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J. exp.
Biol.
109,
229-251 (1984)
229
Printed in Great Britain © The Company of Biologists Limited 1984
FAST CONTINUOUS SWIMMING OF SAITHE
(POLLACHIUS VIRENS): A DYNAMIC ANALYSIS OF
BENDING MOMENTS AND MUSCLE POWER
BY
F. HESS
AND
J. J. VIDELER
Department of
Zoology,
State University
Groningen,
P.O. Box 14, 9750
AAHaren, The Netherlands
Accepted 7
October
1983
SUMMARY
This paper deals with the hydrodynamics and internal dynamics of fish
swimming. Our analysis starts from kinematic data obtained for fast swim-
ming saithe, and treats the
fish
as
a
flexible
elongated body. The distribution
along the body of the lateral bending moment and the bending power
generated inside the fish are computed as well as the power spent on the
water. The computed thrust implies a drag coefficient (based on wetted
surface area) of about 0-007, which is probably an over-estimate. Our major
result is that the bending moment does not travel as a running wave from
head to tail like the lateral body curvature does, but behaves as a standing
wave. The left and right sides produce alternate contractions simultaneously
over the whole body length. This finding is in agreement with myographic
data from the literature.
INTRODUCTION
In the preceding paper we presented a kinematic analysis of the swimming move-
ments of saithe and mackerel (Videler & Hess, 1984). The present paper continues
with a dynamic analysis, using the kinematic results of the first paper as a point of
departure. This analysis is aimed at the hydrodynamic forces between fish and water,
the bending moments inside a fish and the mechanical work done by the fish body
during swimming.
We shall use Lighthill's (1960) hydrodynamic slender-body (or elongated-body)
theory. The fish is assumed to be a streamlined body, ending in
a
vertical trailing edge
of the tail fin. The water flows smoothly along the body surface and the stream lines
leave the body at the trailing edge only. Viscous effects are ignored. Slender-body
theory requires that the transverse dimensions of the body are small compared to its
length and that the cross-section shape varies only gradually along the body in a
lengthwise direction, and is therefore not applicable for mackerel. The tail fin of
mackerel shows
a
sharp increase in height from the caudal peduncle onward (see Fig. 1
of Videler
&
Hess, 1984), whereas in saithe this increase is less abrupt, although still
considerable. To what extent slender-body theory is applicable to swimming saithe
will be discussed in a later section.
Key words: Fish, swimming, hydrodynamics, fish muscles.
230
F. HESS AND J. J. VIDELER
The validity of Lighthill's (1960) theory is restricted to lateral oscillations of the fishj
body with
an
amplitude small
in
comparison
to
the
body length. Although severar
more refined versions
of
slender-body theory have been developed (e.g. Lighthill,
1971,
for
large amplitude motions, Newman
&
Wu, 1972, for interaction between fins
and body),
the
1960 theory has the advantage
of
being relatively simple, and linear.
The linearity
is
essential
to
our
approach, because we represent
the
periodic lateral
motion
of
the fish as a sum
of
several Fourier terms. Linearity implies that
if
a certain
lateral motion
is
considered as the sum
of
two other motions, A and B say, then
the
lateral hydrodynamic force distribution belonging
to
it is
obtained
by
summing
the
force distributions belonging to the motions
A
and
B.
Similarly for the lateral bending
moments inside
the
fish.
The theoretical model presented
in
the
next sections will
be
applied
to the
swim-
ming movements
of
saithe
as
analysed
in
Videler
&
Hess (1984) from
13
film
sequences.
MATHEMATICAL MODEL
We assume that the fish,
to
a good approximation, swims along a straight line
at
a
constant speed. Our moving coordinate system is chosen such that the fish stays close
to the x-axis and occupies a region between x
=
0 (nose) and x
= L
(tail). With respect
to the coordinate frame the fluid has a uniform velocity
U
in the x-direction (Fig.
1).
The z-axis points
in
the lateral direction, and the y-axis downward.
The 'centreline' (physically: the backbone)
of
the fish is described by the equation:
= h(x,t), 0=£x=SL.
(1)
We make the assumption that the fish is slender, which implies that its thickness and
height are much smaller than
its
length
L,
also that |h(x,t)|
<<
L
and that the angle
between centreline and x-axis is small.
In
the following analysis we will treat the fish
as
a
thin flexible
rod
under the influence
of
hydrodynamic forces. We consider only
U—water
Fig.
1.
Schematic dorsal view
of
fish
and
coordinate system.
Dynamic analysis of swimming 231
iateral bending, because it appears to determine completely the variations in body
shape of a swimming fish with the exception of the tail region and extended fins. For
fast-swimming saithe this approach seems to be justified.
Let us look at the forces and moments acting on an arbitrary thin slab of the fish
perpendicular to its backbone (Fig. 2). The slab lies between x and x + 6x and has
length
<5x.
The forces and moments experienced by the slab come from three regions:
the anterior part of the body, the posterior part of the body and the water adjacent to
the slab. We call the lateral force in the z-direction exerted at any section x by the
anterior part on the posterior part F(x). Hence the opposite force exerted by the
posterior part on the anterior part is
—
F(x).
The moment exerted by the anterior part
attempting to turn the posterior part counter-clockwise is called M(x). Hence the
opposite moment exerted by the posterior on the anterior is —
M(x).
The lateral force
exerted by the water per unit of length is L(x). Thus the following lateral forces act
on the slab:
F(x,t) - F(x + 6x,t) + L(x,t)5x.(2)
The moments acting on the slab are those from the anterior and posterior parts as well
as the moment resulting from the pair of lateral forces acting a distance
<5x
apart:
M(x,t) - M(x + <5x,t) - F(x,t)(5x. (3)
The variable t is included as all quantities are time dependent. If
<5x
is small enough
we can replace F(x + 6x,t) by F(x,t) + [3F(x,t)/3x]<5x and M(x + 6x,t) analogously.
The total net force acting on the slab must equal the slab's mass times its lateral
acceleration. Let the fish's body mass per unit length be mt,(x). We then obtain:
,a2h(x,t)
0F(x,t)
axL(x,t) = mb(x)-(4)
where we dropped the common factor
«5x
in all terms. The total net moment acting
on the slab must equal its angular acceleration times the slab's moment of inertia about
its vertical axis. This last quantity is negligibly small for a thin rod, therefore:
L(x)<5x
Fig. 2. Forces and moments acting on a thin slab of fish.
232 F. HESS AND J. J. VIDELER
The 'internal' force F and moment M vanish at the nose and tail ends:
F(O,t)= F(L,t) = 0, ]
M(0,t) = M(L,t) = 0. J
These end conditions apply whenever the fish moves freely in the water. They would
be violated if the fish were attached to some object at either its nose or its tail.
Equation (5) indicates that F = —dM/dx. Substitution into (4) yields:
dy
ay (7)
For L(x,t), the hydrodynamic lateral force per unit length, we take the following
expression, as derived by Lighthill (1960) for his small-amplitude slender-body
theory:
(jU^) {(f£)}
(8)
where ma(x) is the lateral added mass per unit length and it depends on the local cross-
section shape of the fish. The combination of (7) and (8) gives an equation connecting
h(x,t) with the bending moment M(x,t). If the lateral motion h(x,t) is given, together
with L, U, ma(x) and mt,(x), then the second derivative of the bending moment
d2M/dx2, can be obtained and from this follows M(x,t) itself after integrating twice.
However, note that whereas we must satisfy four end conditions (6), we can only
adjust two integration constants. Hence, for an arbitrary lateral motion, at most two
of the four end conditions can be satisfied. This means that only a restricted class of
lateral motions h(x,t) is allowed. This restriction is equivalent to the restriction
imposed by the recoil conditions stated by Lighthill (1960).
It should be pointed out that essentially the same mathematical model was outlined
by Wu (1971), who explicitly took elasticity into account and also by Lighthill (private
communication in 1978), who treated the fish as an elastic beam. In this paper we do
not distinguish between the elastic bending moments and the bending moments
generated by the fish's muscles.
Let us now consider the mechanical energy generated and spent by a thin slab at
an arbitrary section of the fish. As we are interested in the muscle power required to
propel the fish by bending its body, we will first look at the power (= energy per unit
time) exerted by the bending moment
M(x,t).
The mechanical power produced inside
the slab between x and x + dx equals the bending moment times the rate of change
of the slab's curvature:
n*/ \ d 8 h(x.t) £ .„.
M(X't)d~t~dx^6x- (9)
Hence, the power produced per unit length equals:
^ (101
Dynamic analysis of swimming 233
How is this power spent? Firstly, some power is spent on the slab itself by increasing
its kinetic energy:
rdh(xt)12l ah(xt)d2h(x,t) ....
^>>.
(11)
Secondly, the power spent on the water adjacent to the slab equals (force times
velocity):
^M (12)
L(x,t)&cat
These two parts, by virtue of (4), add up to
gF(x.t
ax at
Hence, the power spent per unit length on fish plus water equals:
D/..^_d2M(x,t)ah(x,t)
iZVMV ax2 at
where we used (5). The difference (Pi(x,t)—Pz(x,t)}6x is the power 'exported' by the
slab to the anterior and posterior parts of the fish. At each instant, t, the power
generated in the whole fish must equal the power spent on the whole fish plus water:
ft P,(x,t)dx = ft P2(x,t)dx = P(t). (15)
For the difference Pi
—P2
we have:
P,(x,t)-P2(x,t) = M(x,t)^-ax2 at
(16)
IX
where the function R is defined by:
R(x,t) = M(x,t)^--
R(x,t) is the power transported inside the fish across the section at x from anterior to
posterior. The first term in the right-hand side of (17) is the power exerted by the
anterior part on the posterior part by moment and angular velocity, the second term
is the power exerted by the anterior part on the posterior part by force and lateral
velocity. From (16) it follows:
Jg» {P,(x,t) - P2(x,t)}dx = R(xo,t). (18)
If L is substituted for xo, then (15) shows that R(L,t) must vanish. This agrees with
the end conditions (6): both M and dM/dx vanish at x = L and hence R according to
(17).
In addition to the power transported inside the fish, as represented by R, there is
also power leaving the fish body and entering adjacent water at some places and
234 F. HESS AND J. J.
VIDELER
returning to the body
elsewhere.
The power spent on the water per unit length
is
given
by (12). Hence, the power transport outside the fish across the plane x =
xo
is:
dx. (19)
Periodic motion
We now apply the theory of the previous section to the periodic swimming motion
as described by Videler
&
Hess (1984). The time period of the lateral motion
is
T, and
h(x,t) is represented by a Fourier series:
s
h(x,t) = .2 (aj(x)cosj(ot + bj(x)sinj<wt},
(O=2n/T,
(20)
odd
which can be written in the alternative form:
h(x,t) = .2 w hj(x)co8ja>[t -
TJ(X)],
(21)
where the hj(x) are amplitude functions and the Tj(x) phase functions. The time origin
is chosen such that Ti(L) = 0 by definition.
We define the function f by:
^>
(22)
It is the lateral body curvature function, which we can write as:
f(x,t) = .2 3s {ai"(x)cosj(Wt + bj"(x)sinjart}
= i?.^ fi(x)cosj«o[t -
oj(x)].
(23)
The Fourier coefficients aj(x) and bj(x) are represented by cubic splines are explained
by Videler
&
Hess (1984). We have chosen the end conditions of vanishing curvature
at both nose and tail: f(O,t) = f(L,t) = 0. The numerical values for aj"(x) and bj"(x)
are obtained by the 'spline-on-spline' method.
Let us represent the lateral bending moment M by:
M(x,t) =
2ij5
(pj(x)cosjan + qj(s)sinjcwt}
= 2AsMj(x)cosja)[t-tt(x)]. (24)
As
The pj(x) and qj(x) are Fourier coefficients, the Mj(x) amplitude functions and the
fi(x) phase functions.
If we substitute (24) and (20) for M(x,t) and h(x,t) in equations (7) and (8) we
obtain a set of equations relating the Fourier coefficients aj(x), bj(x) and pj(x), qj(x).
Each of the frequencies (j =
1,3,5)
can be dealt with separately. After carrying out the
necessary differentiations, we obtain:
Dynamic analysis of swimming
235
Pj"(x) = -jW{ma(x) + mb(x)}aj(x) + Ujft){2ma(x)bj'(x) + ma'(x)bj(x)}
+ U2{ma(x)aj"(x) + ma'(x)aj'(x)}> (25)
qj"(x) = -jW{ma(x) + mb(x)}bj(x) - Uj<o{2ma(x)aj'(x) + ma'(x)aj(x)}
+ U2{ma(x)bj"(x) + ma'(x)bj'(x)}.
Integrating twice yields pj(x) and qj(x), which determine the contribution of the jth
frequency to the bending moment. However, the correct solution requires that both
M and M' vanish at the end points, hence that:
for x = 0 and x = L,j =
1,3,5.
(26)
These 'recoil-conditions' will not automatically be satisfied. The next section gives a
method to deal with this problem.
Values for Pi(x,t), Pz(x,t), R(x,t), etc. are obtained by substitution of the ex-
pressions (20) and (24) and their derivatives. The mean bending power per unit
length is:
P:(x) =
4
2iM )(o{Pi(x)W'(x) ~ qj(x)aj"(x)}. (27)
The mean power spent per unit length on fish plus water is:
P2(x) = \ .2 w ja>{Pj"(x)bj(x) - q)"(x)aj(x)} (28)
and the mean internal power transport is:
R(x) = i 2iw j(u{pj(x)bj'(x) - qj(x)ai'(x) - pj'WbjW + qi'(x)aj(x)}. (29)
According to Lighthill (1960) the mean thrust 0for a periodic motion is given by:
and the mean power delivered by the fish:
+
u !
dt J dt ox JX=L
Both 8 and P depend only on what happens at the tail end. The hydrodynamic Froude
efficiency is given by:
rj = 0U/P
.
(32)
If the lateral motion has the form (20), then we have:
6
=
im.(L) 2 ,, {j
V(aj2
+ bj2) - U2(a/2 + bj'2)} (33)
j-l.3.5
P
=
|Uma(L)
2
{j2w2(aj2+
bj2) +
j«uU(bjaj'
-
ajbj')},
(34)
236 F.
HESS
AND J. J. VIDELER
where the values of aj, bj, z\ , b{ are those at x = L.
We note that the mean total power P and the mean thrust 6 are the sums of ths
means for each frequency. The fluctuations within one period, however, are the result
of an interplay between the various frequencies.
Rewriting equation (6) from Lighthill (1960), we find for the instantaneous thrust
i I - (35)
where H is defined by
(36)
The time average of the integral in (35) vanishes and the second term yields (30). The
second term becomes negative as well as positive during one half period. Our numeri-
cal calculations indicate that the first term cancels the strong fluctuations of the second
term only for a small part.
The higher frequency terms contribute relatively very little to the thrust and total
power according to our calculations. We therefore paid most attention to the first
frequency terms; those with j =
1
in the above formulae. For the bending power per
unit length, this term can be written as:
Pi(x,t) = £<wfi(x)Mi(x){sin£»[ai(x) - ^(x)] - sinco[2t - ffi(x) - //i(x)]} . (37)
This quantity fluctuates with a period of T/2. Its mean value is determined by the
first term between the braces, it may be positive or negative depending on the phase
difference between curvature f and bending moment M. The fluctuations are deter-
mined by the second term. Pi changes its sign twice per T/2 (except when
o\—/ii
equals an odd multiple of T/4, then Pi just touches
zero).
Pi has
a
positive mean value
if
kT<ai(x)-/*i(x)<(k+l/2)T, (38)
where
k
is any integer. The higher frequency contributions may influence the fluctua-
tions of Pi , but for any periodic motion Pi(x,t) is negative as well as positive in the
course of one period, because Pi is the product of two quantities, M and dh"/3t,
which both change their signs periodically and generally not simultaneously. This
means that at any cross section of the fish the power associated with the bending
moment fluctuates around its local mean value, inevitably becoming negative or zero
during part of the period.
For the power spent on fish plus water P2(x,t) (equation 14) expressions similar to
those for Pi can be derived. In the periodic case the time average l?2(x) is the mean
differential power spent on the water, because the mean power spent on the body
vanishes.
Recoil correction
We now must regard a crucial but rather inconvenient point: recoil. As mentioned
before, the fish's motion must be such that the end conditions (6) are satisfied. Our
functions h(x,t) do not satisfy these conditions for two main reasons: firstly, the lateraj
Dynamic analysis of swimming 237
(
potion h(x,t) as well as the mass functions mb(x) and ma(x) are not exact, but contain
ome errors. And, secondly, a real fish (such as saithe) is not a true slender body,
hence the hydrodynamic forces computed on the basis of Lighthill's (1960) theory are
not exact. Now we are faced with the problem of applying some sort of recoil correc-
tion to h(x,t) to make it satisfy the conditions (26) for the periodic case. There are
many ways in which this could be done. We use one that is relatively simple, but not
necessarily the best. We allow a certain amount of stiff motion A(t) + xB(t) to be
added to h(x,t). In this manner the curvature f(x,t) is not affected. This stiff motion
is represented by:
A(t) + xB(t) = . xr2j)cosftrt + (r3j + xr4j)sinart}.(39)
(40)
C3r4j,
(41)
For this motion we obtain, in analogy with (24):
Pj"(x) = —j2<w2{ma(x) + mb(x)}(rij + xr2j) + j<oU{2ma(x)r4j
+ ma'(x)(r3j + xr4j)} + U2ma'(x)r2j,
qj"(x) = -jW{ma(x) + mb(x)}(r3j + xr4j) - jeoU{2ma(x)r2j
+ ma'(x)(rij
4-
xr2j)} + U2ma'(x)r4j
for j = 1, 3, 5. After integrating twice, we get:
Pj'(L) = +Cirij +
C2r3j
+
C3r2j
+
C
4
r
4
j,
qj'(L) = -C2rij + Cir3j
—
C4r2j
Pj(L) = +Csrij + C6r3j + C?r2j
-\
qj(L) = —C6rij + Csr3j
—
Csr2j
H
The coefficients Ci through
Cg
can be computed from the functions ma(x) and mb(x)
by numerical integration
Ci = -jW J0L {ma(x) + mb(x)}dx
C2 = jwUma(L)
C3 = -j2ft>2 ft {ma(x) + mb(x)}xdx + Uma(L)
C4 = jft>U {ftma(x)dx + Lma(L)}
Cs = -}W ft ft' {ma(x) + mb(x)}dxdx'
C6
=
]CO\J
ft
ma(x)dx
C7 = -jV [ft ft' (ma(x) + mb(x)}xdxdx' + U2 ft ma(x)dx]
C8
= jwU {ft Jo' ma(x)dxdx' + ft ma(x)xdx}.
We proceed as follows. First we solve equations (7) and (8) for M"(x,t) using the
original h(x,t) as derived from the
film
sequences.
By
twice integrating (25) we obtain
Pj'(L),
qj'(L), pj(L), qj(L). The values atx =
0
vanish automatically. We want to add
the appropriate amount of stiff motion. (Here the linearity of the theory is crucial.)
ffhen the equations (41) must be solved for rij, r2j, r3j and r4j with the left-hand sides
(42)
238 F. HESS AND J. J. VIDELER
set equal to minus the corresponding quantities obtained from h(x,t). We then ad
the resulting stiff motion (39) to h(x,t). A similar treatment of recoil is given b
Lighthill (1970).
This concludes the description of the mathematical model. The numerical calcula-
tions are carried out by evaluating the integrands at 101 equidistant points, and
applying the trapezium rule on each of the
100
segments. The computing programme
are written in Basic and run on an HP9835A computer.
Dimensionless quantities
We shall express the physical quantities in dimensionless form, because this makes
it easier to compare cases differing in size or swimming speed. The resistance or drag
D which the fish must overcome when moving forward is often expressed as:
D = £pU2AwCD, (43)
where
CD
is the dimensionless drag coefficient, and Aw the wetted surface area of the
fish, g
is
the density of water and also the mean density of the fish. Similarly the thrust
coefficient
CT
is connected with the thrust 6:
0 = £pU2AwCT. (44)
If the Reynolds number is high enough (so that viscous effects are restricted to a thin
boundary layer on the body surface),
CD
depends only on the body shape of the fish,
not on absolute size or speed. If thrust and drag are not equal in magnitude, then the
fish accelerates or decelerates:
(45)
where mf is the fish's volume. From this follows:
„ n mf UL a ,.,.
CT~CD =
IA^TF=SP-
(46)
The first factor is the dimensionless shape parameter s, and the second factor is the
dimensionless acceleration parameter /S (see Videler & Hess. 1984). Just as in that
paper we shall use the time period T as a unit of time and the fish length L as a unit
of length. Further we choose the mass of
a
volume L3 of water as a mass unit. In these
units we have: L=l, T = 1, g=
1
and
a)
=
2.K.
To convert to conventional units,
lengths must be multiplied by L, times byT, velocities by LT"1, forces by pL4 T~2,
moments by pLsT~2, powers by pL5T~3, powers per unit length by pL4T~3, mass
per unit length by £>L2, etc.
Body shape of saithe
Fig. 3A shows the dorsal and lateral views of a swimming saithe as drawn from cin£
pictures. Fig. 3B presents graphs for the body mass distribution mb and the
hydrodynamic lateral added mass distribution
nia •
All quantities are made dimension-
less as outlined above. The cross-sectional shape of saithe was determined by measur-
ing three specimens (length about 0-22 m each). Cin6 pictures were used to determine
the shape of the tail fin during regular swimming, and to verify that the other fins wer'a
Dynamic analysis of swimming
239
completely collapsed. The body mass per unit length mt,(x) and the lateral
ded mass per unit length ma(x) were calculated by:
ma = iwh2
'
where
b is
the local body width,
hi
the local body height excluding fins and
h2
including fins. Both formulae are valid for elliptical sections. Lighthill (1970) showed
that deviations from the calculated
ma
values
are likely to be
small.
After
b,
hi,
Ii2
were
determined at some 18 points, smooth functions mt,(x) and rria(x) were obtained by
interpolation with cubic splines.
Some relevant quantities are:
Height of tail
fin
at trailing edge
Added mass at trailing edge
Body volume
Wetted area
Shape parameter, see (45)
Reynolds number (Re)
0-24 [L]
ma(L)
=
mf
=
Aw
=
s
=
between 2
X
0-0452 [L2]
0-0113
[i/
0-401 [L2]
0-0564
10s
and 8
X
We may well pose the question: how closely does a saithe resemble a slender body?
One may think
of
slender-body theory
as an
approximate theory whose resolving
power is limited to details in space which have about the size
of
the cross-sectional
dimensions of the 'slender' body. In our case that means roughly one-quarter of the
fish length, which is not very good. (For eel
it
would be about one-tenth, which
is
much better.)
In
slender-body theory the mean thrust and the mean total power
depend only on what happens at the tail end, but the theory implies that what happens
just ahead of the tail end is not very different. Here 'just ahead' may mean an area as
large as the whole fish tail in saithe. However, the height, for instance, varies strongly
along the tail. From this
it
is clear that the numerical results presented in this paper
should be considered
as
approximate estimates rather than
as
precise quantitative
predictions.
It is quite likely that slender-body theory over-estimates the hydrodynamic forces,
especially at the tail. Firstly, as pointed out by Lighthill (1970), the effective lateral
added mass is smaller if the body wave length
is
not very much (say, at least
five
times)
greater than the body height. Secondly, the tail region of saithe is not slender, strictly
speaking. The tailfin roughly resembles a triangular wing of aspect ratio 4 (Fig. 3A).
For such a wing in steady flow the lift is over-predicted by a factor of 1-8 by slender-
body (or rather slender-wing) theory (Lawrence, 1951). The only remedy would be
to employ some kind
of
unsteady lifting-surface theory,
but
that would involve
tremendous complications in comparison with Lighthill's (1960) elegant slender-body
theory.
RESULTS AND DISCUSSION
The third and fifth frequencies (j
=
3,
5 in
the formulae) each contributed only
ibout
1
% to the power and the thrust (Fig. 6). Therefore, we shall deal only with the
Dynamic analysis of swimming 241
fcst frequency (j = 1). Table
1
lists some results for the
13
sequences and also for the
liverage' saithe obtained by averaging the Fourier coefficients z\ and bj from the 13
sequences (see Videler & Hess, 1984).
Obviously, the stiff-body motion added as a recoil correction was considerable in
all
cases.
Let us first look at the situation before the recoil correction was applied. The
mean total power P and the mean thrust 6 have been computed according to (34) and
(33),
and from these follow the thrust coefficient Or (44) and the Froude efficiency
T}
(32). If we leave the four decelerating cases out,
r}
varied between 0-52 and 0-72.
For 'average' saithe
T]
= 0-63. These values are lower than the estimates made in the
preceding paper (Videler & Hess, 1984). There we looked at the motion of the
posterior part, in particular the quantities hi'(x)/hi(x) and Ti(x). In the present
calculations the trailing edge values were used, which turn out to differ somewhat
from the mean values over, say, the last 10% of the fish length. Although the latter
calculations were carried out with greater precision, we believe the former estimates
to be more realistic.
For a saithe moving with the U and h(x,t) as analysed from the film sequences, the
hydrodynamic forces computed according to slender-body theory would be such that
the end conditions (6), or (26), could not be satisfied. Theoretically, the fish could
only move that way if additional external lateral forces were to act at the nose and tail
Fig. 3. (A) Shape of swimming saithe, lateral and dorsal views. (B) Distribution of body mass per
unit length m
b
(x) (dashed curve) and lateral added mass per unit length m,(x) (drawn curve). (Unit:
pL
2
.)
242F. HESS AND J. J. VIDELER
ends.
We computed these virtual forces. In all cases the external force at the tail
enM
counteracted the computed hydrodynamic force, whereas the additional force at th?
nose end
was
much smaller. Let us take the
case
of
'average'
saithe.
The virtual force on
the nose end had an amplitude 0-002 (pL4T~2) and reached its maximum at t = 0-33
(T).
The virtual lateral force on the
tail
end had an amplitude 0 •
0063,
and its maximum
occurred at t = 0-90. Now, the lateral hydrodynamic force acting on the fish between
x = 0-95 and x = 100 had an amplitude 00061 and reached its maximum at t = 0-36,
that
is
0-54 T earlier than the virtual force. Thus the computed hydrodynamic force on
the last
5
% of the fish length was cancelled for a great part by the virtual force. This
clearly shows that the saithe can only carry out its observed movement if the
hydrodynamic force on the tail is much smaller in reality than as computed.
The stiff-body motion added as recoil correction is indicated in Table 1 by the
values of its amplitude and phase at the nose and tail ends. Fig. 4 provides a com-
parison between the lateral motion before and after recoil correction for 'average'
saithe. The amplitudes at nose and tail ends were hardly affected, but in between the
'corrected' amplitude was higher. The most significant change concerned the tail
region, where hi'(x) was much reduced after the correction. The 'corrected' phase
function Ti(x) equalled -0-043 at the tailing edge rather than zero. The wave speed
V (=
1/TI')
was only marginally increased in the tail region, but its overall value was
higher. Before recoil correction we have V = 1-04, U/V = 0-82, and after recoil cor-
rection V = 1-26, U/V = 0-68 over the posterior half of the fish.
014
012 -
-1-4
Fig. 4. Lateral motion of 'average' saithe before recoil correction (drawn curves) and after recoil
correction (dashed curves). First frequency contribution only. Nose is at x = 0, tail point at x = 1.
Left: amplitude functions h^x) (unit: L). Right: phase functions r,(x) (unit: T).
Dynamic analysis of swimming 243
Values for P, 6, Or and
r\
after recoil correction are listed in Table 1. From the
Observed acceleration
CT~
CD
follows according to (46). This leads to the drag co-
efficient values in the last column of Table 1. The Froude efficiency
r)
ranged from
0-65 to 0*84, or, if the four decelerating cases are left out, from 0-79 to 0-84.
CT
varied
between 0-001 for the decelerating S13 to 0-027 for the rather strongly accelerating
S5.
CD
varied between 0-003 and 0014. For 'average' saithe we find
CT
= 0-009 and,
as the average value for CT—CD = 0-002, we estimate CD = 0-007. The mean total
power P for 'average' saithe was 0-0014 (pL5T~3), which corresponds to about
0-7
W
kg"1
body weight. (For SS it is about 3-5
W
kg"1.)
The calculated thrust 6 fluctuated during each half-period between zero and
approximately twice its average value 6. For 'average' saithe 6
=
0-0013 ±
0-0011.
If
the drag on the fish were constant, such thrust fluctuations would cause oscillations
of the forward speed about its mean value U. According to (46) the fluctuations in U
would have an amplitude:
^ (48)
This gives rise to relative speed fluctuations with amplitude:
AU _ ACTU
U scoL ' (49)
where A Or is the amplitude of the fluctuations in the thrust coefficient. For 'average'
saithe (49) yielded about
2
%, and the maximum thrust occurred at t
—
—0-24 which
was nearly simultaneous with the maximum bending moment, roughly when the tail
point passes the plane z = 0. These fluctuations are somewhat stronger than those
observed in the preceding paper (Videler & Hess, 1984). In the accelerating case S5
we find AU/U
— 5
%, which agrees with the observed value. The computed instant
of maximum thrust was at t =
—0-28,
whereas the kinematic data yielded t = —0-20.
Fig. 5 shows the amplitude Mi(x) and the phase function jUi(x) of the bending
moment M(x,t) for 'average'
saithe.
The bending moment was strongest in the central
part of the fish body. The phase curve was nearly horizontal: fii(x)
—
—025.
This is
by far the most striking result: the bending moment did not travel from head to tail
as a wave but it reached its maximum value nearly simultaneously all along the body!
The muscles on the right side of the body exerted their maximum contraction force
at about the instant when the tail end, in its sweep from left to right, had reached the
plane of symmetry (z = 0).
Fig. 6 shows the mean differential bending power Pi(x) and the mean differential
power imparted to the water
P2(x).
Pi was almost zero in the anterior part because the
fish body hardly bends there, it reached
a
maximum in the central part around
x
= 0-7,
and it was negative in the tail, which contains no muscles. Fig. 6 clearly shows that
the power (Pi) was generated in the region 0-4 < x < 0-9 and spent on the water (P2)
in the tail region x>0-85. Considering the fish as a hydrodynamic propulsion
machine: the central part of the fish body contains the motor and the tail serves as the
propeller.
The fluctuation in the differential bending power Pi(x,t) are not shown, but, as
^plained above, they were very strong. At x = 0-65, where the bending moment was
244F.
HESS
AND J. J.
VIDELER
o
X
S 8"
S 6-
T3
"E.
-1-4
Fig.
5. Lateral body curvature (dashed curves) and lateral bending moment (drawn curves) for
'average'
saithe. First frequency contribution only. Left: amplitude functions fi(x) (unit: L"
1
) and
M|(x)
X
10
4
(unit: pL
5
T~
z
). Right: phase functions ai(x) + 0S and fii(x) (unit: T). Because the
body curvature found for
x
< 0-2 is due to noise in the data rather than to real bending, the dashed
phase curve has no physical meaning in this region.
greatest, Pi fluctuated between zero and twice its mean value and at most other places
the power became negative during part of each period. This implies that most of the
lateral fish muscles used in swimming are periodically stretched while exerting a
contracting force.
How accurate are the computed results presented here? Since the hydrodynamic
forces are over-predicted (at least on the tail), the bending moment M, the differential
bending power Pi, and the thrust 0are likely to be over-predicted as well. We found,
however, strong indications that the standing-wave character of the bending moment
is not a spurious result but a real phenomenon. We have tried several ways to reduce
the hydrodynamic force on the tail. We worked with a lower 'effective' tail height, or
decreased the lateral added mass ma as suggested by Lighthill's (1970) Fig. 2. Also we
have employed a different curvature function f(x,t), which has its amplitude fi(x)
increasing towards the trailing edge instead of decreasing. All these methods yield a
reduced hydrodynamic tail force, a much smaller recoil correction and smaller values
forMi(x), Pi(x), 6,
CT
and hence
CD
. But the curves for /ii(x) remain approximately
horizontal (with the possible exception of the last 10% of the fish length). None of
these methods can be
firmly
justified theoretically, but their results firstly confirm our
main finding that the bending moment does not move along the body as a running
wave, and secondly indicate that the drag coefficient for saithe probably is
CD
—
0-00
j|
Dynamic analysis of swimming 245
Fig. 6. Distribution of mean bending power per unit length
Pi (x)
and mean power spent on the water
per unit length ?2(x) for 'average' saithe. (Unit: pL
4
T ). Drawn curve: Pi(x)x lCr, first frequency
contribution only. Dashed curve: P2(x)XlO
3
, first frequency contribution only. Dotted curves:
Pi(x) X10
3
and Pz(x)X10
3
, sum of first, third and fifth frequency contributions.
This is much lower than the high drag coefficient obtained from computed thrust by
Lighthill (1971) for Leuciscus. For saithe we found no evidence to support his view
'that the viscous drag on the fish while it is swimming must for some reason be many
ti
mes
greater than that which would be associated with gliding motion'. We have not
analysed film sequences of gliding saithe without lateral motion, which could have
provided experimental values for
CD
. For gliding cod (Gadus morhua), however,
Videler (1981) found CD = 0-015,
0-011,
0-011 for one specimen in three gliding
Sequences.
246 F. HESS AND J. J. VIDELER
Our findings are supported by preliminary results of a similar analysis of the swim|
ming motion of eel
{Anguilla
anguilla). Hydrodynamically an eel behaves as
a
slendeii
body to a good approximation. Indeed, the recoil corrections required for eel are
much smaller than for saithe. The bending moment has the same character, although
the phase function
jUi(x)
in eel is not quite so constant as in saithe. The mean differen-
tial bending power Pi has roughly the same shape as in saithe, but the negative peak
in the tail region
is
relatively more pronounced in
eel.
All these results are qualitatively
similar to those presented here for saithe.
In deriving the major result, the 'standing wave' character of the bending moment,
we started from a running wave of body curvature. And indeed, the swimming
strategy of a fish might be to send waves of curvature along its body from head to tail.
However, our findings indicate that a fish may well use the strategy of exerting
bending forces simultaneously throughout its body, alternately using the muscles on
the left side and on the right
side.
The running
wave
in its body shape is then the result
of the interaction with the water flow. If this hypothesis is correct, the running wave
should be absent if a fish starts from stand-still in water or moves in air, provided the
fish produces the same muscle force. Our view is supported by Hertel's (1963) Fig.
169 of a trout starting and swimming, and also by Fig. 2 of Weihs (1973) of a trout
accelerating from stand-still.
The use of lateral muscles in swimming
The results of our dynamic analysis provide new insight into the function of the
lateral muscles for swimming. We shall first give a short description of the relevant
structures of saithe and then discuss the implications of our findings with respect to
muscle function.
Mechanically important parts of the locomotory apparatus used for continuous
swimming include the vertical septum, and left and right lateral muscles, surrounded
by the skin and the tailblade. The anatomy of these structures closely resembles that
for cod, which is described by Wardle & Videler (1980) and Videler (1981). The
vertical septum between the back of the head and the tailblade divides the body into
two lateral halves. It is a sheet of collagenous fibres supported by the vertebral
column. Mechanically the vertebral column can be regarded as an inextensible and
incompressible flexible rod, easily bent in the horizontal plane. The connecting
tissues between the vertebrae give the column self-restoring elastic properties (Sym-
mons,
1979). The lateral muscles are metamerically arranged in myotomes separated
by myosepts, both structures with a complicated geometry. The muscle fibres are
attached to the myosepts and run approximately in the direction of the longitudinal
body axis. Myosepts are attached to the vertical septum and at certain places to the
skin. There is a thin layer of red aerobic muscle fibres on the outside of the myotomes
just under the skin. The bulk of muscle fibres is white and works anaerobically. The
lateral muscles are also firmly attached to the head and on the other end of the fish to
the fin ray heads of the tailblade. From just behind the head (at
x —
0-2) to the position
of the anus (at x = 0-45) the ventral part of the fish contains the abdominal cavity. A
thin layer of lateral muscles supported by ribs surrounds this cavity, and the lateral
bending is restricted in this region. From the anus to the caudal end of the body the
myotomes are bilaterally and dorsoventrally symmetrical.
Dynamic analysis of swimming 247
The skin is a strong structure of layers of collagenous fibres in criss-cross arrange-
ment (Videler, 1975). It is attached to the head and to the vertical septum along the
dorsal and ventral rim and it inserts firmly on to the fin ray heads of the tailblade. The
structure of the joints between the fin rays of the tailfin and the caudal peduncle allows
the
fish
to keep the bending properties of the tailblade under muscular
control.
Details
were given by McCutchen (1970) and Videler (1977, 1981). The curvature of the
tailblade will be the result of elastic properties of the fin rays, controlled by intrinsic
musculature in the peduncle and by lateral musculature via the skin, in interaction
with bending forces exerted by the water.
The body curvature is connected with variations in length of the muscle fibres on
either side of the septum. In our frame of reference, a positive curvature means that
the fibres on the right side are shorter than their resting length, and those on the left
side longer; for negative curvature it is the other way around. A positive bending
moment implies that the muscles on the right side exert a contraction force and those
on the left side are passive or exert a smaller contraction force. We simplify our line
of reasoning by making the assumption that all the contraction forces are exerted by
fibres lying at a distance from the septum d(x) = ib(x) where b
is
the lateral thickness.
This is approximately where the red muscle fibres are situated, and the simplified
situation may not be too unrealistic during swimming at cruising speeds when most
of the bending moment is generated by the red muscles. However, our assumption
mainly serves as an instructive device.
The relative length change A/// of the chosen fibres on the right side of the fish
follows from the curvature h":
Ad(x,t) = -y = h"(x,t)d(x), (50)
where
Ad
is the relative shortening of the fibres. On the left side of the fish A/// has
the opposite sign. If the bending moment is generated by forces in the chosen fibres
then the force Fa(x,t) follows from
Fd(x,t) = M(x,t)/d(x). (51)
If M is positive then the contraction force Fd is exerted by the fibres on the right side,
if
M
is negative then the contraction force —
Fd
is exerted by the left-side
muscles.
The
power produced by the hypothetical muscle fibres per unit length is given by:
Fd(x,t)|-Ad(x,t) = M(x,t)^-h"(x,t) = Pi(x,t). (52)
ot ot
In Fig. 7 the functions Fd ,
Ad
and -^-Ad are plotted as a function of time during one
ot
period for several cross sections along the fish body: x = 0-1(0*1)0-9. At the nose and
tail points (x = 0, x = 1) both Fd and
Ad
vanish. The relative fibre length shortening
Ad(x,t) is represented by the dashed curves. The extreme values reached are plus and
minus
6
%. In the tail region (x = 0-9) the curvature is large (Fig. 5) but the body
thickness is small, and
Ad
varies between plus and minus 4%. The curve at x = 01
is probably caused by noise in the kinematic data, since the fish's head is rigid. The
•entraction force Fd(x,t) is represented by the drawn curves. It does not become very
248F. HESS AND J. J. VIDELER
01
0-2
0-3
0-4
* 0-5
0-6
0-7
0-9
•f >r"i '
1-
0-5
t1-0
Fig. 7. Contraction force, relative length change and contraction speed in outer fibres (see main text
for explanation) during one complete period at nine different sections of 'average' saithe. First
frequency contribution only. Numbers at left indicate x-positions of sections. At x = 0 (nose) and
x = 1 (tailpoint) all curves vanish. Drawn curves: contraction force in outer fibres, if positive then on
the right side, if negative then on the left side. One vertical division equals 002pL
4
T~
z
. Dashed
curves: relative shortening of outer fibres, if positive then the right-side fibres are shortened and the
left-side fibres are lengthened. One vertical division equals 0
-
02 (=2% length change). Dotted
curves: contraction speed (that is the rate of change of relative shortening), if positive then the right-
side fibres shorten. One vertical division equals
0*1
T"
1
. The bending power P|(x,t) at each section
is obtained by multiplying the drawn curve by the dotted curve.
Dynamic analysis of swimming 249
knall in the tail region like the bending moment does, because d(x)
is
very small here.
" Let us now look at what happens in the region where the amplitude Mi(x) and also
the mean differential bending power Pi(x) are maximal: at x = 0-6 or x = 0-7. Be-
tween the instants t = 0-5 and t = 1, Fd (drawn curves) is positive and the right-side
fibres exert a contraction force. The right-side fibres shorten (see dashed curves)
and reach their neutral length more or less when Fd is maximal. The contraction
speed (stippled curves) reaches its maximum nearly at the same instant. Hence, the
contraction force and the contraction speed have nearly the same phase. The power
output (determined by the product of stippled curve and solid curve) is positive
during almost the complete half period. Between t = 0 and t =
0*5,
Fd is negative,
which means that the left-side fibres exert
a
positive contraction force — Fd . The right-
side fibres lengthen (dashed curve falls), hence the left-side fibres contract. The
drawn curve and the stippled curve also have the same sign during most of this half
period. Therefore the power output is nearly always positive. At x = 0*6, Pi(x,t) is
negative only for 0-45 < t < 0-50 and 0-95 < x < 1-00.
In the tail region the situation is completely different. Take the section at x = 0-9.
For
0-53
< t < 1-03, Fd(x,t) is positive, but the right-side
fibres
first
lengthen and then
shorten. The time intervals with positive power output and with negative power
output are equally important. The mean differential power output Pi(x) nearly