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Three-dimensional numerical simulation of hydrodynamic interactionsbetween pectoral-fin vortices and body undulation in a swimming fish

Cheng-Lun Yu,1 Shang-Chieh Ting,2 Meng-Kao Yeh,1 and Jing-Tang Yang2,a)1Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan2Department of Mechanical Engineering, National Taiwan University, Taipei 10617, Taiwan

(Received 2 February 2011; accepted 28 July 2011; published online 26 September 2011)

We investigated numerically the hydrodynamic interactions between pectoral-fin vortices and body

undulation in a fish swimming with carangiform locomotion at a Reynolds number of 3.3 104; thethree-dimensional, viscous, incompressible, Navier-Stokes equations were solved with a

finite-volume method. For a fish swimming with the pectoral fins abducted, we characterized the

wake flow structures, forces, and power consumption with respect to various Strouhal numbers. The

numerical results reveal that a pair of vortices is formed immediately behind the abducted pectoral

fins of a swimming fish. There exist hydrodynamic interactions between the pectoral-fin vortices

and the undulating fish body. For Strouhal numbers in a range 0.20.8, the body undulation impedes

the shedding of pectoral-fin vortices, resulting in vortices closely attached to the pectoral fins. In

contrast, for Strouhal number 0.1, the pectoral-fin vortices are shed from the pectoral fins and driftdownstream. The low-pressure suction forces arising from the shed pectoral-fin vortices facilitate

lateral movements of the fish body, decreasing the power consumption. This phenomenon indicates

the possibility for an actual fish to harvest energy from the shed pectoral-fin vortices.VC

2011American Institute of Physics. [doi:10.1063/1.3640080]

I. INTRODUCTION

The invention and development of energy-saving devices

or techniques are currently the focuses of many research

fields. In research concerning fish-swimming hydro-

dynamics15

and biomimetic autonomous fish robotics,6,7

energy-saving mechanisms have also received much attention.

A common objective is to decrease the power consumption of

a fish robot, so as to facilitate a protracted operation. From a

biologically inspired perspective, the propulsive performance

of an underwater vehicle of human manufacture can be signif-

icantly improved on introducing the swimming principles

employed by a fish into the design, so mimicking a live fish.

The reason is that, through evolution by natural selection, fish

exhibit exceptional propulsive efficiency that is superior to

that of contemporary underwater vehicles.

Mechanisms of vortex control for saving locomotive

energy were originally proposed in the context of fish school-

ing behavior. The hydrodynamics in a fish school were inves-

tigated by Weihs and Webb;8 they suggested that fish could

make effective use of the environmental vortices by means of

a tactical arrangement of their relative positions. A reversed

Karman vortex street911 (see Fig. 1) is generally shed by a

fish swimming upstream, and the direction of fluid jets

formed inside this reversed Karman vortex street is opposite

to the direction of a fish swimming downstream; thereby

the downstream fish must keep a greater distance from the

upstream fish to avoid impacting this jet. In contrast, the

direction of upward oriented flow formed by a reversed Kar-

man vortex street is identical to the direction of a downstream

fish that is situated laterally, propelling the downstream fish

forward. In terms of such impacts of fluid jets on downstream

fish, a diamond-shaped fish school is typically recognized as

an optimal configuration to economize the overall energy

consumption for the schooling fish8 (see Fig. 1).

There seemed to be insufficient empirical evidence of an

energy-saving mechanism in fish utilizing environmental

vortices until Liao et al.1214 used the digital particle-image

velocimetry (DPIV) to reveal the hydrodynamics of a fish

(trout) swimming behind an upstream D-shaped obstacle.

They experimentally found that a fish slaloms betweenKarman vortex streets generated by the upstream D-shaped

obstacle rather than to swim through them. A fish is able to

decrease effectively its muscular activity through exploiting

the oncoming Karman vortex street, harvesting the kinetic

energy of the vortices.

Drucker and Lauder1517 also used DPIV to study the

hydrodynamic interactions of vortices between the upstream

dorsal fin and the downstream caudal fin in a swimming fish.

Their experimental results indicated that the vortices shed

from dorsal fin enhanced the propulsive efficiency of the

caudal fin. Akhtar et al.18 used two-dimensional computa-

tional fluid dynamics (CFD) to verify this hypothesis; they

simplified the upstream dorsal fin and downstream caudal finas two foils in a tandem arrangement undergoing pitch and

heaving motions. Their numerical results revealed that vorti-

ces shed from an upstream foil can initiate the formation of a

strong leading-edge stall vortex on the downstream foil. The

thrust and propulsive efficiency of the downstream foil can

be enhanced because this leading-edge stall vortex offers the

downstream foil a forward suction force. Exploitation of

environmental vortices shed from upstream fins is evidently

a significant mechanism for a swimming fish to decrease

power consumption.

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected].

1070-6631/2011/23(9)/091901/12/$30.00 VC 2011 American Institute of Physics23, 091901-1

PHYSICS OF FLUIDS 23, 091901 (2011)

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According to Breder,19 the modes of undulatory fish

swimming are classifiable as types BCF (body and/or caudal

fin) and MPF (median and/or paired fins), in terms of the be-

havioral characteristics of locomotion. The BCF mode is fur-

ther classified as carangiform (e.g., mullet), sub-carangiform

(e.g., trout), thunniform (e.g., tuna), and anguilliform (e.g.,

eel). Webb2022 pointed out that abducted pectoral fins (seeFig. 2) can generate trimming forces for posture control in a

BCF-mode swimming fish. The abducted pectoral fins, how-

ever, also shed vortices downstream; we term the vortices so

generated the pectoral-fin vortices. The hydrodynamic

interactions between pectoral-fin vortices and the body undu-

lation in a swimming fish remain unclear.

In this work, our objective was to explore numerically

the hydrodynamic interactions between pectoral-fin vortices

and the body undulation in a fish swimming with carangi-

form locomotion. For the cases that we simulated, the

Strouhal number (St) was varied (i.e., 0, 0.1, 0.2, 0.4, 0.6,

and 0.8). The Strouhal number is defined as fA/U, in which f

denotes the undulating frequency of the fish body wave, A

the peak-to-peak amplitude of tail beating, and U the fish

swimming velocity. We examined the simulated flow fields,

energy expenditure, and force production of a swimming fish

to unveil the hydrodynamic interactions between pectoral-fin

vortices and body undulation. Our results provide a biome-

chanical and biophysical foundation for the design of

energy-saving mechanisms adaptable in biomimetic vehicles

mimicking an undulatory swimming fish.

II. PHYSICAL MODEL AND NUMERICAL METHOD

A. Physical model

For our numerical simulation, a fish with rigid, abducted

pectoral fins was modeled, as shown in Fig. 3(a). The Carte-

sian coordinate system shown in Fig. 3(a) corresponds to a

frame of reference fixed at the fish, with x the longitudinal

(anterior-posterior) coordinate, y the lateral coordinate, and z

the vertical coordinate. In particular, the origin of the Carte-

sian coordinate system is placed at the snout of the fish; in

this simulation, the fish was considered to swim at a station-

ary point situated within a computational domain subject to a

background uniform free-stream velocity (U) (corresponding

to the swimming velocity of the fish but with an opposite

orientation);2326L represents the length of the fish ( 0.1 m),as shown in Fig. 3(b).

In the numerical simulation, in this work, we did not

consider the fluid-structure interaction problem. We assumed

also that the body length of the fish model remains constant

during undulation; only a lateral (i.e., y-direction) undulation

of the fish body is allowed. A lateral undulation of the body-

wave traveling backward from the snout to the tail of the fish

is prescribed with a formula of this form,

yx; t ax sin 2px

k

t

T

h i; (1)

in which t denotes time, k denotes the wavelength of fishundulation, T denotes the period of fish undulation; in all

simulations k is the body length ( L), which is in the range0.891.1 L observed in most fish swimming with carangi-form locomotion;27 a(x) depicts the amplitude envelope of

the lateral motion of the body wave (see Fig. 4(a)) and is

expressed here in a quadratic form,

ax C0 C1x C2x2; (2)

in which coefficients C0, C1, and C2 are solvable according to

kinematic data associated with a fish swimming with car-

angiform locomotion.28 The results indicate that C0 0.002,

C1 0.12, and C2 2, with a(0) 0.002 m, a(0.05) 0.001 m, and a(0.1) 0.01 m. The undulatory movementof the fish was realized via a mesh deformation complying

with Eq. (1), forming a wave traveling backward along the

fish body, as shown in Fig. 4(b).

The simulations carried out in this study pertain to a fish

that executes steady straight-line swimming with a constant

forward velocity. Typically, for fishes in a status of steady

straight-line swimming, their pectoral fins are kept abducted

and motionless with a fixed inclination, rather than undergo

remarkable deformation. Large flexibility and deformation

are usually observed only in fishes executing maneuvers

such as braking and turning. Thereby, it is considered that

the flexibility of the pectoral fins is negligible for the casewe studied which pertains to a fish in a steady straight-line

swimming status.

The numerical method employed in this work took no

account of the fluid-structure interaction problem; that is, the

swimming fish simulated in this work is subject to a

tethered condition without self-propulsion. Such tethered

treatment requires a virtual stationary pivot to be attached to

the fish model; the pivot sometimes exerts a tethering force

on the fish model to ensure force balance necessary for the

condition of steady swimming. Despite these limitations of

the computational modeling, the numerical results of the

analysis are expected to yield useful insight into theFIG. 2. (Color online) A fish swimming with pectoral fins abducted.

FIG. 1. (Color online) Schematic illustration of the mechanism of vortexcontrol for saving locomotive energy in a diamond-shaped fish school. The

circles with arrows denote the reversed Karman vortex street shed by the

fish swimming upstream. The blue arrows represent fluid jets formed inside

the reversed Karman vortex street. The block arrows represent upward ori-

ented flow formed by the reversed Karman vortex street. Dashed lines are

drawn to highlight the diamond shape of the fish school.

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hydrodynamic interactions between pectoral-fin vortices and

body undulation in a swimming fish.

B. Numerical method

We employed the three-dimensional viscous incompres-

sible Navier-Stokes equations as governing equations,

r u 0; (3)

@u

@t u ug

ru

1

qrp tr2u g; (4)

in which u is velocity, ug is mesh-grid velocity, q denotes

density, p denotes pressure, t is the fluid kinematic viscosity,

and g denotes the body forces per unit mass.

The computational domain is illustrated in Fig. 3(b).

The computational domain around the fish body was a

cylinder with a hemispherical end at the head of the body.

The boundary conditions are as follows. A boundary condi-

tion of uniform inlet flow velocity was applied for the inlet

boundary at the left side (Fig. 3(b)) and a boundary condition

of constant pressure on the outlet boundary at the right side.

The no-slip surface of the fish was set with ub uf; ub and ufare, respectively, the velocities of the fish body and the fluid.

All exterior boundaries of the computational domain that

were treated as no-slip surfaces had u 0 imposed.The governing Navier-Stokes equations were made dis-

crete with the finite-volume method; a second-order Crank-

Nicolson scheme was applied for discrete time and a

second-order upwind scheme for discrete space. These dis-

cretized governing Navier-Stokes equations were solved

with commercial software (CFD-RC). With the SIMPLEC

algorithm, we treated the pressure-velocity coupling, satis-

fying the continuity equation. Space discretization of the

FIG. 3. (Color online) Schematic diagram illustrating arrangement of the physical model and computational domain.

FIG. 4. (a) Amplitude envelope of the

body wave. (b) The undulatory body of

the fish model.

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computational domain was performed on a block-structured

mesh; the mesh grids were locally refined and concentrated

near the fish body and the wake region. To enable the fish

models to undergo undulatory motions complying with that

prescribed by equation (1), grid deformation methods for

computation of unsteady flow were exploited. For the three-

dimensional fish model, the boundary displacements were

realized through the standard transfinite interpolation (TFI)

re-meshing scheme.29,30

We have conducted convergence tests to ensure the

insensitivity of the computed solutions to the size of the

mesh grids and the time step. For our simulation, the mesh

grids amounted to 3.2 105; the dimensionless time step(i.e., Dt/T) was 0.02. A finer mesh with 6.4 105 grid pointsand a smaller dimensionless time step, 0.01, were also tested

in our simulation. In the convergence tests, the net longitudi-

nal force coefficient (CD) and net lateral force coefficient

(CL) were evaluated for the entire fish model (consisting of

both the body and pectoral fins) (Fig. 5(a)) and for solely the

pectoral fins (Fig. 5(b)). The dashed CD curve (for coarse

mesh-grids) agrees satisfactorily with the solid CD curve (forfine mesh-grids). The dashed CL curve also agrees satisfacto-

rily with the solid CL curve. These results of the convergence

tests ensured that our numerical simulation rendered solu-

tions independent of both the grid size and the time step.

C. Simulation parameters

In our simulation, the two non-dimensional parameters

that characterize hydrodynamic performance of a swimming

fish with pectoral fins abducted are the Reynolds number

(Re) for the flow and the Strouhal number (St) for the body

undulation, defined as follows:

Re UL=t; (5)

St fA=U; (6)

in particular, A is twice the undulation amplitude of the tail,

i.e., A 0.02 m (see Fig. 4 (a)).The Reynolds number associated with a fish swimming

with carangiform locomotion is typically greater than

104.1,24 In this work, the Reynolds number associated with

all the simulated cases was set as 3.3 104. We altered theStrouhal number to study the hydrodynamic interactions

between pectoral-fin vortices and the body undulation in a

swimming fish. The Strouhal number varied as 0, 0.1, 0.2,

0.4, 0.6, and 0.8. Because the uniform free-stream velocity

(U) and peak-to-peak amplitude of tail beating of a fish (A)

were fixed, the Strouhal number was just adjusted by the tail

beat frequency (f) of a fish.

D. Performance parameters

According to our results of numerical simulation, the

hydrodynamic forces and power consumption of the undulat-

ing fish are evaluated as follows.23,24 The friction and pressure

forces acting on the fish were evaluated respectively on inte-

grating the viscous stress and pressure around the fish surface.

The net longitudinal force (FD) acting on the fish body (i.e.,

along the x-axis) corresponds to a sum of longitudinal compo-

nents of friction force (FF) and pressure force (FPr), i.e.,

FD FF FPr. By definition (see Fig. 3(a)), the forces FF,FPr, and FD functionally act as thrust forces when they have

negative values, pushing the fish forward. The lateral compo-

nents of the friction and pressure forces are, respectively,denoted FFl and FPrl; the net lateral force FL equals FFl FPrl.These hydrodynamic forces were evaluated and normalized as

dimensionless force coefficients as follows:25,26

CF FF

1=2qU2L2; CPr

FPr1=2qU2L2

; and

CD FD

1=2qU2L2;

CFl FFl

1=2qU2L2; CPrl

FPrl1=2qU2L2

; and

CL FL

1=2qU2L2

:

(7)

For a swimming fish, the power (PS) required to perform a

lateral undulation of the body wave is defined as

PS

pn vs dS; (8)

in which p is the pressure acting on the fish surface, n is the

normal vector of the surface element of the fish body, vs is

the lateral (i.e., y-component) velocity of the surface element

of the fish body, and dS is the differential surface element.

FIG. 5. The independence of the grid and of the time step during one undulation cycle for flow over a swimming fish with the pectoral fins abducted at

St 0.8. The net longitudinal force coefficient (CD) and net lateral force coefficient (CL) were evaluated (a) for the entire fish model consisting of both thebody and pectoral fins and (b) for solely the pectoral fins. Dashed lines: mesh number 3.2 105, dimensionless time step 0.02; solid lines: mesh number6.4 105, dimensionless time step 0.01. The dashed lines agree satisfactorily with their corresponding solid lines.



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The coefficient of power consumption, CP, is accordingly

defined as

CP PS

1=2qU3L2: (9)

The contribution from viscous stress forces to the power con-sumption is omitted from Eq. (8), because, in practice, the

contribution from viscous stress forces is small and negligi-

ble relative to the contribution from the pressure force.

III. RESULTS AND DISCUSSION

In our simulation, the time-dependent hydrodynamic

forces and power consumption of the fish varied periodically

after the first six simulated undulation cycles; the time-

dependent and cycle-averaged quantities presented below

were evaluated after the tenth undulation cycle.

A. Lateral force versusStrouhal number

Our numerical results reveal that the Strouhal number

evidently affects the lateral force because of hydrodynamic

interactions between the pectoral-fin vortices and the undu-

lating fish body. In Fig. 6, we exhibit the variation of the net

lateral force coefficient (CL) within an undulation cycle for

various Strouhal numbers. A phase difference p exists

between the net lateral force coefficient (CL) at St 0.1 andthat at other Strouhal numbers (i.e., St 0.2, 0.4, 0.6, and0.8). This condition implies that a fish swimming at varied

Strouhal number with the pectoral fins abducted is subjected

to hydrodynamic interactions of varied types between the

ambient fluid and the undulating fish body.

In Fig. 7, we exhibit that, at time instant t/T 0.2, thefish tail is beating upward (i.e., towards the y direction) forall Strouhal numbers except St 0; note that St 0 corre-sponds to a static fish without body undulation (see Fig.

7(a)). For St 0.1, the net lateral force coefficient (CL) has,however, a small positive value (i.e., with the y orienta-tion) at time instant t/T 0.2 (see Fig. 6). In brief, the lateral

force and the movement of the beating tail have an identical

orientation. This condition indicates that the ambient fluid

provides the fish with a suction force that facilitates upward

undulatory movement; alternatively stated, the lateral force

serves as a suction force benefiting the body undulation.

In contrast, for Strouhal numbers in the range 0.20.8,

the lateral force coefficients (CL) all correspond to negative

values (i.e., with the y orientation), as shown in Fig. 6.Hence, for St 0.20.8, the lateral force and the movement ofthe beating tail have opposite orientations at time instant

t/T 0.2. For this situation, the undulating fish body provides

FIG. 6. Variation of the net lateral force coefficient (CL) within an undula-

tion cycle for various Strouhal numbers.

FIG. 7. (Color online) The undulating

movement of a fish swimming at various

Strouhal numbers with the pectoral fins

abducted, corresponding to time instantt/T 0.2. The blue solid arrow denotesthe moving direction of the undulating

fish body. The outline of the fish body is

highlighted in blue.



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an upward propelling force on the ambient fluid, accordingly

obtaining a downward reaction force. These findings indicate

that a fish swimming at varied Strouhal numbers with the pec-

toral fins abducted is subjected to hydrodynamic interactions

of varied types between the ambient fluid and the undulating

fish body.

B. Shedding of pectoral-fin vortices

Our numerical results indicate that a pair of vortices is

formed immediately behind the abducted pectoral fins of a

swimming fish. The Strouhal number is a crucial dimension-

less parameter that governs the shedding of pectoral-fin vor-

tices. A decreased Strouhal number enables the shedding of

the pectoral-fin vortices.

The hydrodynamic interaction between the ambient fluid

and the undulating fish body was analyzed on examining the

vorticity (xz) and pressure contours of flow fields observed

on the horizontal midplane (z 0) intersecting the fish body.The vorticity (xz) is defined as

xz @v

@x

@u

@y

; (10)

in which u and v are velocity components in x and y direc-

tions, respectively.

When a fish swims with the pectoral fins abducted, a

pair of vortices is formed immediately behind the pectoral

fins. The shear layers (subjected to strong vorticity) originat-

ing from the fin tips roll up and evolve into the pectoral-fin

vortices that structurally dominate the near-body wake

behind the pectoral fins. In Fig. 8, we exhibit the periodic

and symmetric shedding of pectoral-fin vortices within a

shedding cycle for St 0. The pectoral-fin vortices are shedsymmetrically, which is dynamically dissimilar to the asym-

metric shedding of a Karman vortex street.911 The region

immediately behind the abducted pectoral fins is separated

symmetrically by the static fish body, which cancels the

instability between the pectoral-fin vortices, accordingly

making pectoral-fin vortices shed symmetrically. The func-

tion of the static fish body is functionally analogous to a

splitter plate situated behind a bluff body for control of the

dynamics of vortex shedding.31,32

In Fig. 9, we exhibit the periodic and asymmetric shed-

ding of pectoral-fin vortices within an undulation cycle for

St 0.1. Referring to Figs. 9(a)9(c), the upper flank (i.e.,the right flank) of the posterior fish body gradually forms a

convex surface, whereas the opposite lower flank (i.e., the

left flank) gradually forms a concave surface. This concave

surface of the arched fish body is able to suck in the ambient

fluid because a region of negative pressure is formed23,24

(see Figs. 10(a)10(c)), which is favorable for the shedding

of pectoral-fin vortices. Suction forces yielded from the low-

pressure region enhance the detachment (i.e., shedding) of

pectoral-fin vortices. The pressure contours shown in Fig. 10

correspond to those of the fish flow fields shown in Fig. 9,

which is also for St 0.1.For the succeeding lateral tail-beat towards the opposite

direction (see Figs. 9(d)9(f)), a region of negative pressure

occurs again downstream of the upper abducted pectoral fin

FIG. 8. (Color online) Vorticity con-

tours (xz) of flow fields observed on the

horizontal midplane (z 0) intersectinga fish swimming at St 0. The blackdashed ellipses with arrows denote the

pectoral-fin vortices and their direction

of rotation. (a)(f) Six sequential images

illustrating the periodic and symmetric

shedding of pectoral-fin vortices.



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(see Figs. 10(d)10(f)). In brief, regions of negative pressure

occur alternately adjacent to the upper and lower flanks of

the posterior fish body because of periodic undulation. As a

result, a region of asymmetric negative pressure behind theabducted pectoral fins constantly occurs, which facilitates

the temporally and spatially asymmetric shedding of

pectoral-fin vortices.

Referring to Figs. 9(a)9(c), the pectoral-fin vortex is

shed from the upper abducted pectoral fin and drifts down-

stream. Through scrutiny of the corresponding pressure con-

tours of the flow fields and the kinematic movements of the

fish (see Figs. 10(a)10(c)), we found that the low-pressure

suction force arising from the pectoral-fin vortices is able to

facilitate the upward undulating movement of the posterior

fish body.

Referring to Figs. 9(d)9(f), the pectoral-fin vortex is

shed from the lower abducted pectoral fin and drifts down-

stream. Similarly, the low-pressure suction force arising

from the shed pectoral-fin vortex in turn facilitates the down-

ward undulating movement of the posterior fish body (see

Figs. 10(d)10(f)). In terms of these findings, it is clear that,

for St 0.1, the pectoral-fin vortex is an ambient flowmotion of a kind that provides lateral suction forces assisting

undulation of the posterior fish body.

In contrast, for Strouhal numbers in the range 0.20.8,

the pectoral-fin vortices are not shed downstream. The rea-

son is that rapid undulation of the posterior fish body at

increased frequencies (or St) produces a locally high pres-

sure in the region downstream of the pectoral-fin vortices.

This downstream high-pressure region adversely suppresses

the detachment of pectoral-fin vortices, resulting in vortices

closely attached behind the abducted pectoral fins. The struc-

tural characteristics of flow fields behind the pectoral fins aresimilar for Strouhal numbers in the range 0.20.8, but subtle

differences remain recognizable from the results of the asso-

ciated performance parameters.

To address in detail the physical mechanism underlying

the close attachment of vortices to the abducted pectoral fins,

we show a representative case of the vorticity and pressure

contours of flow fields for one undulation cycle at St 0.8.Referring to Figs. 11(a)11(c), the upper flank of the poste-

rior fish body gradually forms a convex surface, whereas the

opposite corresponding lower flank gradually forms a con-

cave surface. As noted above, this concave surface of the

arched fish body causes a negative pressure in the region

downstream of the lower abducted pectoral fin. Contrary to

expectation, however, the pectoral-fin vortex does not shed

off downstream, but remains closely attached to the lower

abducted pectoral fin.

The pressure contours of flow fields shown in Fig. 12

correspond to those of the flow fields shown in Fig. 11. A

locally high pressure is induced in the downstream region

(the dark patches) because of high-frequency body undula-

tion (see Figs. 12(d)12(f)). Although suction forces arising

from the low pressure in the downstream concave region

enhance the detachment of the lower pectoral-fin vortex, the

locally high pressure caused by the succeeding lateral tail-

beat adversely impedes the detachment of the pectoral-fin


tours (xz) of fish flow fields observed on

the horizontal midplane (z 0) for St 0.1. The blue solid arrow denotesthe moving direction of the undulating

fish body. The black dashed ellipses

with arrows denote the pectoral-fin vorti-

ces and their direction of rotation. (a)(f)

Six sequential images illustrating the

periodic and asymmetric shedding of

pectoral-fin vortices.



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FIG. 10. (Color online) Pressure con-

tours of the fish flow fields shown in

Fig. 9, for a fish swimming at St 0.1.The blue solid arrow denotes the direc-

tion of fish body movement.


tours (xz) of flow fields observed on the

horizontal midplane (z 0) intersectinga fish swimming at St 0.8. The bluesolid arrow denotes the moving direction

of the undulating fish body. The blackdashed ellipses with arrows denote the

pectoral-fin vortices and their direction

of rotation.



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vortex. Alternatively stated, there is insufficient time for the

pectoral-fin vortex to be detached because the locally high

pressure in the downstream region is rapidly created by the

succeeding lateral tail-beat, suppressing the vortex shedding.

The vortex thus remains closely attached to the lower

abducted pectoral fin (see also Fig. 11). To conclude, for a

fish swimming at Strouhal numbers in the range 0.20.8, the

high-frequency undulation of the fish body actively provides

lateral propelling forces to the ambient fluid. Accordingly,

the locally high pressure is induced in the downstream region

to suppress the shedding of pectoral-fin vortices.

C. Forces versusStrouhal number

Our numerical results indicate that an increased Strouhal

number causes an increased cycle-averaged friction-force

coefficient (CF) and a decreased cycle-averaged pressure-

force coefficient (CPr) in a fish swimming with the pectoral

fins abducted. This correlation has been reported in numeri-

cal treatments of the pectoral fins as adducted (or took no

account of the pectoral fins).24,25

The cycle-averagedfriction-force coefficient (CF) and pressure-force coefficient

(CPr) in a fish swimming with the pectoral fins abducted are,

notably, invariably greater than those in a fish swimming

with the pectoral fins adducted.

The friction force is dominated primarily by the fluid ve-

locity relative to the surface of a fish body. For that reason,

the friction force in a swimming fish with the pectoral fins

adducted depends largely on the velocity gradient resulting

from the velocity disparity between the body-wave velocity

(V) and the uniform free-stream velocity (U), as shown in Fig.

13(a). In contrast, the friction force in a fish swimming with

the pectoral fins abducted is generated because of the velocity

gradient resulting from the velocity disparity between the

body-wave velocity (V) and the velocity of the recirculating

flows (i.e., the pectoral-fin vortices) near the fish-body surface

(Uvortex), as shown in Fig. 13(b); V and Uvortex have opposite

directions. Accordingly, for a fish swimming with the pectoral

fins abducted, a larger velocity gradient is generally incurred,

producing a larger friction force. This condition accounts for

the fact that a fish swimming with the pectoral fins abducted

is capable of yielding a larger cycle-averaged friction-force

coefficient (CF) than a fish swimming with the pectoral fins

adducted, as shown in Fig. 13(c).

The pressure force was evaluated on integrating the sur-

face pressure around the undulatory body of a swimming fish

(see Fig. 14(a)). When a fish swims with the pectoral fins

abducted, high pressure is established in the regions immedi-

ately before the abducted fins because of the impact of theincoming free stream, whereas regions subjected to low pres-

sure are formed immediately behind the abducted fins

because of the existence of pectoral-fin vortices (see Fig.

14(b)). Large pressure (or form) drag forces are consequently

caused. This phenomenon does not occur in a fish swimming

with the pectoral fins adducted. The additional large pressure

drag caused by the abducted pectoral fins in a swimming fish

FIG. 12. (Color online) Pressure con-

tours of the fish flow fields shown in Fig.

11. The blue solid arrow denotes the

direction of fish body movement.



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is accordingly capable of rendering a larger cycle-averaged

pressure-force coefficient (CPr) than for a fish swimming

with the pectoral fins adducted, as shown in Fig. 14(c).

In nature, there is a great diversity of the pectoral-fin

geometries and fin inclination among distinct fish species. If

the geometry and inclination of the pectoral fins of the fish

model alter, the computational results pertaining to the fish

flow fields and force generation would correspondingly

change. A further systematic study is required to address the

effects associated with the geometry and inclination of the

pectoral fins.

D. Energy-saving mechanism

An energy-saving mechanism is found in a swimming

fish that sheds pectoral-fin vortices downstream. A decreased

Strouhal number enables the shedding of pectoral-fin vorti-

ces, consequently causing shed pectoral-fin vortices to drift

downstream. The low-pressure suction forces stemming

from the shed pectoral-fin vortices facilitate lateral undulat-

ing movements of the fish body, decreasing the power con-

sumption, which is a beneficial and significant mechanism

for a swimming fish to save locomotive energy.

To elucidate the energy-saving mechanism, we show an

illustrative and representative case (Fig. 15) for a fish swim-

ming at St 0.1. In Fig. 15(a), a pectoral-fin vortex was shedfrom the upper abducted pectoral fin and drifted downstream.

Through scrutiny of the pressure contour of the fish flow

field and kinematic movements of the swimming fish, we

found that the low-pressure suction forces stemming from

the shed pectoral-fin vortex facilitate upward movement of

the posterior fish body (see Fig. 15(a)). We further inspected

the distribution of power-consumption coefficient (CP)

within the section alongside the upper (i.e., right) surfacethat is in direct contact with the upper pectoral-fin vortex.

The power-consumption coefficients (CP) within this section

all have negative values that signify negative work output

(see Fig. 15(b)). From an energetic perspective, this condi-

tion suggests that the undulatory body is capable of harvest-

ing kinetic energy from the shed pectoral-fin vortex, which

might serve as a beneficial and significant mechanism for a

swimming fish to save locomotive energy.

As mentioned above, abducted pectoral fins are evi-

dently hydrodynamically unfavorable for swimming propul-

sion in a fish because of an increased friction force and

pressure drag force. A fish is, nevertheless, sometimes com-

pelled to swim with the pectoral fins abducted, so as to gen-

erate required trimming forces for posture stabilization. In

such circumstances, our findings indicate that it is practical

for a fish to harvest kinetic energy sophisticatedly from the

shed pectoral-fin vortices, given that the body undulation is

appropriately synchronized (or phase-locked) with the shed-

ding of pectoral-fin vortices. Hence, a fish is likely to be a

swimmer capable of implementing the energy-saving action

described above.

E. A biohydrodynamic analogy

We propose a biohydrodynamic analogy (Fig. 16)

between a fish swimming with the pectoral fins abducted and

a fish swimming behind an upstream D-shaped obstacle.

Through examination of the energy-saving mechanism per-

taining to pectoral-fin vortices and that pertaining to a Karman

vortex street shed by an upstream D-shaped obstacle,1214 we

found that, as long as there exist environmental vortices, a

fish can readily initiate energy-saving actions. Although the

manners of operation of these energy-saving actions vary, the

exploitation of environmental vortices is common in fish.

As schematically illustrated in Fig. 16, the shedpectoral-fin vortices drift downstream. The low-pressure

FIG. 13. (Color online) Vorticity contours (xz) of flow

fields observed on the horizontal midplane (z 0) inter-secting a fish swimming with the pectoral fins (a)

adducted and (b) abducted. The brighter and darker col-

ors of the contour represent positive and negative vor-

ticity values, respectively. The blue and black solid

arrows denote, respectively, the directions of body-

wave velocity (V) and background free-stream velocity

(U). The black dashed ellipses with arrows denote the

pectoral-fin vortices and their direction of rotation, with

Uvortex representing the velocity of the recirculating

flow near the fish-body surface. (c) Cycle-averaged fric-

tion-force coefficient (CF) versus Strouhal number.



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suction forces arising from the shed pectoral-fin vortices

facilitate lateral movements of the fish body, decreasing the

pertinent power consumption (see Fig. 16(a)). The fish body

posterior to the abducted pectoral fins is behaviorally similar

to a fish swimming between vortices generated from an

upstream D-shaped obstacle (see Fig. 16(b)) because both

cases slalom between the incoming vortices (Fig. 16). This

effect is proposed as a biohydrodynamic analogy.

In terms of this biohydrodynamic analogy, the abducted

pectoral fins act like a vortex generator and are hydrody-

namically similar to an upstream D-shaped obstacle. Other-

wise, the fish body posterior to the abducted pectoral fins

acts like a vortex adapter and is similar to a fish slaloming

behind an upstream D-shaped obstacle (Fig. 16). This biohy-

drodynamic analogy indicates that a fish can readily initiate

energy-saving actions; although the manners of operation of

energy-saving actions vary, an exploitation of environmental

vortices is common in fishes. The associated energy-saving

mechanism serves as a potentially useful biomechanical

guide for the design of future biomimetic vehicles in a per-spective of diminishing the power consumption.

FIG. 14. (Color online) Pressure contours of flow fieldsobserved on the horizontal midplane (z 0) intersectinga fish swimming with pectoral fins (a) adducted and (b)

abducted. The brighter and darker colors of the contour,

respectively, represent negative and positive pressure

values. (c) Cycle-averaged pressure-force coefficient

(CPr) versus Strouhal number.

FIG. 15. (Color online) Energy-saving mechanism in a fish swimming with

the pectoral fins abducted. (a) Pressure contour of the fish flow field

observed on the horizontal midplane (z 0). The brighter and darker colorsof the contour, respectively, represent negative and positive pressure values.

The blue dashed arrow denotes the direction of local movement of the undu-

lating fish body. The black dashed ellipses with arrows denote the pectoral-

fin vortices and their direction of rotation. The two vertical, dashed, and

black lines specify the borders of the section alongside the upper surface at

which the power-consumption coefficient (CP) exhibited in (b) is evaluated.

FIG. 16. (Color online) Schematic illustrations for the proposed biohydro-

dynamic analogy. (a) A fish swimming with the pectoral fins abducted. (b) A

fish swimming behind an upstream D-shaped obstacle. (a) and (b) The black

dashed ellipses with arrows denote the shed vortices and their direction of

rotation. The blue solid arrows denote the direction of motion of the fish

body. The black dashed rectangles highlight the vortex generator; the blue

dashed rectangles highlight the vortex adapter. Both abducted pectoral fins

and the D-shaped obstacle act like a vortex generator that sheds vortices

downstream. Both the fish body posterior to the abducted pectoral fins and

the fish slaloming behind the D-shaped obstacle act like a vortex adapter.



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IV. CONCLUSION

In this work, we numerically explored the hydrodynamic

interactions between pectoral-fin vortices and body undulation

in a swimming fish. Our numerical results indicate that a pair

of vortices is formed immediately behind the abducted pecto-

ral fins of a swimming fish. There exist hydrodynamic interac-

tions between the pectoral-fin vortices and the undulating fish

body. For Strouhal numbers in the range 0.20.8, the bodyundulation impedes the shedding of pectoral-fin vortices,

resulting in vortices becoming closely attached to the pectoral

fins. In contrast, for Strouhal number 0.1, the pectoral-finvortices are shed from the pectoral fins and drift downstream.

The low-pressure suction forces arising from the shed

pectoral-fin vortices facilitate lateral movements of the fish

body, decreasing the pertinent power consumption. This effect

is conjectured as a beneficial and significant mechanism for a

swimming fish to save locomotive energy because kinetic

energy is harvested from shed pectoral-fin vortices.

Although abducted pectoral fins are hydrodynamically

unfavorable for swimming propulsion, a fish is sometimes

compelled to swim with the pectoral fins abducted, so as tostabilize its swimming posture. Our numerical results reveal

that, in such circumstances, it is practical for a fish to harvest

kinetic energy sophisticatedly from the shed pectoral-fin vor-

tices, provided that the body undulation is appropriately

synchronized (or phase-locked) with the shedding of

pectoral-fin vortices, rendering negative power-consumption

coefficients (CP) signifying negative work output.

We propose a biohydrodynamic analogy between a fish

swimming with the pectoral fins abducted and a fish swim-

ming behind an upstream D-shaped obstacle. Both the

abducted pectoral fins and the D-shaped obstacle act like a

vortex generator shedding vortices downstream. Acting like

a vortex adapter, the posterior fish body behind abducted

pectoral fins and the fish behind a D-shaped obstacle slalom

between the incoming vortices. A fish can readily initiate

energy-saving actions; although the manners of operation of

energy-saving actions vary, the exploitation of environmen-

tal vortices is common in fishes. The energy-saving mecha-

nism revealed in this work provides a useful biomechanical

foundation for the design of future biomimetic vehicles with

a view to diminish power consumption.

ACKNOWLEDGMENT

The National Science Council of the Republic of Chinapartially supported this work under Contract Nos. NSC 96-

2628-E-002-256-MY3 and NSC 96-2628-E-002-258-MY3.

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