring radius

  •  
  • R0

    vortex core radius

  •  
  • \({\bar{R}}_{0}\)

    mean core radius of the two vortices in a pair

  •  
  • rc

    Pearson product moment correlation coefficient

  •  
  • re

    ratio of energy EB to energy ES

  •  
  • Re

    Reynolds number

  •  
  • Re1

    Reynolds number in the burst phase

  •  
  • Re2

    Reynolds number in the coast phase

  •  
  • S

    wetted surface area of the carp

  •  
  • t

    time (general)

  •  
  • t0

    duration of the accelerating period of the burst phase

  •  
  • t1

    duration of the burst phase

  •  
  • t2

    duration of the coast phase

  •  
  • u

    velocity component parallel to x″ axis

  •  
  • U

    swimming speed (general)

  •  
  • Uf

    final speed of burst, also the initial speed of coast

  •  
  • Ui

    initial speed of burst

  •  
  • Ui1

    initial speed of burst, fitted value

  •  
  • Ui2

    initial speed of coast, fitted value

  •  
  • Um

    maximum speed during burst phase

  •  
  • US

    speed of steadily swimming carp

  •  
  • v

    velocity component parallel to y″ axis

  •  
  • V

    body volume of the carp

  •  
  • v1

    mean speed in burst phase

  •  
  • v2

    mean speed in coast phase

  •  
  • VT

    tangential velocity component about the curve C

  •  
  • x, y, z

    original coordinates of experimental setup

  •  
  • x′, y′

    Cartesian coordinates of the swimming fish

  •  
  • x″, y″

    Cartesian coordinates of vortex ring

  •  
  • α

    jet angle of vortex ring

  •  
  • β

    turn angle of the carp

  •  
  • Γ

    circulation of single vortex

  •  
  • \({\bar{{\Gamma}}}\)

    mean absolute value of the circulations of a pair of vortices

  •  
  • ϵ

    dimensionless mean core radius

  •  
  • μ

    dynamic viscosity of water

  •  
  • ρ

    density of water

  •  
  • φ

    momentum angle of vortex ring

We thank Dr U. K. Müller for her help in calculating the impulse of vortex rings. We also thank Dr L. Li for his help in improving the English in our paper. Three anonymous reviewers contributed greatly to the manuscript. This work is supported by National Natural Science Foundation of China 10332040.

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