AMER. ZOOL., 29:151-160 (1989)
Design Features and Mechanics of Axial Locomotion in Fish1
DANIEL WEIHS
Department of Aeronautical Engineering,
Technion—Israel Institute of Technology, Haifa 32000, Israel
SYNOPSIS. Locomotion is the result of transfer of momentum from the fish musculature
to the surrounding water. The present paper discusses some basic principles of this momentum transfer and shows the effects of various adaptations of body shape and fin shape,
size and positioning.
Muscles take up a large part of the fish body volume in many cases. The effects of
distribution of muscle mass on external shape, and drag (with its reciprocal influence on
the muscular system) are analysed. Fins provide an effective means of momentum transfer,
by allowing large amounts of water to be moved by small body masses. Fin shape, variable
flexibility and positioning all interact to influence thrust producing performance. A framework for understanding the various combinations of fins, their shapes and motion is
presented. Reasons for shifting the center of propulsion to the rear part of the fish, in
anguilliform, and much more so in carangiform swimmers are discussed. This shift is
shown to result from considerations of propulsive efficiency. Double-tailed fin configurations, defined as dorsal and ventral fins placed at the same longitudinal positions so as
to produce a "continuous" fin are analysed. Examples of both fast starters (such as esocids)
and cruising species (scombrids, etc.) are used to point out the advantages of such fin
placement.
Much thought has been given to the
ramifications
of this type of motion on the
The basic mode of aquatic locomotion
driven by axial musculature is the wiggling shape and distribution of muscle along the
motion produced by sending waves of side- fish body, i.e., the shape and placement of
ways displacement of the vertebral column myomeres relative to the backbone in crossin the caudal direction. These waves are section and in longitudinal position. Thus
produced by phased contractions of the Alexander (1969) obtained some very
trunk muscles, extensively discussed else- important results pertaining to the
where (Alexander, 1967; Lighthill, 1975; W-shaped myomeres in some of the more
Webb and Weihs, 1983; McClellan, 1989; advanced teleosts. In these fish, the trajecSigvardt, 1989). This description fits both tories followed by muscle fibers in adjacent
transient and periodic (formerly described myotomes provide a helical pattern. Alexas steady) motions. The difference between ander showed that this arrangement results
these two types is that in transient motion in bending the body with very little shortonly one or at most several waves are sent ening of the sarcomeres, and also that the
down the body, while in periodic motion, contraction of all white fibers is approxia large number of essentially identical waves mately equal in these configurations. The
is produced at regular time intervals. The teleost helical pattern might also result in
amplitude of the displacement waves usu- more rapid body-mass motion for a given
ally grows as the wave progresses towards rate of muscular contraction. However, this
the tail. This effect is especially pro- more rapid motion is accompanied by a
nounced in transient motions such as rapid smaller bending moment. Thus, while some
starting (Fig. la, b) where the amplitude of the characteristics of the muscular sysreaches one-half body length {i.e., the tail tems are now well understood, many unanmotion is essentially perpendicular to the swered questions and unproven hypothedirection of motion of the body as a whole). ses still remain (Wainwright, 1983). In the
present paper I concentrate on body shape
and motion, and their influence on
1
momentum and energy transfer. As menFrom the Symposium on Axial Movement Systems:
Biomechanics and Neural Control presented at the
tioned above, propulsive motions typical of
Annual Meeting of the American Society of Zooloperiodic (steady) swimming are a series of
INTRODUCTION
gists, 27-30 December 1986, at Nashville, Tennessee.
151
152
DANIEL WEIHS
Swimming
Direction
lo
"added mass mode," and the "vorticity
mode," respectively. The former is basically an unsteady fluid-dynamical phenomenon independent of viscosity, by which a
solid moving through the water pushes and
pulls the neighboring water masses, causing them to be accelerated to roughly the
body speed. This so-called added mass of
a body moving through water at speed U
is obtained from the energy E required to
move the body. Writing the energy as
(1)
head
K has the dimensions of mass. The coefficient K can thus be obtained from (1) when
E and U are measured. It can be written
as
Swimming
K = Mf + M,
Direction
(2)
so that
Ib
FIG. 1. Typical transverse oscillation patterns for la.
periodic swimming and lb. transient motion. Figure
la is of Scomber scombrus adapted from Videler (1985)
and Figure lb is of Salmo, from Weihs (1973). Lines
are instantaneous tracings of body centerline during
the motion, with head on right side.
transverse waves of displacement of the
vertebral column, with amplitude minimal
at the vicinity of the fish center of mass
(see Fig. la). This is advantageous from the
point of view of propulsive efficiency, as
any sideways motions of the center of mass
do not contribute to the goal of transport
but cost energy.
Basically, the organism requires propulsive energy to counteract viscous effects.
These effects are highly dependent on the
Reynolds number, which is the ratio of
inertial to viscous effects, calculated from
the ratio of speed of motion times a typical
length, over the kinematic viscosity of
water. In the present paper we deal only
with motions at high Reynolds numbers
(> 1,000) where viscous effects are limited
to thin layers next to the body, and in the
wake.
MOMENTUM AND ENERGY TRANSFER
Forces are transmitted to the water by
two main modes. I shall call these the
I
M.IP
(3)
where Mf is the fish mass. M, is then the
added mass. For simple shapes such as circular cylinders, spheres, cones, etc. M, can
be calculated directly, enabling the estimation of the energy required for motion
without actually taking physiological or
other measurements.2 Thus, fish shapes are
commonly approximately described as
combinations of cylindrical, elliptical and
conical parts.
The added mass per unit length of an
elliptical cylinder moving perpendicularly
to its longitudinal axis is a measure of the
hydrodynamic effects of the motion of a
slab of an elongated fish performing propulsive movements (Lighthill, 1969). This
can be shown (Lighthill, 1975) to be
M,=-S2P
(4)
where S is the section long axis and p is the
water density. We thus see that the added
mass is the mass of a circular slab of water
of diameter S and unit length. If we now
2
This is the mechanical energy. Obviously, to obtain
the energy actually expended by the fish musculature,
muscular efficiency must be known.
AXIAL LOCOMOTION IN FISH
define the hydrodynamic efficiency 17, of
energy transfer from the fish muscle to the
surrounding water as
E,
E
E,
r + E,
(5)
we see that the more slender the elliptical
shape, the larger 77, will be, as Ei is constant
and Ef becomes smaller.
Following this argument, we now see that
the highest transfer efficiency for given
motion is obtained when the ellipse degenerates to a straight line, i.e., the minor axis
tends to zero. This can be seen as the mathematical description of a fin, which in crosssection is roughly a thin straight line. We
then arrive at the engineering rationale for
the development of fins—which are mechanisms for efficiently transferring energy
to the surrounding water.
The relationship of slenderness of crosssection shapes to transfer efficiency is also
applied in the inverse form, i.e., to facilitate
or hinder motion when a given force (or
energy) is exerted. I previously mentioned
that sideways motions of the center of mass
are superfluous from the point of view of
locomotion. These motions result from
recoils caused by the nonuniform instantaneous distribution of forces along the fish.
To minimize motion, and thus energy
transfer (loss—in this case), massive crosssections, such as the shoulder region should
be elliptical with the major axis vertical, so
that a given force acting on them will result
in small motions. Observations of various
species, especially those that oscillate most
of their body length, show this distribution
of body mass, with deep cross-sections at
the shoulder and mid-body. Species with
more rigid bodies, swimming in the carangiform mode where only the rear third
of the body is involved in locomotory
motions, often have round cross-sections
in the mid-body region. These species
require less recoil motions and thus can
stay with the circular shape—which is better from the point of view of viscous drag,
as surface area is minimized for given volume.
In the previous paragraph, we saw how
unnecessary recoil motions are minimized
by deepening the cross-section. However,
153
where large motions which do not contribute to thrust are required, such as head
motions in turning maneuvers or the sideways motions of the caudal peduncle, efficiency dictates cross-sections with the major
axis horizontal. Beautiful examples of the
changing direction of ellipticity associated
with body sections not contributing to
thrust can be seen in pelagic selachians
(Weihs, 1981) where the head is flattened
horizontally, the mid-body vertically and
the rear part horizontally again up to the
tail which is vertically oriented.
We now examine the vorticity mode of
transmitting forces to the water. This effect
is the result of viscous interactions between
the flow and body, but can be analysed with
good precision by inviscid means. When
the rear end of a submerged shape is sharp,
viscosity causes the flow to leave from both
sides of the sharp edge in a direction parallel to the surface (see Fig. 2). Thus if, as
in Fig. 2b, the flow incident on the body
was at a finite angle to the rear edge surfaces (this angle is called the angle of attack
in the fluid-mechanics literature) a change
in water flow direction is produced. This
change can be understood as an addition
of a component of velocity normal to the
incident flow. This added velocity represents a change in momentum, which the
body feels as a force in the opposite direction, usually called the lift force. In a well
rounded body (Fig. 2a), on the other hand,
no such change in flow direction is caused
and no normal (lift) forces are produced.
These lift forces are perpendicular to the
direction of motion of the body (or the
direction of oncoming flow if the coordinate system is taken to move with the body).
This hydrodynamic lift can be directed
sideways (as in vertical fins) or even downwards, and should not be confused with
hydrostatic lift, better designated buoyancy. As we have seen the main requirement for lift production is moving a body
at incidence to the surfaces connecting to
the sharp trailing edge. Thus, if a body is
to be designed to produce lift efficiently, it
should be as thin as possible (so as not to
waste energy dragging masses around), and
the angle of incidence should be relatively
small so that the added masses are small.
There are additional hydrodynamic rea-
154
DANIEL WEIHS
(a)
(b)
FIG. 2. Flow around rounded (a) and sharp edged (b) cross-sections moving at incidence to their major axis.
U, is incoming velocity, Uj wake velocity, V the change in velocity due to the lift L on the sharp-edged shape,
and D the hydrodynamic resistance (drag).
sons for keeping the angle of attack small,
as at large angles the flow can separate off
the surface facing downstream, reducing
the lift effect and producing additional
hydrodynamic drag. From the requirements above, we see that fins are very good
lifting surfaces especially at high swimming
speeds. There, even large sideways motions
will only result in a small angle of attack
due to the large component of forward
motion.
The vorticity, or lift mode of producing
hydrodynamic forces is much more efficient than the added mass mode. It can be
shown that the forces produced are up to
5 times as large, for given effort in the
vorticity mode. Therefore, all rapidly
swimming species have adopted the lift
technique of producing thrust, by developing hydrodynamically sophisticated caudal fins with related changes in the rear
part of the body and peduncle area, to be
described later.
FINS
In the previous section, I showed that
fins are effective transmitters of momentum, in both the added mass, or drag mode,
and the vorticity, or lift mode. Fish and
other marine organisms ranging from
mammals (cetaceans, etc.) to invertebrates
(cephalopods) have thus developed a large
variety of fins for different purposes. Fins
can be categorised in different ways; by
shape, structure, mobility and positioning.
In this section I will attempt to show how
hydromechanical principles correlate these
characteristics with fin function.
Starting off with fin shape we come across
everything from slight extensions of the
body profile in ribbon form as in eels, and
chimeras, usually dorsoventrally placed, to
highly extended, very narrow lunate caudal and pectoral fins as in some scombrids.
In terms of fin aspect ratio (defined as span
squared over fin area) one can find ribbon
fins with ratios of less than 0.1 and pectoral
and caudal fins of aspect ratio of approximately 10. This range covers two completely differing hydrodynamic regimes
(Lighthill, 1975; Yates, 1983): 1) low-aspect
ratio, slender surfaces which are inefficient
in terms of lift production and thus mainly
used in the added mass mode; and 2) high
(»1) aspect ratio fins which have high lift/
drag ratios, at small angles of attack, used
in the lift mode.
Between these two extremes are a great
variety of fins with aspect ratio approximately 1. In many cases, dorsal and ventral
fins of this group are of approximately
square shape. These are frequently retractable serving mainly as added mass augmentors during maneuvers of transient
AXIAL LOCOMOTION IN FISH
type. Fixed fins of triangular form are also
common, especially in the rear dorsal area.
These again mainly increase added mass,
but also have other purposes such as swimming drag reduction and the formation of
a second tail, which I will discuss more fully
later. Adipose fins may serve for drag
reduction (Aleyev, 1977). These as well as
finlets in scombrids and some selachians
are usually limp and tend to be dragged at
a phase lag to the body. Here transverse
flow over the sideways moving body is
directed to delay separation of flow on the
downstream side, thus reducing drag.
Pectoral fins are usually triangular or
spatulate, with the narrow part attached to
the body. These are almost always highly
flexible and movable, both in translation
and base rotation. This again is an adaptation for better hydrodynamical action as
pectorals serve for locomotion and maneuvering by sculling and pushing motions. In
both activities the parts of the fins moving
at high speed are the most effective. The
motion is transferred to the fin through its
base so that velocities relative to the fish
grow linearly with distance from the base.
Thus, while the base needs to be strong to
be able to transmit the forces, it does not
contribute to the forces itself and the surface area can be, and is, small. Further away
from the base, the fin chord is much larger,
thus utilising the high-speed areas.
In ribbon type fins, the main propulsive
motions are produced in many cases by
running waves of sideways oscillation down
the fins, pushing quantities of water backwards or forwards as required. Angular
motion here is mostly less than 90°, higher
forces being obtained by higher frequency.
Next we look at fin structure. Fin-rays
are a most versatile combination of
strength, flexibility and the thinness
required from the hydrodynamic considerations of the previous section. The combination of rigid (or voluntarily rigid) rods
and a highly flexible connecting tissue
characteristic of teleost fins can be used to
transfer momentum in different ways.
Keeping the rays rigid and spreading the
fins in the direction perpendicular to the
rays (we will call this the chordwise
155
direction3) a wide fin is produced enabling
large added mass coefficients for periodic
propulsion. Chordwise spreading can be
regulated such that the wide configuration
is obtained for a limited time compressing
the fin afterwards. This behavior is found
in rapid accelerations or changes of direction in the caudal fins of many fresh water
fish. It can also be effective for pectoral
propulsion, where the power stroke is performed with wide fins, then compressing
the fin for the return stroke. In this way,
large forces are produced when required
while the losses incurred during the return
stroke are minimized. For the same reason,
the fin plane is also rotated from an essentially vertical orientation in the power
stroke to horizontal during the return in
the case of pectoral fins. All of the examples in this paragraph are "added mass"
type motions.
Further applications of the possibility of
changing rigidity are also seen in the spanwise (parallel to the rays) direction. There,
rigidity can be relaxed in the caudal fin
when smaller forces are required. Also,
nonplanar curved shapes, acting as airfoils
or funnels in the vorticity and added mass
modes, respectively, can be produced by a
combination of chordwise and spanwise
adjustment of the stiffness of the fin-rays.
Some species have foregone the advantages of flexibility, by fusion of the caudal
fin rays, so that a massive rigid lunate body
with air-foil shaped cross-section is produced. These are species whose lifestyle
includes fast swimming for most of their
adult life, in the open ocean where tight
maneuvering is usually not necessary. Thus,
Kishinouye (1923) mentions that South-Sea
islanders used to sharpen thunnid tailfins
and use them as weapons.
Structural adaptations of fin-rays are
extremely interesting but complex. Here I
describe only the points of connection of
the fins to the fish body. The fin mass itself
does not include musculature so that all the
5
The chordwise direction for lifting surfaces is usually defined as the direction of fluid flow over the
surface, while the spanwise direction is laterally perpendicular to the former.
156
DANIEL WEIHS
LIGAMENT
COO
I collogaflout
liuu«
Ibon*
• — fin ray
AREA SHOWN
FIG. 3. Schematic sections through the caudal peduncles of cod, mackerel and long-fin tunny. Adapted from
Videler (1985). H: ray heads.
forces and couples transmitted by the fin
have to pass through the body attachment.
The design of the attachment point is complicated by the fact that it has to fulfill two
functions—transfer of forces, and adjustment of positioning of the fin-ray heads for
varying flexibility. These are conflicting
requirements, requiring ingenious solutions. In an exhaustive study of the joint
between the caudal fin and peduncle of
teleosts, Videler (1977, 1985) has shown
how the end of the vertebral column is
designed for efficient and massive force
transfer. The distal end of the column is a
triangular plate called the ural fan (see Fig.
3). In cod, and other species where flexibility of the caudal fin is important, the ural
fan has a rounded cartilaginous cushion
and a short overlap with the ray heads (H).
This allows for large angular changes in
ray head orientation. Mackerel have
greater overlap of the ray heads and fan,
with correspondingly smaller flexibility. An
extreme case of rigid attachment can be
seen in the tunny peduncle. In that case,
the collagenous interface is completely
gone and the ray heads contact the fan
bone directly.
Fin motion relative to the ray head
attachment point and body is another
parameter of great significance. I have
already mentioned the rotating and compressing of the pectoral fin during the
recovery stroke of the rowing cycle, to conserve energy. Other such motions include
bending of the caudal fin relative to the
peduncle during active swimming. This
bending is required to obtain the phase
difference between the sideways motion
(heave—Yates, 1983) of the peduncle and
the angle of inclination (pitch) of the fin to
the direction of motion. When this phase
difference is one-quarter cycle (90°) the
entire beat cycle produces forward thrust
(Lighthill, 1969) by the vorticity mode, i.e.,
at high efficiency. The possibilty of bending also allows the fish to choose the optimal pitching axis for the caudal fin, which
under various conditions is anywhere from
close to the trailing edge to near the quarter-chord position (Lighthill, 1970).
The caudal fin can also be rotated somewhat around the fish longitudinal axis producing a vertical component of force during the sideways motions, as the tail is then
moved horizontally while oriented at an
angle to the vertical. This motion can provide a vertical force to counter the forces
and couples produced by pectoral fins. Also
the difference in positioning of the center
AXIAL LOCOMOTION IN FISH
of mass and center of buoyancy in most
fish results in couples tending to raise the
fish head (Aleyev, 1977; Weihs, 1987). The
vertical force produced by the caudal fin
can cause a couple cancelling this tendency, at relatively small energetic cost as
the force required is small, the distance
between the caudal fin and fish center of
mass being large.
Fin positioning on the body is the last
major parameter influencing function we
shall examine here. Starting from the rear,
we see that a vertical caudal fin has developed in almost all nektonic (i.e., self-propelling) species. This development may be
a result of locomotory patterns in small,
elongate cylindrical animals. The first
movements of larval stages take place at
low Reynolds numbers. In this domain of
Reynolds numbers, force transfer by viscosity is dominant so that waves of transverse displacement are required. As animals grow, their speed grows and the
Reynolds number regime changes (Webb
and Weihs, 1986) leading to the added mass
and vorticity modes of propulsion.
Body shape and muscular distribution
having been set in the pattern for transverse wave motion, the natural improvement was to increase cross-section depth in
the body parts moving the most (furthest
from the center of mass), resulting in the
caudal fin. This configuration has several
advantages over its possible competitor in
terms of distance from the center of mass,
a head fin. First, in terms of drag and propulsive efficiency, it is always better to
change the relative speed of the water as
little as possible during passage through
the propulsion system. The caudal propulsor moves in an environment that (in
part at least) is moving with the fish (the
boundary layers), while a hypothetical head
fin would move in undisturbed waters—
i.e., at higher relative velocity. This interactive effect is further developed in
advanced species that have a "second tail"
(see next section). A second difference is
in the inherent stability. A head fin (or
canard—as it is called in aeronautical engineering) would produce large sideways
deflections of the body during forward
motion, as a result of chance disturbances.
157
Stability analyses show that this sensitivity
is much less so for tail-fin configurations.
This difference is mainly significant for
burst-and-coast swimmers (Weihs and
Webb, 1983), who spend a major part of
the swimming cycle coasting in a stretched
straight configuration, and for lunging
species, who need to be sure that no
unplanned deviations occur in the direction of motion.
Once the caudal fin developed, this
became the main propulsive element for
most fast swimmers. This means that thrust
from the central parts of the body was less
necessary, and the deepened cross-sections
could be given up for efficiency. Thus, dorsal and ventral fins remained only over
parts of the body, with three main uses.
First, these fins serve as producers of
sideways forces for turning. These fins are
retractable or fold down in many species.
Second, dorso-ventral fins help reduce
unnecessary recoil side motions. Third,
they act as secondary propulsors, working
in interaction with the caudal fin.
In addition to the dorso-ventral fins, most
nektonic species have developed one or
more horizontally oriented pairs of fins. In
many species, the front paired-pectoral fins
serve as propulsors. These are usually sedentary species with high maneuverability
requirements. The horizontal fins produce
vertical forces by the vorticity principle and
so there usually are two sets of paired fins
distanced from the center of mass in such
a manner that any couples produced are
cancelled out. The front pair can act differentially, while the rear pair usually act
as a unit, passively in many cases.
FIN INTERACTIONS AND THE
DOUBLE-TAIL CONFIGURATION
As shown in the previous sections, the
forces produced by the axial muscular system are most efficiently applied to locomotion by means of fins. Various fin forms
and locations were shown to be specifically
related to, and in many cases to be optimized for, given functions. All processes
of momentum transfer and force (or couple) production have an associated energy
loss—by imparting motion to the water
surrounding the fish, i.e., producing a wake.
158
DANIEL WEIHS
This loss is intrinsic to locomotion in water
which is set in motion when forces act on
it. Water has sufficiently low viscosity that
its motion will be sustained for a long time
(relative to the time required for a fish to
pass a given point in space), before being
dissipated. Thus, a possibility arises of raising the net efficiency of the propulsive system by interactions, and coordinated
motions between fins, such as the dorsoventral and caudal fins.
Some aspects of interactions have been
recognised and considered theoretically in
the past, mainly by Lighthill (1970) and
Wu (1971) who included the wakes from
dorsal and ventral fins as a continuation of
those fins in their slender-body theories.
This model is required if such theory is to
be used for fish, as classical slender-body
theory does not permit cross-section reduction in the downstream direction. The
interaction in this case is limited to the
reabsorption of the wakes from median
dorso-ventral fins into the flow around the
caudal fin, so that only one vortex sheet
(that of the caudal fin) constitutes the fish
wake far downstream. This implies an
increase in efficiency of locomotion in the
sense that the wake is a measure of energy
lost to the fish, and any cancellation or
reabsorption reduces this loss. It also means
that changes in cross-section caudal to the
section of maximum depth do not affect
the added mass mode of locomotion.
The existence of optimal combinations
of oscillations of hydrofoils in tandem has
been examined, with technological applications in mind (Sparenberg, 1984). Quantitative estimates of the possible gains of
having two oscillating fins are being studied at the present (Yates and Su, in preparation) showing that tens of percents of
the lost energy on the forward fin may be
utilised, improving the total performance
of the fish for given muscular effort. The
analysis presented by Yates and Su is simplified by assuming the fish has dorso-ventral symmetry (which is a relatively good
approximation for many gadids and thunnids) and performs small amplitude motions
(as the solution is obtained by superposition).
Some of the conclusions of this study are
summarized here. First, the existence of a
finite separation, in the longitudinal direction between the dorso-ventral fin and the
tail is always beneficial in terms of propulsion efficiency. This may help to explain
why most rapidly moving fish have evolved
a portion of their body length close to the
tail free of dorso-ventral fins. The latter
explanation by Yates and Su is obviously
only one of the reasons for the development of the caudal peduncle. A probably
more significant contribution to efficiency
resulting from the existence of the peduncle is the fact that the amplitude of oscillation grows rapidly as one moves along
the vertebral column in the peduncle area.
In view of the great advantage in using the
vorticity mode of momentum transfer
mentioned in the previous section, the contribution of the peduncle to added mass
propulsion is "sacrificed" by having a horizontally thickened section, gaining ease of
movement by lowering the resistance and
the required recoil motions. This leads to
large amplitude motions of the caudal fin
and vorticity forces large enough to make
the combination worthwhile.
Another result from the Yates and Su
study is that the tail fin should be at least
as deep as the dorso-ventral fins, otherwise
some of the wake from the latter is not reutilised. The third conclusion relevant to
our subject is that the efficiency of the
dorso-ventral fin pair interaction with the
tail becomes higher as the distance between
them grows. Thus, a double-tail configuration is obtained, the real tail and the
dorso-ventral fin combination, which we
call the second (or front) tail. For high tail
beat frequencies, an optimal (in terms of
propulsive efficiency and thrust) distance
between the forward and rear tails is found,
of about 0.4 of the body length. As mentioned before, these results are strictly
applicable only to small amplitude motions
of the caudal fin. Even so, taking measurements from the plates of scombrids, from
Joseph et al. (1980) we see that many species
have a distance between the tails compatible with the Yates and Su optimum (Fig.
This type of analysis, when further
developed to include realistic distributions
AXIAL LOCOMOTION IN FISH
of oscillation amplitudes may lead to a
method of estimation of optimal cruising
speeds for different species. Observation
of the relation between tail beat frequency
and forward speed, and distribution of
amplitudes are the only data needed for
calculating the efficiency and thrust at the
various speeds. At each speed, (obtained
from the nondimensional frequency
parameter a which is oscillation frequency
times 2ir, multiplied by fish length and
divided by the speed), a different distance
8* (see Fig. 4) is optimal. The speed at which
the measured C* is optimal is thus most
probably the fish's voluntary cruising speed.
In a different context, Weihs and Webb
(1983) have argued that the rearward
migration of a dorso-ventral pair of fins in
esocids many serve to produce, in the
hydrodynamic sense, a larger composite fin,
i.e., a double tail. It now seems from the
above that in this case the main function
of the dorsal and ventral fins is to enable
increasing the thrust for rapid acceleration. The interactive advantage follows
from the close proximity of the dorsal and
ventral fins to the tail which stabilizes the
flow over the caudal fin. This occurs even
at large angles of attack where the flow
would have otherwise separated and caused
loss of thrust. This principle has recently
been recognised by aircraft designers,
where this configuration is known as the
"close-coupled canard." The lift produced
by a thin lifting surface such as the caudal
fin is proportional to sin a, where a is the
angle of attack.4 This relation is true up to
some critical angle where the flow on the
lee surface separates and becomes chaotic,
causing the force on the fin to drop. However, if vorticity is fed into the flow on the
lee surface this flow breakdown can be relegated to much higher angles of attack.
This vorticity is produced on an auxiliary
surface, which can be much smaller (and
thus require less energy to operate), and
be at a smaller angle of attack. The dorsal
and ventral fins take the role of the auxiliary (canard) airfoil producing the vortex
wake that stabilizes the flow on the caudal
4
At high Reynolds numbers—see Introduction.
159
i.o
0.9
-
-
0 8
0.7
-0.5
-1.0
FIG. 4. Propulsive efficiency TJ (=T/(E/U)) and non] of a double-tailed fish,
pSU!
as a function of S*, the distance between tails normalised by fork-length, (p: water density; S: caudal fin
area; T: thrust.) The solid lines show theoretical calculations of Yates and Su (personal communication)
while the dashed vertical lines show the range of 8*
measured from the plates in Joseph el al. (1980) for
17 scombroids.
dimensional thrust C
fin. This function is in addition to the thrust
produced by the "auxiliary tail" itself, as
previously suggested by Weihs and Webb
(1983).
The description above is largely theoretical at this point. Detailed study of the
flow around the canard fin (the rearward
shifted dorsal and ventral fins) and the caudal fin is required to establish that the proposed mechanisms are actually utilised by
rapidly starting fish. Flow visualisation such
as dyes or particles, and high speed photography of at least 200 frames/sec are
needed to clarify this point.
CONCLUDING REMARKS
Various adaptations for transfer of
momentum between the fish and surrounding water are discussed. These are
just some of the features of the locomotory
160
DANIEL WEIHS
system, comprising the muscles, propulsors
and neural controls which can be profitably
analysed from a physical-engineering
standpoint.
The importance of fins, both as extensions of the body especially adapted for
locomotion, and as appendages utilised
independently (as in pectoral propulsion)
for the improved efficiency of cruising and
maneuvering swimming has been highlighted.
Various other subjects were not mentioned here, but deserve much more attention than they have received up to now.
These include drag reducing mechanisms,
both active and passive, and the sensory
system with its active feedback connections
to the neural drivers of the muscular system.
Lighthill, M.J. 1970. Aquatic animal propulsion of
high hydromechanical efficiency. J. Fluid Mech.
44:265-301.
Lighthill, M. J. 1975. Mathematical biofiuiddynamics.
SIAM, Philadelphia.
McClellan, A. D. 1989. Control of locomotion in a
lower vertebrate, the lamprey: Brainstem command systems and spinal cord regeneration. Amer.
Zool. 29:37-51.
Sigvardt, K. A. 1989. Spinal mechanisms in the control of lamprey swimming. Amer. Zool. 29:1935.
Sparenberg,J. A. 1984. Elements of hydrodynamics propulsion. Martinus Nyhoff, The Hague.
Videler.J.J. 1977. Mechanical properties of fish tail
joints. Fortsch. Zool. 24:183-194.
Videler, J. J. 1985. Fish swimming movements: A
study of one element of behavior. Neth. J. Zool.
35:170-185.
Wainwright, S. A. 1983. To bend a fish. In P. W.
Webb and D. Weihs (eds.), Fish biomechanics, pp.
68-91. Praeger, New York.
Webb, P. W. and D. Weihs. (eds.) 1983. Fish biomechanics. Praeger, New York.
Webb, P. W. and D. Weihs. 1986. Functional locoACKNOWLEDGMENTS
motor morphology of early life history stages of
This study was supported by the Fund
fishes. Trans. Amer. Fish. Soc. 115:115-127.
for Promotion of Research at Technion. I Weihs, D. 1973. The mechanism of rapid starting of
slender fish. Biorheology 10:343-350.
am grateful to Paul Webb for his comD. 1981. Body section variations in sharks—
ments on a previous version of this paper. Weihs,
an adaptation for efficient swimming. Copeia
1981:217-219.
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